Calculus

James Stewart

Chapter 3

Differentiation Rules - all with Video Answers

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+ 1 more 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
$$\lim _{h \rightarrow 0} \frac{2.7^{4}-1}{h} \quad \text { and } \quad \lim _{h \rightarrow 0} \frac{2.8^{*}-1}{h}$$
correct to two decimal places. What can you conclude about the value of $e ?$

Sean Perry

Sean Perry

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 fact allows you to do this?
(b) What types of functions are $f(x)=e^{x}$ and $g(x)=x^{2} ?$ Compare the differentiation formulas for $f$ and $g$.
(c) Which of the two functions in part (b) grows more rapidly when $x$ is large?

Carson Merrill

Numerade Educator

Problem 3

Differentiate the function.
$$f(x)=186.5$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 4

Differentiate the function.
$$f(x)=\sqrt{30}$$

Clarissa Noh

Numerade Educator

Problem 5

Differentiate the function.
$$f(t)=2-\frac{2}{3} t$$

John Irizar

Numerade Educator

Problem 6

Differentiate the function.
$$F(x)=\frac{3}{4} x^{8}$$

Clarissa Noh

Numerade Educator

Problem 7

Differentiate the function.
$$f(x)=x^{3}-4 x+6$$

Amrita Bhasin

Numerade Educator

Problem 8

Differentiate the function.
$$f(t)=\frac{1}{2} t^{6}-3 t^{4}+t$$

Clarissa Noh

Numerade Educator

Problem 9

Differentiate the function.
$$f(t)=\frac{1}{4}\left(t^{4}+8\right)$$

Amrita Bhasin

Numerade Educator

Problem 10

Differentiate the function.
$$h(x)=(x-2)(2 x+3)$$

Clarissa Noh

Numerade Educator

Problem 11

Differentiate the function.
$$A(s)=-\frac{12}{s^{5}}$$

Amrita Bhasin

Numerade Educator

Problem 12

Differentiate the function.
$$B(y)=c y^{-6}$$

Clarissa Noh

Numerade Educator

Problem 13

Differentiate the function.
$$g(t)=2 t^{-3 / 4}$$

Amrita Bhasin

Numerade Educator

Problem 14

Differentiate the function.
$$h(t)=\sqrt[4]{t}-4 e^{t}$$

Clarissa Noh

Numerade Educator

Problem 15

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

Frank Lin

Numerade Educator

Problem 16

Differentiate the function.
$$y=\sqrt{x}(x-1)$$

Vishnu Rathnam

Numerade Educator

Problem 17

Differentiate the function.
$$F(x)=\left(\frac{1}{2} x\right)^{5}$$

Amrita Bhasin

Numerade Educator

Problem 18

Differentiate the function.
$$f(x)=\frac{x^{2}-3 x+1}{x^{2}}$$

Clarissa Noh

Numerade Educator

Problem 19

Differentiate the function.
$$y=\frac{x^{2}+4 x+3}{\sqrt{x}}$$

Amrita Bhasin

Numerade Educator

Problem 20

Differentiate the function.
$$g(u)=\sqrt{2} u+\sqrt{3 u}$$

Clarissa Noh

Numerade Educator

Problem 21

Differentiate the function.
$$y=4 \pi^{2}$$

Amrita Bhasin

Numerade Educator

Problem 22

Differentiate the function.
$$y=a e^{x}+\frac{b}{y}+\frac{c}{v^{2}}$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 23

Differentiate the function.
$$u=\sqrt[5]{t}+4 \sqrt{t^{5}}$$

Amrita Bhasin

Numerade Educator

Problem 24

Differentiate the function.
$$v=\left(\sqrt{x}+\frac{1}{\sqrt[3]{x}}\right)^{2}$$

Clarissa Noh

Numerade Educator

Problem 25

Differentiate the function.
$$z=\frac{A}{y^{10}}+B e^{y}$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 26

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

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 27

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

Heather Eichman

Heather Eichman

Numerade Educator

Problem 28

Find an equation of the tangent line to the curve at the given point.
$$y=x^{4}+2 x^{2}-\quad(1,2)$$

Clarissa Noh

Numerade Educator

Problem 29

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

Amrita Bhasin

Numerade Educator

Problem 30

Find equations of the tangent line and normal line to the curve at the given point.
$$y=(1+2 x)^{2}, \quad (1,9)$$

Clarissa Noh

Numerade Educator

Problem 31

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 32

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 33

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

Gregory Higby

Numerade Educator

Problem 34

Find $f^{\prime}(x)$. Compare the graphs of $f$ and $f^{\prime}$ and use them to explain why your answer is reasonable.
$$f(x)=3 x^{5}-20 x^{3}+50 x$$

Gregory Higby

Numerade Educator

Problem 35

Find $f^{\prime}(x)$. Compare the graphs of $f$ and $f^{\prime}$ and use them to explain why your answer is reasonable.
$$f(x)=3 x^{15}-5 x^{3}+3$$

