Solution Manual For University Calculus Early Transcendentals 3rd Edition, Global Edition By Joel R. Hass

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Solution Manual For University Calculus Early Transcendentals 3rd Edition, Global Edition By Joel R. Hass

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SAMPLE QUESTIONS

 

Chapter 1 Functions

Functions

In Exercises 16, find the domain and range of each function.

  1. (x) = 18 + x2 2. (x) = 1 2x
  2. F(x) = 25x + 10 4. g(x) = 22 3x
  3. (t) = 4

3 t

  1. G(t) = 5

t2 4

In Exercises 7 and 8, which of the graphs are graphs of functions of x,

and which are not? Give reasons for your answers.

  1. a.

x

y

0

b.

x

y

0

  1. a.

x

y

0

b.

x

y

0

Finding Formulas for Functions

  1. Express the area and perimeter of an equilateral triangle as a

function of the triangles side length x.

  1. Express the side length of a square as a function of the length d of

the squares diagonal. Then express the area as a function of the

diagonal length.

  1. Express the edge length of a cube as a function of the cubes

diagonal length d. Then express the surface area and volume of

the cube as a function of the diagonal length.

  1. A point P in the first quadrant lies on the graph of the function

(x) = 2x. Express the coordinates of P as functions of the

slope of the line joining P to the origin.

  1. Consider the point (x, y) lying on the graph of the line

2x + 4y = 5. Let L be the distance from the point (x, y) to the

origin (0, 0). Write L as a function of x.

  1. Consider the point (x, y) lying on the graph of y = 2x 3. Let

L be the distance between the points (x, y) and (4, 0). Write L as a

function of y.

Functions and Graphs

Find the natural domain and graph the functions in Exercises 1520.

  1. (x) = 5 2x 16. (x) = 1 2x x2
  2. g(x) = 20 x 0 18. g(x) = 2-x
  3. F(t) = t> 0 t 0 20. G(t) = 1> 0 t 0
  4. Find the domain of y = x + 7

12 22 25

.

  1. Find the range of y = 2 + x2

x2 + 4 .

  1. Graph the following equations and explain why they are not

graphs of functions of x.

  1. 0 y 0 = x b. y2 = x2
  2. Graph the following equations and explain why they are not

graphs of functions of x.

  1. 0 x 0 + 0 y 0 = 1 b. 0 x + y 0 = 1

Piecewise-Defined Functions

Graph the functions in Exercises 2528.

  1. (x) = e

x, 0 x 1

2 x, 1 6 x 2

  1. g(x) = e

1 x, 0 x 1

2 x, 1 6 x 2

  1. F(x) = e

4 x2, x 1

x2 + 2x, x 7 1

  1. G(x) = e

1>x, x 6 0

x, 0 x

Find a formula for each function graphed in Exercises 2932.

  1. a.

x

y

0

1

2

(1, 1)

b.

t

y

0

2

1 2 3 4

  1. a.

x

y

2 5

2

(2, 1)

b.

_1

x

y

3

1 2

2

1

_2

_3

_1

(2, _1)

  1. a.

x

y

3

1

(_1, 1) (1, 1)

b.

x

y

1

2

(_2, _1) (1, _1) (3, _1)

  1. a.

x

y

0

1

T T2

(T, 1)

b.

t

y

0

A

T

_A

T2

3T

2

2T

The Greatest and Least Integer Functions

  1. For what values of x is
  2. :x; = -1? b.
  3. What real numbers x satisfy the equation :x; =
  4. Does <-x= = -:x; for all real x? Give reasons for your answer.
  5. Graph the function

(x) = e

:x;, x 0

Why is (x) called the integer part of x?

Increasing and Decreasing Functions

Graph the functions in Exercises 3746. What symmetries, if any, do

the graphs have? Specify the intervals over which the function is

increasing and the intervals where it is decreasing.

  1. y = -x3 38. y =

1

x2

  1. y =

1x

  1. y = 1

0 x 0

  1. y = 20 x 0 42. y = 2-x
  2. y = x3>8 44. y = -42x
  3. y = -x3>2 46. y = (-x)2>3

Even and Odd Functions

In Exercises 4758, say whether the function is even, odd, or neither.

Give reasons for your answer.

  1. (x) = -8 48. (x) = x-5
  2. g(x) = 93 3 50. (x) = x2 + x
  3. g(x) = x3 + x 52. g(x) = x4 + 32 1
  4. g(x) = 1

x2 4

  1. g(x) = x

x2 1

  1. h(t) = 1

t3 + 1

  1. h(t) = _ t3 _
  2. h(t) = 5t 1 58. h(t) = 2 _ t _ + 1

Theory and Examples

  1. The variable s is proportional to t, and s = 15 when t = 105.

Determine t when s = 40.

  1. Kinetic energy The kinetic energy K of a mass is proportional

to the square of its velocity y. If K = 12,960 joules when

y = 18 m>sec, what is K when y = 10 m>sec?

  1. The variables r and s are inversely proportional, and r = 7 when

s = 4. Determine s when r = 10.

  1. Boyles Law Boyles Law says that the volume V of a gas at

constant temperature increases whenever the pressure P decreases,

so that V and P are inversely proportional. If P = 14.7 lb>in2

when V = 1000 in3, then what is V when P = 23.4 lb>in2?

  1. A box with an open top is to be constructed from a rectangular

piece of cardboard with dimensions 14 in. by 22 in. by cutting out

equal squares of side x at each corner and then folding up the

sides as in the figure. Express the volume V of the box as a function

of x.

x

x

x

x

x

x

x

x

22

14

  1. The accompanying figure shows a rectangle inscribed in an isosceles

right triangle whose hypotenuse is 2 units long.

  1. Express the y-coordinate of P in terms of x. (You might start

by writing an equation for the line AB.)

  1. Express the area of the rectangle in terms of x.

x

y

_1 0 x 1

A

B

P(x, ?)

In Exercises 65 and 66, match each equation with its graph. Do not

use a graphing device, and give reasons for your answer.

  1. a. y = x4 b. y = x7 c. y = x10

x

y

f

g

h

0

  1. a. y = 5x b. y = 5x c. y = x5

x

y

f

h

g

0

  1. a. Graph the functions (x) = x>2 and g(x) = 1 + (4>x) to-

gether to identify the values of x for which

x

2 7 1 + 4x

.

  1. Confirm your findings in part (a) algebraically.
  2. a. Graph the functions (x) = 3>(x 1) and g(x) = 2>(x + 1)

together to identify the values of x for which

3

x 1 6 2

x + 1

.

  1. Confirm your findings in part (a) algebraically.
  2. For a curve to be symmetric about the x-axis, the point (x, y) must

lie on the curve if and only if the point (x, -y) lies on the curve.

Explain why a curve that is symmetric about the x-axis is not the

graph of a function, unless the function is y = 0.

  1. Three hundred books sell for $40 each, resulting in a revenue of

(300)($40) = $12,000. For each $5 increase in the price, 25

fewer books are sold. Write the revenue R as a function of the

number x of $5 increases.

  1. A pen in the shape of an isosceles right triangle with legs of

length x ft and hypotenuse of length h ft is to be built. If fencing

costs $2/ft for the legs and $8/ft for the hypotenuse, write the

total cost C of construction as a function of h.

