CalculusBC Ipart~ CALCULUSBC SECTION I, Part A Time-55 minutes Number of questions-28 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAM. Directions: Solve each of the following problems, using the available space for scratch work. After examining the form of the choices, decide which is the best ofthe choices given and fill in the corresponding oval on the answer sheet. No credit will be given for anything written in the exam book. Do not spend too much time on anyone problem. In this exam: (1) Unless otherwisespecified, the domainofa function I is assumedto bethe setof all real numbers x for which I(x) is a real number. (2) The inverse of a trigonometric function I may be indicated using the inverse function notation I-lor with the. prefix "are" (e.g., sin-1 x = arcsin x ). UllIIuthorized copying or "'UN of any part of thle page Ie Illegal. GO ON TO THE NEXT PAGE. 57
Calculus E 1. At time t ~ 0, a particle moving in the xy-plane has velocity vector given by vct} =(t 2, 5t), What is the acceleration vector of the particle at time t = 3? (B) (6,5) (C) (2,0) (D)../306 (E) J6f 2 (B) ex + C 2 (C) xe x + C (E) e 2x + C 3 I, sinxcosx,,1m IS x-+o X (A) -I (B) 0 (C) 1 (D) 1& 4 (E) nonexistent 58 Unauthortzed copying or NU" of any part of this JJ89lIla Illegal. GO ON TO THE NEXT PA
Calculus BC 00 n 4. Consider the series L;. Ifthe ratio test is applied to the series, which of the following inequalities results, n=l n. implying that the series converges? (A) lim-;<1 n~oo n. (B) lim n! < 1 n~oo e (C) rlm--< n + 1 1 n~oo e (D) lim _e_ < 1 n~oon + 1 (E) lim e < 1 n~oo (n + I)! 5. Which ofthe following gives the length of the path described by the parametric equations x =sin(t 3 ) and y =e 5t from t =0 to t = 1r? (A) J: ~sin2 (t 3 ) + e lot dt (B) J: ~cos2 (t 3 ) + e lot dt (C) J: ~9t4 cos 2 (t 3 ) + 25e lot dt (D) J: ~3t2 cos(t 3 ) + 5e 5t dt (E) J: ~cos2 (3t 2 ) + e lot dt Unauthorized copying or reu.. of any part of thi8 pege I. Illegal. GO ON TO THE NEXT PAGE. 59
Calculus E x 2-4 j(x)= x-2 ifx~2 { 1 ifx=2 6. Let f be the function defined above. Which ofthe following statements about f are true? I. f has a limit at x = 2. II. f is continuous at x = 2. ill.f is differentiable at x = 2. (A) I only (B) II only (C) ill only (D) I and II only (E) I, II, and ill 7. Given that y(l) = -3 and : =2x + y, what is the approximation for y(2) ifeuler's method is used with a step size of 0.5, starting at x = 1? (A) -5 (B) -4.25 (C) -4 (D) -3.75 (E) -3.5 60 Unauthorized copying or leu..of any part of thi. page I. Illegal. GO ON TO THE NEXT PAl
Calculus Be Ipalt~ x 2 3 5 8 13 f(x) 6-2 -1 3 9 8. The function f is continuous on the closed interval [2,13] and has values as shown in the table above. Using the intervals [2,3], [3,5], [5,8], and [8,13], what is the approximation of f 13 2 f(x) dx obtained from a left Riemann sum? (A) 6 (B) 14 (C) 28 (D) 32 (E) 50 Graph off 9. The graph of the piecewise linear function f is shown in the figure above. If g(x) = IX f(t) dt, which of the -2 following values is greatest? (A) g(-3) (B) g(-2) (C) g(o) (D) g(l) (E) g(2) Unauthorized copying or reuse of any J)8rt of ttll. page la Illegal. GO ON TO THE NEXT PAGE. 61
