AP Calculus BC AP Exam Problems Chapters 1 3 Precalculus .

AP Calculus BCAP Exam ProblemsChapters 1 – 3Precalculus Review1. If f is a continuous function defined for all real numbers x and if the maximum value of f(x) is 5and the minimum value of f(x) is 7 , then which of the following must be true?I.The maximum value of f xII.The maximum value of f ( x ) is 7.III.The minimum value of f xA) I onlyB) II only is 5. is 0.C) I and II onlyD) II and III onlyE) I, II, and III2. If f g( x ) ln x 2 4 , f ( x ) ln x 2 , and g( x ) 0 for all real x, then g( x ) A)1x 42B)1x 42x2 4C)D) x 2 43. What is the domain of the function f given by f ( x ) A) x : x 3 B) x : x 2 x : x 2 D) x : x 2 and x 3 C) 1 4. If ln x ln 2 , then x x 11A) 2B)C) eee5. If f ( x ) A)x 1xD) 2eE) x 2x2 4?x 3E) x : x 2 and x 3 E) e2x, then the inverse function, f 1 , is given by f 1( x ) x 1B)x 1xC)x1 xD)xx 1E) x6. Which of the following does NOT have a period of ? 1 A) f ( x ) sin x 2 B) f ( x ) sin xC) f ( x ) sin2 xD) f ( x ) tan x1E) f ( x ) tan2 x

AP Calculus BCAP Exam ProblemsChapters 1 – 37. The graph of which of the following equations has y 1 as an asymptote?A) y ln xB) y sin xC) y x2xD) y E) y e xx 1x 18. Which of the following is continuous for all real numbers x?I. y x2II. y e x3A) NoneB) I onlyy tan xIII.C) II onlyD) I and IIE) I and IIIx2 49. If f ( x ) is continuous for all real numbers and if f ( x ) when x 2 , then f ( 2) x 2A) 4B) 2C) 1D) 0E) 210. If h( x ) f g( x ) , where f ( x ) 3x 2 1 and g( x ) x , then h( x ) A) 3x 2 xC) 3x 2 x 1B) 3x 2 1D) 3 x 1E) 3x 2 111. The fundamental period of 2cos 3x isA)2 3B) 2 12. If the graph of y C) 6 D) 2E) 3ax bhas a horizontal asymptote y 2 and a vertical asymptote x 3 ,x cthen a c A) 5B) 1C) 0xf(x)D) 101E) 51k2213. The function f is continuous on the closed interval [0, 2] and has values given in the table1above. The equation f ( x ) must have at least two solutions in the interval [0, 2] if k 2A) 0B)12C) 1D) 22E) 3

AP Calculus BCAP Exam ProblemsChapters 1 – 314. Let f be the function defined by the following. sin x , x 0 2 x , 0 x 1f (x) 2 x 1 x 2 x 3 x 2For what values of x is f NOT continuous?A) 0 onlyB) 1 onlyC) 2 onlyD) 0 and 2 onlyE) 0, 1, and 215. Which of the following functions is continuous at x 1?I. ln xII. e xIII. ln e x 1 A) I onlyB) II onlyC) I and II only16. Let f be the function given by f ( x ) x 1 x 2 4 continuous for all real numbers x?A) NoneB) 1 onlyD) II and III onlyC) 2 onlyx2 aE) I, II, and III. For what positive values of a is fD) 4 onlyE) 1 and 4 onlyD) 4E) nonexistentLimits and Continuity17. If f ( x ) 2x 2 1 , then limx 0A) 0B) 1f ( x ) f (0)isx2C) 2for 0 x 2 ln x18. If f ( x ) 2then lim f ( x ) isx 2 x ln2 for 2 x 4A) ln 2B) ln 8C) ln 16D) 4E) nonexistentC) 1D) 4E) nonexistent4n219. Find lim 2n n 10,000nA) 0B)12,5003

