James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 2.7 Derivatives and Rates of Change 141
y
P{a, f(a)}
Q{x, ƒ}
ƒ-f(a)
a number m, then we define the tangent t to be the line through P with slope m. (This
amounts to saying that the tangent line is the limiting position of the secant line PQ as Q
approaches P. See Figure 1.)
x-a
0 a x
x
y
t
Q
Q
1 Definition The tangent line to the curve y − f sxd at the point Psa, f sadd is
the line through P with slope
provided that this limit exists.
m − lim
x l a
f sxd 2 f sad
x 2 a
In our first example we confirm the guess we made in Example 2.1.1.
P
Q
ExamplE 1 Find an equation of the tangent line to the parabola y − x 2 at the
point Ps1, 1d.
SOLUtion Here we have a − 1 and f sxd − x 2 , so the slope is
0 x
FIGURE 1
m − lim
x l1
f sxd 2 f s1d
x 2 1
− lim
x l1
sx 2 1dsx 1 1d
x 2 1
− lim
x l1
x 2 2 1
x 2 1
− lim
x l1
sx 1 1d − 1 1 1 − 2
Point-slope form for a line through the
point sx 1, y 1d with slope m:
y 2 y 1 − msx 2 x 1d
Using the point-slope form of the equation of a line, we find that an equation of the
tangent line at s1, 1d is
y 2 1 − 2sx 2 1d or y − 2x 2 1 n
TEC Visual 2.7 shows an animation
of Figure 2.
We sometimes refer to the slope of the tangent line to a curve at a point as the slope
of the curve at the point. The idea is that if we zoom in far enough toward the point, the
curve looks almost like a straight line. Figure 2 illustrates this procedure for the curve
y − x 2 in Example 1. The more we zoom in, the more the parabola looks like a line. In
other words, the curve becomes almost indistinguishable from its tangent line.
2
1.5
1.1
(1, 1)
(1, 1)
(1, 1)
0 2
0.5 1.5
0.9 1.1
FIGURE 2 Zooming in toward the point (1, 1) on the parabola y − x 2
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