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

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Section 2.3 • The Derivative of a Polynomial Function; The Derivative of y = e and y = ln x 197
Use the point-slope form of an equation of a line to find an equation of the
normal line.
1
y − (−5) = (x − 1)
2
1 1 1 1 11
y = (x − 1) − 5 = x − − 5 = x −
2 2 2 2 2
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1 11
The line y = x − is normal to the graph of f at the point (1, −5).
2 2
(f) The graphs of f , the tangent line, and the normal line to f at (1, −5) are shown
f210, 14g 3 f210, 5g
in Figure 26. Because we are graphing the tangent line and the normal line, which are
2
4
Figure 26 f (x) = 2x − 6x + 2x − 3 perpendicular to each other, we use a square screen to obtain Figure 26.
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NOW WORK Problem 33 and AP Practice Problems 2, 5, 8, 11, and 12.
In some applications, we need to solve equations or inequalities involving the
derivative of a function.

EXAMPLE 6 Solving Equations and Inequalities Involving Derivatives
3
2
(a) Find the points on the graph of f (x) = 4x − 12x + 2, where f has a horizontal
tangent line.
′
(b) Where is f (x) > 0? Where is f (x) < 0?
′
Solution
(a) The slope of a horizontal tangent line is 0. Since the derivative of f equals the slope
of the tangent line, we need to find the numbers x for which f (x) = 0.
′
2
′
f (x) = 12x − 24x = 12x(x − 2)
12x(x − 2) = 0 f (x) = 0
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x = 0 or x = 2 Solve.
y At the points (0, f (0)) = (0, 2) and (2, f (2)) = (2, −14), the graph of the
5 3 2
function f (x) = 4x − 12x + 2 has horizontal tangent lines.
(0, 2)
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(b) Since f (x) = 12x(x − 2) and we want to solve the inequalities f (x) > 0
′
′
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2 2 4 x and f (x) < 0, we use the zeros of f , 0 and 2, and form a table using the
intervals (−∞, 0), (0, 2), and (2, ∞). See Table 2.
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TABLE 2
Interval (−∞, 0) (0, 2) (2, ∞)
10
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Sign of f (x) = 12x(x − 2) Positive Negative Positive
15 (2, 14) We conclude f (x) > 0 on (−∞, 0) ∪ (2, ∞) and f (x) < 0 on (0, 2).
′
′
3
2
Figure 27 f (x) = 4x − 12x + 2 Figure 27 shows the graph of f and the two horizontal tangent lines.
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NOW WORK Problem 37 and AP Practice Problem 3.
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4 Differentiate the Exponential Function y = e and the Natural
Logarithm Function y = ln x

None of the differentiation rules developed so far allow us to find the derivative of an
x
exponential function. To differentiate f (x) = a , we need to return to the definition of
a derivative.
We begin by making some general observations about the derivative
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NEED TO REVIEW? Exponential functions of f (x) = a , a > 0 and a 6= 1. We then use these observations to find the derivative
are discussed in Section P.5, pp. 43–46. x
of the exponential function y = e .
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