Single Variable Calculus: Early Transcendentals

James Stewart, Daniel K. Clegg, Saleem Watson, Lothar Redlin

Chapter 3

Differentiation Rules - all with Video Answers

Educators

Section 1

Derivatives of Polynomials and Exponential Functions

Problem 1

(a) How is the number $e$ defined?
(b) Use a calculator to estimate the values of the limits

$$
\lim _{h \rightarrow 0} \frac{2.7^h-1}{h} \quad \text { and } \quad \lim _{h \rightarrow 0} \frac{2.8^h-1}{h}
$$

correct to two decimal places. What can you conclude about the value of $e$ ?

James Kiss

James Kiss

Numerade Educator

Problem 2

(a) Sketch, by hand, the graph of the function $f(x)=e^x$, paying particular attention to how the graph crosses the $y$-axis. What is the slope of the tangent line at that point?
(b) What types of functions are $f(x)=e^x$ and $g(x)=x^e$ ? Compare the differentiation formulas for $f$ and $g$.
(c) Which of the two functions in part (b) grows more rapidly when $x$ is large?

Sarah Parrigin

Numerade Educator

Problem 3

Differentiate the function.
$g(x)=4 x+7$

Gregory Higby

Numerade Educator

Problem 4

Differentiate the function.
$g(t)=5 t+4 t^2$

Gregory Higby

Numerade Educator

Problem 5

Differentiate the function.
$f(x)=x^{75}-x+3$

Gregory Higby

Numerade Educator

Problem 6

Differentiate the function.
$g(x)=\frac{7}{4} x^2-3 x+12$

Clarissa Noh

Numerade Educator

Problem 7

Differentiate the function.
$f(t)=-2 e^t$

Gregory Higby

Numerade Educator

Problem 8

Differentiate the function.
$F(t)=t^3+e^3$

Gregory Higby

Numerade Educator

Problem 9

Differentiate the function.
$W(v)=1.8 v^{-3}$

Gregory Higby

Numerade Educator

Problem 10

Differentiate the function.
$r(z)=z^{-5}-z^{1 / 2}$

Gregory Higby

Numerade Educator

Problem 11

Differentiate the function.
$f(x)=x^{3 / 2}+x^{-3}$

Gregory Higby

Numerade Educator

Problem 12

Differentiate the function.
$V(t)=t^{-3 / 5}+t^4$

Gregory Higby

Numerade Educator

Problem 13

Differentiate the function.
$s(t)=\frac{1}{t}+\frac{1}{t^2}$

Gregory Higby

Numerade Educator

Problem 14

Differentiate the function.
$r(t)=\frac{a}{t^2}+\frac{b}{t^4}$

Gregory Higby

Numerade Educator

Problem 15

Differentiate the function.
$y=2 x+\sqrt{x}$

Harsh Gadhiya

Numerade Educator

Problem 16

Differentiate the function.
$h(w)=\sqrt{2} w-\sqrt{2}$

Gregory Higby

Numerade Educator

Problem 17

Differentiate the function.
$g(x)=\frac{1}{\sqrt{x}}+\sqrt[4]{x}$

Gregory Higby

Numerade Educator

Problem 18

Differentiate the function.
$W(t)=\sqrt{t}-2 e^t$

Gregory Higby

Numerade Educator

Problem 19

Differentiate the function.
$f(x)=x^3(x+3)$

Gregory Higby

Numerade Educator

Problem 20

Differentiate the function.
$F(t)=(2 t-3)^2$

Gregory Higby

Numerade Educator

Problem 21

Differentiate the function.
$y=3 e^x+\frac{4}{\sqrt[3]{x}}$

Clarissa Noh

Numerade Educator

Problem 22

Differentiate the function.
$S(R)=4 \pi R^2$

Jack Gage

Numerade Educator

Problem 23

Differentiate the function.
$f(x)=\frac{3 x^2+x^3}{x}$

Gregory Higby

Numerade Educator

Problem 24

Differentiate the function.
. $y=\frac{\sqrt{x}+x}{x^2}$

Clarissa Noh

Numerade Educator

Problem 25

Differentiate the function.
$G(r)=\frac{3 r^{3 / 2}+r^{5 / 2}}{r}$

Gregory Higby

