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Thomas Calculus

George B. Thomas, Jr.

Chapter 3

Derivatives

Educators


Problem 1

In Exercises $1-4,$ use the grid and a straight edge to make a rough
estimate of the slope of the curve (in $y$ -units per $x$ -unit) at the points
$P_{1}$ and $P_{2}$ .

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Problem 2

Use the grid and a straight edge to make a rough
estimate of the slope of the curve (in $y$ -units per $x$ -unit) at the points
$P_{1}$ and $P_{2}$ .

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Problem 3

Use the grid and a straight edge to make a rough
estimate of the slope of the curve (in $y$ -units per $x$ -unit) at the points
$P_{1}$ and $P_{2}$ .

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Problem 4

Use the grid and a straight edge to make a rough
estimate of the slope of the curve (in $y$ -units per $x$ -unit) at the points
$P_{1}$ and $P_{2}$ .

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Problem 5

In Exercises $5-10,$ find an equation for the tangent to the curve at the
given point. Then sketch the curve and tangent together.
\begin{equation}
y=4-x^{2}, \quad(-1,3)
\end{equation}

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Problem 6

Find an equation for the tangent to the curve at the given point. Then sketch the curve and tangent together.
\begin{equation}
y=(x-1)^{2}+1, \quad(1,1)
\end{equation}

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Problem 7

Find an equation for the tangent to the curve at the given point. Then sketch the curve and tangent together.
\begin{equation}
y=2 \sqrt{x}, \quad(1,2)
\end{equation}

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Problem 8

Find an equation for the tangent to the curve at the given point. Then sketch the curve and tangent together.
\begin{equation}
y=\frac{1}{x^{2}}, \quad(-1,1)
\end{equation}

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Problem 9

Find an equation for the tangent to the curve at the given point. Then sketch the curve and tangent together.
\begin{equation}
y=x^{3}, \quad(-2,-8)
\end{equation}

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Problem 10

Find an equation for the tangent to the curve at the given point. Then sketch the curve and tangent together.
\begin{equation}
y=\frac{1}{x^{3}}, \quad\left(-2,-\frac{1}{8}\right)
\end{equation}

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Problem 11

In Exercises $11-18,$ find the slope of the function's graph at the given point. Then find an equation for the line tangent to the graph there.
\begin{equation}
f(x)=x^{2}+1, \quad(2,5)
\end{equation}

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Problem 12

Find the slope of the function's graph at the given point. Then find an equation for the line tangent to the graph there.
\begin{equation}
f(x)=x-2 x^{2}, \quad(1,-1)
\end{equation}

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Problem 13

Find the slope of the function's graph at the given point. Then find an equation for the line tangent to the graph there.
\begin{equation}
g(x)=\frac{x}{x-2}, \quad(3,3)
\end{equation}

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Problem 14

Find the slope of the function's graph at the given point. Then find an equation for the line tangent to the graph there.
\begin{equation}
g(x)=\frac{8}{x^{2}}, \quad(2,2)
\end{equation}

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Problem 15

Find the slope of the function's graph at the given point. Then find an equation for the line tangent to the graph there.
\begin{equation}
h(t)=t^{3}, \quad(2,8)
\end{equation}

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Problem 16

Find the slope of the function's graph at the given point. Then find an equation for the line tangent to the graph there.
\begin{equation}
h(t)=t^{3}+3 t, \quad(1,4)
\end{equation}

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Problem 17

Find the slope of the function's graph at the given point. Then find an equation for the line tangent to the graph there.
\begin{equation}
f(x)=\sqrt{x}, \quad(4,2)
\end{equation}

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Problem 18

Find the slope of the function's graph at the given point. Then find an equation for the line tangent to the graph there.
\begin{equation}
f(x)=\sqrt{x+1},(8,3)
\end{equation}

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Problem 19

In Exercises $19-22,$ find the slope of the curve at the point indicated.
\begin{equation}
y=5 x-3 x^{2}, \quad x=1
\end{equation}

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Problem 20

Find the slope of the curve at the point indicated.
\begin{equation}
y=x^{3}-2 x+7, \quad x=-2
\end{equation}

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Problem 21

Find the slope of the curve at the point indicated.
\begin{equation}
y=\frac{1}{x-1}, \quad x=3
\end{equation}

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Problem 22

Find the slope of the curve at the point indicated.
\begin{equation}
y=\frac{x-1}{x+1}, \quad x=0
\end{equation}

