Calculus: Analytic Geometry and Calculus, with Vectors

286 Functions, graphs, and numbers

for which xo - S < x < xo + S lies entirely on or above L or lies entirely
on or below L. To emphasize that tangent lines were defined as we
defined them because of custom and not because of logical necessity,
we can imagine that a man from Mars might come to our earth with a
language identical with ours except that his meanings of the terms "line
of support" and "tangent line" could be obtained by interchanging ours.
This man from Mars might wonder why on earth we study our tangent
lines instead of his. The problems at the end of this section may provide
reasons.
To be honorable, we must show that the remark made in Section 3.7
about tangents to curves is in agreement with the ideas of this section.
Putting z(t) = 0 gives the assertion that if r(t) is the vector OP running
from the origin to a particle P which traverses a curve C as t increases,
and if

(5.16) r(t) = x(t)i + y(t)j,

where x and y are differentiable functions oft for which r'(t) 0 0, then,

for each t, the vector

(5.161) r'(t) = x'(t)i + y'(t)j

is tangent to the path. In case the particle P always lies on the graph

of the equation y = f(x), we always have y(t) = f(x(t)). Therefore,

(5.162) r(t) = x(t)i + f(x(t))j,

and differentiating with the aid of the chain rule gives the result that,

at each time t, the vector

(5.163) r'(t) = x'(t) [i + f'(x(t))jI

is tangent to the graph. The hypothesis that r'(t) 0 0 implies that

x'(t) 0. Since x'(t) is a nonzero scalar, our result is equivalent to the

statement that, for each x, the vector

rxvty i I havingits tailat the point (x,y) on the graph is

f'(x)i

(5.164) i + f (x)j

Figure 5.165

tangent to the graph at the point. With or with-

out the aid of Figure 5.165, we can see that this

vector lies on the line through (x,y) having slope

f'(x). Thus the tangent line obtained by use of vectors is the same as

the tangent line obtained by use of slopes.

The remainder of the text of this section is devoted to a useful theorem

which is, from our present point of view, thoroughly difficult. The

theorem is important because it gives precise information that is very

often used. The proof of the theorem shows that we must learn more