6 EXAM II CALCULUS BC SECTION I PART A Time-55 minutes Number of questions-8 A CALCULATOR MAY NOT BE USE ON THIS PART OF THE EXAMINATION irections: 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 box. o not spend too much time on anyone problem. In this test: (1) Unless otherwise specified, the domain of a function / is assumed to be the set of all real numbers x for which lex) is a real number. () The inverse of a trigonometric function/may be indicated using the inverse function notation /-1 or with the prefix "arc" (e.g., sin -Ix = arcsin x ). 1 x 1. Jxe dx o I (A) t(e-1) (B) (e 1) (C) e () e (E) e (A) 0 (B) 1 (C) () 3 (E) nonexistent Copyright 011 Venture Publishing
No Calculators BC EXAM II Seeton I Part A 7 3. If cos X ey and 0 < x < n, then dy is dx (A) -tanx (B) -cotx (C) tanx () cotx (E) cscx 4. Ify Arc sin( e x ), then dy dx e x (A) ~1- e 4x (B) (C) () (E) x e ~1- e 4x e x ~1 + e 4x x e 1 -e 4x x e ~e4x - 1 x 1 5. If f(x) = f ~ dt, then /,(1) = 3 t + 1 (A) 0 (C) ~ () v (E) undefmed Copyright 011 Venture Publishing
8 BC EXAM II Section I Part A Multiple-Choice 6. Which ofthe following represents the graph ofthe polar curve r 3csc8? (A) 3 1 (B) (C) 3 1 () (E) 1 3-1 - -3 7. If g(x) tan (e x ), then g'(x)= (A) e X tan(e x )sec (e x ) (B) tan(e x )sec (ex) (C) tan (ex)sec(e x ) () ex sec (e x ) (E) e x tan ( ex) 8. f 1 ~x - x + 1 dx = 0 1 1 (A) -1 (B) (C) 0 () (E) 1 Copyright 011 Venture Publishing
No Calculators BC EXAM II Seeton I Part A 9 9. The coefficient of x 3 in the Taylor series for I(x) = ~ 1-1/ dt about x=o is Jo e (A) -1 (B) -i (C) i () t (E) 1 10. F and G are two functions whose derivatives exist for all real x; F'(x) < 0 and G'(x) > 0 for all real x. Which ofthe following could be true about the graphs of y = F(x) and y = G(x)? I. they do not intersect II. they intersect once III. they intersect more than once (A) I only (B) II only (C) III only () I and II only (E) II and III only 11. The length ofthe curve determined by the parametric equations x sin t and y = t from t ::;;: 0 to t = 1t is 3t (A) f ~cos t + 1 dt o 3t. (B) f ~sin t+ 1 dt o 3t (C) f..j cos t +1 dt o 3t () f..jsin t + 1 dt o 3t (E) f..jl- cos t dt o Copyright 011 Venture Publishing
30 BC EXAM II Section I Part A Multiple-Choice 1. lim x-o e 3x _ 1 sin X (A) 0 (B) 1 (C) () 3 (E) nonexistent 13. The slope ofthe line tangent to the graph of In(x + y) = x at the point where x = 1 is (A) 0 (B) 1 (C) e 1 () e - 1 (E) e x 14. At x = 0, which ofthe following is true ofthe function f(x) = sin x + e-? (A) f is increasing (B) f is decreasing (C) f is discontinuous () The graph of f is concave up (E) The graph of f is concave down Copyright 011 Venture Publishing
No Calculators BC EXAM II Seeton I Part A 31 15. The radius ofconvergence ofthe series ~ n + 1. (x - 3)n is ;:1 n + 1 n (A) 4 (B) 3 (C) () 1 (E) 0 16. A particle moves along the curve x y = at time t> O. If dy = 8 when x = -1, what is the dt value of dx at that time? dt (A) - (B) -1 (C) 0 () 1 (E) b b 17. If f f(x) dx = 3 and f g(x) dx,which ofthe following must be true? a I. f(x) > g(x) for all a so x so b b II. f [f(x) - g(x)] dx = 1 a b III. f [f(x)' g(x)] dx = 6 a a (A) I only (8) II only (C) III only () II and III only (E) I, II, III Copyright 011 Venture Publishing
3 BC EXAM II Section I Part A Multiple-Choice 18. Consider the curve in the xy-plane represented by x ~ and y = In t for t> O. The slope of t the line tangent to the curve at the point where x 1 is (A) 1 (B) t (C) 0 () t (E) 1 19 If dy xy, then y could be. dx -1 (A) x + 1 1 (B) -+ 1 x - (C) x + 1 () 3e x / x (E) 3e / + 1 0. f x d x + x (A) x Inlx+1 + C (B) x+ Inlx+1 + C (C) x- lnlx+1 + C () x- Inlx+1 + C (E) x - Arc tan x + C Copyright 011 Venture Publishing