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Problem 23

Growth of yeast cells $\operatorname{In}$ a controlled laboratory experiment,
yeast cells are grown in an automated cell culture system that
counts the number $P$ of cells present at hourly intervals. The number after $t$ hours is shown in the accompanying figure.
\begin{equation}
\begin{array}{l}{\text { a. Explain what is meant by the derivative } P^{\prime}(5) . \text { What are its }} \\ {\text { units? }} \\ {\text { b. Which is larger, } P^{\prime}(2) \text { or } P^{\prime}(3) ? \text { Give a reason for your }} \\ {\text { answer. }}\end{array}
\end{equation}
\begin{equation}
\begin{array}{l}{\text { c. The quadratic curve capturing the trend of the data points }} \\ {\text { (see Section } 1.4 ) \text { is given by } P(t)=6.10 t^{2}-9.28 t+16.43} \\ {\text { Find the instantaneous rate of growth when } t=5 \text { hours. }}\end{array}
\end{equation}

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Problem 24

Effectiveness of a drug On a scale from 0 to 1, the effectiveness $E$ of a pain-killing drug $t$ hours after entering the bloodstream is displayed in the accompanying figure.
\begin{equation}
\begin{array}{l}{\text { a. At what times does the effectiveness appear to be increasing? }} \\ {\text { What is true about the derivative at those times? }} \\ {\text { b. At what time would you estimate that the drug reaches its }} \\ {\text { maximum effectiveness? What is true about the derivative at }} \\ {\text { that time? What is true about the derivative as time increases }} \\ {\text { in the } 1 \text { hour before your estimated time? }}\end{array}
\end{equation}

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Problem 25

At what points do the graphs of the functions in Exercises 25 and 26 have horizontal tangents?
\begin{equation}
f(x)=x^{2}+4 x-1
\end{equation}

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Problem 26

At what points do the graphs of the functions in Exercises 25 and 26 have horizontal tangents?
\begin{equation}
g(x)=x^{3}-3 x
\end{equation}

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Problem 27

Find equations of all lines having slope $-1$ that are tangent to the
curve $y=1 /(x-1)$ .

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Problem 28

Find an equation of the straight line having slope 1$/ 4$ that is tangent to the curve $y=\sqrt{x} .$

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Problem 29

Object dropped from a tower An object is dropped from the
top of a $100-$ m-high tower. Its height above ground after $t$ sec is
$100-4.9 t^{2}$ m. How fast is it falling 2 sec after it is dropped?

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Problem 30

Speed of a rocket $A t t$ sec after liftoff, the height of a rocket is
3$t^{2}$ ft. How fast is the rocket climbing 10 sec after liftoff?

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Problem 31

Circle's changing area What is the rate of change of the area
of a circle $\left(A=\pi r^{2}\right)$ with respect to the radius when the radius
is $r=3 ?$

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Problem 32

Ball's changing volume
ume of a ball $\left(V=(4 / 3) \pi r^{3}\right)$ with respect to the radius when
the radius is $r=2 ?$

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Problem 33

Show that the line $y=m x+b$ is own tangent line at any
point $\left(x_{0}, m x_{0}+b\right)$

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Problem 34

Find the slope of the tangent to the curve $y=1 / \sqrt{x}$ at the point
where $x=4 .$

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Problem 35

Does the graph of
\begin{equation}
f(x)=\left\{\begin{array}{ll}{x^{2} \sin (1 / x),} & {x \neq 0} \\ {0,} & {x=0}\end{array}\right.
\end{equation}
have a tangent at the origin? Give reasons for your answer.

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Problem 36

Does the graph of
\begin{equation}
g(x)=\left\{\begin{array}{ll}{x \sin (1 / x),} & {x \neq 0} \\ {0,} & {x=0}\end{array}\right.
\end{equation}
have a tangent at the origin? Give reasons for your answer.
We say that a continuous curve $y=f(x)$ has a vertical tangent at the
point where $x=x_{0}$ if the limit of the difference quotient is $\infty$ or $-\infty$ .
For example, $y=x^{1 / 3}$ has a vertical tangent at $x=0$ (see accompanying figure):
\begin{equation}
\begin{aligned} \lim _{h \rightarrow 0} \frac{f(0+h)-f(0)}{h} &=\lim _{h \rightarrow 0} \frac{h^{1 / 3}-0}{h} \\ &=\lim _{h \rightarrow 0} \frac{1}{h^{2 / 3}}=\infty \end{aligned}
\end{equation}
However, $y=x^{2 / 3}$ has no vertical tangent at $x=0$ (see next figure):
\begin{equation}
\begin{aligned} \lim _{h \rightarrow 0} \frac{g(0+h)-g(0)}{h} &=\lim _{h \rightarrow 0} \frac{h^{2 / 3}-0}{h} \\ &=\lim _{h \rightarrow 0} \frac{1}{h^{1 / 3}} \end{aligned}
\end{equation}

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Problem 37

Does the graph of
\begin{equation}
f(x)=\left\{\begin{array}{cc}{-1,} & {x<0} \\ {0,} & {x=0} \\ {1,} & {x>0}\end{array}\right.
\end{equation}
have a vertical tangent at the origin? Give reasons for your answer.

