ap-calculus-ab 1998 Q2

ap-calculus-ab · USA · full-exam Not Maths
2. Do you regularly speak or hear the language at home? ◯ Yes \end{tabular} \hline \end{tabular}
\multicolumn{17}{|l|}{INDICATE YOUR ANSWERS TO THE EXAM QUESTIONS IN THIS SECTION. IF A QUESTION HAS ONLY FOUR ANSWER OPTIONS, DO NOT MARK OPTION (E). YOUR ANSWER SHEET WILL BE SCORED BY MACHINE. USE ONLY NO. 2 PENCILS TO MARK YOUR ANSWERS ON PAGES 2 AND 3 (ONE RESPONSE PER QUESTION). AFTER YOU HAVE DETERMINED YOUR RESPONSE, BE SURE TO COMPLETELY FILL IN THE OVAL CORRESPONDING TO THE NUMBER OF THE QUESTION YOU ARE ANSWERING. STRAY MARKS AND SMUDGES COULD BE READ AS ANSWERS, SO ERASE CAREFULLY AND COMPLETELY. ANY IMPROPER GRIDDING MAY AFFECT YOUR GRADE.}
1(A)(B)(C)(D)(E)26(A)(B)(c)(D)(E)51(A)(B)(C)(D)
2(A)(B)(C)(D)(E)27(A)(B)(c)(D)(E)52(A)(B)(C)(D)
3(A)(B)(C)(D)(E)28(A)(B)(C)(D)(E)53(A)(B)(C)(D)
4(A)(B)(C)(D)(E)29(A)(B)(C)(D)(E)54(A)(B)(C)(D)
5(A)(B)(C)(D)(E)30(A)(B)(C)(D)(E)55(A)(B)(C)(D)
6(A)(B)(C)(D)(E)31(A)(B)(C)(D)(E)56(A)(B)(C)(D)
7(A)(B)(C)(D)(E)32(A)(B)(C)(D)(B)57(A)(B)(C)(D)
8(A)(B)(C)(D)(E)33(A)(B)(C)(D)(E)58(A)(B)(C)(D)
9(A)(B)(C)(D)(E)34(A)(B)(C)(D)(E)59(A)(B)(C)(D)
10(A)(B)(C)(D)(E)35(A)(B)(C)(D)(E)60(A)(B)(C)(D)
11(A)(B)(c)(D)(E)36(A)(B)(C)(D)(E)61(A)(B)(C)(D)
12(A)(B)(C)(D)(E)37(A)(B)(C)(D)(E)62(A)(B)(C)(D)
13(A)(B)(c)(D)(E)38(A)(B)(C)(D)(E)63(A)(B)(C)(D)
14(A)(B)(C)(D)(E)39(A)(B)(C)(D)(E)64(A)(B)(C)(D)
15(A)(B)(C)(D)(E)40(A)(B)(C)(D)(E)65(A)(B)(C)(D)
16(A)(B)(C)(D)(E)41(A)(B)(C)(D)(E)66(A)(B)(C)(D)
17(A)(B)(C)(D)(E)42(A)(B)(C)(D)(E)67(A)(B)(C)(D)
18(A)(B)(C)(D)(E)43(A)(B)(C)(D)(E)68(A)(B)(C)(D)
19(A)(B)(C)(D)(E)44(A)(B)(C)(D)(E)69(A)(B)(C)(D)
20(A)(B)(C)(D)(E)45(A)(B)(C)(D)(E)70(A)(B)(C)(D)
21(A)(B)(C)(D)(E)46(A)(B)(C)(D)(E)71(A)(B)(C)(D)
22(A)(B)(C)(D)(E)47(A)(B)(C)(D)(E)72(A)(B)(C)(D)
23(A)(B)(C)(D)(E)48(A)(B)(C)(D)(E)73(A)(B)(C)(D)
24(A)(B)(C)(D)(E)49(A)(B)(C)(D)(E)74(A)(B)(C)(D)
25(A)(B)(C)(D)(E)50(A)(B)(C)(D)(E)75(A)(B)(c)(D)