No Calculators BC EXAM II Seeton I Part A 33 1. Let / be a function with /() = 4 and derivative f'(x) = ~x3 + 1. Using a tangent line approximation to the graph of / at x, estimate /(.). (A) 4.0 (B) 4. (C) 4.4 () 4.6 (E) 4.8. A region in the plane is bounded by y Jx, the x-axis, the line x m and the line x m where m > O. A solid is fonned by revolving the region about the x-axis. The volume ofthis solid (A) is independent of m (B) increases as m increases (C) decreases as m increases () increases until m = t, then decreases (E) is none ofthe above Copyright 0 II Venture Publishing
34 BC EXAM II Section I Part A Multiple-Choice 3. Ifa particle moves in the xy-plane so that at time t> 0 its position vector is \ sin(3t - ;), 3t ), then at time t ~ the velocity vector is (A) (-3, 3Jt) (B) (-1,3Jt) (C) (-1,Jt) () (3,Jt) (E) (3,3Jt) 4. Which ofthe following series converge? I~~ II. III..t:r n + 1 (A) I only (B) II only (C) III only () I and II only (E) II and III only x 5. F and G are differentiable functions such that F(x) f G(t) dt. If F(a) 3 and o F(b) = 3, where 0 < a < b, which ofthe following must be true? (A) G(x) =0 for some x such that a < x < b (B) G(x) = 0 for all x such that a < x < b (C) G(x) > ofor all x such that a < x < b () F(x) ~ 0 for all x such thata < x < b (E) F(x) = 0 for some x such that a < x < b Copyright 011 Venture Publishing
No Calculators BC EXAM II Seeton I Part A 35 6. f xe- x dx (A) 1 (B) 1 - J. (C) J. 1 () 1 + J. (E) e e e e 7. Ifthe average rate ofchange ofa function f over the interval from x = to x = + h is given by te h -4 cos(h), then /,() (A) -1 (B) 0 (C) 1 () (E) 3 8. The slope field shown in the figure at the right represents solutions to a certain differential equation. Which ofthe following could be a specific solution to that differential equation? (A) y=e- x (B) y=sm x (C) y=-jx --//// I I I I --//// I I I I --//// I I I I --//// I I I I --//// I I I I --//// I I I I --//// I I I I --//// I I I I --/// / I I I I () y = In x (E) y = eo. 5x Copyright 011 Venture Publishing
36 EXAM II CALCULUS BC SECTION I PART B Time-50 minutes Number of questions-17 A GRAPHING CALCULATOR IS REQUIRE FOR SOME QUESTIONS ON THIS PART OF THE EXAMINATION irections: Solve each of the following problems, using the available space for scratchwork. After examining the form ofthe choices, decide which is the best of the choices given and fill in the box. o not spend too much time on anyone problem. In this test: (1) The ~ numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number that best approximates the exact numerical value. () 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. (3) The inverse ofa trigonometric functionfmay be indicated using the inverse function notation f- I or with the prefix "arc" (e.g., sin -Ix = arcsin x ). 1. Let J(x) = (1 + X)I/3. The Taylor polynomial ofdegree for the functionj about x = 0 is (A) 1+ x-x (B) 1+ t + x; () 1+ t + ~ (E) None ofthe above.. Which of the following are true about the function J if its derivative is defmed by f'(x) = (x-l)(4-x)? I. J is decreasing for all x < 4. II. J has a local maximum atx = 1. III. The graph of J is concave up for all 1 < x < 3 (A) I only (B) II only (C) III only () II and III only (E) I, II and III Copyright 011 Venture Publishing
Calculators EXAM II Section I Part B 37 3. A tangent line drawn to the graph of y 4x 3 at the point (1,) forms a right 1 + x triangle with the coordinate axes. The area ofthe triangle is (A) 3.0 (B) 3.5 (C) 4.0 () 4.5 (E) 5.0 4. The graph ofthe second derivative /" for a function / is shown below. If / is increasing atx -1, which ofthe following statements must be true? I. /,() == /,(4) II. f'e4) > /,(-1) IlL /,(4) > 0 (A) I only (B) II only (C II and III only () I and III only (E) I, II and III 5. Suppose a particle is moving along a coordinate line and its position at time t is given by s(t) = 9t For what value of t in the interval [1,4] is the instantaneous velocity t + equal to the average velocity? (A).199 (B).09 (C).19 ().9 (E).39 Copyright 011 Venture Publishing
