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

142 Chapter 2 Limits and Derivatives
Q{a+h, f(a+h)}
y
t
P{a, f(a)}
h f(a+h)-f(a)
0 a a+h
x
FIGURE 3
There is another expression for the slope of a tangent line that is sometimes easier to
use. If h − x 2 a, then x − a 1 h and so the slope of the secant line PQ is
m PQ −
f sa 1 hd 2 f sad
h
(See Figure 3 where the case h . 0 is illustrated and Q is to the right of P. If it happened
that h , 0, however, Q would be to the left of P.)
Notice that as x approaches a, h approaches 0 (because h − x 2 a) and so the expression
for the slope of the tangent line in Definition 1 becomes
2
m − lim
h l 0
f sa 1 hd 2 f sad
h
ExamplE 2 Find an equation of the tangent line to the hyperbola y − 3yx at the
point s3, 1d.
SOLUtion Let f sxd − 3yx. Then, by Equation 2, the slope of the tangent at s3, 1d is
m − lim
h l 0
f s3 1 hd 2 f s3d
h
− lim
h l 0
3
3 1 h 2 1
h
− lim
h l 0
3 2 s3 1 hd
3 1 h
h
x+3y-6=0
y
0
3
y=
x
(3, 1)
x
− lim
h l 0
2h
hs3 1 hd − lim
h l 0 2 1
3 1 h − 2 1 3
Therefore an equation of the tangent at the point s3, 1d is
y 2 1 − 2 1 3 sx 2 3d
which simplifies to x 1 3y 2 6 − 0
FIGURE 4
The hyperbola and its tangent are shown in Figure 4.
n
position at
time t=a
0 s
f(a+h)-f(a)
f(a)
FIGURE 5
f(a+h)
position at
time t=a+h
Velocities
In Section 2.1 we investigated the motion of a ball dropped from the CN Tower and
defined its velocity to be the limiting value of average velocities over shorter and shorter
time periods.
In general, suppose an object moves along a straight line according to an equation of
motion s − f std, where s is the displacement (directed distance) of the object from the
origin at time t. The function f that describes the motion is called the position function
of the object. In the time interval from t − a to t − a 1 h the change in position is
f sa 1 hd 2 f sad. (See Figure 5.)
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