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Mathematics LibreTexts

12-4. Derivatives

Derivatives
In this section, you will:
  • Find the derivative of a function.
  • Find instantaneous rates of change.
  • Find an equation of the tangent line to the graph of a function at a point.
  • Find the instantaneous velocity of a particle.

The average teen in the United States opens a refrigerator door an estimated 25 times per day. Supposedly, this average is up from 10 years ago when the average teenager opened a refrigerator door 20 times per day 1.

It is estimated that a television is on in a home 6.75 hours per day, whereas parents spend an estimated 5.5 minutes per day having a meaningful conversation with their children. These averages, too, are not the same as they were 10 years ago, when the television was on an estimated 6 hours per day in the typical household, and parents spent 12 minutes per day in meaningful conversation with their kids.

What do these scenarios have in common? The functions representing them have changed over time. In this section, we will consider methods of computing such changes over time.

Finding the Average Rate of Change of a Function

The functions describing the examples above involve a change over time. Change divided by time is one example of a rate. The rates of change in the previous examples are each different. In other words, some changed faster than others. If we were to graph the functions, we could compare the rates by determining the slopes of the graphs.

A tangent line to a curve is a line that intersects the curve at only a single point but does not cross it there. (The tangent line may intersect the curve at another point away from the point of interest.) If we zoom in on a curve at that point, the curve appears linear, and the slope of the curve at that point is close to the slope of the tangent line at that point.

[link] represents the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x )= x 3 −4x.  We can see the slope at various points along the curve.

slope at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=−2  is 8
slope at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=−1  is –1
slope at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=2  is 8

 

<figure class="small" id="CNX_Precalc_Figure_12_04_001"> <figcaption>Graph showing tangents to curve at –2, –1, and 2.</figcaption> Graph of f(x) = x^3 - 4x with tangent lines at x = -2 with a slope of 8, at x = -3 with a slope of -1, and at x=2 with a slope of 8.</figure>

Let’s imagine a point on the curve of function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>f</mi><mtext> </mtext></annotation-xml></semantics></math>at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a  as shown in [link]. The coordinates of the point are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> ( a,f(a) ).  Connect this point with a second point on the curve a little to the right of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a, with an x-value increased by some small real number<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> h.  The coordinates of this second point are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> ( a+h,f(a+h) )  for some positive-value<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> h.

<figure class="small" id="CNX_Precalc_Figure_12_04_002"> <figcaption>Connecting point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>a</mi><mtext> </mtext></annotation-xml></semantics></math>with a point just beyond allows us to measure a slope close to that of a tangent line at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a.</figcaption> Graph of an increasing function that demonstrates the rate of change of the function by drawing a line between the two points, (a, f(a)) and (a, f(a+h)).</figure>

We can calculate the slope of the line connecting the two points<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> (a,f(a))  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> (a+h,f(a+h)), called a secant line, by applying the slope formula,

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>slope = </mtext><mfrac/></mrow></annotation-xml></semantics></math> change in y change in x

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>slope = </mtext><mfrac/></mrow></annotation-xml></semantics></math> change in y change in x

We use the notation<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> m sec  to represent the slope of the secant line connecting two points.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>m</mi></msub></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> sec = f(a+h)−f(a) (a+h)−(a)        = f(a+h)−f(a) a +h− a

The slope<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> m sec  equals the average rate of change between two points<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> (a,f(a))  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> (a+h,f(a+h)).

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>m</mi></msub></mrow></annotation-xml></semantics></math> sec = f( a+h )−f( a ) h
The Average Rate of Change between Two Points on a Curve

The average rate of change (AROC) between two points<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> (a,f(a))  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> (a+h,f(a+h))  on the curve of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>f</mi><mtext> </mtext></annotation-xml></semantics></math>is the slope of the line connecting the two points and is given by

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>AROC</mtext><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> f( a+h )−f( a ) h
Finding the Average Rate of Change

Find the average rate of change connecting the points<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> ( 2,−6 )  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> ( −1,5 ).

We know the average rate of change connecting two points may be given by

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>AROC</mtext><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> f( a+h )−f( a ) h .

If one point is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> ( 2,−6 ), or<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> ( 2,f( 2 ) ), then<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( 2 )=−6.

The value<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>h</mi><mtext> </mtext></annotation-xml></semantics></math>is the displacement from<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mn>2</mn><mtext> </mtext></annotation-xml></semantics></math>to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> −1, which equals<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> −1−2=−3.

For the other point,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( a+h )  is the y-coordinate at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> a+h , which is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> 2+(−3)  or<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> −1, so<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(a+h)=f(−1)=5.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext>AROC</mtext><mo>=</mo><mfrac/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> f(a+h)−f(a) h            = 5−(−6) −3            = 11 −3            =− 11 3

Find the average rate of change connecting the points<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> ( −5,1.5 )  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> (−2.5,9).

3

Understanding the Instantaneous Rate of Change

Now that we can find the average rate of change, suppose we make<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>h</mi><mtext> </mtext></annotation-xml></semantics></math>in [link] smaller and smaller. Then<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> a+h  will approach<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>a</mi><mtext> </mtext></annotation-xml></semantics></math>as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>h</mi><mtext> </mtext></annotation-xml></semantics></math>gets smaller, getting closer and closer to 0. Likewise, the second point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> ( a+h,f(a+h) )  will approach the first point,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> ( a,f(a) ).  As a consequence, the connecting line between the two points, called the secant line, will get closer and closer to being a tangent to the function at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a, and the slope of the secant line will get closer and closer to the slope of the tangent at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a.  See [link].

<figure class="small" id="CNX_Precalc_Figure_12_04_003"> <figcaption>The connecting line between two points moves closer to being a tangent line at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a.</figcaption> Graph of an increasing function that contains a point, P, at (a, f(a)). At the point, there is a tangent line and two secant lines where one secant line is connected to Q1 and another secant line is connected to Q2.</figure>

Because we are looking for the slope of the tangent at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a, we can think of the measure of the slope of the curve of a function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>f</mi><mtext> </mtext></annotation-xml></semantics></math>at a given point as the rate of change at a particular instant. We call this slope the instantaneous rate of change, or the derivative of the function at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a.  Both can be found by finding the limit of the slope of a line connecting the point at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a  with a second point infinitesimally close along the curve. For a function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>f</mi><mtext> </mtext></annotation-xml></semantics></math>both the instantaneous rate of change of the function and the derivative of the function at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a  are written as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f'(a), and we can define them as a two-sided limit that has the same value whether approached from the left or the right.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>f</mi></msup></mrow></annotation-xml></semantics></math> ′ (a)= lim h→0 f( a+h )−f( a ) h

The expression by which the limit is found is known as the difference quotient.

Definition of Instantaneous Rate of Change and Derivative

The derivative, or instantaneous rate of change, of a function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>f</mi><mtext> </mtext></annotation-xml></semantics></math>at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a , is given by

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo>'</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><munder/></mrow></annotation-xml></semantics></math> lim h→0 f( a+h )−f( a ) h

The expression<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( a+h )−f( a ) h  is called the difference quotient.

We use the difference quotient to evaluate the limit of the rate of change of the function as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>h</mi><mtext> </mtext></annotation-xml></semantics></math>approaches 0.

Derivatives: Interpretations and Notation

The derivative of a function can be interpreted in different ways. It can be observed as the behavior of a graph of the function or calculated as a numerical rate of change of the function.