  1. Industrial costs A power plant sits next to a river where the

river is 800 ft wide. To lay a new cable from the plant to a location

in the city 2 mi downstream on the opposite side costs $180

per foot across the river and $100 per foot along the land.

P x Q

Power plant

City

800 ft

2 mi

NOT TO SCALE

  1. Suppose that the cable goes from the plant to a point Q on the

opposite side that is x ft from the point P directly opposite the

plant. Write a function C(x) that gives the cost of laying the

cable in terms of the distance x.

  1. Generate a table of values to determine if the least expensive

location for point Q is less than 2000 ft or greater than 2000 ft

from point P.

  1. a.

x

y

3

1

(_1, 1) (1, 1)

b.

x

y

1

2

(_2, _1) (1, _1) (3, _1)

  1. a.

x

y

0

1

T T2

(T, 1)

b.

t

y

0

A

T

_A

T2

3T

2

2T

The Greatest and Least Integer Functions

  1. For what values of x is
  2. :x; = -1? b.
  3. What real numbers x satisfy the equation :x; =
  4. Does <-x= = -:x; for all real x? Give reasons for your answer.
  5. Graph the function

(x) = e

:x;, x 0

Why is (x) called the integer part of x?

Increasing and Decreasing Functions

Graph the functions in Exercises 3746. What symmetries, if any, do

the graphs have? Specify the intervals over which the function is

increasing and the intervals where it is decreasing.

  1. y = -x3 38. y =

1

x2

  1. y =

1x

  1. y = 1

0 x 0

  1. y = 20 x 0 42. y = 2-x
  2. y = x3>8 44. y = -42x
  3. y = -x3>2 46. y = (-x)2>3

Even and Odd Functions

In Exercises 4758, say whether the function is even, odd, or neither.

Give reasons for your answer.

  1. (x) = -8 48. (x) = x-5
  2. g(x) = 93 3 50. (x) = x2 + x
  3. g(x) = x3 + x 52. g(x) = x4 + 32 1
  4. g(x) = 1

x2 4

  1. g(x) = x

x2 1

  1. h(t) = 1

t3 + 1

  1. h(t) = _ t3 _
  2. h(t) = 5t 1 58. h(t) = 2 _ t _ + 1

Theory and Examples

  1. The variable s is proportional to t, and s = 15 when t = 105.

Determine t when s = 40.

  1. A point P in the first quadrant lies on the graph of the function

(x) = 2x. Express the coordinates of P as functions of the

slope of the line joining P to the origin.

  1. Consider the point (x, y) lying on the graph of the line

2x + 4y = 5. Let L be the distance from the point (x, y) to the

origin (0, 0). Write L as a function of x.

  1. Consider the point (x, y) lying on the graph of y = 2x 3. Let

L be the distance between the points (x, y) and (4, 0). Write L as a

function of y.

Functions and Graphs

Find the natural domain and graph the functions in Exercises 1520.

  1. (x) = 5 2x 16. (x) = 1 2x x2
  2. g(x) = 20 x 0 18. g(x) = 2-x
  3. F(t) = t> 0 t 0 20. G(t) = 1> 0 t 0
  4. Find the domain of y = x + 7

12 22 25

.

  1. Find the range of y = 2 + x2

x2 + 4 .

  1. Graph the following equations and explain why they are not

graphs of functions of x.

  1. 0 y 0 = x b. y2 = x2
  2. Graph the following equations and explain why they are not

graphs of functions of x.

  1. 0 x 0 + 0 y 0 = 1 b. 0 x + y 0 = 1

Piecewise-Defined Functions

Graph the functions in Exercises 2528.

  1. (x) = e

x, 0 x 1

2 x, 1 6 x 2

  1. g(x) = e

1 x, 0 x 1

2 x, 1 6 x 2

  1. F(x) = e

4 x2, x 1

x2 + 2x, x 7 1

  1. G(x) = e

1>x, x 6 0

x, 0 x

Find a formula for each function graphed in Exercises 2932.

  1. a.

x

y

0

1

2

(1, 1)

b.

t

y

0

2

1 2 3 4

  1. a.

x

y

2 5

2

(2, 1)

b.

_1

x

y

3

1 2

2

1

_2

_3

_1

(2, _1)

  1. Kinetic energy The kinetic energy K of a mass is proportional

to the square of its velocity y. If K = 12,960 joules when

y = 18 m>sec, what is K when y = 10 m>sec?

  1. The variables r and s are inversely proportional, and r = 7 when

s = 4. Determine s when r = 10.

  1. Boyles Law Boyles Law says that the volume V of a gas at

constant temperature increases whenever the pressure P decreases,

so that V and P are inversely proportional. If P = 14.7 lb>in2

when V = 1000 in3, then what is V when P = 23.4 lb>in2?

  1. A box with an open top is to be constructed from a rectangular

piece of cardboard with dimensions 14 in. by 22 in. by cutting out

equal squares of side x at each corner and then folding up the

sides as in the figure. Express the volume V of the box as a function

of x.

x

x

x

x

x

x

x

x

22

14

  1. The accompanying figure shows a rectangle inscribed in an isosceles

right triangle whose hypotenuse is 2 units long.

  1. Express the y-coordinate of P in terms of x. (You might start

by writing an equation for the line AB.)

  1. Express the area of the rectangle in terms of x.

x

y

_1 0 x 1

A

B

P(x, ?)

In Exercises 65 and 66, match each equation with its graph. Do not

use a graphing device, and give reasons for your answer.

  1. a. y = x4 b. y = x7 c. y = x10

x

y

f

g

h

0

  1. a. y = 5x b. y = 5x c. y = x5

x

y

f

h

g

0

  1. a. Graph the functions (x) = x>2 and g(x) = 1 + (4>x) to-

gether to identify the values of x for which

x

2 7 1 + 4x

.

  1. Confirm your findings in part (a) algebraically.
  2. a. Graph the functions (x) = 3>(x 1) and g(x) = 2>(x + 1)

together to identify the values of x for which

3

x 1 6 2

x + 1

.

  1. Confirm your findings in part (a) algebraically.
  2. For a curve to be symmetric about the x-axis, the point (x, y) must

lie on the curve if and only if the point (x, -y) lies on the curve.

Explain why a curve that is symmetric about the x-axis is not the

graph of a function, unless the function is y = 0.

  1. Three hundred books sell for $40 each, resulting in a revenue of

(300)($40) = $12,000. For each $5 increase in the price, 25

fewer books are sold. Write the revenue R as a function of the

number x of $5 increases.

  1. A pen in the shape of an isosceles right triangle with legs of

length x ft and hypotenuse of length h ft is to be built. If fencing

costs $2/ft for the legs and $8/ft for the hypotenuse, write the

total cost C of construction as a function of h.

  1. Industrial costs A power plant sits next to a river where the

river is 800 ft wide. To lay a new cable from the plant to a location

in the city 2 mi downstream on the opposite side costs $180

per foot across the river and $100 per foot along the land.

