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Yet Another Calculus Text

A Short Introduction with

Infinitesimals

Dan Sloughter

Department of Mathematics

Furman University

May 10, 2007

ii

c Dan Sloughter 2007

Preface

I intend this book to be, firstly, a introduction to calculus based on the hy-

perreal number system. In other words, I will use infinitesimal and infinite

numbers freely. Just as most beginning calculus books provide no logical jus-

tification for the real number system, I will provide none for the hyperreals.

The reader interested in questions of foundations should consult books such as

Abraham Robinson’s Non-standard Analysis or Robert Goldblatt’s Lectures on

the Hyperreals.

Secondly, I have aimed the text primarily at readers who already have some

familiarity with calculus. Although the book does not explicitly assume any

prerequisites beyond basic algebra and trigonometry, in practice the pace is

too fast for most of those without some acquaintance with the basic notions of

calculus.

iii

iv

PREFACE

Contents

Preface

iii

Contents

v

1

Derivatives

1

1.1

The arrow paradox . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.1.1

Zero divided by zero . . . . . . . . . . . . . . . . . . . . .

1

1.1.2

Adding up zeroes . . . . . . . . . . . . . . . . . . . . . . .

2

1.2

Rates of change . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.3

The hyperreals . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.4

Continuous functions . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.5

Properties of continuous functions

. . . . . . . . . . . . . . . . .

12

1.5.1

Polynomials and rational functions . . . . . . . . . . . . .

14

1.5.2

Trigonometric functions . . . . . . . . . . . . . . . . . . .

15

1.5.3

Compositions . . . . . . . . . . . . . . . . . . . . . . . . .

20

1.5.4

Consequences of continuity . . . . . . . . . . . . . . . . .

22

1.6

The derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

1.7

Properties of derivatives . . . . . . . . . . . . . . . . . . . . . . .

26

1.7.1

Sums and differences . . . . . . . . . . . . . . . . . . . . .

27

1.7.2

Constant multiples . . . . . . . . . . . . . . . . . . . . . .

27

1.7.3

Products

. . . . . . . . . . . . . . . . . . . . . . . . . . .

28

1.7.4

Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . .

30

1.7.5

Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

1.7.6

Composition of functions

. . . . . . . . . . . . . . . . . .

33

1.7.7

Trigonometric functions . . . . . . . . . . . . . . . . . . .

37

1.8

A geometric interpretation of the derivative . . . . . . . . . . . .

39

1.9

Increasing, decreasing, and local extrema

. . . . . . . . . . . . .

40

1.9.1

The mean-value theorem . . . . . . . . . . . . . . . . . . .

41

1.9.2

Increasing and decreasing functions . . . . . . . . . . . . .

43

1.10 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

1.10.1 Optimization on a closed interval . . . . . . . . . . . . . .

46

1.10.2 Optimization on other intervals . . . . . . . . . . . . . . .

49

1.11 Implicit differentiation and rates of change . . . . . . . . . . . . .

52

1.12 Higher-order derivatives . . . . . . . . . . . . . . . . . . . . . . .

57

v

vi

CONTENTS

1.12.1 Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . .

59

1.12.2 Concavity . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

1.12.3 The second-derivative test . . . . . . . . . . . . . . . . . .

60

2

Integrals

63

2.1

Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

2.1.1

The case of constant acceleration . . . . . . . . . . . . . .

67

2.2

Definite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

2.3

Properties of definite integrals . . . . . . . . . . . . . . . . . . . .

72

2.4

The fundamental theorem of integrals . . . . . . . . . . . . . . .

75

2.5

Applications of definite integrals

. . . . . . . . . . . . . . . . . .

79

2.5.1

Area between curves . . . . . . . . . . . . . . . . . . . . .

79

2.5.2

Volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

2.5.3

Arc length . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

2.6

Some techniques for evaluating integrals . . . . . . . . . . . . . .

91

2.6.1

Change of variable . . . . . . . . . . . . . . . . . . . . . .

91

2.6.2

Integration by parts . . . . . . . . . . . . . . . . . . . . .

95

2.6.3

Some integrals involving trigonometric functions . . . . .