  • The derivative of a function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)  at a point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a  is the slope of the tangent line to the curve<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)  at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a.  The derivative of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)  at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a  is written<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f ′ (a).
  • The derivative<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f ′ (a)  measures how the curve changes at the point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> ( a,f(a) ).
  • The derivative<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f ′ (a)  may be thought of as the instantaneous rate of change of the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)  at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a.
  • If a function measures distance as a function of time, then the derivative measures the instantaneous velocity at time<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> t=a.
Notations for the Derivative

The equation of the derivative of a function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x )  is written as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> y ′ = f ′ (x), where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> y=f(x).  The notation<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f ′ (x)  is read as “<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mtext> prime of </mtext><mi>x</mi><mo>.</mo></mrow></annotation-xml></semantics></math>” Alternate notations for the derivative include the following:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>f</mi></msup></mrow></annotation-xml></semantics></math> ′ (x)= y ′ = dy dx = df dx = d dx f(x)=Df(x)

The expression<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f ′ (x)  is now a function of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi></annotation-xml></semantics></math>; this function gives the slope of the curve<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> y=f( x )  at any value of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x.  The derivative of a function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x )  at a point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a  is denoted<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f ′ (a).

Given a function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f, find the derivative by applying the definition of the derivative.

  1. Calculate<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( a+h ).
  2. Calculate<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( a ).
  3. Substitute and simplify<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( a+h )−f( a ) h .
  4. Evaluate the limit if it exists:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f ′ (a)= lim h→0 f( a+h )−f( a ) h .
Finding the Derivative of a Polynomial Function

Find the derivative of the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)= x 2 −3x+5  at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a.

We have:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msup><mi>f</mi></msup></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> ′ (a)= lim h→0 f( a+h )−f( a ) h                   Definition of a derivative 

Substitute<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(a+h)= (a+h) 2 −3(a+h)+5  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(a)= a 2 −3a+5.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msup><mi>f</mi></msup></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> ′ (a)= lim h→0 (a+h)(a+h)−3(a+h)+5−( a 2 −3a+5) h         = lim h→0 a 2 +2ah+ h 2 −3a−3h+5− a 2 +3a−5 hEvaluate to remove parentheses.         = lim h→0 a 2 +2ah+ h 2 −3a −3h +5 − a 2 +3a −5 h Simplify.         = lim h→0 2ah+ h2 −3h h         = lim h→0 h (2a+h−3) h Factor out an h.         =2a+0−3 Evaluate the limit.         =2a−3

Find the derivative of the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)=3 x 2 +7x  at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>f</mi></msup></mrow></annotation-xml></semantics></math> ′ (a)=6a+7

Finding Derivatives of Rational Functions

To find the derivative of a rational function, we will sometimes simplify the expression using algebraic techniques we have already learned.

Finding the Derivative of a Rational Function

Find the derivative of the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)= 3+x 2−x  at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msup><mi>f</mi></msup></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> ′ (a)= lim h→0 f( a+h )−f( a ) h         = lim h→0 3+( a+h ) 2−( a+h ) −( 3+a 2−a ) h Substitute f(a+h) and f(a)        = limh→0 (2−(a+h))(2−a)[ 3+( a+h ) 2−( a+h ) −( 3+a 2−a ) ] (2−(a+h))(2−a)(h)Multiply numerator and denominator by (2−(a+h))(2−a)        = lim h→0 ( 2−( a+h ) ) (2−a)( 3+( a+h ) ( 2−( a+h ) ) )−(2−(a+h)) ( 2−a ) ( 3+a 2−a ) ( 2−( a+h ) )(2−a)(h) Distribute        = lim h→0 6−3a+2a− a 2 +2h−ah−6+3a+3h−2a+ a 2 +ah( 2−( a+h ) )( 2−a )(h) Multiply        = lim h→0 5 h ( 2−( a+h ) )( 2−a )( h ) Combine like terms        = lim h→0 5 ( 2−( a+h ))( 2−a ) Cancel like factors        = 5 ( 2−( a+0 ) )( 2−a ) = 5 ( 2−a )( 2−a ) = 5 ( 2−a ) 2 Evaluate the limit

Find the derivative of the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)= 10x+11 5x+4  at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>f</mi></msup></mrow></annotation-xml></semantics></math> ′ (a)= −15 ( 5a+4 ) 2

Finding Derivatives of Functions with Roots

To find derivatives of functions with roots, we use the methods we have learned to find limits of functions with roots, including multiplying by a conjugate.

Finding the Derivative of a Function with a Root

Find the derivative of the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)=4 x  at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=36.

We have

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msup><mi>f</mi></msup></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> ′ (a)= lim h→0 f(a+h)−f(a) h         = lim h→0 4 a+h −4 a h Substitute f(a+h) and f(a)

Multiply the numerator and denominator by the conjugate:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> 4 a+h +4 a 4 a+h +4 a .

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext>   </mtext><msup/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> f ′ (a)= lim h→0 ( 4 a+h −4 a h )⋅( 4 a+h +4 a 4 a+h +4 a )            = lim h→0 ( 16(a+h)−16a h4( a+h +4 a ) ) Multiply.           = lim h→0 ( 16a +16h− 16a h4( a+h +4 a ) ) Distribute and combine like terms.            = lim h→0 ( 16 h h ( 4 a+h +4a ) ) Simplify.            = lim h→0 ( 16 4 a+h +4 a ) Evaluate the limit by letting h=0.            = 16 8 a = 2 a   f ′ (36)= 2 36Evaluate the derivative at x=36.            = 2 6            = 1 3

Find the derivative of the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x )=9 x at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=9.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>3</mn></mfrac></mrow></annotation-xml></semantics></math> 2

Finding Instantaneous Rates of Change

Many applications of the derivative involve determining the rate of change at a given instant of a function with the independent variable time—which is why the term instantaneous is used. Consider the height of a ball tossed upward with an initial velocity of 64 feet per second, given by<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> s(t)=−16 t 2 +64t+6, where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>t</mi><mtext> </mtext></annotation-xml></semantics></math>is measured in seconds and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> s( t )  is measured in feet. We know the path is that of a parabola. The derivative will tell us how the height is changing at any given point in time. The height of the ball is shown in [link] as a function of time. In physics, we call this the “s-t graph.”

<figure class="small" id="CNX_Precalc_Figure_12_04_004">Graph of a negative parabola with a vertex at (2, 70) and two points at (1, 55) and (3, 55).</figure>
Finding the Instantaneous Rate of Change

Using the function above,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> s(t)=−16 t 2 +64t+6,what is the instantaneous velocity of the ball at 1 second and 3 seconds into its flight?

The velocity at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> t=1  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> t=3  is the instantaneous rate of change of distance per time, or velocity. Notice that the initial height is 6 feet. To find the instantaneous velocity, we find the derivative and evaluate it at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> t=1  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> t=3:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msup><mi>f</mi></msup></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> ′ (a)= lim h → 0   f(a+h)−f(a) h         = lim h → 0   −16 (t+h) 2 +64(t+h)+6−(−16 t 2 +64t+6) h Substitute s(t+h) and s(t).        = lim  h → 0 −16 t 2 −32ht− h 2 +64t+64h+6+16 t 2 −64t−6 h Distribute.         = lim h → 0   −32ht− h 2 +64h hSimplify.         = lim h → 0   h (−32t−h+64) h Factor the numerator.         = lim h → 0 −32t−h+64Cancel out the common factor h.     s ′ (t)=−32t+64 Evaluate the limit by letting h=0.

For any value of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>t</mi></annotation-xml></semantics></math>,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> s ′ ( t )  tells us the velocity at that value of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> t.

Evaluate<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> t=1  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> t=3.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msup><mi>s</mi></msup></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> ′ (1)=−32(1)+64=32 s ′ (3)=−32(3)+64=−32

The velocity of the ball after 1 second is 32 feet per second, as it is on the way up.

