The Art and Craft of Problem Solving

9.2 CONVERGENCE AND CONTINUITY 317

since in a small interval of time .1t, the curtain's area will grow by approximately

f(t}.1t. Equation (2) immediately yields the FTC, because if we define F(t) := g(t) +

C, where C is any constant, we have F' (t) = f( t) and

F(b) -F(a) = g(b) -g(a) = lb f(x)dx.

The crux move was to interpret the definite integral dynamically, and then observe

the intuitive relationship between the speed that the area changes and the height of

the curtain. This classic argument illustrates the critical importance of knowing many

possible alternate interpretations of both differentiation and integration.

You may argue that we have not proved FTC rigorously, and indeed (2) deserves

a more careful treatment. After all. the curtain does not grow by exactly f(t).1t. The

exact amount is equal to

jt+.1t

t f(x)dx,

which is equal to f(t).1t +E(t), where E(t) is the area of the "error," shown shaded

below [note that E(t) is negative in this picture].

Hilt

Everything hinges on showing that

lim

(E(t))

=

(^0).
.11->0 .1t
This requires an understanding of continuity; we will prove (3) in 9.2.7.

9.2 Convergence and Continuity

(3)

You already have an intuitive understanding of concepts like limits and continuity , but

in order to tackle interesting problems, you must develop a rigorous wisdom. Luckily,

almost everything stems from one fundamental idea: convergence of sequences. If

you understand this, you can handle limits, and continuity, and differentiation, and

integration. Convergence of sequences is the theoretical foundation of calculus.