Numerade Educator

Problem 26

Differentiate the function.
$G(t)=\sqrt{5 t}+\frac{\sqrt{7}}{t}$

Frank Lin

Numerade Educator

Problem 27

Differentiate the function.
$j(x)=x^{2.4}+e^{2.4}$

Jack Gage

Numerade Educator

Problem 28

Differentiate the function.
$k(r)=e^r+r^r$

Clarissa Noh

Numerade Educator

Problem 29

Differentiate the function.
$F(z)=\frac{A+B z+C z^2}{z^2}$

Gregory Higby

Numerade Educator

Problem 30

Differentiate the function.
$G(q)=\left(1+q^{-1}\right)^2$

Prashansha Kaushik

Prashansha Kaushik

Numerade Educator

Problem 31

Differentiate the function.
$D(t)=\frac{1+16 t^2}{(4 t)^3}$

Gregory Higby

Numerade Educator

Problem 32

Differentiate the function.
$f(v)=\frac{\sqrt[3]{v}-2 v e^z}{v}$

Gregory Higby

Numerade Educator

Problem 33

Differentiate the function.
$P(w)=\frac{2 w^2-w+4}{\sqrt{w}}$

Gregory Higby

Numerade Educator

Problem 34

Differentiate the function.
$y=e^{x+1}+1$

Clarissa Noh

Numerade Educator

Problem 35

Find $d y / d x$ and $d y / d t$.
$y=t x^2+t^3 x$

Gregory Higby

Numerade Educator

Problem 36

Find $d y / d x$ and $d y / d t$.
$y=\frac{t}{x^2}+\frac{x}{t}$

Carson Merrill

Numerade Educator

Problem 37

Find an equation of the tangent line to the curve at the given point.
$y=2 x^3-x^2+2, \quad(1,3)$

Frank Lin

Numerade Educator

Problem 38

Find an equation of the tangent line to the curve at the given point.
$y=2 e^x+x, \quad(0,2)$

Marcella Sippey

Marcella Sippey

Numerade Educator

Problem 39

Find an equation of the tangent line to the curve at the given point.
$y=x+\frac{2}{x}, \quad(2,3)$

Arin Asawa

Numerade Educator

Problem 40

Find an equation of the tangent line to the curve at the given point.
$y=\sqrt[4]{x}-x, \quad(1,0)$

Frank Lin

Numerade Educator

Problem 41

Find equations of the tangent line and normal line to the curve at the given point.
$y=x^4+2 e^x, \quad(0,2)$

Clarissa Noh

Numerade Educator

Problem 42

Find equations of the tangent line and normal line to the curve at the given point.
$y=x^{3 / 2},(1,1)$

Gregory Higby

Numerade Educator

Problem 43

Find an equation of the tangent line to the curve at the given point. Illustrate by graphing the curve and the tangent line on the same screen.
$y=3 x^2-x^3, \quad(1,2)$

Clarissa Noh

Numerade Educator

Problem 44

Find an equation of the tangent line to the curve at the given point. Illustrate by graphing the curve and the tangent line on the same screen.
$y=x-\sqrt{x}, \quad(1,0)$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 45

Find $f^{\prime}(x)$. Compare the graphs of $f$ and $f^{\prime}$ and use them to explain why your answer is reasonable.
$f(x)=x^4-2 x^3+x^2$

Clarissa Noh

Numerade Educator

Problem 46

Find $f^{\prime}(x)$. Compare the graphs of $f$ and $f^{\prime}$ and use them to explain why your answer is reasonable.
$f(x)=x^5-2 x^3+x-1$

Clarissa Noh

Numerade Educator

Problem 47

(a) Graph the function

$$
f(x)=x^4-3 x^3-6 x^2+7 x+30
$$

in the viewing rectangle $[-3,5]$ by $[-10,50]$.
(b) Using the graph in part (a) to estimate slopes, make a rough sketch, by hand, of the graph of $f^{\prime}$. (See Example 2.8.1.)
(c) Calculate $f^{\prime}(x)$ and use this expression to graph $f^{\prime}$. Compare with your sketch in part (b).

