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

x
Section 2.3 • The Derivative of a Polynomial Function; The Derivative of y = e and y = ln x 191

(
4x + 1 if x ≤ 2 Free-Response Questions
PAGE
183 9. At x = 2, the function f (x) = is
PAGE
2
3x − 3 if x > 2 181 11. A rod of length 12 cm is heated at one end. The table below
(A) Both continuous and differentiable. gives the temperature T (x) in degrees Celsius at selected
(B) Continuous but not differentiable. distances x cm from the heated end.
(C) Differentiable but not continuous. x 0 2 5 7 9 12
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(D) Neither continuous nor differentiable.
T (x) 80 71 66 60 54 50
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183 10. The table below lists several values of a function y = f (x).
′
(a) Use the table to approximate T (8).
x 1 2 3 4 5 (b) Using appropriate units, interpret T (8) in the context of
′
f (x) 4 −6 2 1 6 the problem.
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181 12. Oil is leaking from a tank. The amount of oil, in gallons, in the
Suppose f is continuous on the interval (0, 6), except at 3. tank is given by G(t) = 4000 − 3t , where t, 0 ≤ t ≤ 24 is the
2
Suppose f has a derivative at each number in the number of hours past midnight.
interval (0, 6) except at 3 and 4. Which of the following
′
statements must be true? (a) Find G (5) using the definition of the derivative.
(b) Using appropriate units, interpret the meaning of G (5) in
′
I. The graph of f has a vertical tangent line at (4, 1).
the context of the problem.
II. The graph of f has a corner at the point (3, 2).
III. f has a zero in the interval (1, 2).
See the BREAK IT DOWN on page 231 for a stepped out solution to
(A) I only (B) II and III only AP Practice Problem 12.
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(C) III only (D) I, II, and III
Retain Your Knowledge
Multiple-Choice Questions

2 if x ≤ 3 3x + 2 sin(3x)
1. Find lim f (x), for the function f (x) = . 3. Find lim , if it exists.
x→3 + x + 1 if x > 3 x→0 2x
3 5 9
(A) 2 (B) 3 (C) 4 (D) The limit does not exist. (A) (B) (C) (D) The limit does not exist.
2 2 2
2x + 6
2. lim =
x→2 − 2 − x Free-Response Question
3
2
(A) −∞ (B) −2 (C) −1 (D) ∞ 4. f (x) = x − 4x + 2x + 1. Show that there is at least one
number c in the interval [0, 4] for which f (c) = 7.
2.3 The Derivative of a Polynomial Function;
x
The Derivative of y = e and y = ln x
OBJECTIVES When you finish this section, you should be able to:
1 Differentiate a constant function (p. 192)
2 Differentiate a power function; the Simple Power Rule (p. 192)
3 Differentiate the sum and the difference of two functions (p. 195)
x
4 Differentiate the exponential function y = e and
the natural logarithm function y = ln x (p. 197)

Finding the derivative of a function from the definition can become tedious, especially if
the function f is complicated. Just as we did for limits, we derive some basic derivative
formulas and some properties of derivatives that make finding a derivative simpler.
Before getting started, we introduce other notations commonly used for the
derivative f (x) of a function y = f (x). The other most commonly used notations are
′
dy
y ′ Df (x)
dx

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