97

2.6.4

Change of variable revisited . . . . . . . . . . . . . . . . . 104

2.7

The exponential and logarithm functions . . . . . . . . . . . . . . 110

2.7.1

The exponential function

. . . . . . . . . . . . . . . . . . 110

2.7.2

The logarithm function

. . . . . . . . . . . . . . . . . . . 114

2.7.3

Some applications

. . . . . . . . . . . . . . . . . . . . . . 119

129

Index

140

Chapter 1

Derivatives

1.1

In his famous arrow paradox, Zeno contends that an arrow cannot move since

at every instant of time it is at rest. There are at least two logical problems

hidden in this claim.

1.1.1

Zero divided by zero

In one interpretation, Zeno seems to be saying that, since at every instant of

time the arrow has a definite position, and hence does not travel any distance

during that instant of time, the velocity of the arrow is 0. The question is, if an

object travels a distance 0 in time of duration 0, is the velocity of the object 0?

That is, is

0 = 0?

(1.1.1)

0

To answer this question, we need to examine the meaning of dividing one

number by another. If a and b are real numbers, with b = 0, then

a = c

(1.1.2)

b

means that

a = b × c.

(1.1.3)

In particular, for any real number b = 0,

0 = 0

(1.1.4)

b

since b × 0 = 0. Note that if a = 0, then

a

(1.1.5)

0

1

2

CHAPTER 1. DERIVATIVES

is undefined since there does not exist a real number c for which 0 × c is equal

to a. We say that division of a non-zero number by zero is meaningless. On the

other hand,

0

(1.1.6)

0

is undefined because 0 × c = 0 for all real numbers c. For this reason, we say

that division of zero by zero is indeterminate.

The first logical problem exposed by Zeno’s arrow paradox is the problem

of giving determinate meaning to ratios of quantities with zero magnitude. We

shall see that infinitesimals give us one way of giving definite meanings to ratios

of quantities with zero magnitudes, and these ratios will provide the basis for

what we call the differential calculus.

1.1.2

Another possible interpretation of the arrow paradox is that if at every instant

of time the arrow moves no distance, then the total distance traveled by the

arrow is equal to 0 added to itself a large, or even infinite, number of times.

Now if n is any positive integer, then, of course,

n × 0 = 0.

(1.1.7)

That is, zero added to itself a finite number of times is zero. However, if an

interval of time is composed of an infinite number of instants, then we are asking

for the product of infinity and zero, that is,

∞ × 0.

(1.1.8)

One might at first think this result should also be zero; however, more careful

reasoning is needed.

Note that an interval of time, say the interval [0, 1], is composed of an infinity

of instants of no duration. Hence, in this case, the product of infinity and 0

must be 1, the length of the interval. However, the same reasoning applied to

the interval [0, 2] would lead us to think that infinity times 0 is 2. Indeed, as

with the problem of zero divided by 0, infinity times 0 is indeterminate.

Thus the second logical problem exposed by Zeno’s arrow paradox is the

problem of giving determinate meaning to infinite sums of zero magnitudes, or,

in the simplest cases, to products of infinitesimal and infinite numbers.

Since division is the inverse operation of multiplication we should expect a

close connection between these questions. This is in fact the case, as we shall

see when we discuss the fundamental theorem of calculus.

1.2

Rates of change

Suppose x(t) gives the position, at some time t, of an object (such as Zeno’s

arrow) moving along a straight line. The problem we face is that of giving a

1.2. RATES OF CHANGE

3

determinate meaning to the idea of the velocity of the object at a specific instant

of time. We first note that we face no logical difficulties in defining an average

velocity over an interval of time of non-zero length. That is, if a < b, then the

object travels a distance

∆x = x(b) − x(a)

(1.2.1)

from time t = a to time t = b, an interval of time of length ∆t = b − a, and,

consequently, the average velocity of the object over this interval of time is

x(b) − x(a)

∆x

v[a,b] =

=

.

(1.2.2)

b − a

∆t

Example 1.2.1. Suppose an object, such as a lead ball, is dropped from a

height of 100 meters. Ignoring air resistance, the height of the ball above the

earth after t seconds is given by

x(t) = 100 − 4.9t2 meters,

a result first discovered by Galileo. Hence, for example, from time t = 0 to time

t = 2 we have

∆x = x(2) − x(0) = (100 − (4.9)(4)) − 100 = −19.6 meters,

∆t = 2 − 0 = 2 seconds,

and so

19.6

v[0,2] = −

= −9.8 meters/second.

2

For another example, from time t = 1 to time t = 4 we have

∆x = x(4) − x(1) = 21.6 − 95.1 = −73.5,

∆t = 4 − 1 = 3 seconds,

and so

73.5

v[1,4] = −

= −24.5 meters/second.

