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Thomas Calculus 12

George B. Thomas, Jr. Maurice D. Weir, Joel Hass

Chapter 3

Differentiation

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

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 3

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 4

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 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.
$$
y=4-x^{2}, \quad(-1,3)
$$

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Problem 6

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

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Problem 7

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

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Problem 8

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

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Problem 9

In Exercises $5-10$ , find an equation for the tangent to the curve at the given point. Then sketch the curve and tangent together.
$$
y=x^{3}, \quad(-2,-8)
$$

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Problem 10

In Exercises $5-10$ , find an equation for the tangent to the curve at the given point. Then sketch the curve and tangent together.
$$
y=\frac{1}{x^{3}}, \quad\left(-2,-\frac{1}{8}\right)
$$

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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.
$$
f(x)=x^{2}+1, \quad(2,5)
$$

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Problem 12

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.
$$
f(x)=x-2 x^{2}, \quad(1,-1)
$$

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Problem 13

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.
$$
g(x)=\frac{x}{x-2},(3,3)
$$

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Problem 14

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.
$$
g(x)=\frac{8}{x^{2}}, \quad(2,2)
$$

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Problem 15

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.
$$
h(t)=t^{3}, \quad(2,8)
$$

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Problem 16

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.
$$
h(t)=t^{3}+3 t, \quad(1,4)
$$

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Problem 17

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.
$$
f(x)=\sqrt{x},(4,2)
$$

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Problem 18

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.
$$
f(x)=\sqrt{x+1},(8,3)
$$

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Problem 19

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

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Problem 20

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

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Problem 21

In Exercises $19-22,$ find the slope of the curve at the point indicated.
$$
y=\frac{1}{x-1}, \quad x=3
$$

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Problem 22

In Exercises $19-22,$ find the slope of the curve at the point indicated.
$$
y=\frac{x-1}{x+1}, \quad x=0
$$

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Problem 23

At what points do the graphs of the functions in Exercises 23 and 24 have horizontal tangents?
$$
f(x)=x^{2}+4 x-1
$$

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Problem 24

At what points do the graphs of the functions in Exercises 23 and 24 have horizontal tangents?
$$
g(x)=x^{3}-3 x
$$

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Problem 25

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

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Problem 26

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 27

Object dropped from a tower $A$ n 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 28

Speed of a rocket At $t$ 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 29

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 30

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 31

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

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Problem 32

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

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Problem 33

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

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Problem 34

Does the graph of
$$
g(x)=\left\{\begin{array}{ll}{x \sin (1 / x),} & {x \neq 0} \\ {0,} & {x=0}\end{array}\right.
$$
have a tangent at the origin? Give reasons for your answer.
Vertical Tangents We say that a continuous curve $y=f(x)$ has a vertical tangent at the
point where $x=x_{0}$ if $\lim _{k \rightarrow 0}\left(f\left(x_{0}+h\right)-f\left(x_{0}\right)\right) / h=\infty$ or $-\infty$ . For example, $y=x^{1 / 3}$ has a vertical tangent at $x=0$ (see accompanying figure):
$$
\begin{aligned} \lim _{h \rightarrow 0} \frac{f(0+h)-f(0)}{h} &=\lim _{h \rightarrow 0} \frac{h^{1 / 3}-0}{h} \\ &=\lim _{h \rightarrow 0} \frac{1}{h^{2 / 3}}=\infty \end{aligned}
$$
However, $y=x^{2 / 3}$ has $n o$ vertical tangent at $x=0$ (see next figure):
$$
\begin{aligned} \lim _{h \rightarrow 0} \frac{g(0+h)-g(0)}{h} &=\lim _{h \rightarrow 0} \frac{h^{2 / 3}-0}{h} \\ &=\lim _{h \rightarrow 0} \frac{1}{h^{1 / 3}} \end{aligned}
$$
does not exist, because the limit is $\infty$ from the right and $-\infty$ from the left.

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Problem 35

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

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Problem 36

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

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Problem 37

Graph the curves in Exercises $37-46$ .
a. Where do the graphs appear to have vertical tangents?
b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to
Exercises 35 and 36.
$$
y=x^{2 / 5}
$$

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Problem 38

Graph the curves in Exercises $37-46$ .
a. Where do the graphs appear to have vertical tangents?
b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to
Exercises 35 and 36.
$$
y=x^{4 / 5}
$$

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Problem 39

Graph the curves in Exercises $37-46$ .
a. Where do the graphs appear to have vertical tangents?
b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to
Exercises 35 and 36.
$$
y=x^{1 / 5}
$$

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Problem 40

Graph the curves in Exercises $37-46$ .
a. Where do the graphs appear to have vertical tangents?
b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to
Exercises 35 and 36.
$$
y=x^{3 / 5}
$$

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Problem 41

Graph the curves in Exercises $37-46$ .
a. Where do the graphs appear to have vertical tangents?
b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to
Exercises 35 and 36.
$$
y=4 x^{2 / 5}-2 x
$$

