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Calculus

James Stewart

Chapter 2

Derivatives - all with Video Answers

Educators

Section 1

2.1 Exercises

02:10

Problem 1

A curve has equation $y=f(x)$.
(a) Write an expression for the slope of the secant line through the points $P(3, f(3))$ and $Q(x, f(x))$.
(b) Write an expression for the slope of the tangent line at $P$.

Daniel Jaimes
Daniel Jaimes
Numerade Educator
02:08

Problem 2

Graph the curve $y=\sin x$ in the viewing rectangles $[-2,2]$ by $[-2,2],[-1,1]$ by $[-1,1]$, and $[-0.5,0.5]$ by
$[-0.5,0.5]$. What do you notice about the curve as you zoom in toward the origin?

Ryan Pollard
Ryan Pollard
Numerade Educator
05:39

Problem 3

(a) Find the slope of the tangent line to the parabola $y=4 x-x^2$ at the point $(1,3)$
(i) using Definition 1
(ii) using Equation 2
(b) Find an equation of the tangent line in part (a).
(c) Graph the parabola and the tangent line. As a check on your work, zoom in toward the point $(1,3)$ until the parabola and the tangent line are indistinguishable.

Angelina Chavez
Angelina Chavez
Numerade Educator
08:42

Problem 4

(a) Find the slope of the tangent line to the curve $y=x-x^3$ at the point $(1,0)$
(i) using Definition 1
(ii) using Equation 2
(b) Find an equation of the tangent line in part (a).
(c) Graph the curve and the tangent line in successively smaller viewing rectangles centered at $(1,0)$ until the curve and the line appear to coincide.

Benjamin Schreyer
Benjamin Schreyer
Numerade Educator
03:05

Problem 5

Find an equation of the tangent line to the curve at the given point.
$y=4 x-3 x^2, \quad(2,-4)$

Bobby Barnes
Bobby Barnes
University of North Texas
01:45

Problem 6

Find an equation of the tangent line to the curve at the given point.
$y=x^3-3 x+1$,$(2,3)$

Allan Hungria
Allan Hungria
Numerade Educator
02:32

Problem 7

Find an equation of the tangent line to the curve at the given point.
$y=\sqrt{x},(1,1)$

Bobby Barnes
Bobby Barnes
University of North Texas
02:42

Problem 8

Find an equation of the tangent line to the curve at the given point.
$y=\frac{2 x+1}{x+2}, \quad(1,1)$

Heather Eichman
Heather Eichman
Numerade Educator
05:01

Problem 9

(a) Find the slope of the tangent to the curve $y=3+4 x^2-2 x^3$ at the point where $x=a$.
(b) Find equations of the tangent lines at the points $(1,5)$ and $(2,3)$.
(c) Graph the curve and both tangents on a common screen.

Bobby Barnes
Bobby Barnes
University of North Texas
06:37

Problem 10

(a) Find the slope of the tangent to the curve $y=1 / \sqrt{x}$ at the point where $x=a$.
(b) Find equations of the tangent lines at the points $(1,1)$ and $\left(4, \frac{1}{2}\right)$.
(c) Graph the curve and both tangents on a common screen.

Bobby Barnes
Bobby Barnes
University of North Texas
08:08

Problem 11

(a) A particle starts by moving to the right along a horizontal line; the graph of its position function is shown. When is the particle moving to the right? Moving to the left? Standing still?
(b) Draw a graph of the velocity function.

Bobby Barnes
Bobby Barnes
University of North Texas
03:37

Problem 12

Shown are graphs of the position functions of two runners, A and $B$, who run a $100-\mathrm{m}$ race and finish in a tie.
(a) Describe and compare how the runners run the race.
(b) At what time is the distance between the runners the greatest?
(c) At what time do they have the same velocity?

David Mccaslin
David Mccaslin
Numerade Educator
03:24

Problem 13

If a ball is thrown into the air with a velocity of $40 \mathrm{ft} / \mathrm{s}$, its height (in feet) after $t$ seconds is given by $y=40 t-16 t^2$. Find the velocity when $t=2$.

Bobby Barnes
Bobby Barnes
University of North Texas
10:53

Problem 14

If a rock is thrown upward on the planet Mars with a velocity of $10 \mathrm{~m} / \mathrm{s}$, its height (in meters) after $t$ seconds is given by $H=10 t-1.86 t^2$.
(a) Find the velocity of the rock after one second.
(b) Find the velocity of the rock when $t=a$.
(c) When will the rock hit the surface?
(d) With what velocity will the rock hit the surface?

Bobby Barnes
Bobby Barnes
University of North Texas
10:53

Problem 14

If a rock is thrown upward on the planet Mars with a velocity of $10 \mathrm{~m} / \mathrm{s}$, its height (in meters) after $t$ seconds is given by $H=10 t-1.86 t^2$.
(a) Find the velocity of the rock after one second.
(b) Find the velocity of the rock when $t=a$.
(c) When will the rock hit the surface?
(d) With what velocity will the rock hit the surface?