~AI Calculus B( 10. In the xy-plane, what is the slope ofthe line tangent to the graph of x 2 + xy + l =7 at the point (2, I)? (A) 4 3 (B) 5 4 (C) -1 (D) 4 5 (E) 3 4 11. Let R be the region between the graph of y = e-2x and the x-axis for x ~ 3. The area of R is (A) 1 2e 6 (E) infinite 62 Unauthortzed copying or reuse of any part of this page Is Illegal. GO ON TO THE NEXT PAGI
Calculus BC Ipart~ 12. Which ofthe following series converges for all real numbers x? 00 n (A) l:x n n=l 00 n (B) l:\ n=l n 00 n (C) l:}n n=l n (E) 00 l:n!:n n=l e 13. I e x2 + 1 dx = 1 X e 2-1 (A) 2 e 2 + 1 (B) 2 e 2 + 2 (C) 2 (E) 2e 2-8e + 6 3e Unauthorized copying or reum of any part of thl. page I. Illegal. GO ON TO THE NEXT PAGE. 63
Calculus BC 14. The polynomial function I has selected values of its second derivative I" given in the table above. Which of the following statements must be true? (A) I is increasing on the interval (0,2). (B) I is decreasing on the interval (0,2). (c) I has a local maximum at x = 1. (D) The graph ofi has a point of inflection at x = 1. (E) The graph ofi changes concavity in the interval (0,2). 15. If I(x) = (Inx)2, then I"(..re) = (A) 1 (B) 2 (c) _I_ e e 2..re (D) _1 (E).2...re..re 64 Unauthorfzed copying or reu..of any part of thl_ page I_ liieg_" GO ON TO THE NEXT PAGI
Calculus BC Ipart~ 16. What are all values of x for which the series f(+)n converges? n=} X + 1 (A) -1 < x < 1 (B) (C) x> 1 only x ~ 1 only (0) x < -1 and x> 1 only (E) x:5-1 and x ~ 1 17. Let h be a differentiable function, and let j be the function defined by j(x) =h(x 2-3). Which ofthe following is equal to 1'(2)? (A) h'(i) (B) 4h'(I) (C) 4h'(2) (0) h'(4) (E) 4h'(4) Unauthorized copylng or Iltu..of any part of thi. page I."'.gal. GO ON TO THE NEXT PAGE. 65
Calculus BC 18. In the xy-plane, the line x + y = k, where k is a constant, is tangent to the graph of y = x 2 + 3x + 1. What is the value of k? (A) -3 (B) -2 (C) -1 (D) 0 (E) 1 19. f (2x _ ; ~x + 2) dx = 3 (A) Ilnl2x - 3/ + 2lnlx + 21 + c (B) 3lnl2x - 31 + 2lnlx + 21 + c (C) 3lnl2x - 31-2lnlx + 21 + c (D) _ 6 _ 2 + C (2x - 3)2 (x + 2)2 (E) 3 (2x - 3)2 2 +c (x + 2)2 66 Unauthorized copying or reu.. of any part of this page Is Illegal. GO ON TO THE NEXT PAGE
Calculus BC 'IPart~ 20. What is the sum of the series 1 + In 2 + (ln2~)2 +... + (Inn~t +...? (A) In 2 (B) In(1 + In 2) (C) 2 (D) e 2 (E) The series diverges. x(t) 2 1 ~~-+--+---#--+----l~-+-t 21. A particle moves along a straight line. The graph ofthe particle's position x(t) at time t is shown above for o < t < 6. The graph has horizontal tangents at t =1 and t =5 and a point of inflection at t =2. For what values of t is the velocity of the particle increasing? (A) 0 < t < 2 (B) 1 < t < 5 (C) 2 < t < 6 (D) 3 < t < 5 only (E) 1 < t < 2 and 5 < t < 6-2 IUnauthortzed copying or reu..ofi any part of thi. pligit I. II... GO ON TO THE NEXT PAGE. 67
~AI Calculus Be x 0 1 f(x) 2 4 j'(x) 6-3 g(x) -4 3 g'(x) 2-1 22. The table above gives values off, 1', g, and g' for selected values of x. If J~j'(x)g(x)dx =5, then J~f(X)gl(X) dx = (A) -14 (B) -13 (C) -2 (D) 7 (E) 15 23. If f(x) =xsin(2x), which of the following is the Taylor series for f about x =O? x 3 x 5 x 7 (A) x - 2! + 4T - 6! +... (B) 4x 3 16x 5 64x 7 x-2f+4!-6!+'" 8x 3 32x 5 128x 7 (C) 2x--+-----+... 3! 5! 7! 2 2x 4 2x 6 2x 8 (D) 2x -31 +51 -"7!+'" 2 8x 4 32x 6 128x 8 (E) 2x -3!+5!-~+... 68 UnauthorlZ8d copying or nlu.. of MY part of 1hI. page I. Illegal. GO ON TO THE NEXT PAG
