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Mathematics LibreTexts

1.3 Rates of Change and Behavior of Graphs

Since functions represent how an output quantity varies with an input quantity, it is natural to ask about the rate at which the values of the function are changing. 

For example, the function C(t) below gives the average cost, in dollars, of a gallon of gasoline t years after 2000.

t

2

3

4

5

6

7

8

9

C(t)

1.47

1.69

1.94

2.30

2.51

2.64

3.01

2.14

If we were interested in how the gas prices had changed between 2002 and 2009, we could compute that the cost per gallon had increased from $1.47 to $2.14, an increase of  $0.67.  While this is interesting, it might be more useful to look at how much the price changed per year.  You are probably noticing that the price didn’t change the same amount each year, so we would be finding the average rate of change over a specified amount of time.

The gas price increased by $0.67 from 2002 to 2009, over 7 years, for an average of File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image002.gifdollars per year. On average, the price of gas increased by about 9.6 cents each year.

Rate of Change

A rate of change describes how the output quantity changes in relation to the input quantity.  The units on a rate of change are “output units per input units

Some other examples of rates of change would be quantities like:

  • A population of rats increases by 40 rats per week
  • A barista earns $9 per hour (dollars per hour)
  • A farmer plants 60,000 onions per acre
  • A car can drive 27 miles per gallon
  • A population of grey whales decreases by 8 whales per year
  • The amount of money in your college account decreases by $4,000 per quarter

Average Rate of Change

The average rate of change between two input values is the total change of the function values (output values) divided by the change in the input values.

Average rate of change = File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image004.gif=File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image006.gif

 

Example 1

Using the cost-of-gas function from earlier, find the average rate of change between 2007 and 2009

From the table, in 2007 the cost of gas was $2.64.  In 2009 the cost was $2.14.

The input (years) has changed by 2.  The output has changed by $2.14 - $2.64 = -0.50.  The average rate of change is then File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image008.gif = -0.25 dollars per year

 

Try it Now

1. Using the same cost-of-gas function, find the average rate of change between 2003 and 2008

 

Notice that in the last example the change of output was negative since the output value of the function had decreased.  Correspondingly, the average rate of change is negative.

Example 2

Given the function g(t) shown here, find the average rate of change on the interval [0, 3].

At t = 0, the graph shows File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image012.gif

At t = 3, the graph shows File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image014.gif

The output has changed by 3 while the input has changed by 3, giving an average rate of change of:

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image016.gif

 
Example 3

On a road trip, after picking up your friend who lives 10 miles away, you decide to record your distance from home over time.  Find your average speed over the first 6 hours.

 

 
 

t (hours)

0

1

2

3

4

5

6

7

D(t) (miles)

10

55

90

153

214

240

292

300

 

 

 

 

 

 


Here, your average speed is the average rate of change. 

You traveled 282 miles in 6 hours, for an average speed of

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image018.gif= 47 miles per hour

We can more formally state the average rate of change calculation using function notation.

Average Rate of Change using Function Notation

Given a function f(x), the average rate of change on the interval [a, b] is

Average rate of change = File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image020.gif

 

Example 4

Compute the average rate of change of File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image022.gif on the interval [2, 4]

 

We can start by computing the function values at each endpoint of the interval

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image024.gif

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image026.gif

 

Now computing the average rate of change

Average rate of change = File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image028.gif

Try it Now

2. Find the average rate of change of File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image030.gif on the interval [1, 9]

 
Example 5

The magnetic force F, measured in Newtons, between two magnets is related to the distance between the magnets d, in centimeters, by the formula File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image032.gif.  Find the average rate of change of force if the distance between the magnets is increased from 2 cm to 6 cm.

 

We are computing the average rate of change of File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image032.gif on the interval [2, 6]

Average rate of change = File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image035.gif       Evaluating the function

 

 

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image035.gif=

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image037.gif                                                       Simplifying

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image039.gif                                                         Combining the numerator terms

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image041.gif                                                             Simplifying further    

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image043.gif Newtons per centimeter

 

This tells us the magnetic force decreases, on average, by 1/9 Newtons per centimeter over this interval. 

 
Example 6

Find the average rate of change of File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image045.gifon the interval File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image047.gif.  Your answer will be an expression involving a.

