Thomas Calculus

George B. Thomas, Jr.

Chapter 2

Limits and Continuity - all with Video Answers

Educators

+ 4 more educators

Section 1

Rates of Change and Tangents to Curves

Problem 1

Find the average rate of change of the function over the given interval or intervals.

$f(x)=x^{3}+1$
a. $[2,3] \quad$ b. $[-1,1]$

TF

Tyler Fyock

Numerade Educator

Problem 2

Find the average rate of change of the function over the given interval or intervals.

$g(x)=x^{2}-2 x$
a. $[1,3] \quad$ b. $[-2,4]$

Gregory Higby

Numerade Educator

Problem 3

Find the average rate of change of the function over the given interval or intervals.

$h(t)=\cot t$
a. $[\pi / 4,3 \pi / 4] \quad$ b. $[\pi / 6, \pi / 2]$

Melissa Munoz

Numerade Educator

Problem 4

Find the average rate of change of the function over the given interval or intervals.

$g(t)=2+\cos t$
a. $[0, \pi] \quad$ b. $[-\pi, \pi]$

William Semus

Numerade Educator

Problem 5

Find the average rate of change of the function over the given interval or intervals.

$R(\theta)=\sqrt{4 \theta+1} ; \quad[0,2]$

Cinsy Krehbiel

Numerade Educator

Problem 6

Find the average rate of change of the function over the given interval or intervals.

$P(\theta)=\theta^{3}-4 \theta^{2}+5 \theta ; \quad[1,2]$

Ahmad Reda

Numerade Educator

Problem 7

Use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and $(\mathrm{b})$ an equation of the tangent line at $P.$

$y=x^{2}-5, \quad P(2,-1)$

Melissa Price

Numerade Educator

Problem 8

Use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and $(\mathrm{b})$ an equation of the tangent line at $P.$

$y=7-x^{2}, \quad P(2,3)$

Matt Just

Numerade Educator

Problem 9

Use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and $(\mathrm{b})$ an equation of the tangent line at $P.$

$y=x^{2}-2 x-3, \quad P(2,-3)$

Matt Just

Numerade Educator

Problem 10

Use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and $(\mathrm{b})$ an equation of the tangent line at $P.$

$y=x^{2}-4 x, \quad P(1,-3)$

Mj Santos

Numerade Educator

Problem 11

Use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and $(\mathrm{b})$ an equation of the tangent line at $P.$

$y=x^{3}, \quad P(2,8)$

RM

Ricardo Martinez

Numerade Educator

Problem 12

Use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and $(\mathrm{b})$ an equation of the tangent line at $P.$

$y=2-x^{3}, \quad P(1,1)$

Mj Santos

Numerade Educator

Problem 13

Use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and $(\mathrm{b})$ an equation of the tangent line at $P.$

$y=x^{3}-12 x, \quad P(1,-11)$

Matt Just

Numerade Educator

Problem 14

Use the method in Example 3 to find (a) the slope of the curve at the given point $P,$ and $(\mathrm{b})$ an equation of the tangent line at $P.$

$y=x^{3}-3 x^{2}+4, \quad P(2,0)$

OS

Oliver Storseth

Numerade Educator

Problem 15

Speed of a car The accompanying figure shows the time-to-distance graph for a sports car accelerating from a standstill.
a. Estimate the slopes of secants $P Q_{1}, P Q_{2}, P Q_{3},$ and $P Q_{4}$ arranging them in order in a table like the one in Figure $2.6 .$ What are the appropriate units for these slopes?
b. Then estimate the car's speed at time $t=20 \mathrm{sec}$ .

Carson Merrill

Numerade Educator

Problem 16

The accompanying figure shows the plot of distance fallen versus time for an object that fell from the lunar landing module a distance 80 $\mathrm{m}$ to the surface of the moon.
a. Estimate the slopes of the secants $P Q_{1}, P Q_{2}, P Q_{3},$ and $P Q_{4}$ arranging them in a table like the one in Figure $2.6 .$
b. About how fast was the object going when it hit the surface?

Ahmad Reda

Numerade Educator

Problem 17

The profits of a small company for each of the first five years of its operation are given in the following table:
a. Plot points representing the profit as a function of year, and join them by as smooth a curve as you can.
b. What is the average rate of increase of the profits between 2012 and 2014?
c. Use your graph to estimate the rate at which the profits were changing in 2012 .

Carson Merrill

Numerade Educator

Problem 18

Make a table of values for the function $F(x)=(x+2) /(x-2)$ at the points $x=1.2, x=11 / 10, x=101 / 100, x=1001 / 1000$ $x=10001 / 10000,$ and $x=1 .$
a. Find the average rate of change of $F(x)$ over the intervals $[1, x]$ for each $x \neq 1$ in your table.
b. Extending the table if necessary, try to determine the rate of change of $F(x)$ at $x=1.$

Matt Just

Numerade Educator

Problem 19

Let $g(x)=\sqrt{x}$ for $x \geq 0.$
a. Find the average rate of change of $g(x)$ with respect to $x$ over the intervals $[1,2],[1,1.5]$ and $[1,1+h] .$
b. Make a table of values of the average rate of change of $g$ with respect to $x$ over the interval $[1,1+h]$ for some values of $h$ approaching zero, say $h=0.1,0.01,0.001,0.0001,0.00001,$ and $0.000001 .$
c. What does your table indicate is the rate of change of $g(x)$ with respect to $x$ at $x=1 ?$
d. Calculate the limit as $h$ approaches zero of the average rate of change of $g(x)$ with respect to $x$ over the interval $[1,1+h].$

George Stanisic

George Stanisic

Numerade Educator

Problem 20

Let $f(t)=1 / t$ for $t \neq 0.$
a. Find the average rate of change of $f$ with respect to $t$ over the intervals (i) from $t=2$ to $t=3,$ and (ii) from $t=2$ to $t=T$ .
b. Make a table of values of the average rate of change of $f$ with respect to $t$ over the interval $[2, T],$ for some values of $T$ approaching $2,$ say $T=2.1,2.01,2.001,2.0001,2.00001$ and $2.000001 .$
c. What does your table indicate is the rate of change of $f$ with respect to $t$ at $t=2 ?$
d. Calculate the limit as $T$ approaches 2 of the average rate of change of $f$ with respect to $t$ over the interval from 2 to $T=2$ . will have to do some algebra before you can substitute $T=2.$

Carson Merrill

Numerade Educator

Problem 21

The accompanying graph shows the total distance $s$ traveled by a bicyclist after $t$ hours.
a. Estimate the bicyclist's average speed over the time intervals $[0,1],[1,2.5],$ and $[2.5,3.5] .$
b. Estimate the bicyclist's instantaneous speed at the times $t=\frac{1}{2}$ $t=2,$ and $t=3 .$
c. Estimate the bicyclist's maximum speed and the specific time at which it occurs.

Carson Merrill

Numerade Educator

Problem 22

The accompanying graph shows the total amount of gasoline $A$ in the gas tank of an automobile after being driven for $t$ days.
a. Estimate the average rate of gasoline consumption over the time intervals $[0,3],[0,5],$ and $[7,10] .$
b. Estimate the instantaneous rate of gasoline consumption at the times $t=1, t=4,$ and $t=8$ .
c. Estimate the maximum rate of gasoline consumption and the specific time at which it occurs.

Matt Just

Numerade Educator