Jackson Henningfield

Jackson Henningfield

Numerade Educator

Problem 48

(a) Graph the function $g(x)=e^x-3 x^2$ in the viewing rectangle $[-1,4]$ by $[-8,8]$.
(b) Using the graph in part (a) to estimate slopes, make a rough sketch, by hand, of the graph of $g^{\prime}$. (See Example 2.8.1.)
(c) Calculate $g^{\prime}(x)$ and use this expression to graph $g^{\prime}$. Compare with your sketch in part (b).

Jackson Henningfield

Jackson Henningfield

Numerade Educator

Problem 49

Find the first and second derivatives of the function.
$f(x)=0.001 x^5-0.02 x^3$

Gregory Higby

Numerade Educator

Problem 50

Find the first and second derivatives of the function.
$G(r)=\sqrt{r}+\sqrt[3]{r}$

Clarissa Noh

Numerade Educator

Problem 51

Find the first and second derivatives of the function. Check to see that your answers are reasonable by comparing the graphs of $f, f^{\prime}$, and $f^{\prime \prime}$.
$f(x)=2 x-5 x^{3 / 4}$

Clarissa Noh

Numerade Educator

Problem 52

Find the first and second derivatives of the function. Check to see that your answers are reasonable by comparing the graphs of $f, f^{\prime}$, and $f^{\prime \prime}$.
$f(x)=e^x-x^3$

Clarissa Noh

Numerade Educator

Problem 53

The equation of motion of a particle is $s=t^3-3 t$, where $s$ is in meters and $t$ is in seconds. Find
(a) the velocity and acceleration as functions of $t$,
(b) the acceleration after 2 s , and
(c) the acceleration when the velocity is 0 .

Check back soon!

Problem 54

The equation of motion of a particle is $s=t^4-2 t^3+t^2-t$, where $s$ is in meters and $t$ is in seconds.
(a) Find the velocity and acceleration as functions of $t$.
(b) Find the acceleration after 1 s .
(c) Graph the position, velocity, and acceleration functions on the same screen.

Jack Gage

Numerade Educator

Problem 55

Biologists have proposed a cubic polynomial to model the length $L$ of Alaskan rockfish at age $A$ :

$$
L=0.0155 A^3-0.372 A^2+3.95 A+1.21
$$

where $L$ is measured in inches and $A$ in years. Calculate

$$
\left.\frac{d L}{d A}\right|_{A=12}
$$

and interpret your answer.

Cooper Wilkinson

Cooper Wilkinson

Numerade Educator

Problem 56

The number of tree species $S$ in a given area $A$ in a forest reserve has been modeled by the power function

$$
S(A)=0.882 A^{0.842}
$$

where $A$ is measured in square meters. Find $S^{\prime}(100)$ and interpret your answer.

Gregory Higby

Numerade Educator

Problem 57

Boyle's Law states that when a sample of gas is compressed at a constant temperature, the pressure $P$ of the gas is inversely proportional to the volume $V$ of the gas.
(a) Suppose that the pressure of a sample of air that occupies $0.106 \mathrm{~m}^3$ at $25^{\circ} \mathrm{C}$ is 50 kPa . Write $V$ as a function of $P$.
(b) Calculate $d V / d P$ when $P=50 \mathrm{kPa}$. What is the meaning of the derivative? What are its units?

Cooper Wilkinson

Cooper Wilkinson

Numerade Educator

Problem 58

Car tires need to be inflated properly because overinflation or underinflation can cause premature tread wear. The data in the table show tire life $L$ (in thousands of miles) for a certain type of tire at various pressures $P\left(\mathrm{in} \mathrm{lb} / \mathrm{in}^2\right)$.
$$
\begin{array}{|l|l|l|l|l|l|l|l|}
\hline P & 26 & 28 & 31 & 35 & 38 & 42 & 45 \\
\hline L & 50 & 66 & 78 & 81 & 74 & 70 & 59 \\
\hline
\end{array}
$$
(a) Use a calculator or computer to model tire life with a quadratic function of the pressure.
(b) Use the model to estimate $d L / d P$ when $P=30$ and when $P=40$. What is the meaning of the derivative? What are the units? What is the significance of the signs of the derivatives?

