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Calculus Early Transcendental Functions

Ron Larson, Bruce Edwards

Chapter 3

Differentiation - all with Video Answers

Educators

MT

Section 1

The Derivative and the Tangent Line Problem

03:22

Problem 1

Estimate the slope of the graph at the points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$.
(GRAPH CAN'T COPY)

MT
Mason Thomas
Numerade Educator
01:45

Problem 2

Estimate the slope of the graph at the points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$.
(GRAPH CAN'T COPY)

Isabella Cooper
Isabella Cooper
Numerade Educator
02:11

Problem 3

Use the graph shown in the figure.
(GRAPH CAN'T COPY)
Identify or sketch each of the quantities on the figure.
(a) $f(1)$ and $f(4)$
(b) $f(4)-f(1)$
(c) $y=\frac{f(4)-f(1)}{4-1}(x-1)+f(1)$

Isabella Cooper
Isabella Cooper
Numerade Educator
02:12

Problem 4

Use the graph shown in the figure.
(GRAPH CAN'T COPY)
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) $\frac{f(4)-f(1)}{4-1} \quad f^{\prime}(1)$

Isabella Cooper
Isabella Cooper
Numerade Educator
02:54

Problem 5

Use the graph shown in the figure.
(GRAPH CAN'T COPY)
$$f(x)=3-5 x, \quad(-1,8)$$

MT
Mason Thomas
Numerade Educator

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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06:03

Problem 8

Find the slope of the tangent line to the graph of the function at the given point.
$$f(x)=5-x^{2}, \quad(3,-4)$$

MT
Mason Thomas
Numerade Educator

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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05:10

Problem 10

Find the slope of the tangent line to the graph of the function at the given point.
$$h(t)=t^{2}+4 t$$

MT
Mason Thomas
Numerade Educator
02:39

Problem 11

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

Alexandra Fomina
Alexandra Fomina
Numerade Educator

Problem 12

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

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Problem 13

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

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02:36

Problem 14

Find the derivative of the function by the limit process.
$$f(x)=7 x-3$$

MT
Mason Thomas
Numerade Educator

Problem 15

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

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02:59

Problem 16

Find the derivative of the function by the limit process.
.
$$f(x)=5-\frac{2}{3} x$$

MT
Mason Thomas
Numerade Educator

Problem 17

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

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02:23

Problem 18

Find the derivative of the function by the limit process.
$$f(x)=x^{2}-5$$

Melissa Munoz
Melissa Munoz
Numerade Educator

Problem 19

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

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Problem 20

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

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Problem 21

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

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06:29

Problem 22

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

MT
Mason Thomas
Numerade Educator

Problem 23

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

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Problem 24

Find the derivative of the function 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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09:50

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.
$$f(x)=x^{2}+2 x-1$$

MT
Mason Thomas
Numerade Educator

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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08:52

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.
$$f(x)=x^{3}+1, \quad(-1,0)$$

MT
Mason Thomas
Numerade Educator

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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12:11

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.
$$f(x)=\sqrt{x-1}$$

MT
Mason Thomas
Numerade Educator
16:12

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.
$$f(x)=x+\frac{4}{x}, \quad(-4,-5)$$

MT
Mason Thomas
Numerade Educator
07:34

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.
$$f(x)=\frac{6}{x+2}, \quad(0,3)$$

MT
Mason Thomas
Numerade Educator

Problem 33

Find an equation of the line that is tangent to the graph of $f$ and parallel to the given line.

Function
$f(x)=x^{2}$

Line
$2 x-y+1=0$

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

Line
$4 x+y+3=0$

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

Line
$3 x-y+1=0$

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

Line
$3 x-y-4=0$

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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
$f(x)=\frac{1}{\sqrt{x}}$

Line
$x+2 y-6=0$

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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
$f(x)=\frac{1}{\sqrt{x-1}}$

Line
$x+2 y+7=0$

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Problem 39

Sketch the graph of $f^{\prime} .$ Explain how you found your answer.
(GRAPH CAN'T COPY)

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Problem 40

Sketch the graph of $f^{\prime} .$ Explain how you found your answer.
(GRAPH CAN'T COPY)

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Problem 41

Sketch the graph of $f^{\prime} .$ Explain how you found your answer.
(GRAPH CAN'T COPY)

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Problem 42

Sketch the graph of $f^{\prime} .$ Explain how you found your answer.
(GRAPH CAN'T COPY)

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Problem 43

Sketch the graph of $f^{\prime} .$ Explain how you found your answer.
(GRAPH CAN'T COPY)

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Problem 44

Sketch the graph of $f^{\prime} .$ Explain how you found your answer.
(GRAPH CAN'T COPY)