The velocity of the ball after 3 seconds is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> −32  feet per second, as it is on the way down.

The position of the ball is given by<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> s(t)=−16 t 2 +64t+6.  What is its velocity 2 seconds into flight?

0

Using Graphs to Find Instantaneous Rates of Change

We can estimate an instantaneous rate of change at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a  by observing the slope of the curve of the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x )  at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a.  We do this by drawing a line tangent to the function at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a  and finding its slope.

Given a graph of a function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x ),  find the instantaneous rate of change of the function at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a.

  1. Locate<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a  on the graph of the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x ).
  2. Draw a tangent line, a line that goes through<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a  at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>a</mi><mtext> </mtext></annotation-xml></semantics></math>and at no other point in that section of the curve. Extend the line far enough to calculate its slope as
    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mtext>change in </mtext><mi>y</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> change in x .
Estimating the Derivative at a Point on the Graph of a Function

From the graph of the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> y=f( x )  presented in [link], estimate each of the following:

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo>'</mo><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo>'</mo><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>

<figure class="small" id="CNX_Precalc_Figure_12_04_005">Graph of an odd function with multiplicity of two and with two points at (0, 1) and (2, 1).</figure>

To find the functional value,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( a ), find the y-coordinate at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a.

To find the derivative at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f ′ ( a ), draw a tangent line at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a,and estimate the slope of that tangent line. See [link].

<figure class="small" id="CNX_Precalc_Figure_12_04_006">Graph of the previous function with tangent lines at the two points (0, 1) and (2, 1). The graph demonstrates the slopes of the tangent lines. The slope of the tangent line at x = 0 is 0, and the slope of the tangent line at x = 2 is 4.</figure>
  1. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>  is the y-coordinate at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=0.  The point has coordinates<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> ( 0,1 ), thus<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(0)=1.
  2. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>  is the y-coordinate at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=2.  The point has coordinates<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> ( 2,1 ), thus<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(2)=1.
  3. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>f</mi></msup></mrow></annotation-xml></semantics></math> ′ (0)  is found by estimating the slope of the tangent line to the curve at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=0.  The tangent line to the curve at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=0  appears horizontal. Horizontal lines have a slope of 0, thus<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f ′ (0)=0.
  4. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>f</mi></msup></mrow></annotation-xml></semantics></math> ′ (2)  is found by estimating the slope of the tangent line to the curve at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=2.  Observe the path of the tangent line to the curve at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=2.  As the<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>value moves one unit to the right, the<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>y</mi><mtext> </mtext></annotation-xml></semantics></math>value moves up four units to another point on the line. Thus, the slope is 4, so<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f ′ (2)=4.

Using the graph of the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)= x 3 −3x  shown in [link], estimate:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(1),<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f ′ (1),<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(0),and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f ′ (0).

<figure class="small" id="CNX_Precalc_Figure_12_04_007">Graph of the function f(x) = x^3-3x with a viewing window of [-4. 4] by [-5, 7</figure>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>2</mn></mrow></annotation-xml></semantics></math> ,0, 0,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> −3

Using Instantaneous Rates of Change to Solve Real-World Problems

Another way to interpret an instantaneous rate of change at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a  is to observe the function in a real-world context. The unit for the derivative of a function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x )  is

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mtext>output units </mtext></mrow></mfrac></mrow></annotation-xml></semantics></math>  input unit 

Such a unit shows by how many units the output changes for each one-unit change of input. The instantaneous rate of change at a given instant shows the same thing: the units of change of output per one-unit change of input.

One example of an instantaneous rate of change is a marginal cost. For example, suppose the production cost for a company to produce<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>items is given by<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> C( x ), in thousands of dollars. The derivative function tells us how the cost is changing for any value of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>in the domain of the function. In other words,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> C ′ ( x )  is interpreted as a marginal cost, the additional cost in thousands of dollars of producing one more item when<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>items have been produced. For example,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> C ′ ( 11 )  is the approximate additional cost in thousands of dollars of producing the 12th item after 11 items have been produced.<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>C</mi></msup></mrow></annotation-xml></semantics></math> ′ ( 11 )=2.50  means that when 11 items have been produced, producing the 12th item would increase the total cost by approximately $2,500.00.

Finding a Marginal Cost

The cost in dollars of producing<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>laptop computers in dollars is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x )= x 2 −100x.  At the point where 200 computers have been produced, what is the approximate cost of producing the 201st unit?

If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x )= x 2 −100x  describes the cost of producing<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>computers,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f ′ ( x )  will describe the marginal cost. We need to find the derivative. For purposes of calculating the derivative, we can use the following functions:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><msup/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> (x+h) 2 −100(x+h)       f(a)= a 2 −100a

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext>    </mtext><msup/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> f ′ (x)= f(a+h)−f(a) h Formula for a derivative              = (x+h) 2 −100(x+h)−( x 2 −100x) h Substitute f(a+h) and f(a).             = x 2 +2xh+ h 2 −100x−100h− x 2 +100x h Multiply polynomials, distribute.              = 2xh+ h 2 −100h hCollect like terms.              = h (2x+h−100) h Factor and cancel like terms.              =2x+h−100 Simplify.              =2x−100Evaluate when h=0.       f ′ (x)=2x−100 Formula for marginal cost   f ′ (200)=2(200)−100=300 Evaluate for 200 units.

The marginal cost of producing the 201st unit will be approximately $300.

Interpreting a Derivative in Context

A car leaves an intersection. The distance it travels in miles is given by the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( t ), where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>t</mi><mtext> </mtext></annotation-xml></semantics></math>represents hours. Explain the following notations:

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>f</mi></msup></mrow></annotation-xml></semantics></math> ′ (1)=60 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>70</mn></mrow></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>2.5</mn><mo stretchy="false">)</mo><mo>=</mo><mn>150</mn></mrow></annotation-xml></semantics></math>

First we need to evaluate the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(t)  and the derivative of the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f ′ (t), and distinguish between the two. When we evaluate the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(t),we are finding the distance the car has traveled in<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>t</mi><mtext> </mtext></annotation-xml></semantics></math>hours. When we evaluate the derivative<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f ′ (t), we are finding the speed of the car after<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>t</mi><mtext> </mtext></annotation-xml></semantics></math>hours.

  1. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow></annotation-xml></semantics></math>  means that in zero hours, the car has traveled zero miles.
  2. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>f</mi></msup></mrow></annotation-xml></semantics></math> ′ (1)=60  means that one hour into the trip, the car is traveling 60 miles per hour.
  3. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>70</mn></mrow></annotation-xml></semantics></math>  means that one hour into the trip, the car has traveled 70 miles. At some point during the first hour, then, the car must have been traveling faster than it was at the 1-hour mark.
  4. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>2.5</mn><mo stretchy="false">)</mo><mo>=</mo><mn>150</mn></mrow></annotation-xml></semantics></math>  means that two hours and thirty minutes into the trip, the car has traveled 150 miles.

A runner runs along a straight east-west road. The function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( t ) gives how many feet eastward of her starting point she is after<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>t</mi></annotation-xml></semantics></math> seconds. Interpret each of the following as it relates to the runner.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0 )=0 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 10 )=150 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>f</mi></msup></mrow></annotation-xml></semantics></math> ′ ( 10 )=15 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>f</mi></msup></mrow></annotation-xml></semantics></math> ′ ( 20 )=−10 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 40 )=−100

  1. After zero seconds, she has traveled 0 feet.
  2. After 10 seconds, she has traveled 150 feet east.
  3. After 10 seconds, she is moving eastward at a rate of 15 ft/sec.
  4. After 20 seconds, she is moving westward at a rate of 10 ft/sec.
  5. After 40 seconds, she is 100 feet westward of her starting point.

