graphs.

10. Which of the following is a function and which is not? Give a reason in

each case.

CHAPTER 1 Basics 53

(a) f assigns to each person his biological father

(b) g assigns to each man his dog

(c) h assigns to each real number its square root

(d) f assigns to each positive integer its cube

(e) g assigns to each car its driver

(f) h assigns to each toe its foot

(g) f assigns to each rational number the greatest integer that does not

exceed it

(h) g assigns to each integer the next integer

(i) h assigns to each real number its square plus six

11. Graph each of these functions on a separate set of axes. Label your graph.

f (x) = 3x 2 ’ x

(a)

x+2

g(x) =

(b)

x

h(x) = x 3 ’ x2

(c)

f (x) = 3x + 2

(d)

g(x) = √ ’ 2x

x2

(e)

h(x) = x + 3

(f)

12. Calculate each of the following trigonometric quantities.

(a) sin(8π/3)

(b) tan(’5π/6)

(c) sec(7π/4)

(d) csc(13π/4)

(e) cot(’15π/4)

(f) cos(’3π/4)

13. Calculate the left and right sides of the twelve fundamental trigonometric

identities for the values θ = π/3 and ψ = ’π/6, thus con¬rming the

identities for these particular values.

14. Sketch the graphs of each of the following trigonometric functions.

f (x) = sin 2x

(a)

g(x) = cos(x + π/2)

(b)

h(x) = tan(’x + π )

(c)

f (x) = cot(3x + π )

(d)

g(x) = sin(x/3)

(e)

h(x) = cos(’π + [x/2])

(f)

15. Convert each of the following angles from radian measure to degree

measure.

54 CHAPTER 1 Basics

= π/24

(a) θ

= ’π/3

(b) θ

= 27π/12

(c) θ

= 9π/16

(d) θ

=3

(e) θ

= ’5

(f) θ

16. Convert each of the following angles from degree measure to radian

measure.

= 65—¦

(a) θ

= 10—¦

(b) θ

= ’75—¦

(c) θ

= ’120—¦

(d) θ

= π—¦

(e) θ

= 3.14—¦

(f) θ

17. For each of the following pairs of functions, calculate f —¦ g and g —¦ f .

f (x) = x 2 + 2x + 3 g(x) = (x ’ 1)2

(a) √

√

f (x) = x + 1 g(x) = x 2 ’ 2

3

(b)

f (x) = sin(x + 3x 2 ) g(x) = cos(x 2 ’ x)

(c)

f (x) = ex+2 g(x) = ln(x ’ 5)

(d)

f (x) = sin(x 2 + x) g(x) = ln(x 2 ’ x)

(e)

g(x) = e’x

2 2

f (x) = ex

(f)

f (x) = x(x + 1)(x + 2) g(x) = (2x ’ 3)(x + 4)

(g)

18. Consider each of the following as functions from R to R and say whether

the function is invertible. If it is, ¬nd the inverse with an explicit formula.

f (x) = x 3 + 5

(a)

g(x) = x 2 ’ x √

(b)

h(x) = (sgn x) · |x|, where sgn x is +1 if x is positive, ’1 if x is

(c)

negative, 0 if x is 0.

f (x) = x 5 + 8

(d)

g(x) = e’3x

(e)

h(x) = sin x

(f)

f (x) = tan x

(g)

g(x) = (sgn x) · x 2 , where sgn x is +1 if x is positive, ’1 if x is

(h)

negative, 0 if x is 0.

19. For each of the functions in Exercise 18, graph both the function and its

inverse in the same set of axes.

CHAPTER 1 Basics 55

20. Determine whether each of the following functions, on the given domain

S, is invertible. If it is, then ¬nd the inverse explicitly.

f (x) = x 2 , S = [2, 7]

(a)

g(x) = ln x, S = [1, ∞)

(b)

h(x) = sin x, S = [0, π/2]

(c)

f (x) = cos x, S = [0, π]

(d)

g(x) = tan x, S = (’π/2, π/2)

(e)

h(x) = x 2 , S = [’2, 5]

(f)

f (x) = x 2 ’ 3x, S = [4, 7]

(g)

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CHAPTER 2

Foundations of

Calculus

2.1 Limits

The single most important idea in calculus is the idea of limit. More than 2000 years

ago, the ancient Greeks wrestled with the limit concept, and they did not succeed. It

is only in the past 200 years that we have ¬nally come up with a ¬rm understanding

of limits. Here we give a brief sketch of the essential parts of the limit notion.

Suppose that f is a function whose domain contains two neighboring intervals:

f : (a, c) ∪ (c, b) ’ R. We wish to consider the behavior of f as the variable x

approaches c. If f (x) approaches a particular ¬nite value as x approaches c, then

we say that the function f has the limit as x approaches c. We write

lim f (x) = .

x’c

The rigorous mathematical de¬nition of limit is this:

De¬nition 2.1 Let a < c < b and let f be a function whose domain contains

(a, c) ∪ (c, b). We say that f has limit at c, and we write limx’c f (x) = when

this condition holds: For each > 0 there is a δ > 0 such that

|f (x) ’ | <

whenever 0 < |x ’ c| < δ.

It is important to know that there is a rigorous de¬nition of the limit concept, and

any development of mathematical theory relies in an essential way on this rigorous

de¬nition. However, in the present book we may make good use of an intuitive

57

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58 CHAPTER 2 Foundations of Calculus

understanding of limit. We now develop that understanding with some carefully

chosen examples.

EXAMPLE 2.1

De¬ne

3’x if x < 1

f (x) =

x2 + 1 if x > 1

See Fig. 2.1. Calculate limx’1 f (x).

