# Calculus of a Single Variable

## Educators

Problem 1

Estimate the slope of the graph at the points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right) .$

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Problem 2

Estimate the slope of the graph at the points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right) .$

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Problem 3

Use the graph shown in the figure. To print an enlarged copy of the graph, go to the website www.mathgraphs.com.
Identify or sketch each of the quantities on the figure.
(a) $f(1)$ and $f(4)$
(b) $f(4)-f(1)$
(c) $y=\frac{f(4)-f(1)}{4-1}(x-1)+f(1)$

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Problem 4

Use the graph shown in the figure. To print an enlarged copy of the graph, go to the website www.mathgraphs.com.
Insert the proper inequality symbol $(<\text { or }>)$ between the given quantities.
(a) $\frac{f(4)-f(1)}{4-1} \quad \frac{f(4)-f(3)}{4-3}$
(b) $\frac{f(4)-f(1)}{4-1} \quad f^{\prime}(1)$

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Problem 5

Find the slope of the tangent line to the graph of the function at the given point.
$f(x)=3-5 x, \quad(-1,8)$

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Problem 6

Find the slope of the tangent line to the graph of the function at the given point.
$g(x)=\frac{3}{2} x+1, \quad(-2,-2)$

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Problem 7

Find the slope of the tangent line to the graph of the function at the given point.
$g(x)=x^{2}-9, \quad(2,-5)$

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Problem 8

Find the slope of the tangent line to the graph of the function at the given point.
$g(x)=6-x^{2}, \quad(1,5)$

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Problem 9

Find the slope of the tangent line to the graph of the function at the given point.
$f(t)=3 t-t^{2}, \quad(0,0)$

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Problem 10

Find the slope of the tangent line to the graph of the function at the given point.
$h(t)=t^{2}+3, \quad(-2,7)$

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Problem 11

Find the derivative by the limit process.
$f(x)=7$

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Problem 12

Find the derivative by the limit process.
$g(x)=-3$

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Problem 13

Find the derivative by the limit process.
$f(x)=-10 x$

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Problem 14

Find the derivative by the limit process.
$f(x)=3 x+2$

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Problem 15

Find the derivative by the limit process.
$h(s)=3+\frac{2}{3} s$

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Problem 16

Find the derivative by the limit process.
$f(x)=8-\frac{1}{5} x$

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Problem 17

Find the derivative by the limit process.
$f(x)=x^{2}+x-3$

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Problem 18

Find the derivative by the limit process.
$f(x)=2-x^{2}$

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Problem 19

Find the derivative by the limit process.
$f(x)=x^{3}-12 x$

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Problem 20

Find the derivative by the limit process.
$f(x)=x^{3}+x^{2}$

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Problem 21

Find the derivative by the limit process.
$f(x)=\frac{1}{x-1}$

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Problem 22

Find the derivative by the limit process.
$f(x)=\frac{1}{x^{2}}$

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Problem 23

Find the derivative by the limit process.
$f(x)=\sqrt{x+4}$

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Problem 24

Find the derivative by the limit process.
$f(x)=\frac{4}{\sqrt{x}}$

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Problem 25

(a) find an equation of the tangent line to the graph of $f$ at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results.
$f(x)=x^{2}+3, \quad(1,4)$

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Problem 26

(a) find an equation of the tangent line to the graph of $f$ at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results.
$f(x)=x^{2}+3 x+4, \quad(-2,2)$

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Problem 27

(a) find an equation of the tangent line to the graph of $f$ at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results.
$f(x)=x^{3}, \quad(2,8)$

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Problem 28

(a) find an equation of the tangent line to the graph of $f$ at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results.
$f(x)=x^{3}+1$

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Problem 29

(a) find an equation of the tangent line to the graph of $f$ at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results.
$f(x)=\sqrt{x}$

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Problem 30

(a) find an equation of the tangent line to the graph of $f$ at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results.
$f(x)=\sqrt{x-1}, \quad(5,2)$

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Problem 31

(a) find an equation of the tangent line to the graph of $f$ at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results.
$f(x)=x+\frac{4}{x}, \quad(4,5)$

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Problem 32

(a) find an equation of the tangent line to the graph of $f$ at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results.
$f(x)=\frac{1}{x+1}, \quad(0,1)$

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Problem 33

Find an equation of the line that is tangent to the graph of $f$ and parallel to the given line.
Function $\quad$ Line
$f(x)=x^{2} \quad 2 x-y+1=0$

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Problem 34

Find an equation of the line that is tangent to the graph of $f$ and parallel to the given line.
Function $\quad$ Line
$f(x)=2 x^{2} \quad 4 x+y+3=0$

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Problem 35

Find an equation of the line that is tangent to the graph of $f$ and parallel to the given line.
Function $\quad$ Line
$f(x)=x^{3} \quad 3 x-y+1=0$