Finding Points Where a Function’s Derivative Does Not Exist

To understand where a function’s derivative does not exist, we need to recall what normally happens when a function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x )  has a derivative at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a. Suppose we use a graphing utility to zoom in on<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a. If the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x )  is differentiable, that is, if it is a function that can be differentiated, then the closer one zooms in, the more closely the graph approaches a straight line. This characteristic is called linearity.

Look at the graph in [link]. The closer we zoom in on the point, the more linear the curve appears.

<figure class="small" id="CNX_Precalc_Figure_12_04_008">Graph of a negative parabola that is zoomed in on a point to show that the curve becomes linear the closer it is zoomed in.</figure>

We might presume the same thing would happen with any continuous function, but that is not so. The function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)=| x |,for example, is continuous at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=0, but not differentiable at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=0.   As we zoom in close to 0 in [link], the graph does not approach a straight line. No matter how close we zoom in, the graph maintains its sharp corner.

<figure class="small" id="CNX_Precalc_Figure_12_04_009"> <figcaption>Graph of the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)=| x |,with x-axis from –0.1 to 0.1 and y-axis from –0.1 to 0.1.</figcaption> Graph of an absolute function.</figure>

We zoom in closer by narrowing the range to produce [link] and continue to observe the same shape. This graph does not appear linear at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=0.

<figure class="small" id="CNX_Precalc_Figure_12_04_010"> <figcaption>Graph of the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)=| x |,with x-axis from –0.001 to 0.001 and y-axis from—0.001 to 0.001.</figcaption> Graph of an absolute function.</figure>

What are the characteristics of a graph that is not differentiable at a point? Here are some examples in which function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x )  is not differentiable at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a.

In [link], we see the graph of

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo></mrow></mrow></annotation-xml></semantics></math> x 2 , x≤2 8−x, x>2 .

Notice that, as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches 2 from the left, the left-hand limit may be observed to be 4, while as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches 2 from the right, the right-hand limit may be observed to be 6. We see that it has a discontinuity at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=2.

<figure class="small" id="CNX_Precalc_Figure_12_04_011"> <figcaption>The graph of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x )  has a discontinuity at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=2.</figcaption> Graph of a piecewise function where from negative infinity to (2, 4) is a positive parabola and from (2, 6) to positive infinity is a linear line.</figure>

In [link], we see the graph of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)=| x |.  We see that the graph has a corner point at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=0.

<figure class="small" id="CNX_Precalc_Figure_12_04_012"> <figcaption>The graph of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)=| x |  has a corner point at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=0.</figcaption> Graph of an absolute function.</figure>

In [link], we see that the graph of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)= x 2 3  has a cusp at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=0.  A cusp has a unique feature. Moving away from the cusp, both the left-hand and right-hand limits approach either infinity or negative infinity. Notice the tangent lines as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches 0 from both the left and the right appear to get increasingly steeper, but one has a negative slope, the other has a positive slope.

<figure class="small" id="CNX_Precalc_Figure_12_04_013"> <figcaption>The graph of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)= x 2 3  has a cusp at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=0.</figcaption> Graph of f(x) = x^(2/3) with a viewing window of [-3, 3] by [-2, 3].</figure>

In [link], we see that the graph of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)= x 1 3  has a vertical tangent at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=0.  Recall that vertical tangents are vertical lines, so where a vertical tangent exists, the slope of the line is undefined. This is why the derivative, which measures the slope, does not exist there.

<figure class="small" id="CNX_Precalc_Figure_12_04_014"> <figcaption>The graph of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)= x 1 3  has a vertical tangent at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=0.</figcaption> Graph of f(x) = x^(1/3) with a viewing window of [-3, 3] by [-3, 3].</figure>
Differentiability

A function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x )  is differentiable at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a  if the derivative exists at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a,which means that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f ′ (a)  exists.

There are four cases for which a function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x )  is not differentiable at a point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a.

  1. When there is a discontinuity at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a.
  2. When there is a corner point at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a.
  3. When there is a cusp at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a.
  4. Any other time when there is a vertical tangent at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a.
Determining Where a Function Is Continuous and Differentiable from a Graph

Using [link], determine where the function is

  1. continuous
  2. discontinuous
  3. differentiable
  4. not differentiable

At the points where the graph is discontinuous or not differentiable, state why.

<figure class="small" id="CNX_Precalc_Figure_12_04_015">Graph of a piecewise function that has a removable discontinuity at (-2, -1) and is discontinuous when x =1.</figure>

The graph of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x ) is continuous on <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −∞,−2 )∪( −2, 1 )∪( 1,∞ ). The graph of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x ) has a removable discontinuity at <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi><mo>=</mo><mn>−2</mn></mrow></annotation-xml></semantics></math>and a jump discontinuity at <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi><mo>=</mo><mn>1.</mn></mrow></annotation-xml></semantics></math> See [link].

<figure class="small" id="CNX_Precalc_Figure_12_04_016"> <figcaption>Three intervals where the function is continuous</figcaption> Graph of the previous function that shows the intervals of continuity.</figure>

The graph of is differentiable on <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −∞,−2 )∪( −2,−1 )∪( −1,1 )∪( 1,2 )∪( 2,∞ ). The graph of <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math> is not differentiable at <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi><mo>=</mo><mn>−2</mn></mrow></annotation-xml></semantics></math> because it is a point of discontinuity, at <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi><mo>=</mo><mn>−1</mn></mrow></annotation-xml></semantics></math> because of a sharp corner, at <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi><mo>=</mo><mn>1</mn></mrow></annotation-xml></semantics></math> because it is a point of discontinuity, and at <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi><mo>=</mo><mn>2</mn></mrow></annotation-xml></semantics></math> because of a sharp corner. See [link].

<figure class="small" id="CNX_Precalc_Figure_12_04_017"> <figcaption>Five intervals where the function is differentiable</figcaption> Graph of the previous function that not only shows the intervals of continuity but also labels the parts of the graph that has sharp corners and discontinuities. The sharp corners are at (-1, -1) and (2, 3), and the discontinuities are at (-2, -1) and (1, 1).</figure>

Determine where the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> y=f( x )  shown in [link] is continuous and differentiable from the graph.

<figure class="small" id="CNX_Precalc_Figure_12_04_018">Graph of a piecewise function with three pieces.</figure>

The graph of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>f</mi><mtext> </mtext></annotation-xml></semantics></math>is continuous on<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> ( −∞,1 )∪( 1,3 )∪( 3,∞ ).  The graph of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>f</mi><mtext> </mtext></annotation-xml></semantics></math>is discontinuous at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=1  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=3.  The graph of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>f</mi><mtext> </mtext></annotation-xml></semantics></math>is differentiable on<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> ( −∞,1 )∪( 1,3 )∪( 3,∞ ).  The graph of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>f</mi><mtext> </mtext></annotation-xml></semantics></math>is not differentiable at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=1  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=3.

Finding an Equation of a Line Tangent to the Graph of a Function

The equation of a tangent line to a curve of the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x )  at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a  is derived from the point-slope form of a line,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> y=m( x− x 1 )+ y 1 .  The slope of the line is the slope of the curve at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a  and is therefore equal to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f ′ (a), the derivative of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x )  at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a.  The coordinate pair of the point on the line at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a  is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> (a,f(a)).

If we substitute into the point-slope form, we have

The point-slope formula that demonstrates that m = f(a), x1 = a, and y_1 = f(a).

