Essential Calculus

Ron Larson, Robert Hostetler, Bruce H. Edwards

Chapter 2

Differentiation - all with Video Answers

Educators

+ 2 more educators

Section 1

The Derivative and the Tangent Line Problem

Problem 1

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

Tom Greenwood

Tom Greenwood

Numerade Educator

Problem 2

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

Tom Greenwood

Numerade Educator

Problem 3

In Exercises 3 and $4,$ use the graph shown in the figure.
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

Problem 4

Use the graph shown in the figure.
Insert the proper inequality symbol $(<$ 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

Problem 5

In Exercises $5-10$, find the slope of the tangent line to the graph of the function at the given point.
$f(x)=3-2 x, \quad(-1,5)$

David Wellman

Numerade Educator

Problem 6

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

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 7

Find the slope of the tangent line to the graph of the function at the given point.
$g(x)=x^{2}-4, \quad(1,-3)$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 8

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

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 9

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

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 10

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

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 11

In Exercises $11-18$, find the derivative by the limit process.
$f(x)=3$

William Hice

Numerade Educator

Problem 12

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

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 13

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

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 14

Find the derivative by the limit process.
$f(x)=9-\frac{1}{2} x$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 15

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

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 16

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

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 17

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

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 18

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

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 19

In Exercises $19-24$, (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.
$f(x)=x^{2}+1, \quad(2,5)$

David Wellman

Numerade Educator

Problem 20

(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.
$f(x)=x^{3}+1, \quad(1,2)$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 21

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

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 22

(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.
$f(x)=\sqrt{x-1}, \quad(5,2)$

Ahmad Reda

Numerade Educator

Problem 23

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

Ahmad Reda

Numerade Educator

Problem 24

(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.
$f(x)=\frac{1}{x+1}, \quad(0,1)$

Ahmad Reda

Numerade Educator

Problem 25

In Exercises 25 and $26,$ find an equation of the line that is tangent to the graph of $f$ and parallel to the given line.
Function $\quad$ Line
$f(x)=x^{3}$ $3 x-y+1=0$

David Wellman

Numerade Educator

Problem 26

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

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 27

The tangent line to the graph of $y=g(x)$ at the point (5,2) passes through the point (9,0) . Find $g(5)$ and $g^{\prime}(5)$.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 28

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 $h^{\prime}(-1)$.

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 29

In Exercises 29 and 30 , sketch the graph of $f^{\prime}$. Explain how you found your answer.

William Hice

Numerade Educator

Problem 30

In Exercises 29 and 30 , sketch the graph of $f^{\prime}$. Explain how you found your answer.

William Hice

Numerade Educator

Problem 31

Sketch a graph of a function whose derivative is always negative.

Stephen Hobbs

Numerade Educator

Problem 32

Sketch a graph of a function whose derivative is always positive.

Kamalesh Bagrecha

Kamalesh Bagrecha

Numerade Educator

Problem 33

In Exercises $33-36,$ the limit represents $f^{\prime}(c)$ for a function $f$ and a number $c$. Find $f$ and $c$.
$\lim _{\Delta x \rightarrow 0} \frac{[5-3(1+\Delta x)]-2}{\Delta x}$

William Hice

Numerade Educator

Problem 34

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

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 35

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

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 36

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

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 37

In Exercises $37-39$, identify a function $f$ that has the following characteristics. Then sketch the function.
$f(0)=2 ;$
$f^{\prime}(x)=-3,-\infty<x<\infty$

William Hice

Numerade Educator

Problem 38

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

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 39

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

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 40

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

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 41

In Exercises 41 and $42,$ find equations of the two tangent lines to the graph of $f$ that pass through the indicated point.

William Hice

Numerade Educator

Problem 42

In Exercises 41 and $42,$ find equations of the two tangent lines to the graph of $f$ that pass through the indicated point.

William Hice

Numerade Educator

Problem 43

Graphical, Numerical, and Analytic Analysis In Exercises 43 and $44,$ use a graphing utility to graph $f$ on the interval [-2,2] . Complete the table by graphically estimating the slope of the graph at the indicated points. Then evaluate the slopes analytically and compare your results with those obtained graphically.
$$
\begin{array}{|l|l|l|l|l|l|l|l|l|l|}
\hline x & -2 & -1.5 & -1 & -0.5 & 0 & 0.5 & 1 & 1.5 & 2 \\
\hline f(x) & & & & & & & & & \\
\hline f^{\prime}(x) & & & & & & & & & \\
\hline
\end{array}
$$
$f(x)=\frac{1}{4} x^{3}$

William Hice

Numerade Educator

Problem 44

Graphical, Numerical, and Analytic Analysis In Exercises 43 and $44,$ use a graphing utility to graph $f$ on the interval [-2,2] . Complete the table by graphically estimating the slope of the graph at the indicated points. Then evaluate the slopes analytically and compare your results with those obtained graphically.
$$
\begin{array}{|l|l|l|l|l|l|l|l|l|l|}
\hline x & -2 & -1.5 & -1 & -0.5 & 0 & 0.5 & 1 & 1.5 & 2 \\
\hline f(x) & & & & & & & & & \\
\hline f^{\prime}(x) & & & & & & & & & \\
\hline
\end{array}
$$
$f(x)=\frac{1}{2} x^{2}$