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Problem 45

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

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Problem 46

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

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Problem 47

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 48

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 49

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 50

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 51

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 52

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 53

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

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Problem 54

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

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Problem 55

Find equations of the two tangent lines to the graph of $f$ that pass through the indicated point.
(GRAPH CAN'T COPY)
$$f(x)=4 x-x^{2}$$

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Problem 56

Find equations of the two tangent lines to the graph of $f$ that pass through the indicated point.
(GRAPH CAN'T COPY)
$$f(x)=x^{2}$$

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Problem 57

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) $f(x)=x^{2}$
(b) $g(x)=x^{3}$

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Problem 58

The figure shows the graph of $g^{\prime}$.
(a) $g^{\prime}(0)=$
(b) $g^{\prime}(3)=$
(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 59

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 60

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 $f^{\prime}\left(-\frac{1}{2}\right), f^{\prime}(-1), f^{\prime}(-2),$ 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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03:00

Problem 61

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

Isabella Cooper
Isabella Cooper
Numerade Educator
02:47

Problem 62

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

Isabella Cooper
Isabella Cooper
Numerade Educator

Problem 63

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 64

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 65

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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03:05

Problem 66

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

MT
Mason Thomas
Numerade Educator

Problem 67

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

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Problem 68

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 69

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

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04:11

Problem 70

Use the alternative form of the derivative to find the derivative at $x=c$ (if it exists).
$$f(x)=3 / x, \quad c=4$$

MT
Mason Thomas
Numerade Educator

Problem 71

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

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Problem 72

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 73

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

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Problem 74

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

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02:12

Problem 75

Describe the $x$ -values at which $f$ is differentiable.
(GRAPH CAN'T COPY)
$$f(x)=\frac{2}{x-3}$$

MT
Mason Thomas
Numerade Educator
01:57

Problem 76

Describe the $x$ -values at which $f$ is differentiable.
(GRAPH CAN'T COPY)
$$f(x)=\left|x^{2}-9\right|$$

MT
Mason Thomas
Numerade Educator
01:51

Problem 77

Describe the $x$ -values at which $f$ is differentiable.
(GRAPH CAN'T COPY)
$$f(x)=(x+4)^{2 / 3}$$

MT
Mason Thomas
Numerade Educator
02:29

Problem 78

Describe the $x$ -values at which $f$ is differentiable.
(GRAPH CAN'T COPY)
$$f(x)=\frac{x^{2}}{x^{2}-4}$$

MT
Mason Thomas
Numerade Educator
01:33

Problem 79

Describe the $x$ -values at which $f$ is differentiable.
(GRAPH CAN'T COPY)
$$f(x)=\sqrt{x-1}$$

Isabella Cooper
Isabella Cooper
Numerade Educator
01:42

Problem 80

Describe the $x$ -values at which $f$ is differentiable.
(GRAPH CAN'T COPY)
$$f(x)=\left\{\begin{array}{ll}
x^{2}-4, & x \leq 0 \\
4-x^{2}, & x > 0
\end{array}\right.$$

Isabella Cooper
Isabella Cooper
Numerade Educator

Problem 81

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 82

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 83

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 84

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 85

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 86

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 87

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 88

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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02:14

Problem 89

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

Isabella Cooper
Isabella Cooper
Numerade Educator

Problem 90

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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01:10

Problem 91

Use a graphing utility to graph $g(x)=[x] / x .$ Then let $f(x)=[| x] |$ and show that
Use a graphing utility to graph $g(x)=[[x]] / x .$ Then let $f(x)=[[ x]]$ and show that
$\lim _{x \rightarrow 0^{-}} \frac{f(x)-f(0)}{x-0}=\infty \quad$ and $\quad \lim _{x \rightarrow 0^{+}} \frac{f(x)-f(0)}{x-0}=0$
Is $f$ differentiable? Explain.

Carson Merrill
Carson Merrill
Numerade Educator
04:26

Problem 92

Consider the functions $f(x)=x^{2}$ and $g(x)=x^{3}$
(a) Graph $f$ and $f^{\prime}$ on the same set of coordinate axes.
(b) Graph $g$ and $g^{\prime}$ on the same set of coordinate axes.
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 $n \geq 2$
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.

Isabella Cooper
Isabella Cooper
Numerade Educator

Problem 93

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 94

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 95

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 96

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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01:05

Problem 97

Let
$f(x)=\left\{\begin{array}{ll}x \sin \frac{1}{x}, & x \neq 0 \\ 0, & x=0\end{array}\right.$
and
$g(x)=\left\{\begin{array}{ll}x^{2} \sin \frac{1}{x}, & x \neq 0 \\ 0, & x=0\end{array}\right.$
Show that $f$ is continuous, but not differentiable, at $x=0$ Show that $g$ is differentiable at 0 , and find $g^{\prime}(0)$

Carson Merrill
Carson Merrill
Numerade Educator

Problem 98

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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