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Problem 36

Find an equation of the line that is tangent to the graph of $f$ and parallel to the given line.
Function $\quad$ Line
$f(x)=x^{3}+2 \quad 3 x-y-4=0$

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Problem 37

Find an equation of the line that is tangent to the graph of $f$ and parallel to the given line.
Function $\quad$ Line
$f(x)=\frac{1}{\sqrt{x}} \quad x+2 y-6=0$

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Problem 38

Find an equation of the line that is tangent to the graph of $f$ and parallel to the given line.
Function $\quad$ Line
$f(x)=\frac{1}{\sqrt{x-1}} \quad x+2 y+7=0$

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Problem 39

The graph of $f$ is given. Select the graph of $f^{\prime} .$

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Problem 40

The graph of $f$ is given. Select the graph of $f^{\prime} .$

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Problem 41

The graph of $f$ is given. Select the graph of $f^{\prime} .$

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Problem 42

The graph of $f$ is given. Select the graph of $f^{\prime} .$

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Problem 43

The tangent line to the graph of $y=g(x)$ at the point $(4,5)$ passes through the point $(7,0) .$ Find $g(4)$ and $g^{\prime}(4) .$

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Problem 44

The tangent line to the graph of $y=h(x)$ at the point $(-1,4)$ passes through the point $(3,6),$ Find $h(-1)$ and $h^{\prime}(-1)$ .

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Problem 45

Sketch the graph of $f^{\prime} .$ Explain how you found your answer.

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Problem 46

Sketch the graph of $f^{\prime} .$ Explain how you found your answer.

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Problem 47

Sketch the graph of $f^{\prime} .$ Explain how you found your answer.

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Problem 48

Sketch the graph of $f^{\prime} .$ Explain how you found your answer.

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Problem 49

Sketch the graph of $f^{\prime} .$ Explain how you found your answer.

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Problem 50

Sketch the graph of $f^{\prime} .$ Explain how you found your answer.

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Problem 51

Sketch a graph of a function whose derivative is always negative. Explain how you found your answer.

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Problem 52

Sketch a graph of a function whose derivative is always positive. Explain how you found your answer.

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Problem 53

The limit represents $f^{\prime}(c)$ for a function $f$ and a number $c .$ Find $f$ and $c .$
$$\lim _{\Delta x \rightarrow 0} \frac{[5-3(1+\Delta x)]-2}{\Delta x}$$

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Problem 54

The limit represents $f^{\prime}(c)$ for a function $f$ and a number $c .$ Find $f$ and $c .$
$$\lim _{\Delta x \rightarrow 0} \frac{(-2+\Delta x)^{3}+8}{\Delta x}$$

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Problem 55

The limit represents $f^{\prime}(c)$ for a function $f$ and a number $c .$ Find $f$ and $c .$
$$\lim _{x \rightarrow 6} \frac{-x^{2}+36}{x-6}$$

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Problem 56

The limit represents $f^{\prime}(c)$ for a function $f$ and a number $c .$ Find $f$ and $c .$
$$\lim _{x \rightarrow 9} \frac{2 \sqrt{x}-6}{x-9}$$

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Problem 57

Identify a function $f$ that has the given characteristics. Then sketch the function.
$f(0)=2$
$f^{\prime}(x)=-3,-\infty<x<\infty$

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Problem 58

Identify a function $f$ that has the given characteristics. Then sketch the function.
$f(0)=4 ; f^{\prime}(0)=0$
$f^{\prime}(x)<0$ for $x<0$
$f^{\prime}(x)>0$ for $x>0$

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Problem 59

Identify a function $f$ that has the given characteristics. Then sketch the function.
$f(0)=0 ; f^{\prime}(0)=0 ; f^{\prime}(x)>0$ for $x \neq 0$

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Problem 60

Assume that $f^{\prime}(c)=3 .$ Find $f^{\prime}(-c)$ if $(\mathrm{a}) f$ is an odd function and if $(\mathrm{b}) f$ is an even function.

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Problem 61

Find equations of the two tangent lines to the graph of $f$ that pass through the indicated point.
$f(x)=4 x-x^{2}$

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Problem 62

Find equations of the two tangent lines to the graph of $f$ that pass through the indicated point.
$f(x)=x^{2}$

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Problem 63

Use a graphing utility to graph each function and its tangent lines at $x=-1, x=0,$ and $x=1$ . Based on the results, determine whether the slopes of tangent lines to the graph of a function at different values of $x$ are always distinct.
(a) $f(x)=x^{2} \quad$ (b) $g(x)=x^{3}$

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Problem 64

The figure shows the graph of $g^{\prime}$.
(a) $g^{\prime}(0)=\square \quad$ (b) $g^{\prime}(3)= \square$
(c) What can you conclude about the graph of $g$ knowing that $g^{\prime}(1)=-\frac{8}{3} ?$
(d) What can you conclude about the graph of $g$ knowing that $g^{\prime}(-4)=\frac{7}{3} ?$
(e) Is $g(6)-g(4)$ positive or negative? Explain.
(f) Is it possible to find $g(2)$ from the graph? Explain.