38 EXAM II Section I Part B Multiple-Choice ex x 6. If I(x) = - and g(x) --, then /,(x) =g'(x) at x x+l x + 1 (A) 0.563 (B) 0.565 (C) 0.567 () 0.569 (E) 0.571 3 ---- ----~----;----:-,- -1-----'----1. I I I I,, I,. ".,.5 7. A graph ofthe function I is shown at the right.,,, Which ofthe following statements are true? I. 1(1) > /,(3) II. f I(x) dx > /,(3.5) I 'm f( + h) f() > f(.5) - f() III. l1 h-o h.5-1.s I \ I I I I I I I I j I I -1---~-1-----1-----1-----1- I I ---1-----1 I, I, I, I I I I, I I I! I I I I I 3 4 (A) I only (B) II only (C) I and II only () II and III only (E) I, II and III 8. Which ofthe following three improper integrals converge? 00 1 00 I L f dx II. f dx III. f'!" dx o 1 + x I x o x (A) II only (B) I and II only (C) I and III only () II and III only (E) I, II and III I Copyright 011 Venture Publishing
Calculators EXAM II Section I Part B 39 9. The proportion ofstudents that have heard a rumor at time t is modeled by the function P that satisfies the logistics differential equation dj: = 3P(3 - P). What proportion ofthe student population has heard the rumor when it is spreading the fastest? (A) 5% (B) 40% (C) 50% () 60% (E) 75% 10. Ifthe graph of y= f(x) contains the point (0,1), and if dy = x sin(x ), then f(x)= dx y (A).f~ cos(x) (B) -fi cos(x) (C) cos(x) () cos(x) (E) -J - cosx x 11. If g(x) = f (t + 7)/3 dt, then g"(1) = o 6 (B) 4 (C) 13 () 33 (E) 8 Copyright 011 Venture Publishing
40 EXAM II Section I Part B Multiple-Choice 4x 8x 3 16x 4 1. The graph of the function f represented by the Maclaurin series 1+ x + - + - + --+...! 3! 4! intersects the graph of y = x 3 at the point where x = (A) 0.18 (B) 0.337 (C) 0.587 () 0.799 () 0.989 13. The figure at the right shows the graph of J', the derivative ofa function f. The domain of f is the interval -4 s x s 4. Which of the following must be true about the graph of f? I. At the points where x = -3 and x = there are horizontal tangents. II. At the point where x = 1 there is a relative minimum point. III. At the point where x inflection point. -3 there is an graph of the derivative off (A) None (B) II only (C) III only () II and III only (E) I, II and III Copyright 011 Venture Publishing
Calculators EXAM II Section I Part B 41 14. The volume ofthe solid generated by revolving the first quadrant region bounded by the curve y = e x / and the lines x = In 3 and y = 1 about the x-axis is (A).80 (B).83 (C).86 ().89 (E).9 15. The number ofbacteria in a culture is given by N(t) = 001n(t + 36), where t is measured in days. On what day is the change in growth a maximum? (A) 4 (B) 6 (C) 8 () 10 (E) 1 Copyright 0 II Venture Publishing
4 EXAM II Section I Part B Multiple-Choice 16. Ifthe function I is differentiable on the interval [a, b] and following statements are true? a < c < b, which ofthe b c b I. f I(x) dx f I(x) dx + f I(x) dx a a c II. There exists a number d in (a, b) such that I'(d) I(b) b I(a) a III. lim I(x) =I(c) x-c (A) I only (B) II only (C) I and II only () II and III only (~) I, II, III I 7. The conical reservoir shown at the right has diameter 1 feet and height 4 feet. Water is flowing into the reservoir at the constant rate of 10 cubic feet per minute. At the instant when the surface ofthe water is feet above the vertex, the water level is rising at the rate of (A) 0.177 ftlmin (B) 0.354 ftlmin (C) 0.531 ftlmin () 0.708 ftlmin (~) 0.885 ftlmin c-----------~~~--- 6 v 1 4 j Copyright 011 Venture Publishing
