Calculus: Analytic Geometry and Calculus, with Vectors

5.1 Graphs, slopes, and tangents 289

whenever 111 < 82. This gives (5.171) and completes the proof of
Theorem 5.17.
As was remarked, the proof of Theorem 5.17 is difficult because
existence of various things must be proved. To help us understand that
questions involving existence and differentiability of functions can be
significant, we look at an example. Let us assume that y is a differen-
tiable function of x for which

(5.18) x2+y2+sinx+sin y+46 =0.

Differentiating with respect to x with the aid of the chain rule then gives

(5.181) 2x + 2y dx + cos x + cos y dx= 0

or

(5.182) (2y + cos y) dx = - (2x + cos x)

and, when (2y + cos y) 7-5 0, dividing by (2y + cos y) gives a formula for

dy/dx. The formula is illusory, however, because the original assump-
tion is incorrect. The inequalities x2 > 0, y2? 0, sin x > -1, and
sin y > -1 imply that, whatever x and y may be,

(5.183) x2+y2+sin x+sin y+46>44.

Consequently, there are no numbers x and y for which (5.18) is true.
The assumption that there is a differentiable function f, defined over some
interval a < x < b, such that

(5.184) x2 + [f(X)12 + sin x + sin f(x) + 46 = 0 (a < x < b)

is false. This example can help us understand the nature of Theorem

5.17. The theorem is not a weak one which tells what dy/dx must be if

it exists. The theorem sets forth conditions under which dy/dx must

exist and gives a formula which must be correct when these conditions

are satisfied. Proof of a weak theorem can be obtained by mixing a few

words with the calculation

Ay dy

(5.185) dx l.oAx o Ox dx'

-At dt

but this one line is very far from the equivalent of a theorem which sets

forth conditions under which y is a differentiable function of x and the

formula is valid. Examples show that matters involving (5.185) are

not always completely simple. The distance r from Earth to Mars and

the blood pressure p of a particular yogi are both functions of time t, but