3

Note that both of these average velocities are negative because we have taken

the positive direction to be upward from the surface of the earth.

Exercise 1.2.1.

Suppose a lead ball is dropped into a well. Ignoring air resis-

tance, the ball will have fallen a distance x(t) = 16t2 feet after t seconds. Find

the average velocity of the ball over the intervals (a) [0, 2], (b) [1, 3], and (c)

[1, 1.5].

Letting ∆t = b − a, we may rewrite (1.2.2) in the form

x(a + ∆t) − x(a)

v[a,a+∆t] =

.

(1.2.3)

∆t

4

CHAPTER 1. DERIVATIVES

Using (1.2.3), there are two approaches to generalizing the notion of average velocity over an interval to that of velocity at an instant. The most common

approach, at least since the middle of the 19th century, is to consider the effect

on v[a,a+∆t] as ∆t diminishes in magnitude and defining the velocity at time

t = a to be the limiting value of these average velocities. The approach we

will take in this text is to consider what happens when we take a and b to be,

although not equal, immeasurably close to one another.

Example 1.2.2. If we have, as in the previous example,

x(t) = 100 − 4.9t2 meters,

then from time t = 1 to time t = 1 + ∆t we would have

∆x = x(1 + ∆t) − x(1)

= (100 − 4.9(1 + ∆t)2) − 95.1

= 4.9 − 4.9(1 + 2∆t + (∆t)2)

= −9.8∆t − 4.9(∆t)2 meters.

Hence the average velocity over the interval [1, 1 + ∆t] is

∆x

v[1,1+∆t] = ∆t

−9.8∆t − 4.9(∆t)2

=

∆t

= −9.8 − 4.9∆t meters/second.

Note that if, for example, ∆t = 3, then we find

v[1,4] = −9.8 − (4.9)(3) = −9.8 − 14.7 = −24.5 meters/second,

in agreement with our previous calculations.

Now suppose that the starting time a = 1 and the ending time b are different,

but the difference is so small that it cannot be measured by any real number.

In this case, we call dt = b − a an infinitesimal . Similar to our computations

above, we have

dx = x(1 + dt) − x(1) = −9.8dt − 4.9(dt)2 meters,

the distance traveled by the object from time t = 1 to time t = 1 + dt, and

dx

v[1,1+dt] =

= −9.8 − 4.9dt meters/second,

dt

the average velocity of the object over the interval [1, 1 + dt]. However, since dt

is infinitesimal, so is 4.9dt. Hence v[1,1+dt] is immeasurably close to −9.8 meters

per second. Moreover, this is true no matter what the particular value of dt.

Hence we should take the instantaneous velocity of the object at time t = 1 to

be

v(1) = −9.8 meters/second.

1.2. RATES OF CHANGE

5

Exercise 1.2.2.

As in the previous exercise, suppose a lead ball has fallen

x(t) = 16t2 feet in t seconds. Find the average velocity of the ball over the

interval [1, 1 + ∆t] and use this result to obtain the answers to parts (b) and (c)

of the previous exercise.

Exercise 1.2.3.

Find the average velocity of the ball in the previous exercise

over the interval [1, 1 + dt], where dt is infinitesimal, and use the result to find

the instantaneous velocity of the ball at time t = 1.

Example 1.2.3. To find the velocity of the object of the previous examples at

time t = 3, we compute

dx = x(3 + dt) − x(3)

= (100 − 4.9(3 + dt)2 − 55.9

= 44.1 − 4.9(9 + 6dt + (dt2))

= −29.4dt − 4.9(dt)2 meters,

from which we obtain

dx = −29.4 − 4.9dt meters/second.

dt

As above, we disregard the immeasurable −4.9dt to obtain the velocity of the

object at time t = 3:

v(3) = −29.4meters/second.

Exercise 1.2.4.

Find the velocity of the ball in the previous exercise at time

t = 2.

In general, if x(t) gives the position, at time t, of an object moving along a

straight line, then we define the velocity of the object at a time t to be the real

number which is infinitesimally close to

x(t + dt) − x(t) ,

(1.2.4)

dt

provided there is exactly one such number for any value of the nonzero infinites-

imal dt.