P x Q

Power plant

City

800 ft

2 mi

NOT TO SCALE

  1. Suppose that the cable goes from the plant to a point Q on the

opposite side that is x ft from the point P directly opposite the

plant. Write a function C(x) that gives the cost of laying the

cable in terms of the distance x.

  1. Generate a table of values to determine if the least expensive

location for point Q is less than 2000 ft or greater than 2000 ft

from point P.

Choosing a Viewing Window

In Exercises 14, use graphing software to determine which of the

given viewing windows displays the most appropriate graph of the

specified function.

  1. (x) = x4 72 + 6x
  2. 3-1, 14 by 3-1, 14 b. 3-2, 24 by 3-5, 54
  3. 3-10, 104 by 3-10, 104 d. 3-5, 54 by 3-25, 154
  4. (x) = x3 42 4x + 16
  5. 3-1, 14 by 3-5, 54 b. 3-3, 34 by 3-10, 104
  6. 3-5, 54 by 3-10, 204 d. 3-20, 204 by 3-100, 1004
  7. (x) = 5 + 12x x3
  8. 3-1, 14 by 3-1, 14 b. 3-5, 54 by 3-10, 104
  9. 3-4, 44 by 3-20, 204 d. 3-4, 54 by 3-15, 254
  10. (x) = 25 + 4x x2
  11. 3-2, 24 by 3-2, 24 b. 3-2, 64 by 3-1, 44
  12. 3-3, 74 by 30, 104 d. 3-10, 104 by 3-10, 104

Finding a Viewing Window

In Exercises 530, find an appropriate graphing software viewing window

for the given function and use it to display its graph. The window

should give a picture of the overall behavior of the function. There is

more than one choice, but incorrect choices can miss important

aspects of the function.

  1. (x) = x4 43 + 15 6. (x) = x3

3 x2

2 2x + 1

  1. (x) = x5 54 + 10 8. (x) = 43 x4
  2. (x) = x29 x2 10. (x) = x2(6 x3)

T

T

  1. y = 2x 32>3 12. y = x1>3(x2 8)
  2. y = 52>5 2x 14. y = x2>3(5 x)
  3. y = 0 x2 1 0 16. y = 0 x2 x 0
  4. y = x + 3

x + 2

  1. y = 1 1

x + 3

  1. (x) = x2 + 2

x2 + 1

  1. (x) = x2 1

x2 + 1

  1. (x) = x 1

x2 x 6

  1. (x) = 8

x2 9

  1. (x) = 62 15x + 6

42 10x

  1. (x) = x2 3

x 2

  1. y = sin 250x 26. y = 3 cos 60x
  2. y = cos a x

50

b 28. y = 1

10

sin a x

10b

  1. y = x + 1

10

sin 30x 30. y = x2 + 1

50

cos 100x

Use graphing software to graph the functions specified in Exercises 3136.

Select a viewing window that reveals the key features of the function.

  1. Graph the lower half of the circle defined by the equation

x2 + 2x = 4 + 4y y2.

  1. Graph the upper branch of the hyperbola y2 162 = 1.
  2. Graph four periods of the function (x) = tan 2x.
  3. Graph two periods of the function (x) = 3 cot

x

2 + 1.

  1. Graph the function (x) = sin 2x + cos 3x.
  2. Graph the function (x) = sin3 x.

Exercises 1.4

1.5 E xponential Functions

Exponential functions are among the most important in mathematics and occur in a wide

variety of applications, including interest rates, radioactive decay, population growth, the

spread of a disease, consumption of natural resources, the earths atmospheric pressure, temperature

change of a heated object placed in a cooler environment, and the dating of fossils.

In this section we introduce these functions informally, using an intuitive approach. We give

a rigorous development of them in Chapter 7, based on important calculus ideas and results.

Exponential Behavior

When a positive quantity P doubles, it increases by a factor of 2 and the quantity becomes

2P. If it doubles again, it becomes 2(2P) = 22P, and a third doubling gives 2(22P) = 23P.

Continuing to double in this fashion leads us to consider the function (x) = 2x. We call

this an exponential function because the variable x appears in the exponent of 2x. Functions

such as g(x) = 10 x and h(x) = (1>2)x are other examples of exponential functions.

In general, if a _ 1 is a positive constant, the function

(x) = ax, a > 0

Sketching Exponential Curves

In Exercises 16, sketch the given curves together in the appropriate

coordinate plane and label each curve with its equation.

  1. y = 2x, y = 4x, y = 3-x, y = (1>5)x
  2. y = 3x, y = 8x, y = 2-x, y = (1>4)x
  3. y = 2-t and y = -2t 4. y = 3-t and y = -3t
  4. y = ex and y = 1>ex 6. y = -ex and y = -e-x

In each of Exercises 710, sketch the shifted exponential curves.

  1. y = 2x 1 and y = 2-x 1
  2. y = 3x + 2 and y = 3-x + 2
  3. y = 1 ex and y = 1 e-x
  4. y = -1 ex and y = -1

Applying the Laws of Exponents

Use the laws of exponents to simplify the expressions in Exercises

1120.

  1. 497 # 49-6.5 12. 91>3 # 91>6
  2. 44.2

43.7 14. 35>3

32>3

  1. 1641>1222 16. 11322222>2
  2. 223 # 723 18. 12321>2 # 121221>2
  3. a 2

22

b

4

  1. a26

3 b

2

Composites Involving Exponential Functions

Find the domain and range for each of the functions in Exercises

2124.

  1. (x) = 1

2 + ex 22. g(t) = cos (e-t)

  1. g(t) = 21 + 3-t 24. (x) = 3

1 e2x

Applications

In Exercises 2528, use graphs to find approximate solutions.

  1. 2x = 5 26. ex = 4
  2. 3x 0.5 = 0 28. 3 2-x = 0

In Exercises 2936, use an exponential model and a graphing calculator

to estimate the answer in each problem.

  1. Population growth The population of Knoxville is 500,000

and is increasing at the rate of 3.75% each year. Approximately

when will the population reach 1 million?

  1. Population growth The population of Silver Run in the year

1890 was 6250. Assume the population increased at a rate of

2.75% per year.

  1. Estimate the population in 1915 and 1940.
  2. Approximately when did the population reach 50,000?
  3. Radioactive decay The half-life of phosphorus-32 is about

14 days. There are 6.6 grams present initially.

  1. Express the amount of phosphorus-32 remaining as a function

of time t.

  1. When will there be 1 gram remaining?
  2. If Jean invests $2300 in a retirement account with a 6% interest rate

compounded annually, how long will it take until Jeans account

has a balance of $4150?

  1. Doubling your money Determine how much time is required

for an investment to double in value if interest is earned at the rate

of 6.25% compounded annually.

  1. Tripling your money Determine how much time is required

for an investment to triple in value if interest is earned at the rate

of 5.75% compounded continuously.

  1. Cholera bacteria Suppose that a colony of bacteria starts with

1 bacterium and doubles in number every half hour. How many

bacteria will the colony contain at the end of 24 hr?

  1. Eliminating a disease Suppose that in any given year the number

of cases of a disease is reduced by 20%. If there are 10,000

cases today, how many years will it take

  1. to reduce the number of cases to 1000?
  2. to eliminate the disease; that is, to reduce the number of cases

to less than 1?