44 EXAM II Section II Part A Free Response 1. A particle moves along a line in such a way that at time t, 1 :s; t :s; 8, its position is given by t s(t) = f [1- x(cos x)- (In x)(sin x)] dx. 1 (a) Write a formula for the velocity ofthe particle at time t. (b) At what instant does the particle reach it maximum speed? (c) When is the particle moving to the left? (d) Find the total distance traveled by the particle from t = 1 to t = 8. Copyright 0 II Venture Publishing
Calculators EXAM II Section II Part A 45. The following table represents the diameter (feet) ofthe cross section ofa tree at various heights (feet) above the ground. Assume that each cross section is circular. (a) Approximate how fast the diameter ofthe tree is changing ft. above the ground. Indicate units ofmeasure. (b) Use the trapezoid rule to approximate the volume ofthe tree from 14 ft to 30 ft above the ground. Indicate units ofmeasure. (c) The section ofthe tree from feet to 8 feet is used to make a rectangular beam oflength 6 feet. The strength of the beam varies jointly as its width and the square of its height. What should be the width and height ofthe beam in order to have the strongest beam? Copyright 011 Venture Publishing
46 EXAM II Section II Part B Free Response Time - 60 minutes Number of problems - 4 A graphing calculator may NOT be used on this part of tbe examination. uring the timed portion for part B, you may go back and continue to work on the problems k 3. Let functions f and g be defined by f(x) = x and g(x) = x +-, where k is a x positive constant. (a) Find the average value of g on the interval [1,3] if k=. (b) If R is the region between the graphs of f and g on the interval [1,3], find in terms ofk, the volume ofthe solid generated when R is rotated about the x-axis. (c) Set up but d!l nq.t evaluate an integral expression in terms of k for the volume ofthe solid generated when R is rotated about the horizontal line y = -. Copyright 0 II Venture Publishing
No Calculators EXAM II Section II Part B 47 4. Let f be a differentiable function such that fit is continuous and f and I' have the values given in the table below. x 0 I 3 4 5 f(x) 1 17 3 8 9 11 f'(x) 5 1 19 15 13 - Use the infonnation in the table to (a) approximate fll(x) at x =. (b) e~aluate f xl'(x ) dx o 3 (c) evaluate f xflt(x) dx 1 Copyright 011 Venture Publishing
48 EXAM II Section II Part B Free Response 5. Let I be the function defined by I(x) = xe-lex, where k is a positive constant. (a) Find, in terms of k, the x-coordinate ofeach critical point of I. (b) For each critical number x, determine whether I(x) is a relative maximum, relative minimum, or neither. Justify your answer. (c) On what interval(s) is the graph of I concave up? (d) Write an equation ofthe horizontal asymptote for the graph of I. Copyright 0 II Venture Publishing
No Calculators EXAM II Section II Part B 49 6. Let f be the function defmed by f( x) = In(x + 1). (a) Find fn)(o) for n = 1 to n = 3, where fn)is the nth derivative of f. (b) Write the first three nonzero tenns and the general tenn for the Taylor series expansion of f(x) about x = o. (c) etennine the radius of convergence for the series in part (b). Show your reasoning. 0.5 (d) Use the series in part (b) to evaluate f f(x) dx with an error no greater than 0.01. o Copyright 011 Venture Publishing
No Calculators EXAM II Section II Part B 49 6. Let f be the function defmed by f(x) = In(x + 1). (a) Find fn)(o) for n 1 to n = 3, where fn)is the nth derivative of f. (b) Write the ftrst three nonzero tenns and the general tenn for the Taylor series expansion of f(x) about x = O. (c) etennine the radius of convergence for the series in part (b). Show your reasoning. 0.5 (d) Use the series in part (b) to evaluate f f(x) dx with an error no greater than 0.01. o Copyright 011 Venture Publishing