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Sullivan 04 apcalc4e 45342 ch02 166 233 5pp August 7, 2023 12:54

Section 2.1 • Rates of Change and the Derivative 169

NEED TO REVIEW? The point-slope form of (b) We use the result from (a) and the point-slope form of an equation of a line to
a line is discussed in Appendix A.3, p. A-20. obtain an equation of the tangent line. An equation of the tangent line containing the
point (−2, 4) is
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AP EXAM TIP y − 4 = f (−2)[x − (−2)] Point-slope form of an equation of the tangent line.
′
On the exam, you can leave the equation of a y − 4 = −4 · (x + 2) f (−2) = 4; f (−2) = −4
′
line in point-slope form instead of simplifying
to slope-intercept form. y = −4x − 4 Simplify.
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y (c) Since the slope of the tangent line to f at (−2, 4) is −4, the slope of the normal line
1
(22, 4)
4 to f at (−2, 4) is .
y 5 x 2 4
1 9
y 5 2x 1 2 Using the point-slope form of an equation of a line, an equation of the normal
4 2
Normal line line is
1
24 4 x y − 4 = (x + 2)
4
y 5 24x 2 4
1
Tangent line y = x + 9
24 4 2
(d) The graphs of f , the tangent line to the graph of f at the point (−2, 4), and the
Figure 3 f (x) = x 2 normal line to the graph of f at (−2, 4) are shown in Figure 3.
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NOW WORK Problem 11 and AP Practice Problems 1 and 5.

2 Find the Rate of Change of a Function

Everything in nature changes. Examples include climate change, change in the phases
of the Moon, and change in populations. To describe natural processes mathematically,
the ideas of change and rate of change are often used.
Recall that the average rate of change of a function y = f (x) from c to x is given by

f (x) − f (c)
Average rate of change = x 6= c
x − c
The average rate of change describes behavior as it changes over an interval. To
describe behavior at a specific number, or instant, we use the instantaneous rate of
change.

DEFINITION Instantaneous Rate of Change
The instantaneous rate of change of f at c is the limit as x approaches c of the
average rate of change. Symbolically, the instantaneous rate of change of f at c is

f (x) − f (c)
lim
x→c x − c

provided the limit exists.
IN WORDS
• An average rate of change describes
behavior over an interval. The expression “instantaneous rate of change” is often shortened to rate of change.
• An instantaneous rate of change f (x) − f (c)
Using prime notation, the rate of change of f at c is f (c) = lim .
′
describes behavior at a number.
x→c x − c
EXAMPLE 2 Finding a Rate of Change

2
Find the rate of change of the function f (x) = x − 5x at:
(a) c = 2
(b) Any real number c
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