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Problem 65

Consider the function $f(x)=\frac{1}{2} x^{2}$.
(a) Use a graphing utility to graph the function and estimate the values of $f^{\prime}(0), f^{\prime}\left(\frac{1}{2}\right), f^{\prime}(1),$ and $f^{\prime}(2) .$
(b) Use your results from part (a) to determine the values of $f^{\prime}\left(-\frac{1}{2}\right), f^{\prime}(-1),$ and $f^{\prime}(-2)$
(c) Sketch a possible graph of $f^{\prime}$
(d) Use the definition of derivative to find $f^{\prime}(x)$

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Problem 66

Consider the function $f(x)=\frac{1}{3} x^{3}$.
(a) Use a graphing utility to graph the function and estimate the values of $f^{\prime}(0), f^{\prime}\left(\frac{1}{2}\right), f^{\prime}(1), f^{\prime}(2),$ and $f^{\prime}(3) .$
(b) Use your results from part (a) to determine the values of $\quad f^{\prime}\left(-\frac{1}{2}\right), f^{\prime}(-1), f^{\prime}(-2),$ and $f^{\prime}(-3)$
(c) Sketch a possible graph of $f^{\prime}$
(d) Use the definition of derivative to find $f^{\prime}(x)$

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Problem 67

Use a graphing utility to graph the functions $f$ and $g$ in the same viewing window where
$g(x)=\frac{f(x+0.01)-f(x)}{0.01}$
Label the graphs and describe the relationship between them.
$f(x)=2 x-x^{2}$

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Problem 68

Use a graphing utility to graph the functions $f$ and $g$ in the same viewing window where
$g(x)=\frac{f(x+0.01)-f(x)}{0.01}$
Label the graphs and describe the relationship between them.
$f(x)=3 \sqrt{x}$

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Problem 69

Evaluate $f(2)$ and $f(2.1)$ and use the results to approximate $f^{\prime}(2) .$
$f(x)=x(4-x)$

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Problem 70

Evaluate $f(2)$ and $f(2.1)$ and use the results to approximate $f^{\prime}(2) .$
$f(x)=\frac{1}{4} x^{3}$

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Problem 71

Use a graphing utility to graph the function and its derivative in the same viewing window. Label the graphs and describe the relationship between them.
$f(x)=\frac{1}{\sqrt{x}}$

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Problem 72

Use a graphing utility to graph the function and its derivative in the same viewing window. Label the graphs and describe the relationship between them.
$f(x)=\frac{x^{3}}{4}-3 x$

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Problem 73

Use the alternative form of the derivative to find the derivative at $x=c$ (if it exists).
$f(x)=x^{2}-5, \quad c=3$

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Problem 74

Use the alternative form of the derivative to find the derivative at $x=c$ (if it exists).
$g(x)=x(x-1), \quad c=1$

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Problem 75

Use the alternative form of the derivative to find the derivative at $x=c$ (if it exists).
$f(x)=x^{3}+2 x^{2}+1, \quad c=-2$

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Problem 76

Use the alternative form of the derivative to find the derivative at $x=c$ (if it exists).
$f(x)=x^{3}+6 x, \quad c=2$

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Problem 77

Use the alternative form of the derivative to find the derivative at $x=c$ (if it exists).
$g(x)=\sqrt{|x|}, \quad c=0$

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Problem 78

Use the alternative form of the derivative to find the derivative at $x=c$ (if it exists).
$f(x)=2 / x, \quad c=5$

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Problem 79

Use the alternative form of the derivative to find the derivative at $x=c$ (if it exists).
$f(x)=(x-6)^{2 / 3}, \quad c=6$

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Problem 80

Use the alternative form of the derivative to find the derivative at $x=c$ (if it exists).
$g(x)=(x+3)^{1 / 3}, \quad c=-3$

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Problem 81

Use the alternative form of the derivative to find the derivative at $x=c$ (if it exists).
$h(x)=|x+7|, \quad c=-7 \quad 8$

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Problem 82

Use the alternative form of the derivative to find the derivative at $x=c$ (if it exists).
$f(x)=|x-6|, \quad c=6$

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Problem 83

Describe the $x$ -values at which $f$ is differentiable.
$f(x)=\frac{2}{x-3}$

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Problem 84

Describe the $x$ -values at which $f$ is differentiable.
$f(x)=\left|x^{2}-9\right|$

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Problem 85

Describe the $x$ -values at which $f$ is differentiable.
$f(x)=(x+4)^{2 / 3}$

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Problem 86

Describe the $x$ -values at which $f$ is differentiable.
$f(x)=\frac{x^{2}}{x^{2}-4}$

