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# 12.R: Introduction to Calculus (Review)

• • Contributed by Jay Abramson
• Principal Lecturer (School of Mathematical and Statistical Sciences) at Arizona State University
• Sourced from OpenStax CNX

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## 12.1: Finding Limits - Numerical and Graphical Approaches

For the exercises 1-6, use the Figure below. 1) $$\lim \limits_{x \to -1^+}f(x)$$

$$2$$

2) $$\lim \limits_{x \to -1^-}f(x)$$

3) $$\lim \limits_{x \to -1}f(x)$$

does not exist

4) $$\lim \limits_{x \to 3}f(x)$$

5) At what values of $$x$$ is the function discontinuous? What condition of continuity is violated?

Discontinuous at $$x=-1\left (\lim \limits_{x \to a}f(x) \text{ does not exist} \right )$$, $$x=3\left (\text{ jump discontinuity} \right )$$, and $$x=7\left (\lim \limits_{x \to a}f(x) \text{ does not exist} \right )$$.

6) Using the Table below, estimate $$\lim \limits_{x \to 0}f(x)$$.

$$x$$ $$F(x)$$
−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 exercises 7-9, 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 $$x$$ approaches $$a$$. If the function has limit as $$x$$ approaches $$a$$, state it. If not, discuss why there is no limit.

7) $$f(x)=\begin{cases} \left | x \right |-1 & \text{ if } x\neq 1 \\ x^3 & \text{ if } x= 1 \end{cases} a=1$$

8) $$f(x)=\begin{cases} \dfrac{1}{x+1} & \text{ if } x= -2 \\ (x+1)^2 & \text{ if } x\neq -2 \end{cases} a=-2$$

$$\lim \limits_{x \to -2}f(x)=1$$

9) $$f(x)=\begin{cases} \sqrt{x+3} & \text{ if } x<1 \\ -\sqrt{x} & \text{ if } x>1 \end{cases} a=1$$

## 12.2: Finding Limits - Properties of Limits

For the exercises 1-6, find the limits if $$\lim \limits_{x \to c} f(x)=-3$$ and $$\lim \limits_{x \to c} g(x)=5$$.

1) $$\lim \limits_{x \to c} (f(x)+g(x))$$

$$2$$

2) $$\lim \limits_{x \to c} \dfrac{f(x)}{g(x)}$$

3) $$\underset{x \to c}{\lim } (f(x)\cdot g(x))$$

$$-15$$

4) $$\lim \limits_{x \to 0^+} f(x), f(x)=\begin{cases} 3x^2+2x+1 & x>0 \\ 5x+3 & x<0 \end{cases}$$

5) $$\lim \limits_{x \to 0^-} f(x), f(x)=\begin{cases} 3x^2+2x+1 & x>0 \\ 5x+3 & x<0 \end{cases}$$

$$3$$

6) $$\lim \limits_{x \to 3^+} (3x-〚x〛)$$

For the exercises 7-11, evaluate the limits using algebraic techniques.

7) $$\lim \limits_{h \to 0} \left ( \dfrac{(h+6)^2-36}{h} \right )$$

$$12$$

8) $$\lim \limits_{x \to 25} \left ( \dfrac{x^2-625}{\sqrt{x}-5} \right )$$

9) $$\lim \limits_{x \to 1} \left ( \dfrac{-x^2-9x}{x} \right )$$

$$-10$$

10) $$\lim \limits_{x \to 4} \left ( \dfrac{7-\sqrt{12x+1}}{x-4} \right )$$

11) $$\lim \limits_{x \to 3} \left ( \dfrac{\frac{1}{3}+\frac{1}{x}}{3+x} \right )$$

$$-\dfrac{1}{9}$$

## 12.3: Continuity

For the exercises 1-5, use numerical evidence to determine whether the limit exists at $$x=a$$. If not, describe the behavior of the graph of the function at $$x=a$$.

1) $$f(x)=\dfrac{-2}{x-4};\; a=4$$

2) $$f(x)=\dfrac{-2}{(x-4)^2};\; a=4$$

At $$x=4$$, the function has a vertical asymptote.