 

Using the average rate of change formula

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image049.gif                                                 Evaluating the function

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image051.gif                      Simplifying

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image053.gif                                              Simplifying further, and factoring

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image055.gif                                                       Cancelling the common factor a

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image057.gif

 

 

This result tells us the average rate of change between t = 0 and any other point t = a.  For example, on the interval [0, 5], the average rate of change would be 5+3 = 8.

 

Try it Now

3. Find the average rate of change of File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image059.gif on the interval File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image061.gif.

 

Graphical Behavior of Functions

As part of exploring how functions change, it is interesting to explore the graphical behavior of functions.

Increasing/Decreasing

A function is increasing on an interval if the function values increase as the inputs increase.  More formally, a function is increasing if f(b) > f(a) for any two input values a and b in the interval with b>a. The average rate of change of an increasing function is positive.

A function is decreasing on an interval if the function values decrease as the inputs increase. More formally, a function is decreasing if f(b) < f(a) for any two input values a and b in the interval with b>a. The average rate of change of a decreasing function is negative.

Example 7

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image063.jpgGiven the function p(t) graphed here, on what intervals does the function appear to be increasing?

The function appears to be increasing from t = 1  to t = 3, and from t = 4 on. 

In interval notation, we would say the function appears to be increasing on the interval File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image065.gifand the interval File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image067.gif

 

 

Notice in the last example that we used open intervals (intervals that don’t include the endpoints) since the function is neither increasing nor decreasing at t = 1, 3, or 4. 

 
Definition: Local Extrema
  • A point where a function changes from increasing to decreasing is called a local maximum
  • A point where a function changes from decreasing to increasing is called a local minimum.

Together, local maxima and minima are called the local extrema, or local extreme values, of the function.

Example 8
 

Description: Description: (1-x)^2

 

Using the cost of gasoline function from the beginning of the section, find an interval on which the function appears to be decreasing.  Estimate any local extrema using the table.

 

 
 

t

2

3

4

5

6

7

8

9

C(t)

1.47

1.69

1.94

2.30

2.51

2.64

3.01

2.14

 

 

 

 

 

 


It appears that the cost of gas increased from t = 2 to t = 8. It appears the cost of gas decreased from t = 8 to t = 9, so the function appears to be decreasing on the interval  (8, 9).

Since the function appears to change from increasing to decreasing at t = 8, there is local maximum at t = 8.

 
Example 9
 

Description: Description: (1-x)^2

 

Use a graph to estimate the local extrema of the function File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image069.gif.   Use these to determine the intervals on which the function is increasing.

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image071.jpg

Using technology to graph the function, it appears there is a local minimum somewhere between x = 2 and x =3, and a symmetric local maximum somewhere between x = -3 and x = -2.

 

Most graphing calculators and graphing utilities can estimate the location of maxima and minima.  Below are screen images from two different technologies, showing the estimate for the local maximum and minimum.

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image073.jpg                   File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image075.jpg

 

Based on these estimates, the function is increasing on the intervals File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image077.gifand File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image079.gif.  Notice that while we expect the extrema to be symmetric, the two different technologies agree only up to 4 decimals due to the differing approximation algorithms used by each.

Try it Now

4. Use a graph of the function File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image081.gif to estimate the local extrema of the function.  Use these to determine the intervals on which the function is increasing and decreasing.

Concavity

The total sales, in thousands of dollars, for two companies over 4 weeks are shown. 

Company A                                                     Company B

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image083.jpg                           File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image085.jpg

As you can see, the sales for each company are increasing, but they are increasing in very different ways. To describe the difference in behavior, we can investigate how the average rate of change varies over different intervals. Using tables of values,

 

       
 

Company A

Week

Sales

Rate of Change

0

0

 

 

 

5

1

5

 

 

 

2.1

2

7.1

 

 

 

1.6

3

8.7

 

 

 

1.3

4

10

 

 

 

Company B

Week

Sales

Rate of Change

0

0

 

 

 

0.5

1

0.5

 

 

 

1.5

2

2

 

 

 

2.5

3

4.5

 

 

 

3.5

4

8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

From the tables, we can see that the rate of change for company A is decreasing, while the rate of change for company B is increasing

 

 

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image088.gif        

When the rate of change is getting smaller, as with Company A, we say the function is concave down.  When the rate of change is getting larger, as with Company B, we say the function is concave up.