Christy Galilei

Christy Galilei

Numerade Educator

Problem 59

Find the points on the curve $y=x^3+3 x^2-9 x+10$ where the tangent is horizontal.

Gregory Higby

Numerade Educator

Problem 60

For what value of $x$ does the graph of $f(x)=e^x-2 x$ have a horizontal tangent?

HS

Hardik Sheliya

Numerade Educator

Problem 61

Show that the curve $y=2 e^x+3 x+5 x^3$ has no tangent line with slope 2.

Jack Gage

Numerade Educator

Problem 62

Find an equation of the line that is both tangent to the curve $y=x^4+1$ and parallel to the line $32 x-y=15$.

Jonathan Segal

Numerade Educator

Problem 63

Find equations for two lines that are both tangent to the curve $y=x^3-3 x^2+3 x-3$ and parallel to the line $3 x-y=15$.

Gregory Higby

Numerade Educator

Problem 64

At what point on the curve $y=1+2 e^x-3 x$ is the tangent line parallel to the line $3 x-y=5$ ? Illustrate by graphing the curve and both lines.

Clarissa Noh

Numerade Educator

Problem 65

Find an equation of the normal line to the curve $y=\sqrt{x}$ that is parallel to the line $2 x+y=1$.

Frank Lin

Numerade Educator

Problem 66

Where does the normal line to the parabola $y=x^2-1$ at the point $(-1,0)$ intersect the parabola a second time? Illustrate with a sketch.

Clarissa Noh

Numerade Educator

Problem 67

Draw a diagram to show that there are two tangent lines to the parabola $y=x^2$ that pass through the point $(0,-4)$. Find the coordinates of the points where these tangent lines intersect the parabola.

Aman Gupta

Numerade Educator

Problem 68

(a) Find equations of both lines through the point $(2,-3)$ that are tangent to the parabola $y=x^2+x$.
(b) Show that there is no line through the point $(2,7)$ that is tangent to the parabola. Then draw a diagram to see why.

Clarissa Noh

Numerade Educator

Problem 69

Use the definition of a derivative to show that if $f(x)=1 / x$, then $f^{\prime}(x)=-1 / x^2$. (This proves the Power Rule for the case $n=-1$.)

Amrita Bhasin

Numerade Educator

Problem 70

Find the $n$th derivative of each function by calculating the first few derivatives and observing the pattern that occurs.
(a) $f(x)=x^n$
(b) $f(x)=1 / x$

Clarissa Noh

Numerade Educator

Problem 71

Find a second-degree polynomial $P$ such that $P(2)=5$, $P^{\prime}(2)=3$, and $P^{\prime \prime}(2)=2$.

Jack Gage

Numerade Educator

Problem 72

The equation $y^{\prime \prime}+y^{\prime}-2 y=x^2$ is called a differential equation because it involves an unknown function $y$ and its derivatives $y^{\prime}$ and $y^{\prime \prime}$. Find constants $A, B$, and $C$ such that the function $y=A x^2+B x+C$ satisfies this equation. (Differential equations will be studied in detail in Chapter 9.)

TP

Tisha Parikh

Numerade Educator

Problem 73

Find a cubic function $y=a x^3+b x^2+c x+d$ whose graph has horizontal tangents at the points $(-2,6)$ and $(2,0)$.

Wesley Hines

Numerade Educator

Problem 74

Find a parabola with equation $y=a x^2+b x+c$ that has slope 4 at $x=1$, slope -8 at $x=-1$, and passes through the point $(2,15)$.

Megan Jenkins

Numerade Educator

Problem 75

Let

$$
f(x)= \begin{cases}x^2+1 & \text { if } x<1 \\ x+1 & \text { if } x \geqslant 1\end{cases}
$$

Is $f$ differentiable at 1 ? Sketch the graphs of $f$ and $f^{\prime}$.

