Conce pts of Calculus

1 . Origins

Many sorts of calculations press themselves upon us. Thus, given a piece of surface, how does one calculate its area? Or, given a section of a curve, how does one calculate its length or the direction of its tangent line at some point? More generally, how does one calculate the rate at which this or that variable quantity is changing with time? The striking discovery (by Newton and Leibniz) that there were systematic methods to calculate

all

these things, and many more like them, had a major influence on the directions and structure of Mathematics. For a considerable period, more practical calculations of such things tended to dominate conceptual under­

standing, in a way that emphasizes the observation that Mathematics takes its origin in human activities.

The calculation of areas began in Euclidean geometry, with formulas for the areas of triangles and squares. Next in line was the area of a cir­

cular disc. This could be determined by inscribing and circumscribing reg­

ular polygons in the disc; as the number of sides of these polygons increased, the area inside the circle was "pinched" between the (calcul­

able) areas inside the larger and the smaller polygon-and this pinching process produced the area inside the disc as a sort of limit. This method, neatly adapted to the case of the circle, suggested to Archimedes and oth­

ers a search for extensions of the method to calculate other areas-that of an elliptical disk, and that for more irregular figures. Similarly the meas­

urements of length began with the Pythagorean theorem used to deter­

mine the length of a slanted line and hence the perimeter of any polygon.

The circumference of a circle could then be found (§1I1.2) through succes­

sive approximation by inscribed polygons. These and other problems of measurement (of volumes, weights, centers of gravity, and the like) emphasized the ubiquitous role of approximation and may have indicated the need for a systematic understanding of the way in which such

succes-I. Origins 1 5 1 sive approximations can converge to a desired limit-an understanding then provided in the calculus by the general definition of the integral as the limit of a sum.

Also, how does one construct tangents? Drawing the tangent line to a circle at some point

P

is hardly problematical, since the line through

P

and perpendicular to the radius there is evidently the correct tangent-just touching the circle evenly on both sides of

P.

For an ellipse, such as a simple construction will succeed only for tangents perpendicular to the major or minor axis at one end of that axis. At other points, here and on the hyperbola, the parabola or on other curves in the plane, more sophis­

ticated methods, some known to the Greeks, are needed to draw an exact tangent; with time, it became plausible that the tangent line to a curve at a point

P

might be well approximated by the line (the secant) determined by two points

P'

and

P"

on the curve taken close (and then closer) to

P.

The need to locate such tangent lines is suggested not j ust by the geometr­

ical aspects of plane or spatial curves, but also by mechanical situations where moving things fly off "on a tangent"-from a bobsled or a roller coaster, for example. Thus the determination of tangent lines again can suggest their calculation by the limit of some systematic sequence of approximations.

Observations of motion raise analogous questions of approximation.

Terrestrial and heavenly bodies move about, but with velocities which often do not stay constant. There still should be some way of calculating a velocity for such motion-an instantaneous velocity which might, on reflection, be measured by using as approximations suitable average velo­

cities over shorter and shorter time intervals. Similar thoughts might then arise for other measurable quantities which vary with time, but not at a uniform rate. In this way attention might be directed to the calculation in general of such instantaneous rates of change-again by some sort of scheme of successively better approximations.

Such problems can also occur for "rates" which are not time rates but rates relative to some other changing quantity. An elementary example might be the rate at which the area of a circle changes as the radius alters.

Decisive examples arise in Mathematical Economics. There, in analysing the costs of producing more, it is wiser not to consider just the average cost for the production of one widget out of a total of

n

such; but to use the marginal cost-meaning roughly the additional cost of producing the last one of those

n

widgets or, more conceptually, the rate at which the cost of production changes relative to the number

n

of widgets produced.

This example of the use of derivatives may have developed historically much later than the original discovery of the calculus; however, it is in­

trinsically one of the appropriate origins for the idea of a systematic cal­

culation of relative rates of change for variable quantities.

This summary description provides a conceptual (though not a specifically historical) account of the multiple origins of the calculus.

1 52 V I . Concepts of Calculus

2. Integration

It is remarkable that so many different processes of approximating total measured quantities by adding together little bits of these quantities can all be subsumed under

one

process, that of integration. Area, volume, length, pressure, moment of inertia, weight, and the like all can be managed by such sums. To be sure, the usual formal definition of the Riemann Integral is usually presented as the calculation of an area­

specifically the area below a curve y = f(x), above the x axis, and bounded left and right by the ordinates x = a and x =

b.

This area is broken up into the usual thin vertical and rectangular strips of width dx running from x = a to x =

b,

as in Figure

I .

