Calculus: One and Several Variables

Saturnino L. Salas, Garret J. Etgen, Einar Hille

Chapter 3

The Derivative; The Process of Differentation - all with Video Answers

Educators

Section 1

The Derivative

Problem 1

Differentiate the function by forming the difference quotient.
$$\frac{f(x+h)-f(x)}{h}$$
and taking the limit as $h$ tends to 0 .
$$f(x)=2-3 x$$

Leon Druch

Leon Druch

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Problem 2

Differentiate the function by forming the difference quotient.
$$\frac{f(x+h)-f(x)}{h}$$
and taking the limit as $h$ tends to 0 .
$f(x)=k, k$ constant.

Leon Druch

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Problem 3

Differentiate the function by forming the difference quotient.
$$\frac{f(x+h)-f(x)}{h}$$
and taking the limit as $h$ tends to 0 .
$$f(x)=5 x-x^{2}$$

Leon Druch

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Problem 4

Differentiate the function by forming the difference quotient.
$$\frac{f(x+h)-f(x)}{h}$$
and taking the limit as $h$ tends to 0 .
$$f(x)=2 x^{3}+1$$

Leon Druch

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Problem 5

Differentiate the function by forming the difference quotient.
$$\frac{f(x+h)-f(x)}{h}$$
and taking the limit as $h$ tends to 0 .
$$f(x)=x^{4}$$

Leon Druch

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Problem 6

Differentiate the function by forming the difference quotient.
$$\frac{f(x+h)-f(x)}{h}$$
and taking the limit as $h$ tends to 0 .
$$f(x)=1 /(x+3)$$

Leon Druch

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Problem 7

Differentiate the function by forming the difference quotient.
$$\frac{f(x+h)-f(x)}{h}$$
and taking the limit as $h$ tends to 0 .
$$f(x)=\sqrt{x-1}$$

Leon Druch

Numerade Educator

Problem 8

Differentiate the function by forming the difference quotient.
$$\frac{f(x+h)-f(x)}{h}$$
and taking the limit as $h$ tends to 0 .
$$f(x)=x^{3}-4 x$$

Leon Druch

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Problem 9

Differentiate the function by forming the difference quotient.
$$\frac{f(x+h)-f(x)}{h}$$
and taking the limit as $h$ tends to 0 .
$$f(x)=1 / x^{2}$$

Leon Druch

Numerade Educator

Problem 10

Differentiate the function by forming the difference quotient.
$$\frac{f(x+h)-f(x)}{h}$$
and taking the limit as $h$ tends to 0 .
$$f(x)=1 / \sqrt{x}$$

Leon Druch

Numerade Educator

Problem 11

Find $f^{\prime}(c)$ by forming the difference quotient
$$\frac{f(c+h)-f(c)}{h}$$
and taking the limit as $h \rightarrow 0$
$$f(x)=x^{2}-4 x ; c=3$$

Leon Druch

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Problem 12

Find $f^{\prime}(c)$ by forming the difference quotient
$$\frac{f(c+h)-f(c)}{h}$$
and taking the limit as $h \rightarrow 0$
$$f(x)=7 x-x^{2} ; c=2$$

Leon Druch

Numerade Educator

Problem 13

Find $f^{\prime}(c)$ by forming the difference quotient
$$\frac{f(c+h)-f(c)}{h}$$
and taking the limit as $h \rightarrow 0$
$$f(x)=2 x^{3}+1 ; c=1$$

Leon Druch

Numerade Educator

Problem 14

Find $f^{\prime}(c)$ by forming the difference quotient
$$\frac{f(c+h)-f(c)}{h}$$
and taking the limit as $h \rightarrow 0$
$$f(x)=5-x^{4} ; c=-1$$

Leon Druch

Numerade Educator

Problem 15

Find $f^{\prime}(c)$ by forming the difference quotient
$$\frac{f(c+h)-f(c)}{h}$$
and taking the limit as $h \rightarrow 0$
$$f(x)=\frac{8}{x+4} ; c=-2$$

Leon Druch

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Problem 16

Find $f^{\prime}(c)$ by forming the difference quotient
$$\frac{f(c+h)-f(c)}{h}$$
and taking the limit as $h \rightarrow 0$
$$f(x)=\sqrt{6-x} ; c=2$$

Leon Druch

Numerade Educator

Problem 17

Write an equation for the tangent line at $(c, f(c))$
$$f(x)=5 x-x^{2} ; c=4$$

Leon Druch

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Problem 18

Write an equation for the tangent line at $(c, f(c))$
$$f(x)=\sqrt{x} ; c=4$$

Leon Druch

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Problem 19

Write an equation for the tangent line at $(c, f(c))$
$$f(x)=1 / x^{2} ; c=-2$$

Leon Druch

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Problem 20

Write an equation for the tangent line at $(c, f(c))$
$$f(x)=5-x^{3} ; c=2$$

Leon Druch

Numerade Educator

Problem 21

The graph of $f$ is shown below.
(a) At which numbers $c$ is $f$ discontinuous? Which of the discontinuities is removable?
(b) $\mathrm{At}$ which numbers $c$ is $f$ continuous but not differentiable?