Identifying One-to-One Functions Graphically

Which of the functions graphed in Exercises 16 are one-to-one, and

which are not?

1.

x

y

0

y _ _3x3

2.

x

y

_1 0 1

y _ x4 _ x2

  1. y

x

y _ 20 x 0

4.

x

y

y _ int x

5.

x

y

0

y _ 1x

6.

x

y

y _ x1_3

In Exercises 710, determine from its graph if the function is one-toone.

  1. (x) = e

3 x, x 6 0

3, x 0

  1. (x) = e

2x + 6, x -3

x + 4, x 7 -3

  1. (x) = d

1 x

2

, x 0

x

x + 2

, x 7 0

  1. (x) = e

2 x2, x 1

x2, x 7 1

Graphing Inverse Functions

Each of Exercises 1116 shows the graph of a function y = (x).

Copy the graph and draw in the line y = x. Then use symmetry with

respect to the line y = x to add the graph of -1 to your sketch. (It is

not necessary to find a formula for -1.) Identify the domain and

range of -1.

  1. 12.

x

y

0 1

1

y _ f (x) _ 1 , x _ 0

x2 + 1

x

y

0 1

1

y _ f (x) _ 1 _ , x > 0 1x

  1. 14.

x

y

0 p2

p2

_

1

_1

p2

p2

_

y _ f (x) _ sin x,

_ x _

p2

p2

_

y _ f (x) _ tan x,

< x <

x

y

0 p2

p2

_

  1. 16.

x

y

0

6

3

f (x) _ 6 _ 2x,

0 _ x _ 3

x

y

0

1

_1 3

_2

x + 1, _1 _ x _ 0

_2 + x, 0 < x < 3

f (x) _ 2

3

  1. a. Graph the function (x) = 21 x2, 0 x 1. What symmetry

does the graph have?

  1. Show that is its own inverse. (Remember that 22 = x if

x 0.)

  1. a. Graph the function (x) = 1>x. What symmetry does the

graph have?

  1. Show that is its own inverse.

Formulas for Inverse Functions

Each of Exercises 1924 gives a formula for a function y = (x) and

shows the graphs of and -1. Find a formula for -1 in each case.

  1. (x) = x2 + 1, x 0 20. (x) = x2, x 0

x

y

1

0 1

y _ f (x)

y _ f 1(x)

x

y

1

0 1

y _ f 1(x)

y _ f (x)

  1. (x) = x3 1 22. (x) = x2 2x + 1, x 1

x

y

1

_1 1

_1

y _ f (x)

y _ f 1(x)

x

y

1

0 1

y _ f (x)

y _ f 1(x)

  1. (x) = (x + 1)2, x -1 24. (x) = x2>3, x 0

x

y

0

1

_1

_1 1

y _ f (x)

y _ f 1(x)

x

y

0

1

1

y _ f 1(x)

y _ f (x)

Each of Exercises 2536 gives a formula for a function y = (x). In

each case, find -1(x) and identify the domain and range of -1. As a

check, show that ( -1(x)) = -1((x)) = x.

  1. (x) = x5 26. (x) = x4, x 0
  2. (x) = x3 + 1 28. (x) = (1>2)x 7>2
  3. (x) = 1>x2, x 7 0 30. (x) = 1>x3, x _ 0
  4. (x) = x + 3

x 2

  1. (x) =

2x

2x 3

  1. (x) = x2 2x, x 1 34. (x) = (23 + 1)1>5

(Hint: Complete the square.)

  1. (x) = x + b

x 2

, b 7 -2 and constant

  1. (x) = x2 2bx, b 7 0 and constant, x b

Inverses of Lines

  1. a. Find the inverse of the function (x) = mx, where m is a constant

different from zero.

  1. What can you conclude about the inverse of a function

y = (x) whose graph is a line through the origin with a nonzero

slope m?

  1. Show that the graph of the inverse of (x) = mx + b, where m

and b are constants and m _ 0, is a line with slope 1>m and

y-intercept -b>m.

  1. a. Find the inverse of (x) = x + 1. Graph and its inverse

together. Add the line y = x to your sketch, drawing it with

dashes or dots for contrast.

  1. Find the inverse of (x) = x + b (b constant). How is the

graph of -1 related to the graph of ?

  1. What can you conclude about the inverses of functions whose

graphs are lines parallel to the line y = x?

  1. a. Find the inverse of (x) = -x + 1. Graph the line

y = -x + 1 together with the line y = x. At what angle do

the lines intersect?

  1. Find the inverse of (x) = -x + b (b constant). What angle

does the line y = -x + b make with the line y = x?

  1. What can you conclude about the inverses of functions whose

graphs are lines perpendicular to the line y = x?

Logarithms and Exponentials

  1. Express the following logarithms in terms of ln 2 and ln 3.
  2. ln 0.75 b. ln (4>9)
  3. ln (1>2) d. ln23

9

  1. ln 322 f. ln 213.5
  2. Express the following logarithms in terms of ln 5 and ln 7.
  3. ln (1>125) b. ln 9.8
  4. ln 727 d. ln 1225
  5. ln 0.056 f. (ln 35 + ln (1>7))>(ln 25)

Use the properties of logarithms to write the expressions in Exercises

43 and 44 as a single term.

  1. a. ln sin u ln asin u

2 b b. ln (32 9x) + ln a 1

3xb

  1. 1

2

ln (4t4) ln b

  1. a. ln sec u + ln cos u b. ln (8x + 4) 2 ln c
  2. 3 ln23

t2 1 ln (t + 1)

Find simpler expressions for the quantities in Exercises 4548.

  1. a. eln 8.3 b. e-ln 66 c. eln 3x-ln 5y
  2. a. eln (x2+y2) b. e-ln 0.3 c. eln px-ln 2
  3. a. 2 ln 2e b. ln (ln ee) c. ln (e-x2-y2)
  4. a. ln (esec u) b. ln (e(ex)) c. ln (e2 ln x)

In Exercises 4954, solve for y in terms of t or x, as appropriate.

  1. ln y = 4t + 5 50. ln y = -t + 5
  2. ln (y 30) = 5t 52. ln (c 2y) = t
  3. ln (y 4) ln 5 = x + ln x
  4. ln (y2 1) ln (y + 1) = ln (sin x)

In Exercises 55 and 56, solve for k.

  1. a. e3k = 27 b. 35e8k = 175 c. ek>8 = a
  2. a. e5k = 1

4

  1. 80ek = 1 c. e(ln 0.8)k = 0.8

In Exercises 5760, solve for t.

  1. a. e-0.3t = 64 b. ekt = 1

7

  1. e(ln 0.7)t = 0.8
  2. a. e-0.01t = 1000 b. ekt = 1

10

  1. e(ln 2)t = 1

2

  1. e2t = x6 60. e(x2)e(2x+1) = et

Simplify the expressions in Exercises 6164.

  1. a. 5log5 7 b. 8log822 c. 1.3log1.3 75
  2. log4 16 e. log323 f. log4 a1

4b

  1. a. 2log2 3 b. 10log10 (1>2) c. plogp 7
  2. log11 121 e. log121 11 f. log3 a1

9b

  1. a. 2log4 x b. 9log3 x c. log2 (e(ln 2)(sin x))
  2. a. 25log5 (32) b. loge (ex) c. log4 (2ex sin x)

Express the ratios in Exercises 65 and 66 as ratios of natural logarithms

and simplify.