The area of such a strip is altitude f(x ) times base d, hence is written f(x )dx and the sum of them all-and hence the total area desired-is the definite integral

f

b f(x)dx .

a

( I )

O n one reading, the width dx o f each strip is an infinitesimal increment in x (hence the notation dx), there is an infinite number of strips, and so the integral sign (an elongated

S)

represents an infinite sum of infinitesimal quantities, giving the desired area. Another reading, devoid of any such uncertain appeal to the infinitely small, would approximate the desired area by a finite sum of the areas of rectangles of finite width. To this end, subdivide the interval from x = a to x =

b

in

n

intervals with successive endpoints a = Xo

<

X I

<

X2

< . . . <

Xn =

b;

call such a subdivi­

sion o. In the ith interval from Xi - I to Xi the function f (if continuous) will take on a maximum Max;f and a minimum Min;f (Figure 2), and the presumptive true area Ai under this portion of the curve will lie between the areas of the two rectangles which have width Xi - Xi- I and height Min;f and Max;f, respectively, as expressed by the inequalities

(Minif)(Xi - Xi - I )

<

Ai

<

(Max;f)(xi - Xi- I ) .

Adding up these rectangles gives for our subdivision 0 a "lower sum"

L(f, o) and an "upper sum" U(f, o)

y

a b x

Figure I . The Riemann integral.

2. Integration 1 53

T

^Max/

�

Figure 2

n n

L(f,a )

=

�

( Minif)(xi - xi- d <

U(f, a )

=

�

( MaxJ)(xi - xi - d ; (2)

i = 1 i = 1

the desired area under the whole curve must be squeezed between these two sums. To actually express this area, we must then take a limit of these sums for successively finer subdivisions

a,

so let the size of

a

be measured by

I a I

^' the maximum of all the interval lengths

I

Xi - Xi - I

I

. When f is continuous (on the whole interval

a < X < b)

it is then the case (as we will soon see) that for each measure (

>

0 of approximation there is a

�

>

0 such that, whenever

I a I

< �, then

I U(f, a ) - L(f, a ) I

< (.

This implies that L and

U

have a common limit as

I a I

approaches 0, lim

L(f,a)

= lim

U(f,a)

=

f.

b ^{f(x)dx ; (3)} I ^a I ... 0 I ^a I ... 0 ^a

this limit is the definite integral (3) of the function f over the interval

a < X < b.

By its construction, it represents what the area under the curve over this interval should be ; if you wish, it is the definition of this area; for those applications such as pressures and volumes it can also serve as a definition. Such definite integrals, so defined as limits of sums, are still written in the classical notation (3 ) suggesting the infinite sum of infinitesimals; in fact that intuitive view of the matter allows us to easily set up the integral representing all sorts of other quantities: The weight of a thin slab of known but variable density, the water pressure on a slab­

like portion of a dam, or the volume enclosed by a surface of revolution.

Much of the instruction in elementary integral calculus consists of repeti­

tive exercises practising the formulations of such integrals. When they cannot be done by slices or slabs, they can be managed (with more tech­

nique but in a wholly similar spirit) by multiple integrals.

This formal definition of the

Riemann integral

of a continuous function replaces the intuitive idea of successive approximations by the use of lim­

its and of the standard logical

quantifiers

(for all (

>

0 there exist a

�

>

0 such that . . . ) needed for the exact description of these limits.

Various general properties of the integral also follow directly from this

1 54 V I . Concepts of Calculus

definition. Linearity is an example : the definite integral from

a

to

b

of the sum of two continuous functions is the sum of the integrals of these func­

tions separately. Again, if

a

<

b

< c, the integral of a given continuous function from

a

to c is the sum of the integrals from

a

to

b

and from

b

to c. This property is really using the idea of "composing" a path (of integration) from

a

to

b

with a path from

b

to c-an idea already present in group theory, and one which will recur in more elaborate cases of line integrals.

So much for the initial formulation of measurement by integration.

However since we do not really carry out infinite sums or limits of an infinite number of successive finite sums, the real "calculus" of such integrals rests on the connection with differentiation, to which we now turn.

3 . Derivatives

The derivative of a variable quantity

y

with respect to another such quan­

tity

x

on which it depends is to be the instantaneous rate of change of

y

relative to the change in

x.

This description has intuitive appeal, espe­

cially in the special case when

x

is taken to be time. Instantaneous rates are modeled on average rates: For a value of

YI

at time

x I ,

changed to a value

Y2

at a different time

X2 ,

the average rate of change is the ratio

(Y2 - YI )/(X2 - XI )

of change in

y

to change in

x.

The "instantaneous"

aspect might again be formulated by an infinitesimal change

dx

from

x

to

x + dx.