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Problem 38

Does the graph of
\begin{equation}
U(x)=\left\{\begin{array}{ll}{0,} & {x<0} \\ {1,} & {x \geq 0}\end{array}\right.
\end{equation}
have a vertical tangent at the point $(0,1) ?$ Give reasons for your
answer.

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Problem 39

Graph the curves in Exercises $39-48$
\begin{equation}
\begin{array}{l}{\text { a. Where do the graphs appear to have vertical tangents? }} \\ {\text { b. Confirm your findings in part (a) with limit calculations. But }} \\ {\text { before you do, read the introduction to Exercises } 37 \text { and } 38 .}\end{array}
\end{equation}
\begin{equation}
y=x^{2 / 5}
\end{equation}

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Problem 40

Graph the curves.
\begin{equation}\begin{array}{l}{\text { a. Where do the graphs appear to have vertical tangents? }} \\ {\text { b. Confirm your findings in part (a) with limit calculations. But }} \\ {\text { before you do, read the introduction to Exercises } 37 \text { and } 38 .}\end{array}\end{equation}
\begin{equation}
y=x^{4 / 5}
\end{equation}

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Problem 41

Graph the curves.
\begin{equation}\begin{array}{l}{\text { a. Where do the graphs appear to have vertical tangents? }} \\ {\text { b. Confirm your findings in part (a) with limit calculations. But }} \\ {\text { before you do, read the introduction to Exercises } 37 \text { and } 38 .}\end{array}\end{equation}
\begin{equation}
y=x^{1 / 5}
\end{equation}

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Problem 42

Graph the curves.
\begin{equation}\begin{array}{l}{\text { a. Where do the graphs appear to have vertical tangents? }} \\ {\text { b. Confirm your findings in part (a) with limit calculations. But }} \\ {\text { before you do, read the introduction to Exercises } 37 \text { and } 38 .}\end{array}\end{equation}
\begin{equation}
y=x^{3 / 5}
\end{equation}

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Problem 43

Graph the curves.
\begin{equation}\begin{array}{l}{\text { a. Where do the graphs appear to have vertical tangents? }} \\ {\text { b. Confirm your findings in part (a) with limit calculations. But }} \\ {\text { before you do, read the introduction to Exercises } 37 \text { and } 38 .}\end{array}\end{equation}
\begin{equation}
y=4 x^{2 / 5}-2 x
\end{equation}

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Problem 44

Graph the curves.
\begin{equation}\begin{array}{l}{\text { a. Where do the graphs appear to have vertical tangents? }} \\ {\text { b. Confirm your findings in part (a) with limit calculations. But }} \\ {\text { before you do, read the introduction to Exercises } 37 \text { and } 38 .}\end{array}\end{equation}
\begin{equation}
y=x^{5 / 3}-5 x^{2 / 3}
\end{equation}

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Problem 45

Graph the curves.
\begin{equation}\begin{array}{l}{\text { a. Where do the graphs appear to have vertical tangents? }} \\ {\text { b. Confirm your findings in part (a) with limit calculations. But }} \\ {\text { before you do, read the introduction to Exercises } 37 \text { and } 38 .}\end{array}\end{equation}
\begin{equation}
y=x^{2 / 3}-(x-1)^{1 / 3}
\end{equation}

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Problem 46

Graph the curves.
\begin{equation}\begin{array}{l}{\text { a. Where do the graphs appear to have vertical tangents? }} \\ {\text { b. Confirm your findings in part (a) with limit calculations. But }} \\ {\text { before you do, read the introduction to Exercises } 37 \text { and } 38 .}\end{array}\end{equation}
\begin{equation}
y=x^{1 / 3}+(x-1)^{1 / 3}
\end{equation}

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Problem 47

Graph the curves.
\begin{equation}\begin{array}{l}{\text { a. Where do the graphs appear to have vertical tangents? }} \\ {\text { b. Confirm your findings in part (a) with limit calculations. But }} \\ {\text { before you do, read the introduction to Exercises } 37 \text { and } 38 .}\end{array}\end{equation}
\begin{equation}
y=\left\{\begin{array}{ll}{-\sqrt{|x|},} & {x \leq 0} \\ {\sqrt{x},} & {x>0}\end{array}\right.
\end{equation}

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Problem 48

Graph the curves.
\begin{equation}\begin{array}{l}{\text { a. Where do the graphs appear to have vertical tangents? }} \\ {\text { b. Confirm your findings in part (a) with limit calculations. But }} \\ {\text { before you do, read the introduction to Exercises } 37 \text { and } 38 .}\end{array}\end{equation}
\begin{equation}
y=\sqrt{|4-x|}
\end{equation}