FOR QUESTIONS 76-151, SEE PAGE 3.
PAGE 3

\multicolumn{17}{|r|}{BE SURE EACH MARK IS DARK AND COMPLETELY FILLS THE OVAL. IF A QUESTION HAS ONLY FOUR ANSWER OPTIONS, DO NOT MARK OPTION E.}
76(B)(C)(D)(E)101(A)(B)(C)(D)(E)126(A)(B)(C)(D)(E)
77(B)(C)(D)(E)102(A)(B)(C)(D)(E)127(A)(B)(C)(D)(E)
78(B)(C)(D)(E)103(A)(B)(C)(D)(E)128(A)(B)(C)(D)(E)
79(B)(C)(D)(E)104(A)(B)(C)(D)(E)129(A)(B)(C)(D)(E)
80(B)(C)(D)(E)105(A)(B)(C)(D)(E)130(A)(B)(C)(D)(E)
81(B)(C)(D)(E)106(A)(B)(C)(D)(E)131(A)(B)(C)(D)(E)
82(B)(C)(D)(E)107(A)(B)(C)(D)(E)132(A)(B)(C)(D)(E)
83(B)(C)(D)(E)108(A)(B)(C)(D)(E)133(A)(B)(C)(D)(E)
84(B)(C)(D)(E)109(A)(B)(C)(D)(E)134(A)(B)(C)(D)(B)
85(B)(C)(D)(C)110(A)(B)(C)(D)(E)135(A)(B)(C)(D)(E)
86(B)(C)(D)(E)111(A)(B)(C)(D)(E)136(A)(B)(C)(D)(E)
87(B)(C)(D)(E)112(A)(B)(C)(D)(E)137(A)(B)(C)(D)(E)
88(B)(C)(D)(E)113(A)(B)(C)(D)(E)138(A)(B)(C)(D)(C)
89(B)(C)(D)(E)114(A)(B)(C)(D)(E)139(A)(B)(C)(D)(E)
90(B)(C)(D)(E)115(A)(B)(C)(D)(E)140(A)(B)(C)(D)(E)
91(B)(C)(D)(E)116(A)(B)(C)(D)(E)141(A)(B)(C)(D)(C)
92(B)(C)(D)(E)117(A)(B)(C)(D)(E)142(A)(B)(C)(D)(E)
93(B)(C)(D)(E)118(A)(B)(C)(D)(E)143(A)(B)(C)(D)(E)
94(B)(C)(D)(E)119(A)(B)(C)(D)(E)144(A)(B)(C)(D)(E)
95(B)(C)(D)(E)120(A)(B)(C)(D)(E)145(A)(B)(C)(D)(E)
96(B)(C)(D)(E)121(A)(B)(C)(D)(E)146(A)(B)(C)(D)(E)
97(B)(C)(D)(E)12.2(A)(B)(C)(D)(E)147(A)(B)(C)(D)(E)
98(B)(C)(D)(E)123(A)(B)(C)(D)(E)148(A)(B)(c)(D)(E)
99(B)(C)(D)(E)124(A)(B)(C)(D)(E)149(A)(B)(C)(D)(E)
100(B)(C)(D)(E)125(A)(B)(C)(D)(E)150(A)(B)(C)(D)(E)
151(A)(B)(C)(D)(E)