The equation of the tangent line is

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>y</mi><mo>=</mo><mi>f</mi><mo>'</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x−a )+f(a)
The Equation of a Line Tangent to a Curve of the Function f

The equation of a line tangent to the curve of a function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>f</mi><mtext> </mtext></annotation-xml></semantics></math>at a point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a  is

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>y</mi><mo>=</mo><mi>f</mi><mo>'</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x−a )+f(a)

Given a function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f,  find the equation of a line tangent to the function at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a.

  1. Find the derivative of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x )  at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a  using<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f ′ (a)= lim h→0 f( a+h )−f( a ) h .
  2. Evaluate the function at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a.  This is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( a ).
  3. Substitute<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> ( a,f( a ) )  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f ′ ( a )  into<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> y=f'(a)( x−a )+f(a).
  4. Write the equation of the tangent line in the form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> y=mx+b.
Finding the Equation of a Line Tangent to a Function at a Point

Find the equation of a line tangent to the curve<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)= x 2 −4x  at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=3.

Using:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo>'</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><munder/></mrow></annotation-xml></semantics></math> lim h→0 f( a+h )−f( a ) h

Substitute<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(a+h)= (a+h) 2 −4(a+h)  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(a)= a 2 −4a.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mo> </mo><msup/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> f ′ (a)= lim h→0 (a+h)(a+h)−4(a+h)−( a 2 −4a) h          = lim h→0 a 2 +2ah+ h 2 −4a−4h− a 2 +4a h Remove parentheses.         = lim h→0 a 2 +2ah+ h 2 −4a −4h − a 2 +4a h Combine like terms.          = lim h→0 2ah+ h 2 −4h h          = lim h→0 h(2a+h−4) h Factor out h.          =2a+0−4 f ′ (a)=2a−4 Evaluate the limit.   f ′ (3)=2(3)−4=2

Equation of tangent line at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=3:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>y</mi><mo>=</mo><mi>f</mi><mo>'</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>a</mi><mo stretchy="false">)</mo><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> y=f'(3)(x−3)+f(3) y=2(x−3)+(−3) y=2x−9
Analysis

We can use a graphing utility to graph the function and the tangent line. In so doing, we can observe the point of tangency at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=3  as shown in [link].

<figure class="small" id="CNX_Precalc_Figure_12_04_020"> <figcaption>Graph confirms the point of tangency at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=3.</figcaption> Graph of f(x) = x^2-4x with a tangent line at x = 3 which has the equation of y = 2x - 9.</figure>

Find the equation of a tangent line to the curve of the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)=5 x 2 −x+4  at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=2.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>y</mi><mo>=</mo><mn>19</mn><mi>x</mi><mo>−</mo><mn>16</mn></mrow></annotation-xml></semantics></math>

Finding the Instantaneous Speed of a Particle

If a function measures position versus time, the derivative measures displacement versus time, or the speed of the object. A change in speed or direction relative to a change in time is known as velocity. The velocity at a given instant is known asinstantaneous velocity.

In trying to find the speed or velocity of an object at a given instant, we seem to encounter a contradiction. We normally define speed as the distance traveled divided by the elapsed time. But in an instant, no distance is traveled, and no time elapses. How will we divide zero by zero? The use of a derivative solves this problem. A derivative allows us to say that even while the object’s velocity is constantly changing, it has a certain velocity at a given instant. That means that if the object traveled at that exact velocity for a unit of time, it would travel the specified distance.

Instantaneous Velocity

Let the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> s( t )  represent the position of an object at time<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> t.  The instantaneous velocity or velocity of the object at time<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> t=a  is given by

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>s</mi></msup></mrow></annotation-xml></semantics></math> ′ (a)= lim h→0 s( a+h )−s( a ) h
Finding the Instantaneous Velocity

A ball is tossed upward from a height of 200 feet with an initial velocity of 36 ft/sec. If the height of the ball in feet after<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>t</mi><mtext> </mtext></annotation-xml></semantics></math>seconds is given by<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> s(t)=−16 t 2 +36t+200, find the instantaneous velocity of the ball at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> t=2.

First, we must find the derivative<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> s ′ ( t ). Then we evaluate the derivative at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> t=2, using<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> s( a+h )=−16 ( a+h ) 2 +36(a+h)+200  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> s( a )=−16 a 2 +36a+200.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msup><mi>s</mi></msup></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> ′ (a)= lim h→0 s(a+h)−s(a) h         = lim h→0 −16 (a+h) 2 +36(a+h)+200−(−16 a 2 +36a+200) h         = lim h→0 −16( a 2+2ah+ h 2 )+36(a+h)+200−(−16 a 2 +36a+200) h         = lim h→0 −16 a 2 −32ah−16 h 2 +36a+36h+200+16 a 2 −36a−200 h        = lim h→0 −16 a 2 −32ah−16 h 2 +36a +36h +200 +16 a 2 −36a −200 h         = lim h→0 −32ah−16 h 2 +36h h         =lim h→0 h (−32a−16h+36) h         = lim h→0 (−32a−16h+36)         =−32a−16⋅0+36   s ′ (a)=−32a+36   s ′ (2)=−32(2)+36        =−28
Analysis

This result means that at time<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> t=2  seconds, the ball is dropping at a rate of 28 ft/sec.

A fireworks rocket is shot upward out of a pit 12 ft below the ground at a velocity of 60 ft/sec. Its height in feet after<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>t</mi><mtext> </mtext></annotation-xml></semantics></math>seconds is given by<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> s=−16 t 2 +60t−12.  What is its instantaneous velocity after 4 seconds?

–68 ft/sec, it is dropping back to Earth at a rate of 68 ft/s.

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Key Equations

average rate of change <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>AROC</mtext><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> f( a+h )−f( a ) h
derivative of a function <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>f</mi></msup></mrow></annotation-xml></semantics></math> ′ (a)= lim h→0 f( a+h )−f( a ) h

Key Concepts

  • The slope of the secant line connecting two points is the average rate of change of the function between those points. See [link].
  • The derivative, or instantaneous rate of change, is a measure of the slope of the curve of a function at a given point, or the slope of the line tangent to the curve at that point. See [link], [link], and [link].
  • The difference quotient is the quotient in the formula for the instantaneous rate of change:
     
    <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></mfrac></mrow></annotation-xml></semantics></math> a+h )−f( a ) h
  • Instantaneous rates of change can be used to find solutions to many real-world problems. See [link].
  • The instantaneous rate of change can be found by observing the slope of a function at a point on a graph by drawing a line tangent to the function at that point. See [link].
  • Instantaneous rates of change can be interpreted to describe real-world situations. See [link] and [link].
  • Some functions are not differentiable at a point or points. See [link].
  • The point-slope form of a line can be used to find the equation of a line tangent to the curve of a function. See [link].
  • Velocity is a change in position relative to time. Instantaneous velocity describes the velocity of an object at a given instant. Average velocity describes the velocity maintained over an interval of time.
  • Using the derivative makes it possible to calculate instantaneous velocity even though there is no elapsed time. See[link].

Section Exercises

Verbal

How is the slope of a linear function similar to the derivative?

The slope of a linear function stays the same. The derivative of a general function varies according to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x.  Both the slope of a line and the derivative at a point measure the rate of change of the function.

What is the difference between the average rate of change of a function on the interval<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> [ x,x+h ]  and the derivative of the function at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x?

A car traveled 110 miles during the time period from 2:00 P.M. to 4:00 P.M. What was the car's average velocity? At exactly 2:30 P.M., the speed of the car registered exactly 62 miles per hour. What is another name for the speed of the car at 2:30P.M.? Why does this speed differ from the average velocity?