Gregory Higby

Numerade Educator

Problem 36

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

Clarissa Noh

Numerade Educator

Problem 37

Estimate the value of $f^{\prime}(a)$ by zooming in on the graph of $f$ Then differentiate $f$ to find the exact value of $f^{\prime}(a)$ and compare with your estimate.
$$f(x)=3 x^{2}-x^{3}, \quad a=1$$

Gregory Higby

Numerade Educator

Problem 38

Estimate the value of $f^{\prime}(a)$ by zooming in on the graph of $f$ Then differentiate $f$ to find the exact value of $f^{\prime}(a)$ and compare with your estimate.
$$f(x)=1 / \sqrt{x}, \quad a=4$$

Gregory Higby

Numerade Educator

Problem 39

(a) Use a graphing calculator or computer to 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 1 in Section 2.7 .)
(c) Calculate $f^{\prime}(x)$ and use this expression, with a graphing device, to graph $f^{\prime}$ Compare with your sketch in part (b).

Carson Merrill

Numerade Educator

Problem 40

(a) Use a graphing calculator or computer to 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 1 in Section 2.7.
(c) Calculate $g^{\prime}(x)$ and use this expression, with a graphing device, to graph $g^{\prime}$ Compare with your sketch in part (b).

Carson Merrill

Numerade Educator

Problem 41

Find the first and second derivatives of the function.
$$f(x)=10 x^{10}+5 x^{5}-x$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 42

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

Clarissa Noh

Numerade Educator

Problem 43

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 $f^{\prime \prime}.$
$$f(x)=2 x-5 x^{3 /}$$

Carson Merrill

Numerade Educator

Problem 44

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 $f^{\prime \prime}.$
$$f(x)=e^{x}-x^{3}$$

Frank Lin

Numerade Educator

Problem 45

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 $0 .$

Frank Lin

Numerade Educator

Problem 46

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 47

On what interval is the function $f(x)=5 x-e^{x}$ increasing?

Lauren Shelton

Numerade Educator

Problem 48

On what interval is the function $f(x)=x^{3}-4 x^{2}+5 x$ concave upward?

Linda Hand

Numerade Educator

Problem 49

Find the points on the curve $y=2 x^{3}+3 x^{2}-12 x+1$ where the tangent is horizontal.

Caleb Miller

Numerade Educator

Problem 50

For what values of $x$ does the graph of $f(x)=x^{3}+3 x^{2}+x+3$ have a horizontal tangent?

Clarissa Noh

Numerade Educator

Problem 51

Show that the curve $y=6 x^{3}+5 x-3$ has no tangent line with slope 4.

Amrita Bhasin

Numerade Educator

Problem 52

Find an equation of the tangent line to the curve $y=x \sqrt{x}$ that is parallel to the line $y=1+3 x$.

Clarissa Noh

Numerade Educator

Problem 53

Find equations of both lines that are tangent to the curve $y=1+x^{3}$ and parallel to the line $12 x-y=1$.

Amrita Bhasin

Numerade Educator

Problem 54

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 55

Find an equation of the normal line to the parabola $y=x^{2}-5 x+4$ that is parallel to the line $x-3 y=5$.

Justin Avila

Numerade Educator

Problem 56

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

Clarissa Noh

Numerade Educator

Problem 57

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.

Amrita Bhasin

Numerade Educator

Problem 58

(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 59

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 $n=-1.$)

Sam Sohn

Numerade Educator

Problem 60

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 61

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

Jack Gage

Numerade Educator

Problem 62

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 7.)

Carson Merrill

Numerade Educator

Problem 63

(a) In Section 2.8 we defined an antiderivative of $f$ to be a function $F$ such that $F^{\prime}=f .$ Try to guess a formula for an antiderivative of $f(x)=x^{2} .$ Then check your answer by differentiating it. How many antiderivatives does $f$ have?
(b) Find antiderivatives for $f(x)=x^{3}$ and $f(x)=x^{4}.$
(c) Find an antiderivative for $f(x)=x^{n}$, where $n \neq-1$ Check by differentiation.

Carson Merrill

Numerade Educator

Problem 64

Use the result of Exercise 63 (c) to find an antiderivative of each function.
(a) $f(x)=\sqrt{x}$
(b) $f(x)=e^{x}+8 x^{3}$

Carson Merrill

Numerade Educator

Problem 65

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 66

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 67

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).

H M

Numerade Educator

Problem 68

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 69

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 70

A tangent line is drawn to the hyperbola $x y=c$ at a point $P.$
(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.

Clarissa Noh

Numerade Educator

Problem 71

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

Amrita Bhasin

Numerade Educator

Problem 72

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 73

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

John Irizar

Numerade Educator

Problem 74

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