  1. a.

log11 x

log12 x

b.

log5 x

log125 x

c.

logx a

logx5 a

  1. a.

log9 x

log3 x

b.

log210 x

log22 x

c.

loga b

logb a

Arcsine and Arccosine

In Exercises 6770, find the exact value of each expression.

  1. a. sin-1 a1

2b b. sin-1 a 1

22

b c. sin-1 a-23

2

b

  1. a. cos-1 a1

2b b. cos-1 a -1

22

b c. cos-1 a23

2

b

  1. a. arccos (-1) b. arccos (0)
  2. a. arcsin (-1) b. arcsin a-

1

22

b

Theory and Examples

  1. If (x) is one-to-one, can anything be said about g(x) = -(x)? Is

it also one-to-one? Give reasons for your answer.

  1. If (x) is one-to-one and (x) is never zero, can anything be said

about h(x) = 1>(x)? Is it also one-to-one? Give reasons for your

answer.

  1. Suppose that the range of g lies in the domain of so that the

composite _ g is defined. If and g are one-to-one, can anything

be said about _ g? Give reasons for your answer.

  1. If a composite _ g is one-to-one, must g be one-to-one? Give

reasons for your answer.

  1. Find a formula for the inverse function -1 and verify that

( _ -1)(x) = ( -1 _ )(x) = x.

  1. (x) = 100

1 + 2-x b. (x) = 50

1 + 1.1-x

  1. The identity sin-1 x + cos-1 x = P>2 Figure 1.68 establishes

the identity for 0 6 x 6 1. To establish it for the rest of 3-1, 1],

verify by direct calculation that it holds for x = 1, 0, and -1.

Then, for values of x in (-1, 0), let x = -a, a 7 0, and apply

Eqs. (3) and (5) to the sum sin-1 (-a) + cos-1 (-a).

  1. Start with the graph of y = ln x. Find an equation of the graph

that results from

  1. shifting down 3 units.
  2. shifting right 1 unit.
  3. shifting left 1, up 3 units.
  4. shifting down 4, right 2 units.
  5. reflecting about the y-axis.
  6. reflecting about the line y = x.
  7. Start with the graph of y = ln x. Find an equation of the graph

that results from

  1. vertical stretching by a factor of 2.
  2. horizontal stretching by a factor of 3.
  3. vertical compression by a factor of 4.
  4. horizontal compression by a factor of 2.
  5. The equation x2 = 2x has three solutions: x = 2, x = 4, and one

other. Estimate the third solution as accurately as you can by

graphing.

  1. Could xln 2 possibly be the same as 2ln x for x 7 0? Graph the

two functions and explain what you see.

  1. Radioactive decay The half-life of a certain radioactive substance

is 36 hours. There are 12 grams present initially.

  1. Express the amount of substance remaining as a function of

time t.

  1. When will there be 1 gram remaining?
  2. Doubling your money Determine how much time is required

for a $6000 investment to double in value if interest is earned at

the rate of 3.75% compounded annually.

  1. Population growth The population of a city is 280,000 and is

increasing at the rate of 2.75% per year. Predict when the population

will be 560,000.

  1. Radon-222 The decay equation for a certain substance is

known to be y = y0 e-0.0462t, with t in days. About how long will it

take the substance in a sealed sample of air to fall to 73% of its

original value?

 

Chapter 2 Limits and Continuity

Exercises 2.1

Average Rates of Change

In Exercises 16, find the average rate of change of the function over

the given interval or intervals.

  1. (x) = 83 + 8
  2. 35, 74 b. 3-5, 54
  3. g(x) = x2 2x
  4. 31, 34 b. 3-2, 44
  5. h(t) = cot t
  6. 33p>4, 5p>44 b. 3p>3, 3p>24
  7. g(t) = 2 + cos t
  8. 30, p4 b. 3-p, p4
  9. R(u) = 23u + 1; 30, 54
  10. P(u) = u3 4u2 + 5u; 31, 24

Slope of a Curve at a Point

In Exercises 714, use the method in Example 3 to find (a) the slope

of the curve at the given point P, and (b) an equation of the tangent

line at P.

  1. y = x2 5, P(2, -1)
  2. y = 1 52, P(2, -19)
  3. y = x2 2x 3, P(2, -3)
  4. y = x2 4x, P(1, -3)
  5. y = x3, P(2, 8)
  6. y = 2 x3, P(1, 1)
  7. y = x3 12x, P(1, -11)
  8. y = x3 32 + 4, P(2, 0)

Instantaneous Rates of Change

  1. Speed of a car The accompanying figure shows the time-todistance

graph for a sports car accelerating from a standstill.

0 5

200

100

Elapsed time (sec)

Distance (m)

10 15 20

300

400

500

600

650

P

Q1

Q2

Q3

Q4

t

s

  1. Estimate the slopes of secants PQ1, PQ2, PQ3, and PQ4,

arranging them in order in a table like the one in Figure 2.6.

What are the appropriate units for these slopes?

  1. Then estimate the cars speed at time t = 20 sec.
  2. The accompanying figure shows the plot of distance fallen versus

time for an object that fell from the lunar landing module a distance

80 m to the surface of the moon.

  1. Estimate the slopes of the secants PQ1, PQ2, PQ3, and PQ4,

arranging them in a table like the one in Figure 2.6.

  1. About how fast was the object going when it hit the surface?

t

y

0

20

Elapsed time (sec)

Distance fallen (m)

5 10

P

40

60

80

Q1

Q2

Q3

Q4

  1. The profits of a small company for each of the first five years of

its operation are given in the following table:

Year Profit in $1000s

2010 6

2011 27

2012 62

2013 111

2014 174

  1. Plot points representing the profit as a function of year, and

join them by as smooth a curve as you can

  1. What is the average rate of increase of the profits between

2012 and 2014?

  1. Use your graph to estimate the rate at which the profits were

changing in 2012.

  1. 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.

  1. Find the average rate of change of F(x) over the intervals

31, x4 for each x _ 1 in your table.

  1. Extending the table if necessary, try to determine the rate of

change of F(x) at x = 1.

  1. Let g(x) = 2x for x 0.
  2. Find the average rate of change of g(x) with respect to x over

the intervals 31, 24, 31, 1.54 and 31, 1 + h4.

  1. Make a table of values of the average rate of change of g with

respect to x over the interval 31, 1 + h4 for some values of h

approaching zero, say h = 0.1, 0.01, 0.001, 0.0001, 0.00001,

and 0.000001.

  1. What does your table indicate is the rate of change of g(x)

with respect to x at x = 1?

  1. Calculate the limit as h approaches zero of the average rate of

change of g(x) with respect to x over the interval 31, 1 + h4.

  1. Let (t) = 1>t for t _ 0.
  2. Find the average rate of change of with respect to t over the

intervals (i) from t = 2 to t = 3, and (ii) from t = 2 to t = T.

  1. Make a table of values of the average rate of change of with

respect to t over the interval 32, T4 , for some values of T

approaching 2, say T = 2.1, 2.01, 2.001, 2.0001, 2.00001,

and 2.000001.