Calculus BC _ Ipart~ pet) 200 --------------------- 100 _I-----------.t o 24. Which of the following differential equations for a population P could model the logistic growth shown in the figure above? (A) dd'p =O.2P - O.OOlp 2 t. (B) ~ =O.lP - O.OOlp 2 (C) ~ = O.2p 2 - O.oolP (D) ~ = O.lp 2 - O.OOlP (E) ~ = O.lp 2 + O.OOlP ex + d for x ~ 2 f{x) = 2 { X - ex for x> 2 25. Let f be the function defined above, where e and d are constants. Iff is differentiable at x = 2, what is the value of e + d? (A) -4 (B) -2 (C) 0 (D) 2 (E) 4 Unauthorized copying or FeU" of any part of this page Is Illegal. GO ON TO THE NEXT PAGE. 69
~AI Calculus y ----~~--- x 26. Which ofthe following expressions gives the total area enclosed by the polar curve r =sin 2 0 shown in the figure above? (A) 1..fr sin 2 0dO 2 0 (B) f: sin 2 0dO (C) 1..fr sin 4 0dO 2 0 (D) f: sin 4 0 do (E) 2f: sin 4 0 do 70 Unauthorized copying or reuee of any part of till. page I. Illegal. GO ON TO THE NEXT PAC
Calculus BC IPart~ 27. Which of the following could be the slope field for the differential equation : =l - 1? y (A) 2 (B) 2 III/III III/III /1/-... ',...-// / / / / / / / ///1/1/ /1/-... '... '...-/1 /////// /////// /1/-... '...-/1 ------- ------- /1/-... ' '...-/1...... /1/-... ' '...-/1...,...,..., 11/-... ' '...-/1...",... ""'" " 11/-... ' '...-/1 X -2/ 1/ _...,...", """ '2... " "..., 11/-... '...-/1 I -2"""... y...-/1 /2...... /1/-... '...-/1 / ------- ------- /1/-... ' '...-/1 / /////// /////// /1/-... '...-/1 I 1//1/1/ / / / / / / / /1/-... '...-/1 I 1///1/1 ////1// /1/-... '...-/1 I -2-2 y y (C) 2 (D) 2 \,...... // //...-..., \ \... 11 " / / I \,......// //..,,~,\ \ \ \ \,...... 11 / I I \,...... // //...-" \ \... 11 /... I I \,...... // //...... 1 1 \... 11/... I I \,......// //... -..., \ \ \... 111 " I I... \,... -...//... 11/ / I //...-"1 \ \ "... \,,-...// //... -... '1 1 \ \,...... 11/ I / -2 \,,-... / / //... -... '\2-21 \ \ \,.........// //...-"1 \ \ \,.........// //...-,,\ \ \,... -......// //...-"1 1 1 \ "... X \,... -...// //...-..., \, 1 \ "... \,... -...// //...-,,\ I \ \ "... \,... -...// //...-,,\ 1 I \ "... -2-2 y (E) 2 I I I / / I I I I I I I / / III/III / / / / / / / /11/11/ / / / / I / / / / / / / / / III/III /1///// I I I / / / / IIIIIII IIIIIII //,////,,;///// -2...... 2 ""'" \\\\\\\ ""'",\\\\\\ \\\\\\\ \\\\\\\ 1\1\1\\ \ \ \ \ \\ \ \\\\\\\ \\\\\\\ \ 1 \ \ I \ 1 I 1 \ 1 I 1 1-2 x... 11/ I /2... 11 / / I I... 11 / / I I... 11 I I I I... 11 / I I I... 11 / / I I... 11 / / I I X x Unauthorized copying or reu.. of any part of thla page I. Illegal. GO ON TO THE NEXT PAGE. 71
Calculus I 28. In the xy-plane, a particle moves along the parabola y = x 2 - x with a constant speed of 2M units per second. If : > 0, what is the value of : when the particle is at the point (2,2)? (A) 1 3 (B) 2M 3 (C) 3 (D) 6 (E) 6M END OF PART A OF SECTION I 72 Unauthorized copying or reuu of any part of this page Is 1118gal.
