Calculus: Analytic Geometry and Calculus, with Vectors

5.1 Graphs, slopes, and tangents 285

Figure 5.11 Figure 5.12 Figure 5.13

xo to emphasize the fact that it remains fixed throughout our discussion.
Our task is to try to decide what we should mean when we say that
a line T is tangent to G at P(xo,yo). We gain the possibility of making
progress when we choose a number Ax for which Ax n 0, plot the point
P(xo + Ox, yo + Ay) on G, and draw the chord joining our two points on
G. Our first feeble idea can be that T is tangent to G at P(xo,yo) if the
chord is nearly coincident with T whenever .x is near zero. We can, so
far as nonvertical tangents are concerned, improve this idea to gain the
concept that the line T through P(xo,yo) having slope m is tangent to G
at P(xo,yo) if the slope Ay/Ax of the chord is near m whenever Ax is near

0. We know how to express this concept in terms of limits and deriva-
   tives, and we do it in the following definition.
   Definition 5.14 If f'(xo) exists, then the line T through the point (xo,yo)
   having slope f'(xo) is said to be tangent to the graph of y = f(x) at the point
   (xo,yo). If f'(xo) fails to exist, then the graph fails to possess a nonvertical
   tangent at the point (xo,yo).
   From this definition and the point-slope formula for the equation of a
   line, we obtain the following theorem.
   Theorem 5.141 If f'(xo) exists, then the equation

Y - Yo = f'(xo)(x - xo)

is the equation of the tangent to the graph of y = f(x) at the point (xo,yo).
To assist in the development and communication of ideas, it turns out
to be exceptionally useful to agree that if a graph has a nonvertical
tangent at a point (xo,yo), then the slope of this tangent will be called
the slope of the graph at the point (xo,yo). In accordance with this idea,
we adopt the following definition.
Definition 5.15 If f'(xo) exists, then f'(xo) is said to be the slope of the
graph of y = f(x) at the point P(xo,yo).
In order to obtain a full understanding of tangents to graphs, and for
other purposes, it is helpful to know about "lines of support" of graphs
and other point sets that lie in a plane. We confine attention here to
cases in which f is a continuous function defined over a 5 x< b and
Po(xo,yo) is a point on the graph of y = f(x) for which a < xo < b. A
line L through P0 is said to be a line of support of the graph of y = f(x)
if there is a positive number 3 such that the part of the graph of y = f(x)