Marcella Sippey

Marcella Sippey

Numerade Educator

Problem 45

Graphical Reasoning In Exercises 45 and $46,$ 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.
$f(x)=2 x-x^{2}$

William Hice

Numerade Educator

Problem 46

Graphical Reasoning In Exercises 45 and $46,$ 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.
$f(x)=3 \sqrt{x}$

William Hice

Numerade Educator

Problem 47

In Exercises 47 and $48,$ evaluate $f(2)$ and $f(2.1)$ and use the results to approximate $f^{\prime}(2)$
$f(x)=x(4-x)$

David Wellman

Numerade Educator

Problem 48

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

Ahmad Reda

Numerade Educator

Problem 49

Graphical Reasoning In Exercises 49 and 50 , 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.
$f(x)=\frac{1}{\sqrt{x}}$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 50

Graphical Reasoning In Exercises 49 and 50 , 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.
$f(x)=\frac{x^{3}}{4}-3 x$

Marcella Sippey

Marcella Sippey

Numerade Educator

Problem 51

In Exercises 51 and $52,$ consider the functions $f$ and $S_{\Delta x}$ where
$S_{\Delta x}(x)=\frac{f(2+\Delta x)-f(2)}{\Delta x}(x-2)+f(2)$
(a) Use a graphing utility to graph $f$ and $S_{\Delta x}$ in the same viewing window for $\Delta x=1,0.5,$ and 0.1
(b) Give a written description of the graphs of $S$ for the different values of $\Delta x$ in part (a).
$f(x)=4-(x-3)^{2}$

Nick Johnson

Numerade Educator

Problem 52

Consider the functions $f$ and $S_{\Delta x}$ where
$S_{\Delta x}(x)=\frac{f(2+\Delta x)-f(2)}{\Delta x}(x-2)+f(2)$
(a) Use a graphing utility to graph $f$ and $S_{\Delta x}$ in the same viewing window for $\Delta x=1,0.5,$ and 0.1
(b) Give a written description of the graphs of $S$ for the different values of $\Delta x$ in part (a).
$f(x)=x+\frac{1}{x}$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 53

In Exercises $53-58,$ use the alternative form of the derivative to find the derivative at $x=c$ (if it exists).
$f(x)=x^{2}-1, \quad c=2$

William Hice

Numerade Educator

Problem 54

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

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 55

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

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 56

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

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 57

Use the alternative form of the derivative to find the derivative at $x=c$ (if it exists).
$h(x)=|x+5|, \quad c=-5$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 58

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

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 59

In Exercises $59-62,$ describe the $x$ -values at which $f$ is differentiable.

William Semus

Numerade Educator

Problem 60

In Exercises $59-62,$ describe the $x$ -values at which $f$ is differentiable.

William Semus

Numerade Educator

Problem 61

In Exercises $59-62,$ describe the $x$ -values at which $f$ is differentiable.

William Semus

Numerade Educator

Problem 62

In Exercises $59-62,$ describe the $x$ -values at which $f$ is differentiable.

William Semus

Numerade Educator

Problem 63

In Exercises 63-66, use a graphing utility to find the $x$ -values at which $f$ is differentiable.
$f(x)=|x+3|$

David Wellman

Numerade Educator

Problem 64

Use a graphing utility to find the $x$ -values at which $f$ is differentiable.
$f(x)=\frac{2 x}{x-1}$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 65

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

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 66

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

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 67

In Exercises $67-70$, find the derivatives from the left and from the right at $x=1$ (if they exist). Is the function differentiable at $x=1 ?$
$f(x)=|x-1|$

David Wellman

Numerade Educator

Problem 68

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

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 69

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

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 70

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

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 71

In Exercises 71 and 72, determine whether the function is differentiable at $x=2$
$f(x)=\left\{\begin{array}{ll}x^{2}+1, & x \leq 2 \\ 4 x-3, & x>2\end{array}\right.$

David Wellman

Numerade Educator

Problem 72

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

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 73

Graphical Reasoning 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?

Linh Vu

Numerade Educator

Problem 74

Conjecture 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.

Ahmad Reda

Numerade Educator

Problem 75

True or False? In Exercises 75 and $76,$ 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.

Anas Venkitta

Numerade Educator

Problem 76

True or False? 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.

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 77

Let $f(x)=\left\{\begin{array}{ll}x \sin \frac{1}{x}, & x \neq 0 \text { and } g(x)=\left\{\begin{array}{ll}x^{2} \sin \frac{1}{x}, & x \neq 0 \\ 0, & x=0\end{array} .\right.\end{array}\right.$
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

Numerade Educator