Calculus: Analytic Geometry and Calculus, with Vectors

5.1 Graphs, slopes, and tangents 287

mathematics before we can fully comprehend the details. In the worst
of circumstances, we are like a person who cannot swim but is thrown into
the water and given a chance to fight for his life. Most of us will soon be
swimming around in the scientific oceans, and Theorem 5.17 will slowly
metamorphose from an ugly demon to a friendly angel. The theorem is

closely related to the preceding paragraph and to the chain rule, but it is
different from both. Using different notation, it sets forth conditions
under which the first of the two equations

x = fi(t), Y = f2 (t)

can be "solved" for tand the result substituted in the second equation to

obtain y as a function of x. Moreover, the theorem tells how we can

find a formula for the derivative of y with respect to x even though we

cannot work out a useful formula that gives y in terms of x. The useful-

ness of the theoremand the difficulty of the proof are both due to the

fact that the conclusion of the theorem guarantees existence of various

things. If we replace the condition x'(t) > 0 by the condition x'(t) < 0

in the theorem, the intervening details become somewhat different but the

final conclusion (5.171) is valid. We could say that the theorem is a

theorem about elimination of parameters, but in case f2(t) = t so t = y it is

an inverse-function theorem.

Theorem 5.17 Let x(t) and y(t) be continuous over the closed interval

ti < t < t2 and be differentiable over the open interval ti < t < t2 and let

x1(t) > 0 when ti < t < t2. Let x(ti) = a and x(t2) = b. Then a < b,

and to each xo for which a < xo < b there corresponds exactly one to for

which tl < to < t2 and x(to) = xo, and to in turn determines exactly one

yo for which yo = y(to). This correspondence between numbers xo and yo

determines a function f for which yo = f(xo) when a < xo < b, and hence

y = f(x) when a < x < b. Moreover, this function f is differentiable and

the first of the formulas

(5.171) f'(x)

dy

y'(t) dy_ dt

x' (t)' dx dx

dt

is valid when x = x(t) and ti < t < t2. The second is also valid when it is

understood to mean what the first does.

To help us understand the things we do

to prove this theorem, we start sketching

Figure 5.172. We mark the points (ti,a)

and (t2,b) in a tx plane. For a schematic

graph of x(t), we sketch a curve headed

upward to the right because we think it

should be so because x(t) > 0. Theorem

Figure 5.172