\multicolumn{4}{|c|}{ETS USE ONLY}
RWFS
PT1
PT2
PT3
PT4
TOT
EQ
TA1
TA2

DO NOT WRITE IN THIS AREA.
CALCULUS AB
A CALCULATOR CANNOT BE USED ON PART A OF SECTION I. A GRAPHING CALCULATOR FROM THE APPROVED LIST IS REQUIRED ON PART B OF SECTION I AND FOR SECTION II OF THE EXAMINATION. CALCULATOR MEMORIES NEED NOT BE CLEARED. COMPUTERS, NONGRAPHING SCIENTIFIC CALCULATORS, CALCULATORS WITH QWERTY KEYBOARDS, AND ELECTRONIC WRITING PADS ARE NOT ALLOWED. CALCULATORS MAY NOT BE SHARED AND COMMUNICATION BETWEEN CALCULATORS IS PROHIBITED DURING THE EXAMINATION. ATTEMPTS TO REMOVE TEST MATERIALS FROM THE ROOM BY ANY METHOD WILL RESULT IN THE INVALIDATION OF TEST SCORES.
SECTION I
Time - 1 hour and 45 minutes All questions are given equal weight. Percent of total grade - 50 Part A: 55 minutes, 28 multiple-choice questions
A calculator is NOT allowed. Part B: 50 minutes, 17 multiple-choice questions
A graphing calculator is required. Parts A and B of Section I are printed in this examination booklet; Section II, which consists of longer problems, is printed in a separate booklet.
General Instructions
DO NOT OPEN THIS BOOKLET UNTIL YOU ARE INSTRUCTED TO DO SO. INDICATE YOUR ANSWERS TO QUESTIONS IN PART A ON PAGE 2 OF THE SEPARATE ANSWER SHEET. THE ANSWERS TO QUESTIONS IN PART B SHOULD BE INDICATED ON PAGE 3 OF THE ANSWER SHEET. No credit will be given for anything written in this examination booklet, but you may use the booklet for notes or scratchwork. After you have decided which of the suggested answers is best, COMPLETELY fill in the corresponding oval on the answer sheet. Give only one answer to each question. If you change an answer, be sure that the previous mark is erased completely.
Example: What is the arithmetic mean of the numbers 1,3 , and 6 ? Sample Answer
(A) 1
(B) $\frac { 7 } { 3 }$
(C) 3
(D) $\frac { 10 } { 3 }$
(E) $\frac { 7 } { 2 }$
Many candidates wonder whether or not to guess the answers to questions about which they are not certain. In this section of the examination, as a correction for haphazard guessing, one-fourth of the number of questions you answer incorrectly will be subtracted from the number of questions you answer correctly. It is improbable, therefore, that mere guessing will improve your score significantly; it may even lower your score, and it does take time. If, however, you are not sure of the best answer but have some knowledge of the question and are able to eliminate one or more of the answer choices as wrong, your chance of answering correctly is improved, and it may be to your advantage to answer such a question. Use your time effectively, working as rapidly as you can without losing accuracy. Do not spend too much time on questions that are too difficult. Go on to other questions and come back to the difficult ones later if you have time. It is not expected that everyone will be able to answer all the multiple-choice questions.