Section I Answer Key and Percent Answering Correctly Calculus BC Item Correct Percent Correct by Grade Total No. Answer 5 4 3 2 1 Percent Correct 1 B 97 95 93 89 81 93 2 A 96 92 87 80 67 89 3 C 88 77 67 55 36 73 4 D 95 86 74 61 40 80 5 C 94 85 76 66 40 80 6 A 81 61 52 43 35 63 7 D 90 73 58 43 23 69 8 B 89 71 56 39 26 68 9 D 96 85 74 58 33 79 10 B 90 80 72 59 39 76 11 A 75 48 34 27 21 52 12 D 64 35 25 21 16 42 13 B 75 55 40 29 23 55 14 E 63 39 32 27 22 45 15 A 85 63 45 31 17 61 16 D 62 38 27 21 16 42 17 B 79 60 45 37 28 59 18 A 59 33 23 16 13 38 19 A 61 41 30 23 18 43 20 C 49 23 16 12 9 30 21 A 58 35 23 16 10 38 22 E 31 11 7 6 5 18 23 E 80 57 42 28 15 56 Analyzing Your Students' Performance on the Multiple-Choice Section Ifyou give your students the 2008 Calculus AB or Calculus BC Exam for practice, you may want to analyze the results to find overall strengths and weaknesses in their understanding of calculus. The following diagnostic worksheets will help you do this. You are permitted to photocopy and distribute them to your students for completion. 1. In each category, students should insert a check mark for each correct answer. 2. Add together the total number of correct answers for each category. Item Correct Percent Correct by Grade Total Percent No. Answer 5 4 3 2 1 Correct 24 A 63 37 25 19 13 41 25 B 57 27 19 13 12 35 26 D 56 35 25 19 13 38 27 A 83 66 52 41 28 64 28 D 26 5 3 2 2 13 76 D 93 84 72 56 36 77 77 D 99 98 94 87 69 93 78 C 94 81 66 49 26 74 79 D 70 45 36 27 21 49 80 E 70 56 44 33 21 54 81 B 84 68 56 41 24 65 82 E 93 78 65 51 37 75 83 B 93 82 71 59 39 77 84 A 95 86 73 54 28 78 85 C 89 74 62 49 35 71 86 C 73 42 28 19 13 47 87 D 91 68 48 31 18 65 88 A 85 60 43 29 16 59 89 E 79 55 38 24 14 54 90 D 91 79 62 44 22 71 91 D 85 66 47 29 15 61 92 B 90 73 54 33 18 67 3. To compare the student's number of correct answers for a given category with the average number correct for that category, copy the number of correct answers to the "Number Correct" table at the end of the Diagnostic Guide. In addition, under each item, the percent ofap students who answered correctly is shown, so students can analyze their performance on individual items. This information will be helpful irt deciding how students should plan their study time. Please note that one item may appear in several different categories, as questions can cover different topics. 92