The Art and Craft of Problem Solving

Chapter 9

Calculus

In this chapter, we take it for granted that you are familiar with the basic calculus ideas

like limits, continuity, differentiation, integration, and power series. On the other hand,

we assume that you may have have heard of, but not mastered:

• Formal" 0 - e" proofs


• Taylor series with "remainder"


• The mean value theorem
  In contrast to, say, Chapter 7, this chapter is not a systematic, self-contained treatment.
  Instead, we concentrate on just a few important ideas that enhance your understanding
  of how calculus works. Our goal is twofold: to uncover the practical meaning of some
  of the things that you have already studied, by developing useful reformulations of old
  ideas; and to enhance your intuitive understanding of calculus, by showing you some
  useful albeit non-rigorous "moving curtains." The meaning of this last phrase is best
  understood with an example.

9.1 The Fundamental Theorem of Calculus

To understand what a moving curtain is, we shall explore, in some detail, the most

important idea of elementary calculus. This example also introduces a number of

ideas that we will keep returning to throughout the chapter.

Example 9.1.1 What is the fundamental theorem of calculus (FTC), what does it

mean, and why is it true?

Partial Solution: You have undoubtedly learned about the FTC. One formulation

of it says that if f is a continuous function,l then

lb f(x)dx = F(b) -F(a), (1)

where F is any anti derivative of f; i.e., F'(x) = f(x). This is a remarkable state­

ment. The left -hand side of (1) can be interpreted as the area bounded by the graph

I In this chapter we will assume that the domain and range of all functions are subsets of the real numbers.

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