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Problem 22

Exercise 21 for the function $f$ graphed below.

Carson Merrill

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Problem 23

Draw the graph of $f$; indicate where $f$ is not differentiable.
$$f(x)=|x+1|$$

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Problem 24

Draw the graph of $f$; indicate where $f$ is not differentiable.
$$f(x)=|2 x-5|$$

Leon Druch

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Problem 25

Draw the graph of $f$; indicate where $f$ is not differentiable.
$$f(x)=\sqrt{|x|}$$

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Problem 26

Draw the graph of $f$; indicate where $f$ is not differentiable.
$$f(x)=\left|x^{2}-4\right|$$

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Problem 27

Draw the graph of $f$; indicate where $f$ is not differentiable.
$$f(x)=\left\{\begin{aligned}
x^{2}, & x \leq 1 \\
2-x, & x > 1
\end{aligned}\right.$$

Leon Druch

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Problem 28

Draw the graph of $f$; indicate where $f$ is not differentiable.
$$f(x)=\left\{\begin{aligned}
x^{2}-1, & x \leq ? \\
3, & x > 2
\end{aligned}\right.$$

Carson Merrill

Numerade Educator

Problem 29

Find $f^{\prime}(c)$ if it exists.
$$f(x)=\left\{\begin{array}{cl}
4 x, & x < 1 \\
2 x^{2}+2 . & x \geq 1
\end{array} \quad c=1\right.$$

Leon Druch

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Problem 30

Find $f^{\prime}(c)$ if it exists.
$$f(x)=\left\{\begin{array}{cl}
3 x^{2}, & x \leq 1 \\
2 x^{3}+1, & x > 1
\end{array} \quad c=1\right.$$

Leon Druch

Numerade Educator

Problem 31

Find $f^{\prime}(c)$ if it exists.
$$f(x)=\left\{\begin{array}{cl}
x+1, & x \leq-1 \\
(x+1)^{2}, & x > -1 ;
\end{array} \quad c=-1\right.$$

Leon Druch

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Problem 32

Find $f^{\prime}(c)$ if it exists.
$$f(x)=\left\{\begin{array}{cl}
-\frac{1}{2} x^{2}, & x < 3 \\
-3 x, & x \geq 3
\end{array} \quad c=3\right.$$

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Problem 33

Sketch the graph of the derivative of the function indicated.

Carson Merrill

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Problem 34

Sketch the graph of the derivative of the function indicated.

Carson Merrill

Numerade Educator

Problem 35

Sketch the graph of the derivative of the function indicated.

Carson Merrill

Numerade Educator

Problem 36

Sketch the graph of the derivative of the function indicated.

Carson Merrill

Numerade Educator

Problem 37

Sketch the graph of the derivative of the function indicated.

Carson Merrill

Numerade Educator

Problem 38

Sketch the graph of the derivative of the function indicated.

Carson Merrill

Numerade Educator

Problem 39

Show that
$$f(x)=\left\{\begin{array}{ll}
x^{2}, & x \leq 1 \\
2 x, & x > 1
\end{array}\right.$$
is not differentiable at $x=1$

Leon Druch

Numerade Educator

Problem 40

Set
$$f(x)=\left\{\begin{array}{ll}
(x+1)^{2}, & x \leq 0 \\
(x-1)^{2}, & x>0
\end{array}\right.$$
(a) Determine $f^{\prime}(x)$ for $x \neq 0$
(b) Show that $f$ is not differentiable at $x=0$

Leon Druch

Numerade Educator

Problem 41

Find $A$ and $B$ givcn that the function
$$f(x)=\left\{\begin{aligned}
x^{3}, & x \leq 1 \\
A x+B, & x > 1
\end{aligned}\right.$$
is differentiable at $x=1$

Carson Merrill

Numerade Educator

Problem 42

Find $A$ and $B$ given that the function
$$f(x)=\left\{\begin{aligned}
x^{2}-2, & x \leq 2 \\
B x^{2}+A x, & x > 2
\end{aligned}\right.$$
is differentiable at $x=2$

Leon Druch

Numerade Educator

Problem 43

Give an example of a function $f$ that is defined for all real numbers and sa::sfics the given conditions.
$f^{\prime}(x)=0$ for all real $x$

Carson Merrill

Numerade Educator

Problem 44

Give an example of a function $f$ that is defined for all real numbers and sa::sfics the given conditions.
$f^{\prime}(x)=0$ for all $x \leq 0 ; f^{\prime}(0)$ does not exist.

