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Chapter 2

Limits and Derivatives

Educators

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Problem 1

Use the given graph to estimate the value of each derivative. Then sketch the graph of $ f' $:

(a) $ f'(-3) $
(b) $ f'(-2) $
(c) $ f'(-1) $
(d) $ f'(0) $
(e) $ f'(1) $
(f) $ f'(2) $
(g) $ f'(3) $

Eduard S.
Numerade Educator

Problem 2

Use the given graph to estimate the value of each derivative. Then sketch the graph of $ f' $:

(a) $ f'(0) $
(b) $ f'(1) $
(c) $ f'(2) $
(d) $ f'(3) $
(e) $ f'(4) $
(f) $ f'(5) $
(g) $ f'(6) $
(h) $ f'(7) $

Daniel J.
Numerade Educator

Problem 3

Match the graph of each function in (a)-(d) with the graph of its derivative in I-IV. Give reasons for your choices.

Daniel J.
Numerade Educator

Problem 4

Trace or copy the graph of the given function $ f $. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of $ f' $ below it.

DM
David M.
Numerade Educator

Problem 5

Trace or copy the graph of the given function $ f $. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of $ f' $ below it.

DM
David M.
Numerade Educator

Problem 6

Trace or copy the graph of the given function $ f $. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of $ f' $ below it.

Aparna S.
Numerade Educator

Problem 7

Trace or copy the graph of the given function $ f $. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of $ f' $ below it.

DM
David M.
Numerade Educator

Problem 8

Trace or copy the graph of the given function $ f $. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of $ f' $ below it.

Aparna S.
Numerade Educator

Problem 9

Trace or copy the graph of the given function $ f $. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of $ f' $ below it.

Aparna S.
Numerade Educator

Problem 10

Trace or copy the graph of the given function $ f $. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of $ f' $ below it.

Daniel J.
Numerade Educator

Problem 11

Trace or copy the graph of the given function $ f $. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of $ f' $ below it.

Aparna S.
Numerade Educator

Problem 12

Shown is the graph of the population function $ P(t) $ for yeast cells in a laboratory culture. Use the method of Example 1 to graph the derivative $ P'(t) $. What does the graph $ P' $ tell us about the yeast population?

DM
David M.
Numerade Educator

Problem 13

A rechargeable battery is plugged into a charger. The graph shows $C(t),$ the percentage of full capacity that the battery reaches as a function of time $t$ elapsed (in hours).
(a) What is the meaning of the derivative $C^{\prime}(t) ?$
(b) Sketch the graph of $C^{\prime}(t) .$ What does the graph tell you?

DM
David M.
Numerade Educator

Problem 14

The graph (from the US Department of Energy) shows how driving speed affects gas mileage. Fuel economy $ F $ is measured in miles per gallon and speed $ v $ is measured in miles per hour.

(a) What is the meaning of the derivative $ F'(v) $?

(b) Sketch the graph of $ F'(v) $.

(c) At what speed should you drive if you want to save on gas?

DM
David M.
Numerade Educator

Problem 15

The graph shows how the average age of first marriage of Japanese men varied in the last half of the 20th century. Sketch the graph of the derivative function $ M'(t) $. During which years was the derivative negative?

DM
David M.
Numerade Educator

Problem 16

Make a careful sketch of the graph of $ f $ and below it sketch the graph of $ f' $ in the same manner as in Exercises 4-11. Can you guess a formula for $ f'(x) $ from its graph?

$ f(x) = \sin x $

Aparna S.
Numerade Educator

Problem 17

Make a careful sketch of the graph of $ f $ and below it sketch the graph of $ f' $ in the same manner as in Exercises 4-11. Can you guess a formula for $ f'(x) $ from its graph?

$ f(x) = e^x $

Daniel J.
Numerade Educator

Problem 18

Make a careful sketch of the graph of $ f $ and below it sketch the graph of $ f' $ in the same manner as in Exercises 4-11. Can you guess a formula for $ f'(x) $ from its graph?

$ f(x) = \ln x $

Daniel J.
Numerade Educator

Problem 19

Let $ f(x) = x^2 $.

(a) Estimate the values of $ f'(0) $, $ f'(\frac{1}{2}) $, $ f'(1) $, and $ f'(2) $ by using a graphing device to zoom in on the graph of $ f $.