A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION.
Directions: Solve each of the following problems, using the available space for scratchwork. After examining the form of the choices, decide which is the best of the choices given and fill in the corresponding oval on the answer sheet. No credit will be given for anything written in the test book. Do not spend too much time on any one problem.
In this test: Unless otherwise specified, the domain of a function $f$ is assumed to be the set of all real numbers $x$ for which $f ( x )$ is a real number.
  1. What is the $x$-coordinate of the point of inflection on the graph of $y = \frac { 1 } { 3 } x ^ { 3 } + 5 x ^ { 2 } + 24$ ?
    (A) 5
    (B) 0
    (C) $- \frac { 10 } { 3 }$
    (D) - 5
    (E) - 10 [Figure]
  2. The graph of a piecewise-linear function $f$, for $- 1 \leq x \leq 4$, is shown above. What is the value of $\int _ { - 1 } ^ { 4 } f ( x ) d x ?$
    (A) 1
    (B) 2.5
    (C) 4
    (D) 5.5
    (E) 8
  3. $\int _ { 1 } ^ { 2 } \frac { 1 } { x ^ { 2 } } d x =$
    (A) $- \frac { 1 } { 2 }$
    (B) $\frac { 7 } { 24 }$
    (C) $\frac { 1 } { 2 }$
    (D) 1
    (E) $2 \ln 2$
  4. If $f$ is continuous for $a \leq x \leq b$ and differentiable for $a < x < b$, which of the following could be false?
    (A) $f ^ { \prime } ( c ) = \frac { f ( b ) - f ( a ) } { b - a }$ for some $c$ such that $a < c < b$.
    (B) $f ^ { \prime } ( c ) = 0$ for some $c$ such that $a < c < b$.
    (C) $f$ has a minimum value on $a \leq x \leq b$.
    (D) $f$ has a maximum value on $a \leq x \leq b$.
    (E) $\int _ { a } ^ { b } f ( x ) d x$ exists.
  5. $\int _ { 0 } ^ { x } \sin t d t =$
    (A) $\sin x$
    (B) $- \cos x$
    (C) $\cos x$
    (D) $\cos x - 1$
    (E) $1 - \cos x$
  6. If $x ^ { 2 } + x y = 10$, then when $x = 2 , \frac { d y } { d x } =$
    (A) $- \frac { 7 } { 2 }$
    (B) - 2
    (C) $\frac { 2 } { 7 }$
    (D) $\frac { 3 } { 2 }$
    (E) $\frac { 7 } { 2 }$
  7. $\int _ { 1 } ^ { e } \left( \frac { x ^ { 2 } - 1 } { x } \right) d x =$
    (A) $e - \frac { 1 } { e }$
    (B) $e ^ { 2 } - e$
    (C) $\frac { e ^ { 2 } } { 2 } - e + \frac { 1 } { 2 }$
    (D) $e ^ { 2 } - 2$
    (E) $\frac { e ^ { 2 } } { 2 } - \frac { 3 } { 2 }$
  8. Let $f$ and $g$ be differentiable functions with the following properties:
    (i) $g ( x ) > 0$ for all $x$
    (ii) $f ( 0 ) = 1$