Average velocity is 55 miles per hour. The instantaneous velocity at 2:30 p.m. is 62 miles per hour. The instantaneous velocity measures the velocity of the car at an instant of time whereas the average velocity gives the velocity of the car over an interval.

Explain the concept of the slope of a curve at point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x.

Suppose water is flowing into a tank at an average rate of 45 gallons per minute. Translate this statement into the language of mathematics.

The average rate of change of the amount of water in the tank is 45 gallons per minute. If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x )  is the function giving the amount of water in the tank at any time<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>t</mi><mo>,</mo></annotation-xml></semantics></math>then the average rate of change of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x )  between<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> t=a  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> t=b  is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(a)+45(b−a).

Algebraic

For the following exercises, use the definition of derivative<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> lim h→0 f(x+h)−f(x) h  to calculate the derivative of each function.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x )=3x−4

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x )=−2x+1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>f</mi></msup></mrow></annotation-xml></semantics></math> ′ (x)=−2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x )= x 2 −2x+1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x )=2 x 2 +x−3

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>f</mi></msup></mrow></annotation-xml></semantics></math> ′ (x)=4x+1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x )=2 x 2 +5

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x )= −1 x−2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>f</mi></msup></mrow></annotation-xml></semantics></math> ′ (x)= 1 (x−2) 2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x )= 2+x 1−x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x )= 5−2x 3+2x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mo>−</mo><mn>16</mn></mrow></mfrac></mrow></annotation-xml></semantics></math> ( 3+2x ) 2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x )= 1+3x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>3</mn><msup/></mrow></annotation-xml></semantics></math> x 3 − x 2 +2x+5

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>f</mi></msup></mrow></annotation-xml></semantics></math> ′ (x)=9 x 2 −2x+2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>5</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>5</mn><mi>π</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>f</mi></msup></mrow></annotation-xml></semantics></math> ′ (x)=0

For the following exercises, find the average rate of change between the two points.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −2,0 )  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> ( −4,5 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 4,−3 )  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> ( −2,−1 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 3

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,5 )  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> ( 6,5 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 7,−2 )  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> ( 7,10 )

undefined

For the following polynomial functions, find the derivatives.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> x 3 +1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>3</mn><msup/></mrow></annotation-xml></semantics></math> x 2 −7x=6

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>f</mi></msup></mrow></annotation-xml></semantics></math> ′ (x)=−6x−7

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>7</mn><msup/></mrow></annotation-xml></semantics></math> x 2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>3</mn><msup/></mrow></annotation-xml></semantics></math> x 3 +2 x 2 +x−26

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>f</mi></msup></mrow></annotation-xml></semantics></math> ′ (x)=9 x 2 +4x+1

For the following functions, find the equation of the tangent line to the curve at the given point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>on the curve.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><msup/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> x 2 −3x x=3

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> x 3 +1 x=2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>y</mi><mo>=</mo><mn>12</mn><mi>x</mi><mo>−</mo><mn>15</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> x x=9

For the following exercise, find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>k</mi></annotation-xml></semantics></math> such that the given line is tangent to the graph of the function.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> x 2 −kx, y=4x−9

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>k</mi><mo>=</mo><mo>−</mo><mn>10</mn></mrow></annotation-xml></semantics></math>  or<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> k=2

Graphical

For the following exercises, consider the graph of the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>f</mi><mtext> </mtext></annotation-xml></semantics></math>and determine where the function is continuous/discontinuous and differentiable/not differentiable.

 

 
Graph of a piecewise function with three segments. The first segment goes from negative infinity to (-3, -2), an open point; the second segment goes from (-3, 1) to (2, 3), which are both open points; the final segment goes from (2, 2), an open point, to positive infinity.

 

 

 
Graph of a piecewise function with three segments. The first segment goes from negative infinity to (-2, -1), an open point; the second segment goes from (-2, -4), an open point, to (0, 0), a closed point; the final segment goes from (0, 1), an open point, to positive infinity.

 

Discontinuous at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=−2  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=0.  Not differentiable at –2, 0, 2.

Graph of a piecewise function with two segments and an asymptote at x = 3. The first segment, which has a removable discontinuity at x = -2, goes from negative infinity to the asymptote, and the final segment goes from the asymptote to positive infinity.
Graph of a piecewise function with two segments. The first segment goes from (-4, 0), an open point to (5, -2), and the final segment goes from (5, 3), an open point, to positive infinity.

Discontinuous at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=5.  Not differentiable at -4, –2, 0, 1, 3, 4, 5.

For the following exercises, use [link] to estimate either the function at a given value of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>or the derivative at a given value of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mo>,</mo></annotation-xml></semantics></math> as indicated.

<figure class="small" id="CNX_Precalc_Figure_12_04_205">Graph of an odd function with multiplicity of 2 with a turning point at (0, -2) and (2, -6).</figure>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −1 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0 )=−2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 1 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2 )=−6

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>f</mi></msup></mrow></annotation-xml></semantics></math> ′ ( −1 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>f</mi></msup></mrow></annotation-xml></semantics></math> ′ ( −1 )=9

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>f</mi></msup></mrow></annotation-xml></semantics></math> ′ ( 0 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>f</mi></msup></mrow></annotation-xml></semantics></math> ′ (1)

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>f</mi></msup></mrow></annotation-xml></semantics></math> ′ ( 1 )=−3

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>f</mi></msup></mrow></annotation-xml></semantics></math> ′ ( 2 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>f</mi></msup></mrow></annotation-xml></semantics></math> ′ ( 3 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>f</mi></msup></mrow></annotation-xml></semantics></math> ′ ( 3 )=9

Sketch the function based on the information below:

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>f</mi></msup></mrow></annotation-xml></semantics></math> ′ ( x )=2x,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( 2 )=4

Technology

Numerically evaluate the derivative. Explore the behavior of the graph of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)= x 2  around<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=1  by graphing the function on the following domains:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> [ 0.9,1.1 ],<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> [ 0.99,1.01 ],<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> [ 0.999,1.001 ] ,and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> [0.9999, 1.0001] . We can use the feature on our calculator that automatically sets Ymin and Ymax to the Xmin and Xmax values we preset. (On some of the commonly used graphing calculators, this feature may be called ZOOM FIT or ZOOM AUTO). By examining the corresponding range values for this viewing window, approximate how the curve changes at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=1,that is, approximate the derivative at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=1.

Answers vary. The slope of the tangent line near<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=1  is 2.

Real-World Applications

For the following exercises, explain the notation in words. The volume<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(t)  of a tank of gasoline, in gallons,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>t</mi><mtext> </mtext></annotation-xml></semantics></math>minutes after noon.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>600</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo>'</mo><mo stretchy="false">(</mo><mn>30</mn><mo stretchy="false">)</mo><mo>=</mo><mn>−20</mn></mrow></annotation-xml></semantics></math>

At 12:30 p.m., the rate of change of the number of gallons in the tank is –20 gallons per minute. That is, the tank is losing 20 gallons per minute.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>30</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo>'</mo><mo stretchy="false">(</mo><mn>200</mn><mo stretchy="false">)</mo><mo>=</mo><mn>30</mn></mrow></annotation-xml></semantics></math>

At 200 minutes after noon, the volume of gallons in the tank is changing at the rate of 30 gallons per minute.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>240</mn><mo stretchy="false">)</mo><mo>=</mo><mn>500</mn></mrow></annotation-xml></semantics></math>

For the following exercises, explain the functions in words. The height,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>s</mi><mo>,</mo></annotation-xml></semantics></math> of a projectile after<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>t</mi><mtext> </mtext></annotation-xml></semantics></math>seconds is given by<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> s(t)=−16 t 2 +80t.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>s</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>=</mo><mn>96</mn></mrow></annotation-xml></semantics></math>

The height of the projectile after 2 seconds is 96 feet.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>s</mi><mo>'</mo><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>=</mo><mn>16</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>s</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mo>=</mo><mn>96</mn></mrow></annotation-xml></semantics></math>

The height of the projectile at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> t=3  seconds is 96 feet.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>s</mi><mo>'</mo><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mo>=</mo><mn>−16</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>s</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>,</mo><mi>s</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0.</mn></mrow></annotation-xml></semantics></math>

The height of the projectile is zero at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> t=0  and again at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> t=5.  In other words, the projectile starts on the ground and falls to earth again after 5 seconds.