(b) Use symmetry to deduce the values of $ f'(-\frac{1}{2}) $, $ f'(-1) $, and $ f'(-2) $.

(c) Use the results from parts (a) and (b) to guess a formula for $ f'(x) $.

(d) Use the definition of derivative to prove that your guess in part (c) is correct.

DM
David M.
Numerade Educator

Problem 20

Let $ f(x) = x^3 $.

(a) Estimate the values of $ f'(0) $, $ f'(\frac{1}{2}) $, $ f'(1) $, $ f'(2) $, and $ f'(3) $ by using a graphing device to zoom in on the graph of $ f $.

(b) Use symmetry to deduce the values of $ f'(-\frac{1}{2}) $, $ f'(-1) $, $ f'(-2) $, and $ f'(-3) $.

(c) Use the values from parts (a) and (b) to graph $ f' $.

(d) Guess a formula for $ f'(x) $.

(e) Use the definition of derivative to prove that your guess in part (d) is correct.

DM
David M.
Numerade Educator

Problem 21

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.

$ f(x) = 3x - 8 $

Leon D.
Numerade Educator

Problem 22

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.

$ f(x) = mx + b $

Leon D.
Numerade Educator

Problem 23

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.

$ f(t) = 2.5t^2 + 6t $

Leon D.
Numerade Educator

Problem 24

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.

$ f(x) = 4 + 8x - 5x^2 $

Leon D.
Numerade Educator

Problem 25

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.

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

Evelyn C.
Numerade Educator

Problem 26

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.

$ g(t) = \dfrac{1}{\sqrt{t}} $

Ma. Theresa A.
Numerade Educator

Problem 27

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.

$ g(x) = \sqrt{9 - x} $

Ma. Theresa A.
Numerade Educator

Problem 28

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.

$ f(x) = \dfrac{x^2 - 1}{2x - 3} $

Evelyn C.
Numerade Educator

Problem 29

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.

$G(t)=\frac{1-2 t}{3+t}$

Ma. Theresa A.
Numerade Educator

Problem 30

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.

$ f(x) = x^{3/2} $

DM
David M.
Numerade Educator

Problem 31

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.

$ f(x) = x^4 $

Ma. Theresa A.
Numerade Educator

Problem 32

(a) Sketch the graph of $ f(x) = \sqrt{6 - x} $ by starting with the graph of $ y = \sqrt{x} $ and using the transformations of Section 1.3.

(b) Use the graph from part (a) to sketch the graph of $ f' $.

(c) Use the definition of a derivative to find $ f'(x) $. What are the domains of $ f $ and $ f' $?

(d) Use a graphing device to graph $ f' $ and compare with your sketch in part (b).

Linda H.
Numerade Educator

Problem 33

(a) If $ f(x) = x^4 + 2x $, find $ f'(x) $.
(b) Check to see that your answer to part (a) is reasonable by comparing the graphs of $ f $ and $ f' $.

Daniel J.
Numerade Educator

Problem 34

(a) If $ f(x) = x + 1/x $, find $ f'(x) $.
(b) Check to see that your answer to part (a) is reasonable by comparing the graphs of $ f $ and $ f' $.

Daniel J.
Numerade Educator

Problem 35

The unemployment rate $ U(t) $ varies with time. The table gives the percentage of unemployed in the US labor force from 2003 to 2012.

(a) What is the meaning of $ U'(t) $? What are its units?

(b) Construct a table of estimated values for $ U'(t) $.

Daniel J.
Numerade Educator

Problem 36

The table gives the number $ N(t) $, measured in thousands, of minimally invasive cosmetic surgery procedures performed in the United States for various years $ t $.

(a) What is the meaning of $ N'(t) $? What are its units?

(b) Construct a table of estimated values for $ N'(t) $.

(c) Graph $ N $ and $ N' $.

(d) How would it be possible to get more accurate values for $ N'(t) $?

DM
David M.
Numerade Educator

Problem 37

The table gives the height as time passes of a typical pine tree grown for lumber at a managed site.

If $ H(t) $ is the height of the tree after $ t $ years, construct a table of estimated values for $ H' $ and sketch its graph.

DM
David M.
Numerade Educator

Problem 38

Water temperature affects the growth rate of brook trout. The table shows the amount of weight gained by brook trout after 24 days in various water temperatures.

If $ W(x) $ is the weight gain at temperature $ x $, construct a table of estimated values for $ W' $ and sketch its graph. What are the units for $ W'(x) $?