Y

FL

Fig. 2.1

AM

SOLUTION

Observe that, when x is to the left of 1 and very near to 1 then f (x) = 3 ’ x

is very near to 2. Likewise, when x is to the right of 1 and very near to 1 then

TE

f (x) = x 2 + 1 is very near to 2. We conclude that

lim f (x) = 2.

x’1

We have successfully calculated our ¬rst limit. Figure 2.1 con¬rms the conclusion

that our calculations derived.

EXAMPLE 2.2

De¬ne

x2 ’ 4

g(x) = .

x’2

Calculate limx’2 g(x).

SOLUTION

We observe that both the numerator and the denominator of the fraction

de¬ning g tend to 0 as x ’ 2 (i.e., as x tends to 2). Thus the question seems

to be indeterminate.

However, we may factor the numerator as x 2 ’ 4 = (x ’ 2)(x + 2).

As long as x = 2 (and these are the only x that we examine when we

59

CHAPTER 2 Foundations of Calculus

calculate limx’2 ), we can then divide the denominator of the expression

de¬ning g into the numerator. Thus

g(x) = x + 2 for x = 2.

Now

lim g(x) = lim x + 2 = 4.

x’2 x’2

Fig. 2.2

The graph of the function g is shown in Fig. 2.2. We encourage the reader to

use a pocket calculator to calculate values of g for x near 2 but unequal to 2 to

check the validity of our answer. For example,

x g(x) = [x 2 ’ 4]/[x ’ 2]

1.8 3.8

1.9 3.9

1.99 3.99

1.999 3.999

2.001 4.001

2.01 4.01

2.1 4.1

2.2 4.2

We see that, when x is close to 2 (but unequal to 2), then g(x) is close (indeed,

as close as we please) to 4.

x 3 ’ 3x 2 + x ’ 3

You Try It: Calculate the limit limx’3 .

x’3

Math Note: It must be stressed that, when we calculate limx’c f (x), we do not

evaluate f at c. In the last example it would have been impossible to do so. We want

to determine what we anticipate f will do as x approaches c, not what value (if any)

f actually takes at c. The next example illustrates this point rather dramatically.

60 CHAPTER 2 Foundations of Calculus

Fig. 2.3

EXAMPLE 2.3

De¬ne

if x = 7

3

h(x) =

if x = 7

1

Calculate limx’7 h(x).

SOLUTION

It would be incorrect to simply plug the value 7 into the function h and

thereby to conclude that the limit is 1. In fact when x is near to 7 but unequal

to 7, we see that h takes the value 3. This statement is true no matter how close

x is to 7. We conclude that limx’7 h(x) = 3.

You Try It: Calculate limx’4 [x 2 ’ x ’ 12]/[x ’ 4].

2.1.1 ONE-SIDED LIMITS

There is also a concept of one-sided limit. We say that

lim f (x) =

x’c’

if the values of f become closer and closer to when x is near to c but on the left.

In other words, in studying limx’c’ f (x), we only consider values of x that are

less than c.

Likewise, we say that

lim f (x) =

x’c+

if the values of f become closer and closer to when x is near to c but on the right.

In other words, in studying limx’c+ f (x), we only consider values of x that are

greater than c.

61

CHAPTER 2 Foundations of Calculus

EXAMPLE 2.4

Discuss the limits of the function

2x ’ 4 if x < 2

f (x) =

if x ≥ 2

x2

at c = 2.

SOLUTION

As x approaches 2 from the left, f (x) = 2x ’ 4 approaches 0. As x

approaches 2 from the right, f (x) = x 2 approaches 4. Thus we see that f

has left limit 0 at c = 2, written

lim f (x) = 0,

x’2’

and f has right limit 4 at c = 2, written

lim f (x) = 4.

x’2+

Note that the full limit limx’2 f (x) does not exist (because the left and right

limits are unequal).

You Try It: Discuss one-sided limits at c = 3 for the function

±

x 3 ’ x if x < 3

f (x) = 24 if x = 3

4x + 1 if x > 3

All the properties of limits that will be developed in this chapter, as well as the

rest of the book, apply equally well to one-sided limits as to two-sided (or standard)

limits.

2.2 Properties of Limits

To increase our facility in manipulating limits, we have certain arithmetical and

functional rules about limits. Any of these may be veri¬ed using the rigorous de¬-

nition of limit that was provided at the beginning of the last section. We shall state

the rules and get right to the examples.

If f and g are two functions, c is a real number, and limx’c f (x) and

limx’c g(x) exist, then

Theorem 2.1

(a) limx’c (f ± g)(x) = limx’c f (x) ± limx’c g(x);

62 CHAPTER 2 Foundations of Calculus

(b) limx’c (f · g) (x) = (limx’c f (x)) · (limx’c g(x)) ;

limx’c f (x)

f

(x) = provided that limx’c g(x) = 0;

(c) lim

limx’c g(x)

g

x’c

(d) limx’c (± · f (x)) = ± · (limx’c f (x)) for any constant ±.

Some theoretical results, which will prove useful throughout our study of

calculus, are these:

Theorem 2.2

Let a < c < b. A function f on the interval {x : a < x < b} cannot have two

distinct limits at c.

Theorem 2.3

If

lim g(x) = 0

x’c

and

lim f (x) either does not exist or exists and is not zero

x’c

then

f (x)

lim

x’c g(x)

does not exist.

Theorem 2.4 (The Pinching Theorem)

Suppose that f, g, and h are functions whose domains each contain S = (a, c) ∪

(c, b). Assume further that

g(x) ¤ f (x) ¤ h(x)

for all x ∈ S. Refer to Fig. 2.4.

y = h(x)

y = f (x)

y = g (x)

a c b

Fig. 2.4

If

lim g(x) =

x’c

63

CHAPTER 2 Foundations of Calculus

and

lim h(x) =

x’c

then

lim f (x) = .

x’c

EXAMPLE 2.5