3) $$f(x)=\dfrac{-x}{x^2-x-6};\; a=3$$

4) $$f(x)=\dfrac{6x^2+23x+20}{4x^2-25};\; a=-\dfrac{5}{2}$$

removable discontinuity at $$a=-\dfrac{5}{2}$$

5) $$f(x)=\dfrac{\sqrt{x}-3}{9-x};\; a=9$$

For the exercises 6-12, determine where the given function $$f(x)$$ is continuous. Where it is not continuous, state which conditions fail, and classify any discontinuities.

6) $$f(x)=x^2-2x-15$$

continuous on $$(-\infty, \infty)$$

7) $$f(x)=\dfrac{x^2-2x-15}{x-5}$$

8) $$f(x)=\dfrac{x^2-2x}{x^2-4x+4}$$

removable discontinuity at $$x=2$$. $$f(2)$$ is not defined, but limits exist.

9) $$f(x)=\dfrac{x^3-125}{2x^2-12x+10}$$

10) $$f(x)=\dfrac{x^2-\frac{1}{x}}{2-x}$$

discontinuity at $$x=0$$ and $$x=2$$. Both $$f(0)$$ and $$f(2)$$ are not defined.

11) $$f(x)=\dfrac{x+2}{x^2-3x-10}$$

12) $$f(x)=\dfrac{x+2}{x^3+8}$$

removable discontinuity at $$x=-2$$. $$f(-2)$$ is not defined.

## 12.4: Derivatives

For the exercises 1-5, find the average rate of change $$f(x)=\dfrac{f(x+h)-f(x)}{h}$$.

1) $$f(x)=3x+2$$

2) $$f(x)=5$$

$$0$$

3) $$f(x)=\dfrac{1}{x+1}$$

4) $$f(x)=\ln (x)$$

$$f(x)=\dfrac{\ln (x+h)-\ln (x)}{h}$$

5) $$f(x)=e^{2x}$$

For the exercises 6-7, find the derivative of the function.

6) $$f(x)=4x-6$$

$$4$$

7) $$f(x)=5x^2-3x$$

8) Find the equation of the tangent line to the graph of $$f(x)$$ at the indicated $$x$$ value. $f(x)=-x^3+4x;\; x=2 \nonumber$

$$y=-8x+16$$

9) For the following exercise, 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. $f(x)=\dfrac{x}{\left | x \right |} \nonumber$

10) Given that the volume of a right circular cone is $$V=\dfrac{1}{3}\pi r^2h$$ 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

$$12\pi$$

## Practice Test

For the exercises 1-6, use the graph of $$f$$ in the Figure below. 1) $$f(1)$$

$$3$$

2) $$\lim \limits_{x \to -1^+} f(x)$$

3) $$\lim \limits_{x \to -1^-} f(x)$$

$$0$$

4) $$\lim \limits_{x \to -1} f(x)$$

5) $$\lim \limits_{x \to -2} f(x)$$

$$-1$$

6) At what values of $$x$$ is $$f$$ discontinuous? What property of continuity is violated?

For the exercises 7-8, 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 $$x$$ approaches $$a$$. If the function has a limit as $$x$$ approaches $$a$$, state it. If not, discuss why there is no limit.

7) $$f(x)=\begin{cases} \dfrac{1}{3}-3 & \text{ if } x\leq 2 \\ x^3+1 & \text{ if } x>2 \end{cases} a=2$$

$$\lim \limits_{x \to 2^-} f(x)=-\dfrac{5}{2}a$$ and $$\lim \limits_{x \to 2^+} f(x)=9$$

Thus, the limit of the function as $$x$$ approaches $$2$$ does not exist.

8) $$f(x)=\begin{cases} x^3+1 & \text{ if } x<1 \\ 3x^2-1 & \text{ if } x=1\; a=1 \\ -\sqrt{x+3}+4 & \text{ if } x>1 \end{cases}$$

For the exercises 9-11, evaluate each limit using algebraic techniques.

9) $$\lim \limits_{x \to -5} \left ( \dfrac{\frac{1}{5}+\frac{1}{x}}{10+2x} \right )$$

$$-\dfrac{1}{50}$$

10) $$\lim \limits_{h \to 0} \left ( \dfrac{\sqrt{h^2+25}-5}{h^2} \right )$$

11) $$\lim \limits_{h \to 0} \left ( \dfrac{1}{h}-\dfrac{1}{h^2+h} \right )$$

$$1$$

For the exercises 12-13, determine whether or not the given function $$f$$ is continuous. If it is continuous, show why. If it is not continuous, state which conditions fail.