 
Definition: Concavity
  • A function is concave up if the rate of change is increasing. 
  • A function is concave down if the rate of change is decreasing.
  • A point where a function changes from concave up to concave down or vice versa is called an inflection point.
 
Example 10

An object is thrown from the top of a building.  The object’s height in feet above ground after t seconds is given by the function File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image090.gif for File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image092.gif.  Describe the concavity of the graph.

 

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image094.jpgSketching a graph of the function, we can see that the function is decreasing.  We can calculate some rates of change to explore the behavior

 

 
 

t

h(t)

Rate of Change

0

144

 

 

 

-16

1

128

 

 

 

-48

2

80

 

 

 

-80

3

0

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Notice that the rates of change are becoming more negative, so the rates of change are decreasing.  This means the function is concave down.

Example 11
 

Description: Description: (1-x)^2

The value, V, of a car after t years is given in the table below.  Is the value increasing or decreasing?  Is the function concave up or concave down?

 

 
 

t

0

2

4

6

8

V(t)

28000

24342

21162

18397

15994

 

 

 

 

 

 


Since the values are getting smaller, we can determine that the value is decreasing.  We can compute rates of change to determine concavity.

 
 

t

0

2

4

6

8

V(t)

28000

24342

21162

18397

15994

Rate of change

-1829

-1590

-1382.5

-1201.5

 

                     

 

 

 

 

 

 

 


Since these values are becoming less negative, the rates of change are increasing, so  this function is concave up.

 

Try it Now

5. Is the function described in the table below concave up or concave down?

 

 
 

x

0

5

10

15

20

g(x)

10000

9000

7000

4000

0

 

 

 

 

 

 

 

 

Graphically, concave down functions bend downwards like a frown, and

 

concave up function bend upwards like a smile.

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image095.gif

 
Example 12
 

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image097.jpgEstimate from the graph shown the intervals on which the function is concave down and concave up. 

 

On the far left, the graph is decreasing but concave up, since it is bending upwards.  It begins increasing at x = -2, but it continues to bend upwards until about x = -1. 

 

From x = -1 the graph starts to bend downward, and continues to do so until about x = 2.  The graph then begins curving upwards for the remainder of the graph shown.

 

From this, we can estimate that the graph is concave up on the intervals File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image099.gif and File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image101.gif, and is concave down on the interval File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image103.gif.  The graph has inflection points at x = -1 and x = 2.

 

 

 

Try it Now

6. Using the graph from Try it Now 4, File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image081.gif, estimate the intervals on which the function is concave up and concave down.

Behaviors of the Toolkit Functions

We will now return to our toolkit functions and discuss their graphical behavior.

 

Function

Increasing/Decreasing

Concavity

Constant Function

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image105.gif

Neither increasing nor decreasing

 

Neither concave up nor down

Identity Function

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image107.gif

Increasing

Neither concave up nor down

 

Quadratic Function

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image109.gif

Increasing on File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image111.gif

Decreasing on File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image113.gif

Minimum at x = 0

Concave up File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image115.gif

Cubic Function

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image117.gif

 

Increasing

Concave down on File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image113.gif

Concave up on File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image111.gif

Inflection point at (0,0)

Reciprocal

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image120.gif

 

Decreasing File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image122.gif

Concave down on File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image113.gif

Concave up on File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image111.gif

 

Function

Increasing/Decreasing

Concavity

Reciprocal squared

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image124.gif

 

Increasing on File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image113.gif

Decreasing on File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image111.gif

 

Concave up on File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image122.gif

Cube Root

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image126.gif 

 

Increasing

Concave down on File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image111.gif

Concave up on File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image113.gif

Inflection point at (0,0)

Square Root

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image128.gif

 

Increasing on File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image111.gif

Concave down on File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image111.gif

Absolute Value

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image130.gif

Increasing on File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image111.gif

Decreasing on File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image113.gif

 

Neither concave up or down

 

Important Topics of This Section

  • Rate of Change
  • Average Rate of Change
  • Calculating Average Rate of Change using Function Notation
  • Increasing/Decreasing
  • Local Maxima and Minima (Extrema)
  • Inflection points
  • Concavity

 

Try it Now Answers

1. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image132.gif = 0.264 dollars per year.