DM
David M.
Numerade Educator

Problem 39

Let $ P $ represent the percentage of a city's electrical power that is produced by solar panels $ t $ years after January 1, 2000.

(a) What does $ dP/dt $ represent in this context?
(b) Interpret the statement $$ \frac{dP}{dt} \bigg|_{t = 2} = 3.5 $$

Leon D.
Numerade Educator

Problem 40

Suppose $ N $ is the number of people in the United States who travel by car to another state for a vacation this year when the average price of gasoline is $ p $ dollars per gallon. Do you expect $ dN/dp $ to be positive or negative? Explain.

Leon D.
Numerade Educator

Problem 41

The graph of $ f $ is given. State, with reasons, the numbers at which $ f $ is $ not $ differentiable.

JT
Jessie T.
Numerade Educator

Problem 42

The graph of $ f $ is given. State, with reasons, the numbers at which $ f $ is $ not $ differentiable.

Aparna S.
Numerade Educator

Problem 43

The graph of $ f $ is given. State, with reasons, the numbers at which $ f $ is $ not $ differentiable.

DM
David M.
Numerade Educator

Problem 44

The graph of $ f $ is given. State, with reasons, the numbers at which $ f $ is $ not $ differentiable.

Aparna S.
Numerade Educator

Problem 45

Graph the function $ f(x) = x + \sqrt{| x |} $. Zoom in repeatedly, first toward the point $ (-1, 0) $ and then toward the origin. What is different about the behavior of $ f $ in the vicinity of these two points? What do you conclude about the differentiability of $ f $?

Daniel J.
Numerade Educator

Problem 46

Zoom in toward the points $ (1, 0) $, $ (0, 1) $, and $ (-1, 0) $ on the graph of the function
$ g(x) = (x^2 -1)^{2/3} $. What do you notice? Account for what you see in terms of the differentiability of $ g $.

Daniel J.
Numerade Educator

Problem 47

The graphs of a function $ f $ and its derivative $ f' $ are shown. Which is bigger, $ f'(-1) $ or $ f''(1) $?

Aparna S.
Numerade Educator

Problem 48

The graphs of a function $ f $ and its derivative $ f' $ are shown. Which is bigger, $ f'(-1) $ or $ f''(1) $?

Aparna S.
Numerade Educator

Problem 49

The figure shows the graphs of $ f $, $ f' $, and $ f'' $. Identify each curve, and explain your choices.

DM
David M.
Numerade Educator

Problem 50

The figure shows graphs of $ f $, $ f' $, $ f'' $, and $ f''' $. Identify each curve, and explain your choices.

Aparna S.
Numerade Educator

Problem 51

The figure shows the graphs of three functions. One is the position function of a car, one is the velocity of the car, and one is its acceleration. Identify each curve, and explain your choices.

DM
David M.
Numerade Educator

Problem 52

The figure shows the graphs of four functions. One is the position function of a car, one is the velocity of the car, one is its acceleration, and one is its jerk. Identify each curve, and explain your choices.

Daniel J.
Numerade Educator

Problem 53

Use the definition of a derivative to find $ f'(x) $ and $ f''(x) $. Then graph $ f $, $ f' $, and $ f'' $ on a common screen and check to see if your answers are reasonable.

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

Daniel J.
Numerade Educator

Problem 54

Use the definition of a derivative to find $ f'(x) $ and $ f''(x) $. Then graph $ f $, $ f' $, and $ f'' $ on a common screen and check to see if your answers are reasonable.

$ f(x) = x^3 - 3x $

Daniel J.
Numerade Educator

Problem 55

If $ f(x) = 2x^2 - x^3 $, find $ f'(x) $, $ f''(x) $, $ f'''(x) $, and $ f^{(4)}(x) $. Graph $ f $, $ f' $, $ f'' $, and
$ f''' $ on a common screen. Are the graphs consistent with the geometric interpretations of these derivatives?

Leon D.
Numerade Educator

Problem 56

(a) The graph of a position function of a car is shown, where $ s $ is measured in feet and $ t $ in seconds. Use it to graph the velocity and acceleration of the car. What is the acceleration at $ t = 10 $ seconds?

(b) Use the acceleration curve from part (a) to estimate the jerk at $ t = 10 $ seconds. What are the units for jerk?