  1. What does your table indicate is the rate of change of with

respect to t at t = 2?

  1. Calculate the limit as T approaches 2 of the average rate of

change of with respect to t over the interval from 2 to T. You

will have to do some algebra before you can substitute T = 2.

  1. The accompanying graph shows the total distance s traveled by a

bicyclist after t hours.

0 1

10

20

30

40

2 3 4

Elapsed time (hr)

Distance traveled (mi)

t

s

  1. Estimate the bicyclists average speed over the time intervals

30, 14, 31, 2.54 , and 32.5, 3.54 .

  1. Estimate the bicyclists instantaneous speed at the times t = 12

,

t = 2, and t = 3.

  1. Estimate the bicyclists maximum speed and the specific time

at which it occurs.

  1. The accompanying graph shows the total amount of gasoline A in

the gas tank of an automobile after being driven for t days.

  1. Estimate the average rate of gasoline consumption over the

time intervals 30, 34, 30, 54, and 37, 104 .

  1. Estimate the instantaneous rate of gasoline consumption at

the times t = 1, t = 4, and t = 8.

  1. Estimate the maximum rate of gasoline consumption and the

specific time at which it occurs

Exercises 2.2

Limits from Graphs

  1. For the function g(x) graphed here, find the following limits or

explain why they do not exist.

  1. lim

xS1

g(x) b. lim

xS2

g(x) c. lim

xS3

g(x) d. lim

xS2.5

g(x)

3

x

y

2

1

1

y _ g(x)

  1. For the function (t) graphed here, find the following limits or

explain why they do not exist.

  1. lim

tS -2

(t) b. lim

tS -1

(t) c. lim

tS0

(t) d. lim

tS -0.5

(t)

t

s

1

0 1

s _ f (t)

_1

_2 _1

  1. Which of the following statements about the function y = (x)

graphed here are true, and which are false?

  1. lim

xS0

(x) exists.

  1. lim

xS0

(x) = 0

  1. lim

xS0

(x) = 1

  1. lim

xS1

(x) = 1

  1. lim

xS1

(x) = 0

  1. lim

xSc

(x) exists at every point c in (-1, 1).

  1. lim

xS1

(x) does not exist.

x

y

_1 1 2

1

_1

y _ f (x)

  1. Which of the following statements about the function y = (x)

graphed here are true, and which are false?

  1. lim

xS2

(x) does not exist.

  1. lim

xS2

(x) = 2

  1. lim

xS1

(x) does not exist.

  1. lim

xSc

(x) exists at every point c in (-1, 1).

  1. lim

xSc

(x) exists at every point c in (1, 3).

x

y

_1 1 2 3

1

_1

_2

y _ f (x)

Existence of Limits

In Exercises 5 and 6, explain why the limits do not exist.

  1. lim

xS0

x

0 x 0 6. lim

xS1

1

x 1

  1. Suppose that a function (x) is defined for all real values of x

except x = c. Can anything be said about the existence of

limxSc (x)? Give reasons for your answer.

  1. Suppose that a function (x) is defined for all x in 3-1, 1]. Can

anything be said about the existence of limxS0 (x)? Give reasons

for your answer.

  1. If limxS1 (x) = 5, must be defined at x = 1? If it is, must

(1) = 5? Can we conclude anything about the values of at

x = 1? Explain.

  1. If (1) = 5, must limxS1 (x) exist? If it does, then must

limxS1 (x) = 5? Can we conclude anything about limxS1 (x)?

Explain.

Calculating Limits

Find the limits in Exercises 1122.

  1. lim

xS -3

(x2 13) 12. lim

xS3

(-x2 + 8x 7)

  1. lim

tS6

8(t 5)(t 7) 14. lim

xS -1

(23 52 + 3x + 5)

  1. lim

xS2

x + 2

x + 5

  1. lim

sS2>3

(8 3s)(2s 1)

  1. lim

xS-1>4

16x(13x + 16)2 18. lim

yS2

y + 2

y2 + 5y + 6

  1. lim

yS -9

(18 y)7>3 20. lim

zS4

2z2 10

  1. lim

hS0

5

25h + 4 + 4

  1. lim

hS0

24h + 1 1

h

Limits of quotients Find the limits in Exercises 2342.

  1. lim

xS9

x 9

x2 81

  1. lim

xS -3

x + 3

x2 + 4x + 3

  1. lim

xS 8

x2 2x 48

x 8

  1. lim

xS6

x2 4x 12

x 6

  1. lim

tS1

t2 + t 2

t2 1

  1. lim

tS -1

t2 + 3t + 2

t2 t 2

  1. lim

xS -2

-2x 4

x3 + 22 30. lim

yS0

5y3 + 8y2

3y4 16y2

  1. lim

xS1

x-1 1

x 1

  1. lim

xS0

1

x 1 + 1

x + 1

x

  1. lim

uS2

u4 16

u3 8

  1. lim

uS3

u4 81

u3 27

  1. lim

xS25

2x 5

x 25

  1. lim

xS4

4x x2

2 2x

  1. lim

xS87

x 87

2x + 13 10

  1. lim

xS -1

22 + 8 3

x + 1

  1. lim

xS2

22 + 12 4

x 2

  1. lim

xS -2

x + 2

22 + 5 3

  1. lim

xS -24

23 22 47

x + 24

  1. lim

xS4

4 x

5 22 + 9

Limits with trigonometric functions Find the limits in Exercises

4350.

  1. lim

xS0

(2 sin x 1) 44. lim

xSp>4

sin2 x

  1. lim

xS0

sec x 46. lim

xSp>3

tan x

  1. lim

xS0

1 + x + sin x

3 cos x

  1. lim

xS0

(x2 1)(2 cos x)

  1. lim

xS -p

2x + 4 cos (x + p) 50. lim

xS0

27 + sec2 x

Using Limit Rules

  1. Suppose limxS0 (x) = 1 and limxS0 g(x) = -5. Name the

rules in Theorem 1 that are used to accomplish steps (a), (b), and

(c) of the following calculation.

lim

xS0

2(x) g(x)

((x) + 7)2>3 =

lim

xS0

(2(x) g(x))

lim

xS0

((x) + 7)2>3 (a)

=

lim

xS0

2(x) lim

xS0

g(x)

alim

xS0

((x) + 7)b

2>3 (b)

=

2 lim

xS0

(x) lim

xS0

g(x)

alim

xS0

(x) + lim

xS0

7b

2>3 (c)

=

(2)(1) (-5)

(1 + 7)2>3 = 7

4

  1. Let limxS1 h(x) = 5, limxS1 p(x) = 1, and limxS1 r(x) = 2.

Name the rules in Theorem 1 that are used to accomplish steps

(a), (b), and (c) of the following calculation.

lim

xS1

25h(x)

p(x)(4 r(x)) =

lim

xS1

25h(x)

lim

xS1

(p(x)(4 r(x)))

(a)

=

4lim

xS1

5h(x)

alim

xS1

p(x)b alim

xS1

(4 r(x))b

(b)

=

45lim

xS1

h(x)

alim

xS1

p(x)b alim

xS1

4 lim

xS1

r(x)b

(c)