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Problem 42

Graph the curves in Exercises $37-46$ .
a. Where do the graphs appear to have vertical tangents?
b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to
Exercises 35 and 36.
$$
y=x^{5 / 3}-5 x^{2 / 3}
$$

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Problem 43

Graph the curves in Exercises $37-46$ .
a. Where do the graphs appear to have vertical tangents?
b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to
Exercises 35 and 36.
$$
y=x^{2 / 3}-(x-1)^{1 / 3}
$$

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Problem 44

Graph the curves in Exercises $37-46$
a. Where do the graphs appear to have vertical tangents?
b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to Exercises 35 and $36 .$
$$
y=x^{1 / 3}+(x-1)^{1 / 3}
$$

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Problem 45

Graph the curves in Exercises $37-46$
a. Where do the graphs appear to have vertical tangents?
b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to Exercises 35 and 36 .
$$
y=\left\{\begin{array}{ll}{-\sqrt{|x|},} & {x \leq 0} \\ {\sqrt{x},} & {x > 0}\end{array}\right.
$$

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Problem 46

Graph the curves in Exercises $37-46$
a. Where do the graphs appear to have vertical tangents?
b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to Exercises 35 and 36 .
$$
y=\sqrt{|4-x|}
$$

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Problem 47

COMPUTER EXPLORATIONS Use a CAS to perform the following steps for the functions in Exercises $47-50$ :
a. Plot $y=f(x)$ over the interval $\left(x_{0}-1 / 2\right) \leq x \leq\left(x_{0}+3\right)$
b. Holding $x_{0}$ fixed, the difference quotient
$$
q(h)=\frac{f\left(x_{0}+h\right)-f\left(x_{0}\right)}{h}
$$
at $x_{0}$ becomes a function of the step size $h$ . Enter this function into your CAS workspace.
c. Find the limit of $q$ as $h \rightarrow 0$ .
d. Define the secant lines $y=f\left(x_{0}\right)+q \cdot\left(x-x_{0}\right)$ for $h=3,2$ , and $1 .$ Graph them together with $f$ and the tangent line over the interval in part (a).
$$
f(x)=x^{3}+2 x, \quad x_{0}=0
$$

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Problem 48

COMPUTER EXPLORATIONS Use a CAS to perform the following steps for the functions in Exercises $47-50$ :
a. Plot $y=f(x)$ over the interval $\left(x_{0}-1 / 2\right) \leq x \leq\left(x_{0}+3\right)$
b. Holding $x_{0}$ fixed, the difference quotient
$$
q(h)=\frac{f\left(x_{0}+h\right)-f\left(x_{0}\right)}{h}
$$
at $x_{0}$ becomes a function of the step size $h$ . Enter this function into your CAS workspace.
c. Find the limit of $q$ as $h \rightarrow 0$ .
d. Define the secant lines $y=f\left(x_{0}\right)+q \cdot\left(x-x_{0}\right)$ for $h=3,2$ , and $1 .$ Graph them together with $f$ and the tangent line over the interval in part (a).
$$
f(x)=x+\frac{5}{x}, \quad x_{0}=1
$$

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Problem 49

COMPUTER EXPLORATIONS Use a CAS to perform the following steps for the functions in Exercises $47-50$ :
a. Plot $y=f(x)$ over the interval $\left(x_{0}-1 / 2\right) \leq x \leq\left(x_{0}+3\right)$
b. Holding $x_{0}$ fixed, the difference quotient
$$
q(h)=\frac{f\left(x_{0}+h\right)-f\left(x_{0}\right)}{h}
$$
at $x_{0}$ becomes a function of the step size $h$ . Enter this function into your CAS workspace.
c. Find the limit of $q$ as $h \rightarrow 0$ .
d. Define the secant lines $y=f\left(x_{0}\right)+q \cdot\left(x-x_{0}\right)$ for $h=3,2$ , and $1 .$ Graph them together with $f$ and the tangent line over the interval in part (a).
$$
f(x)=x+\sin (2 x), \quad x_{0}=\pi / 2
$$

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Problem 50

COMPUTER EXPLORATIONS Use a CAS to perform the following steps for the functions in Exercises $47-50$ :
a. Plot $y=f(x)$ over the interval $\left(x_{0}-1 / 2\right) \leq x \leq\left(x_{0}+3\right)$
b. Holding $x_{0}$ fixed, the difference quotient
$$
q(h)=\frac{f\left(x_{0}+h\right)-f\left(x_{0}\right)}{h}
$$
at $x_{0}$ becomes a function of the step size $h$ . Enter this function into your CAS workspace.
c. Find the limit of $q$ as $h \rightarrow 0$ .
d. Define the secant lines $y=f\left(x_{0}\right)+q \cdot\left(x-x_{0}\right)$ for $h=3,2$ , and $1 .$ Graph them together with $f$ and the tangent line over the interval in part (a).
$$
f(x)=\cos x+4 \sin (2 x), \quad x_{0}=\pi
$$

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