12) $$f(x)=\sqrt{x^2-4}$$

13) $$f(x)=\dfrac{x^3-4x^2-9x+36}{x^3-3x^2+2x-6}$$

removable discontinuity at

For the exercises 14-16, use the definition of a derivative to find the derivative of the given function at $$x=a$$.

14) $$f(x)=\dfrac{3}{5+2x}$$

15) $$f(x)=\dfrac{3}{\sqrt{x}}$$

$$f'(x)=-\dfrac{3}{2a^{\frac{3}{2}}}$$

16) $$f(x)=2x^2+9x$$

17) For the graph in the Figure below, determine where the function is continuous/discontinuous and differentiable/not differentiable. discontinuous at $$-2,0$$, not differentiable at $$-2,0, 2$$.

For the exercises 18-19, 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.

18) $$f(x)=\left | x-2 \right | - \left | x+2 \right |$$

19) $$f(x)=\dfrac{2}{1+e^{\frac{2}{x}}}$$

not differentiable at $$x=0$$ (no limit)

For the exercises 20-24, explain the notation in words when the height of a projectile in feet, $$s$$, is a function of time $$t$$ in seconds after launch and is given by the function

20) $$s(0)$$

21) $$s(2)$$

the height of the projectile at $$t=2$$ seconds

22) $$s'(2)$$

23) $$\dfrac{s(2)-s(1)}{2-1}$$

the average velocity from

24) $$s(t)=0$$

For the exercises 25-28, use technology to evaluate the limit.

25) $$\lim \limits_{x \to 0}\dfrac{\sin (x)}{3x}$$

$$\dfrac{1}{3}$$

26) $$\lim \limits_{x \to 0}\dfrac{\tan ^2(x)}{2x}$$

27) $$\lim \limits_{x \to 0}\dfrac{\sin (x)(1-\cos (x))}{2x^2}$$

$$0$$

28) Evaluate the limit by hand.

$\lim \limits_{x \to 1}f(x), \text{ where } f(x)=\begin{cases} 4x-7 & x\neq 1 \\ x^2-4 & x= 1 \end{cases} \nonumber$

At what value(s) of $$x$$ is the function below discontinuous?

$f(x)=\begin{cases} 4x-7 & x\neq 1 \\ x^2-4 & x= 1 \end{cases} \nonumber$

For the exercises 29-32, consider the function whose graph appears in Figure. 29) Find the average rate of change of the function from $$x=1$$ to $$x=3$$.

$$2$$

30) Find all values of $$x$$ at which $$f'(x)=0$$.

$$x=1$$

31) Find all values of $$x$$ at which $$f'(x)$$ does not exist.

32) Find an equation of the tangent line to the graph of $$f$$ the indicated point: $$f(x)=3x^2-2x-6,\; x=-2$$

$$y=-14x-18$$

For the exercises 33-34, use the function $$f(x)=x(1-x)^{\frac{2}{5}}$$

33) Graph the function $$f(x)=x(1-x)^{\tfrac{2}{5}}$$ by entering $$f(x)=x\left ((1-x)^2 \right )^{\tfrac{1}{5}}$$ and then by entering $$f(x)=x\left ((1-x)^{\tfrac{1}{5}} \right )^2$$.

34) Explore the behavior of the graph of $$f(x)$$ around $$x=1$$ 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

The graph is not differentiable at $$x=1$$ (cusp).

For the exercises 35-42, find the derivative of each of the functions using the definition: $$\lim \limits_{h \to 0} \dfrac{f(x+h)-f(x)}{h}$$

35) $$f(x)=2x-8$$

36) $$f(x)=4x^2-7$$

$$f'(x)=8x$$

37) $$f(x)=x-\dfrac{1}{2}x^2$$

38) $$f(x)=\dfrac{1}{x+2}$$

$$f'(x)=-\dfrac{1}{(2+x)^2}$$

39) $$f(x)=\dfrac{3}{x-1}$$

40) $$f(x)=-x^3+1$$

$$f'(x)=-3x^2$$
41) $$f(x)=x^2+x^3$$
42) $$f(x)=\sqrt{x-1}$$
$$f'(x)=-\dfrac{1}{2\sqrt{x-1}}$$