Then if

y

depends on

x

by a function

y

⁼

f(x),

the instantane­

ous rate of change is the ratio

[f(x + dx) - f(x») /dx

or, in an evident extension of the

dx

notation,

dy/dx,

the quotient of two infinitesimals.

Such infinitesimals serve for quick calculation. For example, if

f(x)

=

x2

, then the instantaneous rate is

dy

=

(x + dx)2 - x2

=

2xdx

₊

dx2

⁼

2xdx

₌

2x

dx dx dx dx ' ( I )

since the square

(dxi

of an infinitesimal surely vanishes, a t least in com­

parison with its first power. Other rates easily follow. Moreover, if

y

depends on

x

and

x

in turn on time

t,

cancellation of the infinitesimal

dx

gives

dy . dx dx dt dy

dt '

(2)

and one has (but hasn't quite proved) the important (because necessarily most useful) "chain rule" for the derivative of a composite function. Thus calculus with infinitesimals is intuitive and an efficient means of calcula­

tion.

But what

are

these infinitesimals? The archimedean law for the real numbers (§IV.4) says that a positive number, no matter how small, has

4. The Fundamental Theorem of the Integral Calculus 1 55

arbitrarily large multiples-in effect, that a positive number cannot be infinitesimal. With Bishop Berkeley, one may conclude that "infinitesimals are the ghosts of departed quantities."

What remains are limits. For each value

x

⁼

a

and each actual finite increment

x

⁼

a + h

with

h =1=

0 one may form from a given function

y

=

f(x)

the very same ratio

[f(a + h) - f(a)] lh

as before. The deriva­

tive, when it exists, is then defined to be the limit of this ratio as

h =1=

0 approaches 0; with either standard notation

dy/dx

or

f '(x),

this means that the derivative is

[ ^dy 1

⁼

^{J '(a)}

^{= lim}

^f(x

⁺

h) - f(a)

^.

dx

_{x =a} ^{h -D}

h

⁽³⁾

The necessary meticulous ( - 8 definition of the limit in this description must again use the quantifiers (for all ( there exists a 8) to achieve preci­

sion. With this precision (as careful students of the calculus know) one can then prove that the derivatives of

x2

and of

xn

are what they should be and that the desired rule

(2)

for the derivative of a composite function does hold for suitable differentiable functions-not by simple cancellation of infinitesimals, but by limits taken after cancellation.

This discussion of alternative views of the calculus may serve to support our thesis that Mathematics is not just formalism and is not just empiri­

cally convenient ideas, but consists in formalizable intuitive or empirical ideas. Calculus fits this thesis as to the nature of Mathematics because it starts from problems (such as the calculation of areas or of rates of change) and from these problems develops suggestive ideas which ulti­

mately can be fully formalized. For calculus, the initial formalization dealt chiefly with practical rules for finding and manipulating derivatives and integrals. This is not yet a complete formalization, which was first provided in the 1 9th century by the rigorous ( - 8 treatment by limits.

Another thesis asserts that the same intuitive idea can be variously for­

malized. For the calculus, we now know that this is the case. Indeed, the earlier and originally wholly speculative use of infinitesimals can be rigorously formalized in at least two different ways : By using Abraham Robinson's non-standard model of the reals, as presented for example in Keisler's text [ 1 976] on calculus, or by using Lawvere's proposals to use

"elementary topoi" in which the real line

R

is presented not as a field but as a ring in which there is a suitable infinitesimal neighborhood of 0 (see Kock [ 1 98 1 D.

4. The Fundamental Theorem of the Integral Calculus

The essential connection between differentiation and integration is pro­

vided by the theorem of the title of this section. This theorem goes back to a simple intuitive idea: That the total change in a quantity ought to be

1 56 V I . Concepts of Calculus

exactly the sum of the successive small instantaneous changes-an idea which in other forms arises early in geometry, with the rule that the whole is the sum of its parts. Specifically, if the quantity in question is a function

F(t)

of time

t,

the total change from time

t

⁼

a

to a later time

t

=

b

is just the different

F(b) - F(a).

Now suppose this function

F(t)

has at each time

t

a derivative

F'(t)

⁼

f(t).

The instantaneous change in an infinitesimal interval

dt

of time is then the product of this interval

dt

by the instantaneous rate of change

F'(t),

and so the sum of all the successive instantaneous changes is the definite integral

f� f( t )dt.

The suggested theorem then reads

Theorem.

If the function F(t) has a continuous derivative f(t)

⁼

F'(t) on the interval a

�

t

�

b, then

i f(t)dt

b ⁼

F(b) - F(a) .

a ( 1 )

The expression o n the right is often written a s [

F(t)

]

g

, while the function

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