If $h ( x ) = f ( x ) g ( x )$ and $h ^ { \prime } ( x ) = f ( x ) g ^ { \prime } ( x )$, then $f ( x ) =$
(A) $f ^ { \prime } ( x )$
(B) $g ( x )$
(C) $e ^ { x }$
(D) 0
(E) 1 [Figure]
$\begin{cases} 1 : & \text { equation in } h \\ 1 : & \text { answer } \end{cases}$
2. Do you regularly speak or hear the language at home? ◯ Yes \\
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\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|}
\hline
\multicolumn{17}{|l|}{INDICATE YOUR ANSWERS TO THE EXAM QUESTIONS IN THIS SECTION. IF A QUESTION HAS ONLY FOUR ANSWER OPTIONS, DO NOT MARK OPTION (E). YOUR ANSWER SHEET WILL BE SCORED BY MACHINE. USE ONLY NO. 2 PENCILS TO MARK YOUR ANSWERS ON PAGES 2 AND 3 (ONE RESPONSE PER QUESTION). AFTER YOU HAVE DETERMINED YOUR RESPONSE, BE SURE TO COMPLETELY FILL IN THE OVAL CORRESPONDING TO THE NUMBER OF THE QUESTION YOU ARE ANSWERING. STRAY MARKS AND SMUDGES COULD BE READ AS ANSWERS, SO ERASE CAREFULLY AND COMPLETELY. ANY IMPROPER GRIDDING MAY AFFECT YOUR GRADE.} \\
\hline
1 & (A) & (B) & (C) & (D) & (E) & 26 & (A) & (B) & (c) & (D) & (E) & 51 & (A) & (B) & (C) & (D) \\
\hline
2 & (A) & (B) & (C) & (D) & (E) & 27 & (A) & (B) & (c) & (D) & (E) & 52 & (A) & (B) & (C) & (D) \\
\hline
3 & (A) & (B) & (C) & (D) & (E) & 28 & (A) & (B) & (C) & (D) & (E) & 53 & (A) & (B) & (C) & (D) \\
\hline
4 & (A) & (B) & (C) & (D) & (E) & 29 & (A) & (B) & (C) & (D) & (E) & 54 & (A) & (B) & (C) & (D) \\
\hline
5 & (A) & (B) & (C) & (D) & (E) & 30 & (A) & (B) & (C) & (D) & (E) & 55 & (A) & (B) & (C) & (D) \\
\hline
6 & (A) & (B) & (C) & (D) & (E) & 31 & (A) & (B) & (C) & (D) & (E) & 56 & (A) & (B) & (C) & (D) \\
\hline
7 & (A) & (B) & (C) & (D) & (E) & 32 & (A) & (B) & (C) & (D) & (B) & 57 & (A) & (B) & (C) & (D) \\
\hline
8 & (A) & (B) & (C) & (D) & (E) & 33 & (A) & (B) & (C) & (D) & (E) & 58 & (A) & (B) & (C) & (D) \\
\hline
9 & (A) & (B) & (C) & (D) & (E) & 34 & (A) & (B) & (C) & (D) & (E) & 59 & (A) & (B) & (C) & (D) \\
\hline
10 & (A) & (B) & (C) & (D) & (E) & 35 & (A) & (B) & (C) & (D) & (E) & 60 & (A) & (B) & (C) & (D) \\
\hline
11 & (A) & (B) & (c) & (D) & (E) & 36 & (A) & (B) & (C) & (D) & (E) & 61 & (A) & (B) & (C) & (D) \\
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12 & (A) & (B) & (C) & (D) & (E) & 37 & (A) & (B) & (C) & (D) & (E) & 62 & (A) & (B) & (C) & (D) \\
\hline
13 & (A) & (B) & (c) & (D) & (E) & 38 & (A) & (B) & (C) & (D) & (E) & 63 & (A) & (B) & (C) & (D) \\
\hline
14 & (A) & (B) & (C) & (D) & (E) & 39 & (A) & (B) & (C) & (D) & (E) & 64 & (A) & (B) & (C) & (D) \\
\hline
15 & (A) & (B) & (C) & (D) & (E) & 40 & (A) & (B) & (C) & (D) & (E) & 65 & (A) & (B) & (C) & (D) \\
\hline
16 & (A) & (B) & (C) & (D) & (E) & 41 & (A) & (B) & (C) & (D) & (E) & 66 & (A) & (B) & (C) & (D) \\
\hline
17 & (A) & (B) & (C) & (D) & (E) & 42 & (A) & (B) & (C) & (D) & (E) & 67 & (A) & (B) & (C) & (D) \\
\hline
18 & (A) & (B) & (C) & (D) & (E) & 43 & (A) & (B) & (C) & (D) & (E) & 68 & (A) & (B) & (C) & (D) \\
\hline
19 & (A) & (B) & (C) & (D) & (E) & 44 & (A) & (B) & (C) & (D) & (E) & 69 & (A) & (B) & (C) & (D) \\
\hline
20 & (A) & (B) & (C) & (D) & (E) & 45 & (A) & (B) & (C) & (D) & (E) & 70 & (A) & (B) & (C) & (D) \\
\hline
21 & (A) & (B) & (C) & (D) & (E) & 46 & (A) & (B) & (C) & (D) & (E) & 71 & (A) & (B) & (C) & (D) \\
\hline
22 & (A) & (B) & (C) & (D) & (E) & 47 & (A) & (B) & (C) & (D) & (E) & 72 & (A) & (B) & (C) & (D) \\
\hline
23 & (A) & (B) & (C) & (D) & (E) & 48 & (A) & (B) & (C) & (D) & (E) & 73 & (A) & (B) & (C) & (D) \\
\hline
24 & (A) & (B) & (C) & (D) & (E) & 49 & (A) & (B) & (C) & (D) & (E) & 74 & (A) & (B) & (C) & (D) \\
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25 & (A) & (B) & (C) & (D) & (E) & 50 & (A) & (B) & (C) & (D) & (E) & 75 & (A) & (B) & (c) & (D) \\
\hline
\end{tabular}
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FOR QUESTIONS 76-151, SEE PAGE 3.