Cooper Wilkinson

Cooper Wilkinson

Numerade Educator

Problem 76

At what numbers is the following function $g$ differentiable?

$$
g(x)= \begin{cases}2 x & \text { if } x \leqslant 0 \\ 2 x-x^2 & \text { if } 0<x<2 \\ 2-x & \text { if } x \geqslant 2\end{cases}
$$

Give a formula for $g^{\prime}$ and sketch the graphs of $g$ and $g^{\prime}$.

Jack Gage

Numerade Educator

Problem 77

(a) For what values of $x$ is the function $f(x)=\left|x^2-9\right|$ differentiable? Find a formula for $f^{\prime}$.
(b) Sketch the graphs of $f$ and $f^{\prime}$.

Joseph Liao

Numerade Educator

Problem 78

Where is the function $h(x)=|x-1|+|x+2|$ differentiable? Give a formula for $h^{\prime}$ and sketch the graphs of $h$ and $h^{\prime}$.

Farnood Ensan

Numerade Educator

Problem 79

Find the parabola with equation $y=a x^2+b x$ whose tangent line at $(1,1)$ has equation $y=3 x-2$.

Amrita Bhasin

Numerade Educator

Problem 80

Suppose the curve $y=x^4+a x^3+b x^2+c x+d$ has a tangent line when $x=0$ with equation $y=2 x+1$ and a tangent line when $x-1$ with equation $y-2-3 x$. Find the values of $a, b, c$, and $d$.

John Irizar

Numerade Educator

Problem 81

For what values of $a$ and $b$ is the line $2 x+y=b$ tangent to the parabola $y=a x^2$ when $x=2$ ?

Amrita Bhasin

Numerade Educator

Problem 82

Find the value of $c$ such that the line $y=\frac{3}{2} x+6$ is tangent to the curve $y=c \sqrt{x}$.

Clarissa Noh

Numerade Educator

Problem 83

What is the value of $c$ such that the line $y=2 x+3$ is tangent to the parabola $y=c x^2$ ?

Clarissa Noh

Numerade Educator

Problem 84

The graph of any quadratic function $f(x)=a x^2+b x+c$ is a parabola. Prove that the average of the slopes of the tangent lines to the parabola at the endpoints of any interval $[p, q]$ equals the slope of the tangent line at the midpoint of the interval.

Clarissa Noh

Numerade Educator

Problem 85

Let

$$
f(x)= \begin{cases}x^2 & \text { if } x \leq 2 \\ m x+b & \text { if } x>2\end{cases}
$$

Find the values of $m$ and $b$ that make $f$ differentiable everywhere.

Cooper Wilkinson

Cooper Wilkinson

Numerade Educator

Problem 86

Find numbers $a$ and $b$ such that the given function $g$ is differentiable at 1 .

$$
g(x)= \begin{cases}a x^3-3 x & \text { if } x \leq 1 \\ b x^2+2 & \text { if } x>1\end{cases}
$$

Gregory Higby

Numerade Educator

Problem 87

Evaluate $\lim _{x \rightarrow 1} \frac{x^{1000}-1}{x-1}$.

Amrita Bhasin

Numerade Educator

Problem 88

A tangent line is drawn to the hyperbola $x y=c$ at a point $P$ as shown in the figure.
(a) Show that the midpoint of the line segment cut from this tangent line by the coordinate axes is $P$.
(b) Show that the triangle formed by the tangent line and the coordinate axes always has the same area, no matter where $P$ is located on the hyperbola.

Carson Merrill

Numerade Educator

Problem 89

Draw a diagram showing two perpendicular lines that intersect on the $y$-axis and are both tangent to the parabola $y=x^2$. Where do these lines intersect?

Frank Lin

Numerade Educator

Problem 90

Sketch the parabolas $y=x^2$ and $y=x^2-2 x+2$. Do you think there is a line that is tangent to both curves? If so, find its equation. If not, why not?

Frank Lin

Numerade Educator

Problem 91

If $c>\frac{1}{2}$, how many lines through the point $(0, c)$ are normal lines to the parabola $y=x^2$ ? What if $c \leq \frac{1}{2}$ ?

Clarissa Noh

Numerade Educator