=

2(5)(5)

(1)(4 2) = 5

2

  1. Suppose limxS3 (x) = 3 and limxS3 g(x) = -8. Find
  2. lim

xS3

(x)g(x) b. lim

xS3

3(x)g(x)

  1. lim

xS3

((x) + 5g(x)) d. lim

xS3

(x)

(x) g(x)

  1. Suppose limxS4 (x) = 0 and limxS4 g(x) = -3. Find
  2. lim

xS4

(g(x) + 3) b. lim

xS4

x(x)

  1. lim

xS4

(g(x))2 d. lim

xS4

g(x)

(x) 1

  1. Suppose limxSb (x) = 10 and limxSb g(x) = -3. Find
  2. lim

xSb

((x) + g(x)) b. lim

xSb

6(x)

  1. lim

xSb

(x) # g(x) d. lim

xSb

(x)>g(x)

  1. Suppose that limxS -2 p(x) = 4, limxS -2 r(x) = 0, and

limxS -2 s(x) = -3. Find

  1. lim

xS -2

(p(x) + r(x) + s(x))

  1. lim

xS -2

p(x) # r(x) # s(x)

  1. lim

xS -2

(-4p(x) + 5r(x))>s(x)

Limits of Average Rates of Change

Because of their connection with secant lines, tangents, and instantaneous

rates, limits of the form

lim

hS0

(x + h) (x)

h

occur frequently in calculus. In Exercises 5762, evaluate this limit

for the given value of x and function .

  1. (x) = x2, x = 1
  2. (x) = x2, x = -2
  3. (x) = 3x 4, x = 2
  4. (x) = 1>x, x = -2
  5. (x) = 2x, x = 7
  6. (x) = 23x + 1, x = 0

Using the Sandwich Theorem

  1. If 25 22 (x) 25 x2 for -1 x 1, find

limxS0 (x).

  1. If 2 x2 g(x) 2 cos x for all x, find limxS0 g(x).
  2. a. It can be shown that the inequalities

1 x2

6 6 x sin x

2 2 cos x 6 1

hold for all values of x close to zero. What, if anything, does

this tell you about

lim

xS0

x sin x

2 2 cos x

?

Give reasons for your answer.

  1. Graph y = 1 (x2>6), y = (x sin x)>(2 2 cos x), and

y = 1 together for -2 x 2. Comment on the behavior

of the graphs as xS 0.

  1. a. Suppose that the inequalities

1

2 x2

24 6 1 cos x

x2 6 1

2

hold for values of x close to zero. (They do, as you will see in

Section 9.9.) What, if anything, does this tell you about

lim

xS0

1 cos x

x2 ?

Give reasons for your answer.

  1. Graph the equations y = (1>2) (x2>24),

y = (1 cos x)>x2, and y = 1>2 together for -2 x 2.

Comment on the behavior of the graphs as xS 0.

Estimating Limits

You will find a graphing calculator useful for Exercises 6776.

  1. Let (x) = (x2 9)>(x + 3).
  2. Make a table of the values of at the points x = -3.1,

-3.01, -3.001, and so on as far as your calculator can go.

Then estimate limxS -3 (x). What estimate do you arrive at

if you evaluate at x = -2.9, -2.99, -2.999,cinstead?

  1. Support your conclusions in part (a) by graphing near

c = -3 and using Zoom and Trace to estimate y-values on

the graph as xS -3.

  1. Find limxS -3 (x) algebraically, as in Example 7.
  2. Let g(x) = (x2 2) >(x 22).
  3. Make a table of the values of g at the points x = 1.4, 1.41,

1.414, and so on through successive decimal approximations

of 22. Estimate limxS22 g(x).

  1. Support your conclusion in part (a) by graphing g near

c = 22 and using Zoom and Trace to estimate y-values on

the graph as xS 22.

  1. Find limxS22 g(x) algebraically.
  2. Let G(x) = (x + 6)> (x2 + 4x 12).
  3. Make a table of the values of G at x = -5.9, -5.99, -5.999,

and so on. Then estimate limxS -6 G(x). What estimate do

you arrive at if you evaluate G at x = -6.1, -6.01,

-6.001, cinstead?

  1. Support your conclusions in part (a) by graphing G and using

Zoom and Trace to estimate y-values on the graph as

xS -6.

  1. Find limxS -6 G(x) algebraically.
  2. Let h(x) = (x2 2x 3) > (x2 4x + 3).
  3. Make a table of the values of h at x = 2.9, 2.99, 2.999, and

so on. Then estimate limxS3 h(x). What estimate do you

arrive at if you evaluate h at x = 3.1, 3.01, 3.001,c

instead?

  1. Support your conclusions in part (a) by graphing h near

c = 3 and using Zoom and Trace to estimate y-values on the

graph as xS 3.

  1. Find limxS3 h(x) algebraically.
  2. Let (x) = (x2 1) > ( 0 x 0 1).
  3. Make tables of the values of at values of x that approach

c = -1 from above and below. Then estimate limxS -1 (x).

T

T

  1. Support your conclusion in part (a) by graphing near

c = -1 and using Zoom and Trace to estimate y-values on

the graph as xS -1.

  1. Find limxS -1 (x) algebraically.
  2. Let F(x) = (x2 + 3x + 2) > (2 0 x 0 ).
  3. Make tables of values of F at values of x that approach

c = -2 from above and below. Then estimate limxS -2 F(x).

  1. Support your conclusion in part (a) by graphing F near

c = -2 and using Zoom and Trace to estimate y-values on

the graph as xS -2.

  1. Find limxS -2 F(x) algebraically.
  2. Let g(u) = (sin u)>u.
  3. Make a table of the values of g at values of u that approach

u0 = 0 from above and below. Then estimate limuS0 g(u).

  1. Support your conclusion in part (a) by graphing g near

u0 = 0.

  1. Let G(t) = (1 cos t)>t2.
  2. Make tables of values of G at values of t that approach t0 = 0

from above and below. Then estimate limtS0 G(t).

  1. Support your conclusion in part (a) by graphing G near

t0 = 0.

  1. Let (x) = x1>(1-x).
  2. Make tables of values of at values of x that approach c = 1

from above and below. Does appear to have a limit as

xS 1? If so, what is it? If not, why not?

  1. Support your conclusions in part (a) by graphing near c = 1.
  2. Let (x) = (3x 1)>x.
  3. Make tables of values of at values of x that approach c = 0

from above and below. Does appear to have a limit as

xS 0? If so, what is it? If not, why not?

  1. Support your conclusions in part (a) by graphing near c = 0.

Theory and Examples

  1. If x4 (x) x2 for x in 3-1, 14 and x2 (x) x4 for

x 6 -1 and x 7 1, at what points c do you automatically know

limxSc (x)? What can you say about the value of the limit at

these points?

  1. Suppose that g(x) (x) h(x) for all x _ 2 and suppose that

lim

xS2

g(x) = lim

xS2

h(x) = -5.

Can we conclude anything about the values of , g, and h at

x = 2? Could (2) = 0? Could limxS2 (x) = 0? Give reasons

for your answers.

  1. If lim

xS4

(x) 5

x 2 = 1, find lim

xS4

(x).