\section*{PAGE 3}
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|}
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\multicolumn{17}{|r|}{BE SURE EACH MARK IS DARK AND COMPLETELY FILLS THE OVAL. IF A QUESTION HAS ONLY FOUR ANSWER OPTIONS, DO NOT MARK OPTION E.} \\
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76 & (B) & (C) & (D) & (E) & 101 & (A) & (B) & (C) & (D) & (E) & 126 & (A) & (B) & (C) & (D) & (E) \\
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77 & (B) & (C) & (D) & (E) & 102 & (A) & (B) & (C) & (D) & (E) & 127 & (A) & (B) & (C) & (D) & (E) \\
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78 & (B) & (C) & (D) & (E) & 103 & (A) & (B) & (C) & (D) & (E) & 128 & (A) & (B) & (C) & (D) & (E) \\
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79 & (B) & (C) & (D) & (E) & 104 & (A) & (B) & (C) & (D) & (E) & 129 & (A) & (B) & (C) & (D) & (E) \\
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80 & (B) & (C) & (D) & (E) & 105 & (A) & (B) & (C) & (D) & (E) & 130 & (A) & (B) & (C) & (D) & (E) \\
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81 & (B) & (C) & (D) & (E) & 106 & (A) & (B) & (C) & (D) & (E) & 131 & (A) & (B) & (C) & (D) & (E) \\
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82 & (B) & (C) & (D) & (E) & 107 & (A) & (B) & (C) & (D) & (E) & 132 & (A) & (B) & (C) & (D) & (E) \\
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83 & (B) & (C) & (D) & (E) & 108 & (A) & (B) & (C) & (D) & (E) & 133 & (A) & (B) & (C) & (D) & (E) \\
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84 & (B) & (C) & (D) & (E) & 109 & (A) & (B) & (C) & (D) & (E) & 134 & (A) & (B) & (C) & (D) & (B) \\
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85 & (B) & (C) & (D) & (C) & 110 & (A) & (B) & (C) & (D) & (E) & 135 & (A) & (B) & (C) & (D) & (E) \\
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86 & (B) & (C) & (D) & (E) & 111 & (A) & (B) & (C) & (D) & (E) & 136 & (A) & (B) & (C) & (D) & (E) \\
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87 & (B) & (C) & (D) & (E) & 112 & (A) & (B) & (C) & (D) & (E) & 137 & (A) & (B) & (C) & (D) & (E) \\
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88 & (B) & (C) & (D) & (E) & 113 & (A) & (B) & (C) & (D) & (E) & 138 & (A) & (B) & (C) & (D) & (C) \\
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89 & (B) & (C) & (D) & (E) & 114 & (A) & (B) & (C) & (D) & (E) & 139 & (A) & (B) & (C) & (D) & (E) \\
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90 & (B) & (C) & (D) & (E) & 115 & (A) & (B) & (C) & (D) & (E) & 140 & (A) & (B) & (C) & (D) & (E) \\
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91 & (B) & (C) & (D) & (E) & 116 & (A) & (B) & (C) & (D) & (E) & 141 & (A) & (B) & (C) & (D) & (C) \\
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92 & (B) & (C) & (D) & (E) & 117 & (A) & (B) & (C) & (D) & (E) & 142 & (A) & (B) & (C) & (D) & (E) \\
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93 & (B) & (C) & (D) & (E) & 118 & (A) & (B) & (C) & (D) & (E) & 143 & (A) & (B) & (C) & (D) & (E) \\
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94 & (B) & (C) & (D) & (E) & 119 & (A) & (B) & (C) & (D) & (E) & 144 & (A) & (B) & (C) & (D) & (E) \\
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95 & (B) & (C) & (D) & (E) & 120 & (A) & (B) & (C) & (D) & (E) & 145 & (A) & (B) & (C) & (D) & (E) \\
\hline
96 & (B) & (C) & (D) & (E) & 121 & (A) & (B) & (C) & (D) & (E) & 146 & (A) & (B) & (C) & (D) & (E) \\
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97 & (B) & (C) & (D) & (E) & 12.2 & (A) & (B) & (C) & (D) & (E) & 147 & (A) & (B) & (C) & (D) & (E) \\
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98 & (B) & (C) & (D) & (E) & 123 & (A) & (B) & (C) & (D) & (E) & 148 & (A) & (B) & (c) & (D) & (E) \\
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99 & (B) & (C) & (D) & (E) & 124 & (A) & (B) & (C) & (D) & (E) & 149 & (A) & (B) & (C) & (D) & (E) \\
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100 & (B) & (C) & (D) & (E) & 125 & (A) & (B) & (C) & (D) & (E) & 150 & (A) & (B) & (C) & (D) & (E) \\
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 &  &  &  &  &  &  &  &  &  &  & 151 & (A) & (B) & (C) & (D) & (E) \\
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\begin{tabular}{|l|l|l|l|}
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\multicolumn{4}{|c|}{ETS USE ONLY} \\
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 & R & W & FS \\
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PT1 &  &  &  \\
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PT2 &  &  &  \\
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PT3 &  &  &  \\
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PT4 &  &  &  \\
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TOT &  &  &  \\
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EQ &  &  &  \\
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TA1 &  &  &  \\
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TA2 &  &  &  \\
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\end{center}