Example 1.2.4. For our previous example, we find

dx = x(t + dt) − x(t)

= (100 − 4.9(t + dt)2) − (100 − 4.9t2)

= −4.9(t + 2tdt + (dt)2) − 4.9t2

= −9.8tdt − 4.9(dt)2 meters

= (−9.8t − 4.9dt)dt.

6

CHAPTER 1. DERIVATIVES

Hence

dx = −9.8t − 4.9dt meters/second,

dt

and so the velocity of the object at time t is

v(t) = −9.8t meters/second.

In particular,

v(1) = −9.8 meters/second

and

v(3) = −9.8(3) = −29.4 meters/second,

as previously computed.

Exercise 1.2.5.

Find the velocity of the ball in the previous exercise at time

Even more generally, we should recognize that velocity is but a particular

example of a rate of change, namely, the rate of change of the position of an

object with respect to time. In general, given any quantity y as a function of

another quantity x, say y = f (x) for some function f , we may ask about the

rate of change of y with respect to x. If x changes from x = a to x = b and we

let

∆x = b − a

(1.2.5)

and

∆y = f (b) − f (a) = f (a + ∆x) − f (x),

(1.2.6)

then

∆y

f (b) − f (a)

=

(1.2.7)

∆x

b − a

is the average rate of change of y with respect to x; if dx is a nonzero infinites-

imal, then the real number which is infinitesimally close to

dy

f (x + dx) − f (x)

=

(1.2.8)

dx

dx

is the instantaneous rate of change, or, simply, rate of change, of y with respect

to x at x = a. In subsequent sections we will look at this quantity in more

detail, but will consider one more example before delving into technicalities.

Example 1.2.5. Suppose a spherical shaped balloon is being filled with water.

If r is the radius of the balloon in centimeters and V is the volume of the balloon,

then

4

V =

πr3 centimeters3.

3

Since a cubic centimeter of water has a mass of 1 gram, the mass of the water

in the balloon is

4

M =

πr3 grams.

3

1.3. THE HYPERREALS

7

To find the rate of change of the mass of the balloon with respect to the radius

of the balloon, we first compute

4

4

dM =

π(r + dr)3 −

πr3

3

3

4

=

π((r3 + 3r2dr + 3r(dr)2 + (dr)3) − r3)

3

4

=

π(3r2 + 3rdr + (dr)2)dr grams,

3

from which it follows that

dM

4

=

π(3r2 + 3rdr + (dr)2) grams/centimeter.

dr

3

Since both 3rdr and (dr)2 are infinitesimal, the rate of change of mass of the

balloon with respect to the radius of the balloon is

4 π(3r2) = 4πr2 gams/centimeer.

3

For example, when the balloon has a radius of 10 centimeters, the mass of the

water in the balloon is increasing at a rate of

4π(10)2 = 400π grams/centimeter.

It may not be surprising that this is also the surface area of the balloon at that

instant.

Exercise 1.2.6.

Show that if A is the area of a circle with radius r, then

dA = 2πr.

dr

1.3

The hyperreals

We will let R denote the set of all real numbers. Intuitively, and historically, we

think of these as the numbers sufficient to measure geometric quantities. For

example, the set of all rational numbers, that is, numbers expressible as the

ratios of integers, is not sufficient for this purpose since, for example, the length

of the diagonal of a square with sides of length 1 is the irrational number

2.

There are numerous technical methods for defining and constructing the real

numbers, but, for the purposes of this text, it is sufficient to think of them as

the set of all numbers expressible as infinite decimals, repeating if the number

is rational and non-repeating otherwise.

A positive infinitesimal is any number

with the property that

> 0 and

< r for any positive real number r. The set of infinitesimals consists of the

positive infinitesimals along with their additive inverses and zero. Intuitively,

8

CHAPTER 1. DERIVATIVES

these are the numbers which, except for 0, correspond to quantities which are

too small to measure even theoretically. Again, there are technical ways to make

the definition and constrution of infinitesimals explicit, but they lie beyond the

scope of this text.

The multiplicative inverse of a nonzero infinitesimal is an infinite number.

That is, for any infinitesimal

= 0, the number

1

N =

is an infinite number.

The finite hyperreal numbers are numbers of the form r + , where r is a real

number and

is an infinitesimal. The hyperreal numbers, which we denote ∗R,

consist of the finite hyperreal numbers along with all infinite numbers.

For any finite hyperreal number a, there exists a unique real number r for

which a = r +

for some infinit