Bobby Barnes
Bobby Barnes
University of North Texas
04:12

Problem 15

The displacement (in meters) of a particle moving in a straight line is given by the equation of motion $s=1 / t^2$, where $t$ is measured in seconds. Find the velocity of the particle at times $t=a, t=1, t=2$, and $t=3$.

Bobby Barnes
Bobby Barnes
University of North Texas
01:17

Problem 16

The displacement (in meters) of a particle moving in a straight line is given by $s=t^2-8 t+18$, where $t$ is measured in seconds.
(a) Find the average velocity over each time interval:
(i) $[3,4]$
(ii) $[3.5 .4]$
(iii) $[4,5]$
(iv) $[4,4.5]$
(b) Find the instantaneous velocity when $t=4$.
(c) Draw the graph of $s$ as a function of $t$ and draw the secant lines whose slopes are the average velocities in part (a) and the tangent line whose slope is the instantaneous velocity in part (b).

Carson Merrill
Carson Merrill
Numerade Educator
02:20

Problem 17

For the function $g$ whose graph is given, arrange the following numbers in increasing order and explain your reasoning:

$$
0 \quad g^{\prime}(-2) \quad g^{\prime}(0) \quad g^{\prime}(2) \quad g^{\prime}(4)
$$

Bobby Barnes
Bobby Barnes
University of North Texas
02:21

Problem 18

Find an equation of the tangent line to the graph of $y=g(x)$ at $x=5$ if $g(5)=-3$ and $g^{\prime}(5)=4$.

Allan Hungria
Allan Hungria
Numerade Educator
03:44

Problem 19

If an equation of the tangent line to the curve $y=f(x)$ at the point where $a=2$ is $y=4 x-5$, find $f(2)$ and $f^{\prime}(2)$.

FR
Fady Ramsis
Numerade Educator
03:52

Problem 20

If the tangent line to $y=f(x)$ at $(4,3)$ passes through the point $(0,2)$, find $f(4)$ and $f^{\prime}(4)$.

Allan Hungria
Allan Hungria
Numerade Educator
View

Problem 21

Sketch the graph of a function $f$ for which $f(0)=0$, $f^{\prime}(0)=3, f^{\prime}(1)=0$, and $f^{\prime}(2)=-1$.

Abbey Knoll
Abbey Knoll
Numerade Educator
04:05

Problem 22

Sketch the graph of a function $g$ for which

$$
\begin{aligned}
& g(0)=g(2)=g(4)=0, g^{\prime}(1)=g^{\prime}(3)=0, g^{\prime}(0)=g^{\prime}(4)=1, \\
& g^{\prime}(2)=-1, \lim _{x \rightarrow 5^{-}} g(x)=\infty, \text { and } \lim _{x \rightarrow-1^{+}} g(x)=-\infty .
\end{aligned}
$$

Allan Hungria
Allan Hungria
Numerade Educator
03:17

Problem 23

If $f(x)=3 x^2-x^3$, find $f^{\prime}(1)$ and use it to find an equation of the tangent line to the curve $y=3 x^2-x^3$ at the point $(1,2)$.

Bobby Barnes
Bobby Barnes
University of North Texas
03:20

Problem 24

If $g(x)=x^4-2$, find $g^{\prime}(1)$ and use it to find an equation of the tangent line to the curve $y=x^4-2$ at the point $(1,-1)$.

Allan Hungria
Allan Hungria
Numerade Educator
03:12

Problem 25

(a) If $F(x)=5 x /\left(1+x^2\right)$, find $F^{\prime}(2)$ and use it to find an equation of the tangent line to the curve $y=5 x /\left(1+x^2\right)$ at the point $(2,2)$.
(b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.

Heather Eichman
Heather Eichman
Numerade Educator
05:27

Problem 26

(a) If $G(x)=4 x^2-x^3$, find $G^{\prime}(a)$ and use it to find equations of the tangent lines to the curve $y=4 x^2-x^3$ at the points $(2,8)$ and $(3,9)$.
(b) Illustrate part (a) by graphing the curve and the tangent lines on the same screen.

