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

Chapter 2 • Review Exercises 231
Break It Down Preparing for the AP Exam
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Let’s take a closer look at AP Practice Problem 12 from Section 2.2 on page 191.
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12. Oil is leaking from a tank. The amount of oil, in gallons, in the tank is given by G(t) = 4000 − 3t ,
where t, 0 ≤ t ≤ 24 is the number of hours past midnight.
(a) Find G (5) using the definition of the derivative.
= limOnly - Do Not Copy.
′
(b) Using appropriate units, interpret the meaning of G (5) in the context of the problem.
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Step 1 Identify the underlying structure and the The problem is asking for G’(5), the derivative of G at the
related concepts. number 5.
Step 2 Determine the appropriate math rule or Because we want the derivative of G at a number, we use
procedure. Form (1) of the Definition of a Derivative (see p. 179).
Step 3 Apply the math rule or procedure. Using Form (1),
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G(t) − G(5) (4000 − 3t ) − (4000 − 3 · 5 )
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G (5) = lim
t→5 t − 5 t→5 t − 5
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− 3(t − 25) − 3(t − 5)(t + 5)
= lim = lim
t→5 t − 5 t→5 t − 5
= lim[− 3(t + 5)] = − 30
t→5
Step 4 Clearly communicate your answer. (a) G (5) = − 30
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(b) A derivative is a rate of change. In this problem, G (t) equals
′
the rate of change of G with respect t, that is, the rate of change
of the amount of oil, in gallons, with respect to the time in hours.
Because G (5) = − 30, we say the amount of oil in the tank is
′
decreasing at the rate of 30 gallons per hour when t = 5 hours past
midnight, or at 5AM.

AP Review Problems: Chapter 2 Preparing for the AP Exam
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Multiple-Choice Questions
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1. If f (x) = sec x, then f ′ = 4. The graph of the function f is shown below.
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√ Which statement about the function is true?
2 √
(A) (B) 2 (C) 1 (D) 2 y
2
2
y 5 f(x)
2. If a function f is differentiable at c, then f (c) is given by
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f (x) − f (c)
I. lim 22 2 x
x→c x − c
f (x + h) − f (x)
II. lim (A) f is differentiable everywhere.
x→c h
(B) 0 ≤ f (x) ≤ 1, for all real numbers.
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f (c + h) − f (c)
III. lim (C) f is continuous everywhere.
h→0 h
(D) f is an even function.
(A) I only (B) III only
(C) I and II only (D) I and III only 5. The table displays select values of a differentiable
function f . What is an approximate value of f (2)?
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3 dy
3. If y = 2 , then =
x − 5 dx x 1.996 1.998 2.002 2.004
6x 6x f (x) 3.168 3.181 3.207 3.220
(A) (B) −
2
2
(x − 5) 2 (x − 5) 2
(A) 6.5 (B) 0.154 (C) 0.013 (D) 1.5
6x 2x
(C) (D)
2
2
x − 5 (x − 5) 2
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