\section*{DO NOT WRITE IN THIS AREA.}
\section*{CALCULUS AB}
A CALCULATOR CANNOT BE USED ON PART A OF SECTION I. A GRAPHING CALCULATOR FROM THE APPROVED LIST IS REQUIRED ON PART B OF SECTION I AND FOR SECTION II OF THE EXAMINATION. CALCULATOR MEMORIES NEED NOT BE CLEARED. COMPUTERS, NONGRAPHING SCIENTIFIC CALCULATORS, CALCULATORS WITH QWERTY KEYBOARDS, AND ELECTRONIC WRITING PADS ARE NOT ALLOWED. CALCULATORS MAY NOT BE SHARED AND COMMUNICATION BETWEEN CALCULATORS IS PROHIBITED DURING THE EXAMINATION. ATTEMPTS TO REMOVE TEST MATERIALS FROM THE ROOM BY ANY METHOD WILL RESULT IN THE INVALIDATION OF TEST SCORES.

\section*{SECTION I}
Time - 1 hour and 45 minutes\\
All questions are given equal weight.\\
Percent of total grade - 50\\
Part A: 55 minutes, 28 multiple-choice questions\\
A calculator is NOT allowed.\\
Part B: 50 minutes, 17 multiple-choice questions\\
A graphing calculator is required.\\
Parts A and B of Section I are printed in this examination booklet; Section II, which consists of longer problems, is printed in a separate booklet.

\section*{General Instructions}
DO NOT OPEN THIS BOOKLET UNTIL YOU ARE INSTRUCTED TO DO SO.\\
INDICATE YOUR ANSWERS TO QUESTIONS IN PART A ON PAGE 2 OF THE SEPARATE ANSWER SHEET. THE ANSWERS TO QUESTIONS IN PART B SHOULD BE INDICATED ON PAGE 3 OF THE ANSWER SHEET. No credit will be given for anything written in this examination booklet, but you may use the booklet for notes or scratchwork. After you have decided which of the suggested answers is best, COMPLETELY fill in the corresponding oval on the answer sheet. Give only one answer to each question. If you change an answer, be sure that the previous mark is erased completely.

Example:\\
What is the arithmetic mean of the numbers 1,3 , and 6 ?\\
Sample Answer\\
(A) 1\\
(B) $\frac { 7 } { 3 }$\\
(C) 3\\
(D) $\frac { 10 } { 3 }$\\
(E) $\frac { 7 } { 2 }$

Many candidates wonder whether or not to guess the answers to questions about which they are not certain. In this section of the examination, as a correction for haphazard guessing, one-fourth of the number of questions you answer incorrectly will be subtracted from the number of questions you answer correctly. It is improbable, therefore, that mere guessing will improve your score significantly; it may even lower your score, and it does take time. If, however, you are not sure of the best answer but have some knowledge of the question and are able to eliminate one or more of the answer choices as wrong, your chance of answering correctly is improved, and it may be to your advantage to answer such a question.\\
Use your time effectively, working as rapidly as you can without losing accuracy. Do not spend too much time on questions that are too difficult. Go on to other questions and come back to the difficult ones later if you have time. It is not expected that everyone will be able to answer all the multiple-choice questions.