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Problem 49

Use a CAS to perform the following steps for the functions in Exercises $49-52$ :
\begin{equation}\begin{array}{l}{\text { a. Plot } y=f(x) \text { over the interval }\left(x_{0}-1 / 2\right) \leq x \leq\left(x_{0}+3\right)} \\ {\text { b. Holding } x_{0} \text { fixed, the difference quotient }}\end{array}\end{equation}\begin{equation}
q(h)=\frac{f\left(x_{0}+h\right)-f\left(x_{0}\right)}{h}\end{equation}
at $x_{0}$ becomes a function of the step size $h .$ Enter this function into your CAS workspace.
\begin{equation}
\begin{array}{l}{\text { c. Find the limit of } q \text { as } h \rightarrow 0 \text { . }} \\ {\text { d. Define the secant lines } y=f\left(x_{0}\right)+q \cdot\left(x-x_{0}\right) \text { for } h=3,2} \\ {\text { and } 1 . \text { Graph them together with } f \text { and the tangent line over }} \\ {\text { the interval in part (a). }}\end{array}\end{equation}
\begin{equation}
f(x)=x^{3}+2 x, \quad x_{0}=0 \quad \text { 50. } f(x)=x+\frac{5}{x}, \quad x_{0}=1
\end{equation}

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Problem 50

Use a CAS to perform the following steps for the functions:
\begin{equation}\begin{array}{l}{\text { a. Plot } y=f(x) \text { over the interval }\left(x_{0}-1 / 2\right) \leq x \leq\left(x_{0}+3\right)} \\ {\text { b. Holding } x_{0} \text { fixed, the difference quotient }}\end{array}\end{equation}\begin{equation}
q(h)=\frac{f\left(x_{0}+h\right)-f\left(x_{0}\right)}{h}\end{equation}
at $x_{0}$ becomes a function of the step size $h .$ Enter this function into your CAS workspace.
\begin{equation}
\begin{array}{l}{\text { c. Find the limit of } q \text { as } h \rightarrow 0 \text { . }} \\ {\text { d. Define the secant lines } y=f\left(x_{0}\right)+q \cdot\left(x-x_{0}\right) \text { for } h=3,2} \\ {\text { and } 1 . \text { Graph them together with } f \text { and the tangent line over }} \\ {\text { the interval in part (a). }}\end{array}\end{equation}
\begin{equation}
f(x)=x+\sin (2 x), \quad x_{0}=\pi / 2
\end{equation}

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Problem 51

Use a CAS to perform the following steps for the functions:
\begin{equation}\begin{array}{l}{\text { a. Plot } y=f(x) \text { over the interval }\left(x_{0}-1 / 2\right) \leq x \leq\left(x_{0}+3\right)} \\ {\text { b. Holding } x_{0} \text { fixed, the difference quotient }}\end{array}\end{equation}\begin{equation}
q(h)=\frac{f\left(x_{0}+h\right)-f\left(x_{0}\right)}{h}\end{equation}
at $x_{0}$ becomes a function of the step size $h .$ Enter this function into your CAS workspace.
\begin{equation}
\begin{array}{l}{\text { c. Find the limit of } q \text { as } h \rightarrow 0 \text { . }} \\ {\text { d. Define the secant lines } y=f\left(x_{0}\right)+q \cdot\left(x-x_{0}\right) \text { for } h=3,2} \\ {\text { and } 1 . \text { Graph them together with } f \text { and the tangent line over }} \\ {\text { the interval in part (a). }}\end{array}\end{equation}
\begin{equation}
f(x)=x+\sin (2 x), \quad x_{0}=\pi / 2
\end{equation}

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Problem 52

Use a CAS to perform the following steps for the functions:
\begin{equation}\begin{array}{l}{\text { a. Plot } y=f(x) \text { over the interval }\left(x_{0}-1 / 2\right) \leq x \leq\left(x_{0}+3\right)} \\ {\text { b. Holding } x_{0} \text { fixed, the difference quotient }}\end{array}\end{equation}\begin{equation}
q(h)=\frac{f\left(x_{0}+h\right)-f\left(x_{0}\right)}{h}\end{equation}
at $x_{0}$ becomes a function of the step size $h .$ Enter this function into your CAS workspace.
\begin{equation}
\begin{array}{l}{\text { c. Find the limit of } q \text { as } h \rightarrow 0 \text { . }} \\ {\text { d. Define the secant lines } y=f\left(x_{0}\right)+q \cdot\left(x-x_{0}\right) \text { for } h=3,2} \\ {\text { and } 1 . \text { Graph them together with } f \text { and the tangent line over }} \\ {\text { the interval in part (a). }}\end{array}\end{equation}
\begin{equation}
f(x)=\cos x+4 \sin (2 x), \quad x_{0}=\pi
\end{equation}

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