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Problem 87

Describe the $x$ -values at which $f$ is differentiable.
$f(x)=\sqrt{x-1}$

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Problem 88

Describe the $x$ -values at which $f$ is differentiable.
$f(x)=\left\{\begin{array}{ll}{x^{2}-4,} & {x \leq 0} \\ {4-x^{2},} & {x>0}\end{array}\right.$

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Problem 89

Use a graphing utility to graph the function and find the $x$ -values at which $f$ is differentiable.
$f(x)=|x-5|$

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Problem 90

Use a graphing utility to graph the function and find the $x$ -values at which $f$ is differentiable.
$f(x)=\frac{4 x}{x-3}$

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Problem 91

Use a graphing utility to graph the function and find the $x$ -values at which $f$ is differentiable.
$f(x)=x^{2 / 5}$

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Problem 92

Use a graphing utility to graph the function and find the $x$ -values at which $f$ is differentiable.
$f(x)=\left\{\begin{array}{ll}{x^{3}-3 x^{2}+3 x,} & {x \leq 1} \\ {x^{2}-2 x,} & {x>1}\end{array}\right.$

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Problem 93

Find the derivatives from the left and from the right at $x=1$ (if they exist). Is the function differentiable at $x=1 ?$
$f(x)=|x-1|$

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Problem 94

Find the derivatives from the left and from the right at $x=1$ (if they exist). Is the function differentiable at $x=1 ?$
$f(x)=\sqrt{1-x^{2}}$

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Problem 95

Find the derivatives from the left and from the right at $x=1$ (if they exist). Is the function differentiable at $x=1 ?$
$f(x)=\left\{\begin{array}{ll}{(x-1)^{3},} & {x \leq 1} \\ {(x-1)^{2},} & {x>1}\end{array}\right.$

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Problem 96

Find the derivatives from the left and from the right at $x=1$ (if they exist). Is the function differentiable at $x=1 ?$
$f(x)=\left\{\begin{array}{ll}{x,} & {x \leq 1} \\ {x^{2},} & {x>1}\end{array}\right.$

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Problem 97

Determine whether the function is differentiable at $x=2$.
$f(x)=\left\{\begin{array}{ll}{x^{2}+1,} & {x \leq 2} \\ {4 x-3,} & {x>2}\end{array}\right.$

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Problem 98

Determine whether the function is differentiable at $x=2$.
$f(x)=\left\{\begin{array}{ll}{\frac{1}{2} x+1,} & {x<2} \\ {\sqrt{2 x},} & {x \geq 2}\end{array}\right.$

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Problem 99

A line with slope $m$ passes through the point $(0,4)$ and has the equation $y=m x+4 .$
(a) Write the distance $d$ between the line and the point $(3,1)$ as a function of $m .$
(b) Use a graphing utility to graph the function $d$ in part (a). Based on the graph, is the function differentiable at every value of $m ?$ If not, where is it not differentiable?

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Problem 100

Consider the functions $f(x)=x^{2}$ and $g(x)=x^{3}$.
(a) Graph $f$ and $f^{\prime}$ on the same set of axes.
(b) Graph $g$ and $g^{\prime}$ 'on the same set of axes.
(c) Identify a pattern between $f$ and $g$ and their respective derivatives. Use the pattern to make a conjecture about $h^{\prime}(x)$ if $h(x)=x^{n}$ , where $n$ is an integer and $n \geq 2$ .
(d) Find $f^{\prime}(x)$ if $f(x)=x^{4}$ . Compare the result with the conjecture in part (c). Is this a proof of your conjecture? Explain.

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Problem 101

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
The slope of the tangent line to the differentiable function $f$ at the point $(2, f(2))$ is $\frac{f(2+\Delta x)-f(2)}{\Delta x}$

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Problem 102

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
If a function is continuous at a point, then it is differentiable at that point.

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Problem 103

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
If a function has derivatives from both the right and the left at a point, then it is differentiable at that point.

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Problem 104

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
If a function is differentiable at a point, then it is continuous at that point.

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Problem 105

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
Let $f(x)=\left\{\begin{array}{ll}{x \sin \frac{1}{x},} & {x \neq 0} \\ {0,} & {x=0}\end{array} \text { and } g(x)=\left\{\begin{array}{ll}{x^{2} \sin \frac{1}{x},} & {x \neq 0} \\ {0,} & {x=0}\end{array}\right.\right.$nt.

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Problem 106

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
Use a graphing utility to graph the two functions $f(x)=x^{2}+1$ and $g(x)=|x|+1$ in the same viewing window. Use the zoom and trace features to analyze the graphs near the point $(0,1) .$ What do you observe? Which function is differentiable at this point? Write a short paragraph describing the geometric significance of differentiability at a point.

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