\section*{A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION.}
Directions: Solve each of the following problems, using the available space for scratchwork. After examining the form of the choices, decide which is the best of the choices given and fill in the corresponding oval on the answer sheet. No credit will be given for anything written in the test book. Do not spend too much time on any one problem.

In this test: Unless otherwise specified, the domain of a function $f$ is assumed to be the set of all real numbers $x$ for which $f ( x )$ is a real number.

\begin{enumerate}
  \item What is the $x$-coordinate of the point of inflection on the graph of $y = \frac { 1 } { 3 } x ^ { 3 } + 5 x ^ { 2 } + 24$ ?\\
(A) 5\\
(B) 0\\
(C) $- \frac { 10 } { 3 }$\\
(D) - 5\\
(E) - 10\\
\includegraphics[max width=\textwidth, alt={}, center]{680bf20b-97e4-45b6-94c8-f4909ce1c9b0-09_510_620_206_764}
  \item The graph of a piecewise-linear function $f$, for $- 1 \leq x \leq 4$, is shown above. What is the value of $\int _ { - 1 } ^ { 4 } f ( x ) d x ?$\\
(A) 1\\
(B) 2.5\\
(C) 4\\
(D) 5.5\\
(E) 8
  \item $\int _ { 1 } ^ { 2 } \frac { 1 } { x ^ { 2 } } d x =$\\
(A) $- \frac { 1 } { 2 }$\\
(B) $\frac { 7 } { 24 }$\\
(C) $\frac { 1 } { 2 }$\\
(D) 1\\
(E) $2 \ln 2$
  \item If $f$ is continuous for $a \leq x \leq b$ and differentiable for $a < x < b$, which of the following could be false?\\
(A) $f ^ { \prime } ( c ) = \frac { f ( b ) - f ( a ) } { b - a }$ for some $c$ such that $a < c < b$.\\
(B) $f ^ { \prime } ( c ) = 0$ for some $c$ such that $a < c < b$.\\
(C) $f$ has a minimum value on $a \leq x \leq b$.\\
(D) $f$ has a maximum value on $a \leq x \leq b$.\\
(E) $\int _ { a } ^ { b } f ( x ) d x$ exists.
  \item $\int _ { 0 } ^ { x } \sin t d t =$\\
(A) $\sin x$\\
(B) $- \cos x$\\
(C) $\cos x$\\
(D) $\cos x - 1$\\
(E) $1 - \cos x$
  \item If $x ^ { 2 } + x y = 10$, then when $x = 2 , \frac { d y } { d x } =$\\
(A) $- \frac { 7 } { 2 }$\\
(B) - 2\\
(C) $\frac { 2 } { 7 }$\\
(D) $\frac { 3 } { 2 }$\\
(E) $\frac { 7 } { 2 }$
  \item $\int _ { 1 } ^ { e } \left( \frac { x ^ { 2 } - 1 } { x } \right) d x =$\\
(A) $e - \frac { 1 } { e }$\\
(B) $e ^ { 2 } - e$\\
(C) $\frac { e ^ { 2 } } { 2 } - e + \frac { 1 } { 2 }$\\
(D) $e ^ { 2 } - 2$\\
(E) $\frac { e ^ { 2 } } { 2 } - \frac { 3 } { 2 }$
  \item Let $f$ and $g$ be differentiable functions with the following properties:\\
(i) $g ( x ) > 0$ for all $x$\\
(ii) $f ( 0 ) = 1$
\end{enumerate}

If $h ( x ) = f ( x ) g ( x )$ and $h ^ { \prime } ( x ) = f ( x ) g ^ { \prime } ( x )$, then $f ( x ) =$\\
(A) $f ^ { \prime } ( x )$\\
(B) $g ( x )$\\
(C) $e ^ { x }$\\
(D) 0\\
(E) 1\\
\includegraphics[max width=\textwidth, alt={}, center]{680bf20b-97e4-45b6-94c8-f4909ce1c9b0-12_607_878_962_676}\\