Heather Eichman
Heather Eichman
Numerade Educator
01:25

Problem 27

Find $f^{\prime}(a)$.
$f(x)=3 x^2-4 x+1$

Leon Druch
Leon Druch
Numerade Educator
View

Problem 28

Find $f^{\prime}(a)$.
$f(t)=2 t^3+t$

Amelia Hardy
Amelia Hardy
Numerade Educator
01:08

Problem 29

Find $f^{\prime}(a)$.
$f(t)=\frac{2 t+1}{t+3}$

Anthony Han
Anthony Han
Numerade Educator
View

Problem 30

Find $f^{\prime}(a)$.
$f(x)=x^{-2}$

David Mccaslin
David Mccaslin
Numerade Educator
04:05

Problem 31

Find $f^{\prime}(a)$.
$f(x)=\sqrt{1-2 x}$

Leon Druch
Leon Druch
Numerade Educator
View

Problem 32

Find $f^{\prime}(a)$.
$f(x)=\frac{4}{\sqrt{1-x}}$

Amelia Hardy
Amelia Hardy
Numerade Educator
01:04

Problem 33

Each limit represents the derivative of some function $f$ at some number $a$. State such an $f$ and $a$ in each case.
$\lim _{h \rightarrow 0} \frac{(1+h)^{10}-1}{h}$

Carson Merrill
Carson Merrill
Numerade Educator
02:09

Problem 34

Each limit represents the derivative of some function $f$ at some number $a$. State such an $f$ and $a$ in each case.
$\lim _{h \rightarrow 0} \frac{\sqrt[4]{16+h}-2}{h}$

Aayush Gupta
Aayush Gupta
Numerade Educator
00:59

Problem 35

Each limit represents the derivative of some function $f$ at some number $a$. State such an $f$ and $a$ in each case.
$\lim _{x \rightarrow 5} \frac{2^x-32}{x-5}$

Angelina Chavez
Angelina Chavez
Numerade Educator
01:18

Problem 36

Each limit represents the derivative of some function $f$ at some number $a$. State such an $f$ and $a$ in each case.
$\lim _{x \rightarrow \pi / 4} \frac{\tan x-1}{x-\pi / 4}$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
00:27

Problem 37

Each limit represents the derivative of some function $f$ at some number $a$. State such an $f$ and $a$ in each case.
$\lim _{h \rightarrow 0} \frac{\cos (\pi+h)+1}{h}$

Amy Jiang
Amy Jiang
Numerade Educator
01:35

Problem 38

Each limit represents the derivative of some function $f$ at some number $a$. State such an $f$ and $a$ in each case.
$\lim _{t \rightarrow 1} \frac{t^4+t-2}{t-1}$

Aayush Gupta
Aayush Gupta
Numerade Educator
02:01

Problem 39

A particle moves along a straight line with equation of motion $s=f(t)$, where $s$ is measured in meters and $t$ in seconds. Find the velocity and the speed when $t=5$.
$f(t)=100+50 t-4.9 t^2$

Charles Machakwa
Charles Machakwa
Numerade Educator
02:10

Problem 40

A particle moves along a straight line with equation of motion $s=f(t)$, where $s$ is measured in meters and $t$ in seconds. Find the velocity and the speed when $t=5$.
$f(t)=t^{-1}-t$

Charles Machakwa
Charles Machakwa
Numerade Educator
04:04

Problem 41

A warm can of soda is placed in a cold refrigerator. Sketch the graph of the temperature of the soda as a function of time. Is the initial rate of change of temperature greater or less than the rate of change after an hour?

Daniel Jaimes
Daniel Jaimes
Numerade Educator
01:54

Problem 42

A roast turkey is taken from an oven when its temperature has reached $185^{\circ} \mathrm{F}$ and is placed on a table in a room where the temperature is $75^{\circ} \mathrm{F}$. The graph shows how the temperature of the turkey decreases and eventually approaches room temperature. By measuring the slope of the tangent, estimate the rate of change of the temperature after an hour.

Allan Hungria
Allan Hungria
Numerade Educator
02:56

Problem 43

The number $N$ of US cellular phone subscribers (in millions) is shown in the table. (Midyear estimates are given.)
$$
\begin{array}{|c|c|c|c|c|c|c|}
\hline t & 1996 & 1998 & 2000 & 2002 & 2004 & 2006 \\
\hline N & 44 & 69 & 109 & 141 & 182 & 233 \\
\hline
\end{array}
$$
(a) Find the average rate of cell phone growth
(i) from 2002 to 2006
(ii) from 2002 to 2004
(iii) from 2000 to 2002

In each case, include the units.
(b) Estimate the instantaneous rate of growth in 2002 by taking the average of two average rates of change. What are its units?
(c) Estimate the instantaneous rate of growth in 2002 by measuring the slope of a tangent.

