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

Section 2.7 Derivatives and Rates of Change 145
We defined the tangent line to the curve y − f sxd at the point Psa, f sadd to be the line
that passes through P and has slope m given by Equation 1 or 2. Since, by Defini tion 4,
this is the same as the derivative f 9sad, we can now say the following.
The tangent line to y − f sxd at sa, f sadd is the line through sa, f sadd whose slope is
equal to f 9sad, the derivative of f at a.
y
If we use the point-slope form of the equation of a line, we can write an equation of
the tangent line to the curve y − f sxd at the point sa, f sadd:
y=≈-8x+9
0 x
(3, _6)
y=_2x
FIGURE 7
y 2 f sad − f 9sadsx 2 ad
ExamplE 5 Find an equation of the tangent line to the parabola y − x 2 2 8x 1 9 at
the point s3, 26d.
SOLUtion From Example 4 we know that the derivative of f sxd − x 2 2 8x 1 9 at
the number a is f 9sad − 2a 2 8. Therefore the slope of the tangent line at s3, 26d is
f 9s3d − 2s3d 2 8 − 22. Thus an equation of the tangent line, shown in Figure 7, is
y 2 s26d − s22dsx 2 3d or y − 22x n
y
Q{¤, ‡}
Rates of Change
Suppose y is a quantity that depends on another quantity x. Thus y is a function of x and
we write y − f sxd. If x changes from x 1 to x 2 , then the change in x (also called the increment
of x) is
Dx − x 2 2 x 1
P{⁄, fl}
Îy
Îx
0 ⁄ ¤ x
average rate of of change −m PQ
m PQ
instantaneous rate of of change −
slope of of tangent at at P
FIGURE 8
and the corresponding change in y is
The difference quotient
Dy − f sx 2 d 2 f sx 1 d
Dy
Dx − f sx 2d 2 f sx 1 d
x 2 2 x 1
is called the average rate of change of y with respect to x over the interval fx 1 , x 2 g and
can be interpreted as the slope of the secant line PQ in Figure 8.
By analogy with velocity, we consider the average rate of change over smaller and
smaller intervals by letting x 2 approach x 1 and therefore letting Dx approach 0. The limit
of these average rates of change is called the (instantaneous) rate of change of y with
respect to x at x − x 1 , which (as in the case of velocity) is interpreted as the slope of the
tangent to the curve y − f sxd at Psx 1 , f sx 1 dd:
6 instantaneous rate of change − lim
Dx l 0
Dy
Dx − lim
x2 l x 1
f sx 2 d 2 f sx 1 d
x 2 2 x 1
We recognize this limit as being the derivative f 9sx 1 d.
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