  1. If lim

xS -2

(x)

x2 = 1, find

  1. lim

xS -2

(x) b. lim

xS -2

(x)

x

  1. a. If lim

xS2

(x) 5

x 2 = 3, find lim

xS2

(x).

  1. If lim

xS2

(x) 5

x 2 = 4, find lim

xS2

(x).

  1. If lim

xS0

(x)

x2 = 1, find

  1. lim

xS0

(x)

  1. lim

xS0

(x)

x

  1. a. Graph g(x) = x sin (1>x) to estimate limxS0 g(x), zooming in

on the origin as necessary.

  1. Confirm your estimate in part (a) with a proof.
  2. a. Graph h(x) = x2 cos (1>x3) to estimate limxS0 h(x), zooming

in on the origin as necessary.

  1. Confirm your estimate in part (a) with a proof.

COMPUTER EXPLORATIONS

Graphical Estimates of Limits

In Exercises 8590, use a CAS to perform the following steps:

  1. Plot the function near the point c being approached.
  2. From your plot guess the value of the limit.

Exercises 2.3

Centering Intervals About a Point

In Exercises 16, sketch the interval (a, b) on the x-axis with the

point c inside. Then find a value of d 7 0 such that for all

x, 0 6 0 x c 0 6 d 1 a 6 x 6 b.

  1. a = 1, b = 7, c = 5
  2. a = 1, b = 7, c = 2
  3. a = -7>2, b = -1>2, c = -3
  4. a = -7>2, b = -1>2, c = -3>2
  5. a = 4>9, b = 4>7, c = 1>2
  6. a = 2.7591, b = 3.2391, c = 3

Finding Deltas Graphically

In Exercises 714, use the graphs to find a d 7 0 such that for all x

0 6 0 x c 0 6 d 1 0 (x) L 0 6 P.

  1. 14.

2.5

2

1.5

y

x

_1

L _ 2

f (x) _

c _ _1

P _ 0.5

16

_ 9

16

_25

0

_x

2

y _

_x

2

0

y

x

c _

L _ 2

P _ 0.01

y _ 1x

f (x) _ 112

2.01

2

1.99

12

1

2.01

1

1.99

NOT TO SCALE

Finding Deltas Algebraically

Each of Exercises 1530 gives a function (x) and numbers L, c, and

P 7 0. In each case, find an open interval about c on which the inequality

0 (x) L 0 6 P holds. Then give a value for d 7 0 such that for

all x satisfying 0 6 0 x c 0 6 d the inequality 0 (x) L 0 6 P

holds.

  1. (x) = x + 1, L = 5, c = 4, P = 0.01
  2. (x) = 2x 2, L = -6, c = -2, P = 0.02
  3. (x) = 2x + 1, L = 1, c = 0, P = 0.1
  4. (x) = 2x, L = 1>2, c = 1>4, P = 0.1
  5. (x) = 219 x, L = 3, c = 10, P = 1
  6. (x) = 2x 7, L = 4, c = 23, P = 1
  7. (x) = 1>x, L = 1>4, c = 4, P = 0.05
  8. (x) = x2, L = 3, c = 23, P = 0.1
  9. (x) = x2, L = 4, c = -2, P = 0.5
  10. (x) = 1>x, L = -1, c = -1, P = 0.1
  11. (x) = x2 5, L = 11, c = 4, P = 1
  12. (x) = 120>x, L = 5, c = 24, P = 1
  13. (x) = mx, m 7 0, L = 2m, c = 2, P = 0.03
  14. (x) = mx, m 7 0, L = 3m, c = 3, P = c 7 0
  15. (x) = mx + b, m 7 0, L = (m>2) + b,

c = 1>2, P = c 7 0

  1. (x) = mx + b, m 7 0, L = m + b, c = 1,

P = 0.05

Using the Formal Definition

Each of Exercises 3136 gives a function (x), a point c, and a positive

number P. Find L = lim

xSc

(x). Then find a number d 7 0 such

that for all x

0 6 0 x c 0 6 d 1 0 (x) L 0 6 P.

  1. (x) = 2 4x, c = 4, P = 0.03
  2. (x) = -3x 2, c = -1, P = 0.03
  3. (x) = x2 9

x 3

, c = 3, P = 0.03

  1. (x) = x2 + 6x + 5

x + 5

, c = -5, P = 0.05

  1. (x) = 21 x, c = -3, P = 0.5
  2. (x) = 4>x, c = 2, P = 0.4

Prove the limit statements in Exercises 3750.

  1. lim

xS4

(9 x) = 5 38. lim

xS3

(3x 7) = 2

  1. lim

xS9

2x 5 = 2 40. lim

xS0

24 x = 2

  1. lim

xS1

(x) = 1 if (x) = e

x2, x _ 1

2, x = 1

  1. lim

xS -2

(x) = 4 if (x) = e

x2, x _ -2

1, x = -2

  1. lim

xS1

1x

= 1 44. lim

xS23

1

x2 = 1

3

  1. lim

xS -3

x2 9

x + 3 = -6 46. lim

xS1

x2 1

x 1 = 2

  1. lim

xS1

(x) = 2 if (x) = e

4 2x, x 6 1

6x 4, x 1

  1. lim

xS0

(x) = 0 if (x) = e

2x, x 6 0

x>2, x 0

  1. lim

xS0

x sin

1x

= 0

  1. lim

xS0

x2 sin

1x

= 0

x

y

1

_1

_1 0 1

y _ x2

y _ _x2

y _ x2 sin1x

2p

2p

_

Theory and Examples

  1. Define what it means to say that lim

xS0

g(x) = k.

  1. Prove that lim

xSc

(x) = L if and only if lim

hS0

(h + c) = L.

  1. A wrong statement about limits Show by example that the

following statement is wrong.

The number L is the limit of (x) as x approaches c

if (x) gets closer to L as x approaches c.

Explain why the function in your example does not have the

given value of L as a limit as xS c.

  1. Another wrong statement about limits Show by example that

the following statement is wrong.

The number L is the limit of (x) as x approaches c if, given any

P 7 0, there exists a value of x for which 0 (x) L 0 6 P.

Explain why the function in your example does not have the

given value of L as a limit as xS c.

  1. Grinding engine cylinders Before contracting to grind engine

cylinders to a cross-sectional area of 9 in2, you need to know how

much deviation from the ideal cylinder diameter of c = 3.385 in.

you can allow and still have the area come within 0.01 in2 of the

required 9 in2. To find out, you let A = p(x>2)2 and look for the

interval in which you must hold x to make 0 A 9 0 0.01.

What interval do you find?

  1. Manufacturing electrical resistors Ohms law for electrical

circuits like the one shown in the accompanying figure states that

V = RI. In this equation, V is a constant voltage, I is the current

in amperes, and R is the resistance in ohms. Your firm has been

asked to supply the resistors for a circuit in which V will be 120

volts and I is to be 5 { 0.1 amp. In what interval does R have to

lie for I to be within 0.1 amp of the value I0 = 5?

V I R

_

+

T

When Is a Number L Not the Limit of (x) as x u c?

Showing L is not a limit We can prove that limxSc (x) _ L by

providing an P 7 0 suc

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