For the following exercises, the volume<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>V</mi><mtext> </mtext></annotation-xml></semantics></math>of a sphere with respect to its radius<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>r</mi><mtext> </mtext></annotation-xml></semantics></math>is given by<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> V= 4 3 π r 3 .

Find the average rate of change of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>V</mi><mtext> </mtext></annotation-xml></semantics></math>as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>r</mi><mtext> </mtext></annotation-xml></semantics></math>changes from 1 cm to 2 cm.

Find the instantaneous rate of change of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>V</mi><mtext> </mtext></annotation-xml></semantics></math>when<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> r=3 cm.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>36</mn><mi>π</mi></mrow></annotation-xml></semantics></math>

For the following exercises, the revenue generated by selling<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>items is given by<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> R(x)=2 x 2 +10x.

Find the average change of the revenue function as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>changes from<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=10  to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=20.

Find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> R'(10)  and interpret.

$50.00 per unit, which is the instantaneous rate of change of revenue when exactly 10 units are sold.

Find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> R'(15)  and interpret. Compare<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> R'(15)  to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> R'(10),and explain the difference.

For the following exercises, the cost of producing<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>cellphones is described by the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> C(x)= x 2 −4x+1000.

Find the average rate of change in the total cost as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>changes from<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=10 to x=15.

$21 per unit

Find the approximate marginal cost, when 15 cellphones have been produced, of producing the 16th cellphone.

Find the approximate marginal cost, when 20 cellphones have been produced, of producing the 21st cellphone.

$36

Extension

For the following exercises, use the definition for the derivative at a point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> lim x→a f(x)−f(a) x−a , to find the derivative of the functions.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 x 2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>5</mn><msup/></mrow></annotation-xml></semantics></math> x 2 −x+4

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo>'</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>10</mn><mi>a</mi><mo>−</mo><mn>1</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><msup/></mrow></annotation-xml></semantics></math> x 2 +4x+7

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> −4 3− x 2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>4</mn></mfrac></mrow></annotation-xml></semantics></math> ( 3−x ) 2

Chapter Review Exercises

Finding Limits: A Numerical and Graphical Approach

For the following exercises, use [link].

<figure class="small" id="CNX_Precalc_Figure_12_04_207">Graph of a piecewise function with two segments. The first segment goes from (-1, 2), a closed point, to (3, -6), a closed point, and the second segment goes from (3, 5), an open point, to (7, 9), a closed point.</figure>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ −1 + f(x)

2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ −1 − f(x)

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→−1 f(x)

does not exist

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→3 f(x)

At what values of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi></annotation-xml></semantics></math> is the function discontinuous? What condition of continuity is violated?

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtable columnalign="left"><mtr><mtd><mtext>Discontinuous at </mtext><mi>x</mi><mo>=</mo><mo>−</mo><mn>1</mn><mo stretchy="false">(</mo><munder/></mtd></mtr></mtable></annotation-xml></semantics></math> lim x→ a  f(x) does not exist),x=3 (jump discontinuity), and x=7 ( lim x→ a  f(x) does not exist).

Using [link], estimate<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> lim x→0 f(x).

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>x</mi></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>
−0.1 2.875
−0.01 2.92
−0.001 2.998
0 Undefined
0.001 2.9987
0.01 2.865
0.1 2.78145
0.15 2.678

3

For the following exercises, with the use of a graphing utility, use numerical or graphical evidence to determine the left- and right-hand limits of the function given as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> a.   If the function has limit as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> a,  state it. If not, discuss why there is no limit.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo></mrow></mrow></annotation-xml></semantics></math> | x |−1, if x≠1 x 3 , if x=1   a=1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo></mrow></mrow></annotation-xml></semantics></math> 1 x+1 , if x=−2 (x+1) 2 , if x≠−2   a=−2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> x→−2 f(x)=1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo></mrow></mrow></annotation-xml></semantics></math> x+3 , if x<1 − x 3 , if x>1   a=1

Finding Limits: Properties of Limits

For the following exercises, find the limits if<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> lim x→c f( x )=−3  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> lim x→c g( x )=5.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→c ( f(x)+g(x) )

2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→c f(x) g(x)

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→c ( f(x)⋅g(x) )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>−15</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ 0 + f(x),f(x)={ 3 x 2 +2x+1 5x+3    x>0 x<0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ 0 − f(x),f(x)={ 3 x 2 +2x+1 5x+3    x>0 x<0

3

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ 3 + ( 3x−〚x〛 )

For the following exercises, evaluate the limits using algebraic techniques.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> h→0 ( ( h+6 ) 2 −36 h )

12

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→25 ( x 2 −625 x −5 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→1 ( − x 2 −9x x )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>10</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable><mtr><mtd><mrow><mi>lim</mi></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> x→4 7− 12x+1 x−4

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→−3 ( 1 3 + 1 x 3+x )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 9

Continuity

For the following exercises, use numerical evidence to determine whether the limit exists at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a.  If not, describe the behavior of the graph of the function at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> −2 x−4 ; a=4

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> −2 ( x−4 ) 2 ; a=4

At<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=4,the function has a vertical asymptote.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> −x x 2 −x−6 ; a=3

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 6 x 2 +23x+20 4 x 2 −25 ; a=− 5 2

removable discontinuity at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> a=− 5 2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> x −3 9−x ; a=9

For the following exercises, determine where the given function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)  is continuous. Where it is not continuous, state which conditions fail, and classify any discontinuities.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> x 2 −2x−15

continuous on<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> (−∞,∞)

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> x 2 −2x−15 x−5

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> x 2 −2x x 2 −4x+4

removable discontinuity at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=2.<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(2)  is not defined, but limits exist.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> x 3 −125 2 x 2 −12x+10

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> x 2 − 1 x 2−x

discontinuity at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=0  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=2.  Both<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(0)  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(2)  are not defined.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> x+2 x 2 −3x−10

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> x+2 x 3 +8

removable discontinuity at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=–2. f(–2)  is not defined.

Derivatives

For the following exercises, find the average rate of change<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x+h)−f(x) h .

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>5</mn></mrow></annotation-xml></semantics></math>

0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 x+1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ln</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mi>ln</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>h</mi><mo stretchy="false">)</mo><mo>−</mo><mi>ln</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></annotation-xml></semantics></math> h

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> e 2x

For the following exercises, find the derivative of the function.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>6</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>=</mo><mn>4</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>5</mn><msup/></mrow></annotation-xml></semantics></math> x 2 −3x

Find the equation of the tangent line to the graph of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x )  at the indicated<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>value.


 
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><msup/></mrow></annotation-xml></semantics></math> x 3 +4x;<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=2.

 

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>16</mn></mrow></annotation-xml></semantics></math>

For the following exercises, with the aid of a graphing utility, explain why the function is not differentiable everywhere on its domain. Specify the points where the function is not differentiable.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> x | x |

Given that the volume of a right circular cone is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> V= 1 3 π r 2 h  and that a given cone has a fixed height of 9 cm and variable radius length, find the instantaneous rate of change of volume with respect to radius length when the radius is 2 cm. Give an exact answer in terms of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>π</mi></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>12</mn><mi>π</mi></mrow></annotation-xml></semantics></math>

Practice Test

For the following exercises, use the graph of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>f</mi><mtext> </mtext></annotation-xml></semantics></math>in [link].