Daniel J.
Numerade Educator

Problem 57

Let $ f(x) = \sqrt[3]{x} $.
(a) If $ a \neq 0 $, use Equation 2.7.5 to find $ f'(a) $.
(b) Show that $ f'(0) $ does not exist.
(c) Show that $ y = \sqrt[3]{x} $ has a vertical tangent line at $ (0, 0) $. (Recall the shape of the graph of
$ f $. See Figure 1.2.13)

AD
Anupa D.
Numerade Educator

Problem 58

(a) If $ g(x) = x^{2/3} $, show that $ g'(0) $ does not exist.
(b) If $ a \neq 0 $, find $ g'(a) $.
(c) Show that $ y = x^{2/3} $ has a vertical tangent line at $ (0, 0) $.
(d) Illustrate part (c) by graphing $ y = x^{2/3} $.

Daniel J.
Numerade Educator

Problem 59

Show that the function $ f(x) = | x - 6 | $ is not differentiable at 6. Find a formula for $ f' $ and sketch its graph.

Leon D.
Numerade Educator

Problem 60

Where is the greatest integer function $ f(x) = [ x ] $ not differentiable? Find a formula for $ f' $ and sketch its graph.

Darshan M.
Numerade Educator

Problem 61

(a) Sketch the graph of the function $ f(x) = x | x | $.
(b) For what values of $ x $ is $ f $ differentiable?
(c) Find a formula for $ f' $.

Daniel J.
Numerade Educator

Problem 62

(a) Sketch the graph of the function $ g(x) = x + | x | $.
(b) For what values of $ x $ is $ g $ differentiable?
(c) Find a formula for $ g' $.

Daniel J.
Numerade Educator

Problem 63

Recall that a function $ f $ is called \textit{even} if $ f(-x) = f(x) $ for all $ x $ in its domain and \textit{odd} if $ f(-x) = -f(x) $ for all such $ x $. Prove each of the following.
(a) The derivative of an even function is an odd function.
(b) The derivative of an odd function is an even function.

Daniel J.
Numerade Educator

Problem 64

The left-hand and right-hand derivatives of $f$ at $a$ are defined by
$$\begin{array}{c}{f_{-}^{\prime}(a)=\lim _{h \rightarrow 0^{-}} \frac{f(a+h)-f(a)}{h}} \\ {\text { and } \quad f_{+}^{\prime}(a)=\lim _{h \rightarrow 0^{-}} \frac{f(a+h)-f(a)}{h}}\end{array}$$
if these limits exist. Then $f^{\prime}(a)$ exists if and only if these one-sided derivatives exist and are equal. (a) Find $f^{\prime}-(4)$ and $f^{\prime}+(4)$ for the function
$$f(x)=\left\{\begin{array}{l}{0} \\ {5-x} \\ {\frac{1}{5-x}}\end{array}\right.$$
if $x \leq 0$
if $0< x<4 $
if $x \geqslant 4$
(b) Sketch the graph of $f$ .
(c) Where is $f$ discontinuous?
(d) Where is $f$ not differentiable?

Will E.
Numerade Educator

Problem 65

Nick starts jogging and runs faster and faster for 3 minutes, then he walks for 5 minutes. He stops at an intersection for 2 minutes, runs fairly quickly for 5 minutes, then walks for 4 minutes.
$$\begin{array}{l}{\text { (a) Sketch a possible graph of the distance } s \text { Nick has covered }} \\ {\text t \text { minutes. }} \\ {\text { (b) Sketch a graph of } d s / d t}\end{array}$$

Pawan Y.
Numerade Educator

Problem 66

When you turn on a hot-water faucet, the temperature $T$ of the water depends on how long the water has been running.
$$\begin{array}{l}{\text { (a) Sketch a possible graph of } T \text { as a function of the time } t} \\ {\text { that has elapsed since the faucet was turned on. }} \\ {\text { (b) Describe how the rate of change of } T \text { with respect to } t} \\ {\text { varies as } t \text { increases. }} \\ {\text { (c) Sketch a graph of the derivative of } T .}\end{array}$$

Carson M.
Numerade Educator

Problem 67

Let $\ell$ be the tangent line to the parabola $y=x^{2}$ at the point
$(1,1)$ . The angle of inclination of $\ell$ is the angle $\phi$ that $\ell$
makes with the positive direction of the $x$ -axis. Calculate $\phi$
correct to the nearest degree.

Madi S.
Numerade Educator