9 Dec 2008

The Antiderivative in Edwards & Penney

presentation of Edwards & Penney's work, by presentation of Edwards & Penney's work, by Corry Shores
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Edwards & Penney's Calculus is an incredibly-impressive, comprehensive, and understandable book. I highly recommend it.

[the following will not stray from Edwards & Penney's procedure, so it is largely quotation.]

The differential equation of the simplest kind takes the form:

where f is a given (known) function and the function y (x) is unknown. When we antidifferentiate, we find a function from its derivative, which is the opposite of differentiation. So if we find a function y (x) whose derivative is f (x), such that y' (x) = f (x), then we call y (x) an antiderivative of f (x).

An antiderivative of the function f is a function F such that

F ' (x) = f (x)

where f (x) is defined.

We see a set of examples of functions paired with their antiderivatives in the table below:

The flow diagram below depicts the operations of differentiation and antidifferentiation, beginning with the same function f and going in opposite directions.

The next flow chart shows differentiation "undoing" the result of antidifferentiation -- the derivative of the antiderivative of f (x) is the original function f (x).

Example: For the function

is an antiderivative of f (x), as are the functions

In fact,

is an antiderivative of

for any choice of the constant C.
As we see, a single function has very many antiderivatives, even though a function can only have one derivative. If F (x) is an antiderivative of f (x), then so is F (x) + C for any choice of the constant C. But consider the converse of this statement: If F (x) is one antiderivative of f (x) on the interval I, then every antiderivative of f (x) on I is of the form F (x) + C.

For this reason, the graphs for any two antiderivatives

of the same function f (x) on the same interval I are "parallel" in the sense depicted in the diagrams below:

What we see is that constant C is the vertical distance between the curves y = F (x) and y = F (x) + C for each x in I. This is the geometric interpretation of the following theorem.

Theorem 1: The Most General Antiderivative
If F ' (x) = f (x) at each point of the open interval I, then every antiderivative G of f on I has the form
G (x) = F (x) + C,
where C is constant.

So if F is some single antiderivative of f on the interval I, the most general antiderivative of f on I has the form F (x) + C, as given in the above equation. The collection of all antiderivatives of the function f (x) is called the indefinite integral of f with respect to x and is denoted by

On the basis of the above theorem, we write then:

where F (x) is any particular antiderivative of f (x). Therefore,

The integral symbol

is made as though it were an elongated capital S. It is the Medieval S which Leibniz used to abbreviate the Latin summa (sum). The combination

... dx

we take as one symbol, and we fill in the ... with the formula of the function whose antiderivative we seek. We may consider the differential dx as specifying the independent variable x both in the function f (x) and in its antiderivatives.

from Edwards & Penney: Calculus. New Jersey: Prentice Hall, 2002, p.272b.257a.

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