Carson Merrill
Carson Merrill
Numerade Educator
03:45

Problem 44

The number $N$ of locations of a popular coffeehouse chain is given in the table. (The numbers of locations as of October 1 are given.)
$$
\begin{array}{|c|c|c|c|c|c|}
\hline \text { Year } & 2004 & 2005 & 2006 & 2007 & 2008 \\
\hline N & 8569 & 10,241 & 12,440 & 15,011 & 16,680 \\
\hline
\end{array}
$$
(a) Find the average rate of growth
(i) from 2006 to 2008
(ii) from 2006 to 2007
(iii) from 2005 to 2006

In each case, include the units.
(b) Estimate the instantaneous rate of growth in 2006 by taking the average of two average rates of change. What are its units?
(c) Estimate the instantaneous rate of growth in 2006 by measuring the slope of a tangent.
(d) Estimate the intantaneous rate of growth in 2007 and compare it with the growth rate in 2006 . What do you conclude?

Heather Eichman
Heather Eichman
Numerade Educator
03:19

Problem 45

The cost (in dollars) of producing $x$ units of a certain commodity is $C(x)=5000+10 x+0.05 x^2$.
(a) Find the average rate of change of $C$ with respect to $x$ when the production level is changed
(i) from $x=100$ to $x=105$
(ii) from $x=100$ to $x=101$
(b) Find the instantaneous rate of change of $C$ with respect to $x$ when $x=100$. (This is called the marginal cost. Its significance will be explained in Section 2.7.)

Heather Eichman
Heather Eichman
Numerade Educator
06:31

Problem 46

If a cylindrical tank holds 100,000 gallons of water, which can be drained from the bottom of the tank in an hour, then Torricelli's Law gives the volume $V$ of water remaining in the tank after $t$ minutes as

$$
V(t)=100,000\left(1-\frac{1}{60} t\right)^2 \quad 0 \leqslant t \leqslant 60
$$

Find the rate at which the water is flowing out of the tank (the instantaneous rate of change of $V$ with respect to $t$ ) as a function of $t$. What are its units? For times $t=0,10,20,30,40,50$, and 60 min , find the flow rate and the amount of water remaining in the tank. Summarize your findings in a sentence or two. At what time is the flow rate the greatest? The least?

Heather Eichman
Heather Eichman
Numerade Educator
04:24

Problem 47

The cost of producing $x$ ounces of gold from a new gold mine is $C=f(x)$ dollars.
(a) What is the meaning of the derivative $f^{\prime}(x)$ ? What are its units?
(b) What does the statement $f^{\prime}(800)=17$ mean?
(c) Do you think the values of $f^{\prime}(x)$ will increase or decrease in the short term? What about the long term? Explain.

Bobby Barnes
Bobby Barnes
University of North Texas
05:49

Problem 48

The number of bacteria after $t$ hours in a controlled laboratory experiment is $n=f(t)$.
(a) What is the meaning of the derivative $f^{\prime}(5)$ ? What are its units?
(b) Suppose there is an unlimited amount of space and nutrients for the bacteria. Which do you think is larger, $f^{\prime}(5)$ or $f^{\prime}(10)$ ? If the supply of nutrients is limited, would that affect your conclusion? Explain.

Benjamin Schreyer
Benjamin Schreyer
Numerade Educator
02:37

Problem 49

Let $T(t)$ be the temperature ( in ${ }^{\circ} \mathrm{F}$ ) in Phoenix $t$ hours after midnight on September 10,2008. The table shows values of this function recorded every two hours. What is the meaning of $T^{\prime}(8)$ ? Estimate its value.
$$
\begin{array}{|c|c|c|c|c|c|c|c|c|}
\hline t & 0 & 2 & 4 & 6 & 8 & 10 & 12 & 14 \\
\hline T & 82 & 75 & 74 & 75 & 84 & 90 & 93 & 94 \\
\hline
\end{array}
$$

Benjamin Schreyer
Benjamin Schreyer
Numerade Educator
01:45

Problem 50

The quantity (in pounds) of a gourmet ground coffee that is sold by a coffee company at a price of $p$ dollars per pound is $Q=f(p)$.
(a) What is the meaning of the derivative $f^{\prime}(8)$ ? What are its units?
(b) Is $f^{\prime}(8)$ positive or negative? Explain.

Heather Eichman
Heather Eichman
Numerade Educator
08:40

Problem 51

The quantity of oxygen that can dissolve in water depends on the temperature of the water. (So thermal pollution influences the oxygen content of water.) The graph shows how oxygen solubility $S$ varies as a function of the water temperature $T$.
(a) What is the meaning of the derivative $S^{\prime}(T)$ ? What are its units?
(b) Estimate the value of $S^{\prime}(16)$ and interpret it.

SK
Samantha Kuijpers
Numerade Educator
03:58

Problem 52

The graph shows the influence of the temperature $T$ on the maximum sustainable swimming speed $S$ of Coho salmon.
(a) What is the meaning of the derivative $S^{\prime}(T)$ ? What are its units?
(b) Estimate the values of $S^{\prime}(15)$ and $S^{\prime}(25)$ and interpret them.

Benjamin Schreyer
Benjamin Schreyer
Numerade Educator