93

Section 3.8

.

Applications of Calculus

3.8

.

Applications of Calculus

In the Applications of Calculus section at the end of the previous

chapter, I discussed how the slope-of-the-tangent-line idea related to the

calculus concept of a derivative, and the branch of calculus known as

differential calculus. The other main branch of calculus, integral calculus,

has to do with the area-under-the-curve concept discussed in section 3.5 of

this chapter. Again there is a concept, a notation, and a bag of tricks for

doing things symbolically rather than graphically. In calculus, the area

under the v-t graph between t=t

1

and t=t

2

is notated like this:

areaunderthecurve=

.

x=vdt

t1

t2

The expression on the right is called an integral, and the s-shaped symbol,

the integral sign, is read as “integral of....”

Integral calculus and differential calculus are closely related. For in-

stance, if you take the derivative of the function x(t), you get the function

v(t), and if you integrate the function v(t), you get x(t) back again. In other

words, integration and differentiation are inverse operations. This is known

as the fundamental theorem of calculus.

On an unrelated topic, there is a special notation for taking the deriva-

tive of a function twice. The acceleration, for instance, is the second (i.e.

double) derivative of the position, because differentiating x once gives v, and

then differentiating v gives a. This is written as

a=d

2

x

dt

2

.

The seemingly inconsistent placement of the twos on the top and bottom

confuses all beginning calculus students. The motivation for this funny

notation is that acceleration has units of m/s

2

, and the notation correctly

suggests that: the top looks like it has units of meters, the bottom seconds

2

.

The notation is not meant, however, to suggest that t is really squared.