9.Iff(x) = 2 x 2 1 √ t 3 + 1 dt , then f ′ (1) = A)0B)13C)23D)√2E)undefined
CHAPTER 4. INTEGRALS 131 10.If b a f ( x ) dx = 3 and b a g ( x ) dx = − 2, then which of the following must be true? I. f ( x ) > g ( x ) for all a ≤ x ≤ b II. b a [ f ( x ) + g ( x )] dx = 1 III. b a [ f ( x ) g ( x )] dx = − 6 11.The graph offis shown below. Approximate 3 − 3 f ( x ) dx using the trapezoid rule with 3 equal subdivisions. 1 2 3 4 f 12.If k 0 sec 2 x 1 + tan x dx = ln 2, then the value of k is
132 The AP CALCULUS PROBLEM BOOK 13.The graph of the functionfon the interval [−4,4] is shown below. 4 − 4 | f ( x ) | dx = A)1B)2C)5D)8E)9 − 4 − 3 − 2 − 1 1 2 3 4 − 3 − 2 − 1 1 2 3 14.The acceleration of a particle moving along thex-axis at timet >0 is given bya(t) =1t2.Whent= 1 second, the particle is atx= 2 and has velocity−1 unit per second. Ifx(t) is theparticle’s position, then the position whent=eseconds is 15.The area enclosed by the two curvesy=x2−4 andy=x−4 is given by 1 0 ( x − x 2 ) dx 1 0 ( x 2 − x ) dx 2 0 ( x − x 2 ) dx 2 0 ( x 2 − x ) dx 4 0 ( x 2 − x ) dx
CHAPTER 4. INTEGRALS 133 A.P. Calculus Test Four Section Two Free-Response Calculators Allowed Time—45 minutes Number of Questions—3 Each of the three questions is worth 9 points. The maximum possible points earned on this section is 27, which represents 50% of the total test score. There is no penalty for guessing. • SHOW ALL YOUR WORK. You will be graded on the methods you use as well as the accuracy of your answers. Correct answers without supporting work may not receive full credit. • Write all work for each problem in the space provided. Be sure to write clearly and legibly. Erased or crossed out work will not be graded. • Justifications require that you give mathematical (non-calculator) reasons and that you clearly identify functions, graphs, tables, or other objects that you use. • You are permitted to use your calculator to solve an equation or graph a func- tion without showing work. However, you must clearly indicate the setup of your problem. • Your work must be expressed in mathematical notation rather than calculator syn- tax. For example, 5 1 x 2 dx may not be written as fnInt(X^2,X,1,5) . • Unless otherwise specified, answers (numeric or algebraic) need not be simplified. If your answer is given as a decimal approximation, it should be correct to three places after the decimal point. Good Luck! NAME:
134 The AP CALCULUS PROBLEM BOOK 1. The temperature on New Year’s Day in Bu ff alo, New York, is given by T ( h ) = − A − B cos π h 12 , where T is the temperature in degrees Fahrenheit and h is the number of hours from midnight (0 ≤ h ≤ 24).
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