<figure class="small" id="CNX_Precalc_Figure_12_04_208">Graph of a piecewise function with two segments. The first segment goes from negative infinity to (-1, 0), an open point, and the second segment goes from (-1, 3), an open point, to positive infinity.</figure>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>

3

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ −1 + f(x)

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ −1 − f(x)

0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→−1 f(x)

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→−2 f(x)

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>−1</mn></mrow></annotation-xml></semantics></math>

At what values of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>f</mi><mtext> </mtext></annotation-xml></semantics></math>discontinuous? What property of continuity is violated?

For the following exercises, with the use of a graphing utility, use numerical or graphical evidence to determine the left- and right-hand limits of the function given as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> a.  If the function has a limit as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> a,state it. If not, discuss why there is no limit

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo></mrow></mrow></annotation-xml></semantics></math> 1 x −3, if x≤2 x 3 +1,if x>2   a=2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→ 2 − f(x)=− 5 2 a  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> lim x→ 2 + f(x)=9  Thus, the limit of the function as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>approaches 2 does not exist.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo></mrow></mrow></annotation-xml></semantics></math> x 3 +1, if x<1 3 x 2 −1, if x=1 − x+3 +4, if x>1   a=1

For the following exercises, evaluate each limit using algebraic techniques.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→−5 ( 1 5 + 1 x 10+2x )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 50

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> h→0 ( h 2 +25 −5 h 2 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> h→0 ( 1 h − 1 h 2 +h )

1

For the following exercises, determine whether or not the given function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>f</mi><mtext> </mtext></annotation-xml></semantics></math>is continuous. If it is continuous, show why. If it is not continuous, state which conditions fail.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt/></mrow></annotation-xml></semantics></math> x 2 −4

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> x 3 −4 x 2 −9x+36 x 3 −3 x 2 +2x−6

removable discontinuity at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=3

For the following exercises, use the definition of a derivative to find the derivative of the given function at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 5+2x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo>'</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 2 a 3 2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><msup/></mrow></annotation-xml></semantics></math> x 2 +9x

For the graph in [link], determine where the function is continuous/discontinuous and differentiable/not differentiable.

<figure class="small" id="CNX_Precalc_Figure_12_04_209">Graph of a piecewise function with three segments. The first segment goes from negative infinity to (-2, -1), an open point; the second segment goes from (-2, -4), an open point, to (0, 0), a closed point; the final segment goes from (0, 1), an open point, to positive infinity.</figure>

discontinuous at –2,0, not differentiable at –2,0, 2.

For the following exercises, with the aid of a graphing utility, explain why the function is not differentiable everywhere on its domain. Specify the points where the function is not differentiable.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> x−2 |−| x+2 |

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 1+ e 2 x

not differentiable at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=0  (no limit)

For the following exercises, explain the notation in words when the height of a projectile in feet,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> s, is a function of time<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>t</mi><mtext> </mtext></annotation-xml></semantics></math>in seconds after launch and is given by the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> s(t).

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>s</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>s</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>

the height of the projectile at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> t=2  seconds

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>s</mi><mo>'</mo><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mi>s</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>−</mo><mi>s</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow></annotation-xml></semantics></math> 2−1

the average velocity from<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> t=1 to t=2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>s</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow></annotation-xml></semantics></math>

For the following exercises, use technology to evaluate the limit.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→0 sin(x) 3x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>1</mn></mfrac></mrow></annotation-xml></semantics></math> 3

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→0 tan 2 (x) 2x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><munder><mrow><mi>lim</mi></mrow></munder></mrow></annotation-xml></semantics></math> x→0 sin(x)(1−cos(x)) 2 x 2

0

Evaluate the limit by hand.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable><mtr><mtd><mrow><mi>lim</mi></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> x→1 f(x), where  f(x)={ 4x−7 x≠1 x 2 −4 x=1

At what value(s) of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>is the function below discontinuous?

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo></mrow></mrow></annotation-xml></semantics></math> 4x−7  x≠1 x 2 −4  x=1

For the following exercises, consider the function whose graph appears in [link].

<figure class="small" id="CNX_Precalc_Figure_12_04_210">Graph of a positive parabola.</figure>

Find the average rate of change of the function from<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=1 to x=3.

2

Find all values of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>at which<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f'(x)=0.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>x</mi><mo>=</mo><mn>1</mn></mrow></annotation-xml></semantics></math>

Find all values of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>x</mi><mtext> </mtext></annotation-xml></semantics></math>at which<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f'(x)  does not exist.

Find an equation of the tangent line to the graph of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>f</mi><mtext> </mtext></annotation-xml></semantics></math>the indicated point:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)=3 x 2 −2x−6,  x=−2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mn>14</mn><mi>x</mi><mo>−</mo><mn>18</mn></mrow></annotation-xml></semantics></math>

For the following exercises, use the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)=x ( 1−x ) 2 5.

Graph the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)=x ( 1−x ) 2 5  by entering<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)=x ( ( 1−x ) 2 ) 1 5  and then by entering<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)=x ( ( 1−x ) 1 5 ) 2.

Explore the behavior of the graph of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f(x)  around<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=1  by graphing the function on the following domains, [0.9, 1.1], [0.99, 1.01], [0.999, 1.001], and [0.9999, 1.0001]. Use this information to determine whether the function appears to be differentiable at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=1.

The graph is not differentiable at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=1  (cusp).

For the following exercises, find the derivative of each of the functions using the definition:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> lim h→0 f(x+h)−f(x) h

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>8</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>4</mn><msup/></mrow></annotation-xml></semantics></math> x 2 −7

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>f</mi></msup></mrow></annotation-xml></semantics></math> ' (x)=8x

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 2 x 2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 x+2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>f</mi></msup></mrow></annotation-xml></semantics></math> ' (x)=− 1 ( 2+x ) 2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 x−1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><msup/></mrow></annotation-xml></semantics></math> x 3 +1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>f</mi></msup></mrow></annotation-xml></semantics></math> ' (x)=−3 x 2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup/></mrow></annotation-xml></semantics></math> x 2 + x 3

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt/></mrow></annotation-xml></semantics></math> x−1

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo>'</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 2 x−1

Glossary

average rate of change
the slope of the line connecting the two points<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> (a,f(a))  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> (a+h,f(a+h))  on the curve of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x );  it is given by<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext>AROC</mtext><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> f( a+h )−f( a ) h .
derivative
the slope of a function at a given point; denoted<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f ′ (a),at a point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a  it is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f ′ (a)= lim h→0 f( a+h )−f( a ) h ,providing the limit exists.
differentiable
a function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f( x )  for which the derivative exists at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a.  In other words, if<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f ′ ( a )  exists.
instantaneous rate of change
the slope of a function at a given point; at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> x=a  it is given by<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> f ′ (a)= lim h→0 f( a+h )−f( a ) h .
instantaneous velocity
the change in speed or direction at a given instant; a function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> s( t )   represents the position of an object at time<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mi>t</mi><mo>,</mo></annotation-xml></semantics></math>and the instantaneous velocity or velocity of the object at time<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> t=a  is given by<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtext> </mtext><mrow/></annotation-xml></semantics></math> s ′ (a)= lim h→0 s( a+h )−s( a ) h .
secant line
a line that intersects two points on a curve
tangent line
a line that intersects a curve at a single point