Carson Merrill

Numerade Educator

Problem 45

Give an example of a function $f$ that is defined for all real numbers and sa::sfics the given conditions.
$f^{\prime}(x)$ exists for all $x \neq-1 ; f^{\prime}(-1)$ docs not exist.

Carson Merrill

Numerade Educator

Problem 46

Give an example of a function $f$ that is defined for all real numbers and sa::sfics the given conditions.
$f^{\prime}(x)$ exists for all $x, t \pm 1 ;$ neither $f^{\prime}(1)$ nor $f^{\prime}(-1)$ exists.

Carson Merrill

Numerade Educator

Problem 47

Give an example of a function $f$ that is defined for all real numbers and sa::sfics the given conditions.
$$f^{\prime}(1) \quad 2 \text { and } f(1)$$

Carson Merrill

Numerade Educator

Problem 48

Give an example of a function $f$ that is defined for all real numbers and sa::sfics the given conditions.
$f^{\prime}(x)=1$ for $x < 0$ and $f^{\prime}(x)=-1$ for $x \because 0$

Carson Merrill

Numerade Educator

Problem 49

$$\text { Set } f(x)=\left\{\begin{array}{ll}
x^{2}-x, & x \leq 2 \\
2 x-2, & x > 2
\end{array}\right.$$
(a) Show that $f$ is continuous at 2
(b) Is $f$ differentiable at $2 ?$

Carson Merrill

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Problem 50

$$\text { Let } f(x)=x \sqrt{x}, x \geq 0 . \text { Calculate } f^{\prime}(x) \text { for } \operatorname{cach} x$$

Carson Merrill

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Problem 51

$$\text { Sel } f(x)=\left\{\begin{aligned}
1-x^{2}, & x \leq 0 \\
x^{2}, & x > 0
\end{aligned}\right.$$
(a) Is $f$ differentiable at $0 ?$
(b) Sketcb the graph of $f$

Carson Merrill

Numerade Educator

Problem 52

Set
$$f(x)=\left\{\begin{array}{ll}
x, & x \text { rational } \\
0, & x \text { irrational. }
\end{array} \quad y(x)=\left\{\begin{array}{ll}
x^{2}, & x \text { rational } \\
0 . & x \text { irrational. }
\end{array}\right.\right.$$
(a) Show that $f$ is not differentiable at 0 .
(b) Show that $g$ is differentiable at 0 and give $g^{\prime}(0)$

Carson Merrill

Numerade Educator

Problem 53

Write an equation for the normal line at ( $c$. $f(c)$ ) given that the tangent line at this point
(a) is horizontal:
(b) has slope $f^{\prime}(c) \neq 0$
(c) is vertical.

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Problem 54

All the normals through a circular arc pass through one point. What is this point?

Leon Druch

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Problem 55

As you saw in Example 7 , the line $y-2=\frac{1}{4}(x-4)$ is tangent to the graph of the square-root function at the point (4,2) Write an equation for the normal line through this point.

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Problem 56

(A follow-up to Exercise 55 ) Sketch the graph of the squareroot function displaying both the tangent and the normal at the point (4,2)

Leon Druch

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Problem 57

The lines tangent and normal to the graph of the squaring function at the point (3,9) intersect the $x$ -axis at points $s$ units apart. What is $\left.s^{\prime}\right\rangle$

Carson Merrill

Numerade Educator

Problem 58

The graph of the function $f(x)=\sqrt{1-x^{2}}$ is the upper half of the unit circle. On that curve (see the figure below) we have marked a point $P(x, y)$
(a) What is the slope of the normal at $P ?$ Express your answer in terms of $x$ and $y$
(b) Deduce from (a) the slope of the tangent at $P .$ Express your answer in terms of $x$ and $y$
(c) Confirm your answer in (b) by calculating
$$f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{\sqrt{1-(x+h)^{2}}-\sqrt{1-x^{2}}}{h}$$

Carson Merrill

Numerade Educator

Problem 59

Let $f(x)=\left\{\begin{aligned} x \sin (1 / x), & x \neq 0 \\ 0, & x=0 \end{aligned} \text { and } g(x)=x f(x)\right.$
The graphs of $f$ and $g$ are indicated in the figures below.
(a) Show that $f$ and $g$ are both continuous at $0 .$
(b) sinow that $f$ is not differentiable at 0
(c) Show that $g$ is differentiable at 0 and give $g^{\prime}(0)$