 

2. Average rate of change = File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image134.gif

 

3. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image136.gif

    File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image138.gif

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image140.jpg

4.  Based on the graph, the local maximum appears to occur at (-1, 28), and the local minimum occurs at (5,-80).  The function is increasing onFile:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image142.gif and decreasing on File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image144.gif.

 

 

5.  Calculating the rates of change, we see the rates of change become more negative, so the rates of change are decreasing.  This function is concave down.

 

 
 

x

0

5

10

15

20

g(x)

10000

9000

7000

4000

0

Rate of change

-1000

-2000

-3000

-4000

 

                     

 

 

 

 

 

 

 


6. Looking at the graph, it appears the function is concave down on File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image146.gif and concave up on File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image101.gif.

Section 1.3 Exercises

1.  The table below gives the annual sales (in millions of dollars) of a product.  What was the average rate of change of annual sales…
a) Between 2001 and 2002?           b) Between 2001 and 2004?

year

1998

1999

2000

2001

2002

2003

2004

2005

2006

sales

201

219

233

243

249

251

249

243

233

2.  The table below gives the population of a town, in thousands.  What was the average rate of change of population…
a) Between 2002 and 2004?           b) Between 2002 and 2006?

year

2000

2001

2002

2003

2004

2005

2006

2007

2008

population

87

84

83

80

77

76

75

78

81

 

File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image149.jpg

3.  Based on the graph shown, estimate the average rate of change from x = 1 to x = 4.

 

4. Based on the graph shown, estimate the average rate of change from x = 2 to x = 5.

 

 

 

Find the average rate of change of each function on the interval specified.

5. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image151.gif on [1, 5]                                     6. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image153.gif on [-4, 2]

7. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image155.gif on [-3, 3]                             8. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image157.gif on [-2, 4]

9. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image159.gif on [-1, 3]                            10. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image161.gif on [-3, 1]         

 

Find the average rate of change of each function on the interval specified.  Your answers will be expressions involving a parameter (b or h).

11. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image163.gif on [1, b]                           12. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image165.gif on [4, b]  

13. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image167.gif on [2, 2+h]                         14. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image169.gif on [3, 3+h]

15. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image171.gif on [9, 9+h]                            16. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image173.gif on [1, 1+h]

17. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image175.gif on [1, 1+h]                              18. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image177.gif on [2, 2+h]

19. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image179.gif on [x, x+h]                        20. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image181.gif on [x, x+h]

 

For each function graphed, estimate the intervals on which the function is increasing and decreasing.

 

21. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image183.jpg     22. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image185.jpg

 

23. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image187.jpg  24. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image189.jpg

 

For each table below, select whether the table represents a function that is increasing or decreasing, and whether the function is concave up or concave down.

25.

x

f(x)

1

2

2

4

3

8

4

16

5

32

 

26.

x

g(x)

1

90

2

70

3

80

4

75

5

72

 

27.

x

h(x)

1

300

2

290

3

270

4

240

5

200

 

28.

x

k(x)

1

0

2

15

3

25

4

32

5

35

 

 

 

 

 

 

 

 

 

 

 

29.

x

f(x)

1

-10

2

-25

3

-37

4

-47

5

-54

 

30.

x

g(x)

1

-200

2

-190

3

-160

4

-100

5

0

 

31.

x

h(x)

1

-100

2

-50

3

-25

4

-10

5

0

 

32.

x

k(x)

1

-50

2

-100

3

-200

4

-400

5

-900

 

 

 

 

 

 

 

 

 

 

 

                                 

 

For each function graphed, estimate the intervals on which the function is concave up and concave down, and the location of any inflection points.

 

33. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image191.jpg              34. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image193.jpg

 

35. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image187.jpg              36. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image195.jpg

 

Use a graph to estimate the local extrema and inflection points of each function, and to estimate the intervals on which the function is increasing, decreasing, concave up, and concave down.

 

37. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image197.gif                                 38. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image199.gif

39. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image201.gif                                           40. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image203.gif

41. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image205.gif            42. File:C:/Users/DELMAR~1/AppData/Local/Temp/OICE_831AB0AC-7707-43BD-BC8E-41ACF4C7C811.0/msohtmlclip1/01/clip_image207.gif

 

 

 

 

Contributors

 

  • David Lippman (Pierce College)
  • Melonie Rasmussen (Pierce College)