Name______________________Period ____ AP Calculus BC 2003 Calculus BC Multiple Choice Exam Part A 1) If sin(3 )y x= then dydx= (A) 3cos(3 )x− (B) cos(3 )x− (C) 1cos(3 )3x− (D) cos(3 )x (E) 3cos(3 )x 2) 20cos 2lim2xxe x xx x→− −− is (A) 12− (B) 0 (C) 12 (D) 1 (E) nonexistent 3) 5(3 1)x dx+ =∫ (A) 6(3 1)18xC++ (B) 6(3 1)6xC++ (C) 6(3 1)2xC++ (D) 263( )22xxC++ (E) 253( )2xx C+ + PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com4) For 0 13t≤ ≤ an object travels along an elliptical path given by the parametric equations 3cosx t= and 4siny t=. At the point where 13t=, the object leaves the path and travels along the line tangent to the path at that point. What is the slope of the line on which the object travels? (A) 43− (B) 34− (C) 4tan133− (D) 43tan13− (E) 34tan13− 5) Let ( )y f x= be the solution to the differential equation dyx ydx= + with the initial condition (1) 2.f= What is the approximation for (2)f if Euler’s method is used, starting at 1x= with a step size of 0.5? (A) 3 (B) 5 (C) 6 (D) 10 (E) 12 6) What are all values of p for which 211pdxx∞∫ converges? (A) 1p< − (B) 0p> (C) 12p> (D) 1p> (E) There are no values of p for which this integral converges. PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com7) The position of a particle moving in the xy-plane is given by the parametric equations 3 23x t t= − and 3 22 3 12 .y t t t= − − For what values of t is the particle at rest? (A) 1only− (B) 0only (C) 2only (D) 1 2and only− (E) 1, 0, 2and− 8) ()2 3cosx x dx=∫ (A) ( )31sin3x C− + (B) ( )31sin3x C+ (C) ( )33sin3xx C− + (D) ( )33sin3xx C+ (E) 3 4sin3 4x xC + 9) If 3( ) ln( 4 ),xf x x e−= + + then (0)f′ is (A) 25− (B) 15 (C) 14 (D) 25 (E) nonexistent PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com10) What is the value of 1123nnn∞+=∑ ? (A) 1 (B) 2 (C) 4 (D) 6 (E) The series diverges. 11) The Maclaurin series for 11x− is 0.nnx∞=∑ Which of the following is a power series expansion for 221xx− ? (A) 2 4 6 81 x x x x+ + + + + ⋅⋅⋅ (B) 2 3 4 5x x x x+ + + + ⋅⋅⋅ (C) 2 3 4 52 3 4x x x x+ + + + ⋅⋅⋅ (D) 2 4 6 8x x x x+ + + + ⋅⋅⋅ (E) 2 4 6 8x x x x− + − + ⋅⋅⋅ 12) The rate of change of the volume, ,V of water in a tank with respect to time, ,t is directly proportional to the square root of the volume. Which of the following is a differential equation that describes this relationship? (A) ( )V t k t= (B) ( )V t k V= (C) dVk tdt= (D) dV kdtV= (E) dVk Vdt= PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com13) The graph of a function f is shown above. At which value of x is f continuous, but not differentiable? (A) a (B) b (C) c (D) d (E) e −3 3−33xy 14) Shown above is a slope field for which of the following differential equations? (A) dy xdx y= (B) 22dy xdx y= (C) 3dy xdx y= (D) 2dy xdx y= (E) 32dy xdx y= xyabcdeOd333−3−xyPDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com15) The length of a curve from 1x= to 4x= is given by 4411 9 .x dx+∫ If the curve contains the point (1,6), which of the following could be an equation for this curve? (A) 23 3y x= + (B) 35y x= + (C) 36y x= + (D) 36y x= − (E) 516 95 5y x x= + + 16) If the line tangent to the graph of the function f at the point (1,7) passes through the point ( 2, 2),− − then (1)f′ is (A) 5− (B) 1 (C) 3 (D) 7 (E) undefined 17) A curve C is defined by the parametric equations 24 1x t t= − + and 3.y t= Which of the following is an equation of the line tangent to the graph of C at the point ( 3,8)−? (A) 3x= − (B) 2x= (C) 8y= (D) 27( 3) 810y x= − + + (E) 12( 3) 8y x= + + PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com1 2 3 4 5 6 7−3−2−11234567xy Graph of f 18) The graph of the function f shown in the figure above has horizontal tangents at 3x= and 6.x= If 20( ) ( ) ,xg x f t dt=∫ what is the value of (3)g′? (A) 0 (B) 1− (C) 2− (D) 3− (E) 6− PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com19) A curve has slope 2 3x+ at each point ( , )x y on the curve. Which of the following is an equation for this curve if it passes through the point (1,2) ? (A) 5 3y x= − (B) 21y x= + (C) 23y x x= + (D) 23 2y x x= + − (E) 22 3 3y x x= + − 20) A function f has Maclaurin series given by 4 5 6 3.2! 3! 4! ( 1)!nx x x xn++ + +⋅⋅⋅+ +⋅⋅⋅+ Which of the following is an expression for ( )f x ? (A) ()23 sin 3x x x− + (B) ()2cos 1x− + (C) ()2 2cosx x x− + (D) 2 3 2xx e x x− − (E) 221xe x− − 21) The number of moose in a national park is modeled by the function M that satisfies the logistic differential equation 0.6 (1 ),200dM MMdt= − where t is the time in years and (0) 50.M= What is lim ( )tM t→∞? (A) 50 (B) 200 (C) 500 (D) 1000 (E) 2000 PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com22) What are all values of p for which the infinite series 11pnnn∞=+∑ converges? (A) 0p> (B) 1p≥ (C) 1p> (D) 2p≥ (E) 2p> 23) ()sin 6x x dx=∫ (A) cos(6 ) sin(6 )x x x C− + + (B) 1cos(6 ) sin(6 )6 36xx x C− + + (C) 1cos(6 ) sin(6 )6 6xx x C− + + (D) 1cos(6 ) sin(6 )6 36xx x C+ + (E) 6 cos(6 ) sin(6 )x x x C− + 24) Which of the following series diverge? 0311sin2.1..1nnnnnnIIIneIIIeπ∞=∞=∞= +∑∑∑ (A) III only (B) I and II only (C) I and III only (D) II and III only (E) I, II, and III PDF Created with deskPDF PDF Writer - Trial :: http://www.docudesk.com2 5 10 14( ) 12 28 34 30xf x 25) The function f is continuous on the closed interval [2,14] and has values as shown in the table above. Using the subintervals [2,5], [5,10], and [10,14], what is the …
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