Carson Merrill

Numerade Educator

Problem 60

This is an alternative definition of derivative which has advantages in certain situations. Convince yourself of the equivalence of both definitions by calculating $f^{\prime}(c)$ by both methods.
$$f(x)=x^{3}+1 ; c=2$$

Leon Druch

Numerade Educator

Problem 61

This is an alternative definition of derivative which has advantages in certain situations. Convince yourself of the equivalence of both definitions by calculating $f^{\prime}(c)$ by both methods.
$$f(x)=x^{2}-3 x ; c=1$$

Leon Druch

Numerade Educator

Problem 62

This is an alternative definition of derivative which has advantages in certain situations. Convince yourself of the equivalence of both definitions by calculating $f^{\prime}(c)$ by both methods.
$$f(x)=\sqrt{1+x} ; c=3$$

Leon Druch

Numerade Educator

Problem 63

This is an alternative definition of derivative which has advantages in certain situations. Convince yourself of the equivalence of both definitions by calculating $f^{\prime}(c)$ by both methods.
$$f(x)=x^{1 / 3} ; c=-1$$

Carson Merrill

Numerade Educator

Problem 64

This is an alternative definition of derivative which has advantages in certain situations. Convince yourself of the equivalence of both definitions by calculating $f^{\prime}(c)$ by both methods.
$$f(x)=\frac{1}{x+2} ; c=0$$

Leon Druch

Numerade Educator

Problem 65

Set $f(x)=x^{5 / 2}$ and consider the difference quotient
$$D(h)=\frac{f(2+h)-f(2)}{h}$$
(a) Use a graphing utility to graph $D$ for $h \neq 0$, Estimate $\int(2)$ to three decimal piaces front the graph.
(b) Create a table of values to cstimate $\lim _{k \rightarrow 0} D(h) .$ Estiniate $f^{\prime}(2)$ io three decimal places from the table.
(c) Compare your results from (a) and (b).

Carson Merrill

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Problem 66

Exercise 65 with $f(x)=x^{2 / 3}$

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Problem 67

Use the definition of the derivative with a CAS to find $f^{\prime}(x)$ in general and $f^{\prime}(c)$ in particular.
(a) $f(x)=\sqrt{5 x-4} ; c=3$
(b) $f(x)=2-x^{2}+4 x^{4}-x^{6} ; c=-2$
(c) $f(x)=\frac{3-2 x}{2+3 x} ; c=-1$

Leon Druch

Numerade Educator

Problem 68

Use a CAS to evaluate, if possible,
$$f^{\prime}(c)=\lim _{h \rightarrow 0} \frac{f(c+h)-f(c)}{h}$$
(a) $f(x)=|x-1|+2 ; c=1$
(b) $f(x)=(x+2)^{y / 3}-1 ; c=-2$
(c) $f(x)=(x-3)^{2 / 3}+3 ; c=3$

Carson Merrill

Numerade Educator

Problem 69

Let $f(x)=5 x^{2}-7 x^{3}$ on [-1.1]
(a) Use a graphing utility to draw the graph of $f$
(b) Use the trace function to approximate the points on the graph where the tangent line is horizontal.
(c) Use a CAS to find $f^{\prime}(x)$
(d) Use a solver to solve the equation $f^{\prime}(x)=0$ and compare what you find to what you found in (b).

Carson Merrill

Numerade Educator

Problem 70

Exercise 69 with $f(x)=x^{3}+x^{2}-4 x+3$ on [-2,2]

Leon Druch

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Problem 71

Set $f(x)=4 x-x^{3}$
(a) Use a CAS to find $f^{\prime}\left(\frac{3}{2}\right)$. Then find equations for the tangent $T$ and the normal $N$ at the point $\left(\frac{3}{2}, f\left(\frac{3}{2}\right)\right)$
(b) Use a graphing utility to display $N, T,$ and the graph of
$f$ in one figure.
(c) Note that $T$ is a good approx:mation to the graph of $f$ for $x$ close to $\frac{3}{2} .$ Determine the interval on which the vertical scparation between $T$ and the gragh of $f$ is of absolute value less than 0.01

Carson Merrill

Numerade Educator

Problem 72

If $f(x)=x,$ then $f^{\prime}(x)=1 \cdot x^{0}=1$
If $f(x)=x^{2},$ then $f^{\prime}(x)=2 x^{\prime}-2 x$
(a) Show that
$$f(x)=x^{3}, \quad \text { then } \quad f^{\prime}(x)=3 x^{2}$$
(b) Prove by induction that for each positive integer $n$
$$f(x)=x^{n} \quad \text { has derivative } \quad f^{\prime}(x)=n x^{n-1}$$

Carson Merrill

Numerade Educator