AP Calculus BCAP Exam ProblemsChapters 1 – 320. If lim f ( x ) L , where L is a real number, which of the following must be true?x aA) f (a) exists.B) f ( x ) is continuous at x a.C) f ( x ) is defined at x a.D) f (a) LE) None of the above3n3 5n21. Find lim 3n n 2n2 1A) 5B) 2C) 1D) 3E) nonexistentC)14D) 1E) nonexistentC)16a2D) 0E) nonexistent1 cos 0 2sin 2 22. Find limA) 0B)18x 2 a2isx a x 4 a 423. If a 0 , then limA)1a2B)12a2Derivatives24. If f ( x ) e x , which of the following is equal to f (e ) ?e x hh 0 he x h eeB) limh 0hee h eh 0hx he 1D) limh 0hA) lim25. The limC) limtan3 x h tan 3x h 0A) 0B) 3sec2 (3x )hee h eeh 0hE) limisC) sec2 (3x )D) 3cot(3x )E) nonexistent4

AP Calculus BCAP Exam ProblemsChapters 1 – 326. If f is a differentiable function, then f (a) is given by which of the following?I. limh 0f (a h) f (a)hA) I onlyB) II onlyII. limx af ( x ) f (a )x aC) I and II onlyIII. limx af ( x h) f ( x )hD) I and III onlyE) I, II, and IIIsin( x h) sin xh 0h27. Find limA) 0B) 1C) sin xD) cos xE) nonexistent28. If lim f ( x ) 7 , which of the following must be true?x 3I.II.III.f is continuous at x 3.f is differentiable at x 3.f (3) 7A) NoneB) II onlyC) III onlyD) I and III onlyE) I, II, and III29. Which of the following functions shows that the statement “If a function is continuous at x 0,then it is differentiable at x 0” is false?A) f ( x ) xB) f ( x ) x 4 13C) f ( x ) x3D) f ( x ) x1E) f ( x ) x 3343 x2if x 330. At x 3, the function given by f ( x ) is 6 x 9 if x 3A) undefinedB) continuous but not differentiableC) not continuous and differentiableD) neither continuous nor differentiableE) both continuous and differentiable31. If f ( x ) x , then f (5) A) 032. FindB)15C) 1252D) 5E)D) 2E) 6d 1 1 x 2 at x 1 3dx xx A) 6B) 4C) 05

AP Calculus BCAP Exam ProblemsChapters 1 – 333. If f ( x ) 2x , then f (2) A)14B)1222C)2D) 1E)D) 6E) 8334. If f ( x ) x 2 , then f (4) A) 6B) 3C) 335. If y cos2 x sin2 x , then y E) 2 cos x sin x A) 1C) 2sin(2x )B) 0D) 2 cos x sin x 36. If f ( x ) sin x , then f 3 A) 12B)12C)37. If y tan x cot x , thenA)2e2 x1 e2e 2 xB)1 e4x4xD)32E)3dy dxC) sec x csc xD) sec2 x csc2 xA) sec x csc xB) sec x csc x38. If y tan 1 e2 x , then22E) sec2 x csc2 xdy dxC)D)e2x1 e4xE)11 e4x611 e4x

AP Calculus BCAP Exam ProblemsChapters 1 – 339. If y tan 1 cos x , thenA)dy dx sin x1 cos2 xC) sec 1(cos x ) 2B) sec 1(cos x ) sin x240. If y A)B)D) E)11 cos2 xE)32x1cos x 2 1 13dy, then 24 xdx 6 x4 x2 2 3x4 x2 2C)D) 6x4 x2 2 34 x2 2 41. The value of the derivative of y 3x2 842x 11B) C) 02A) 1D)at x 0 is12E) 142. What is the instantaneous rate of change at x 2 of the function f given by f ( x ) A) 243. If f ( x ) B)16C)12D) 2E) 6x , then f tan x 4 A) 2B)44. If y x 2e x , thenA) 2xe xB) x x 2e x 12C) 1 2D) 2 1E) 1 2dy dxC) xe x x 2 E) 2x eD) 2x e x7x2 2?x 1

AP Calculus BCAP Exam ProblemsChapters 1 – 345. If f ( x ) x 1 sin x , then f (0) 2A) 2B) 1C) 0D) 1E) 246. If f and g are differentiable functions such that g( x ) e f ( x ) and g ( x ) h( x )e f ( x ) , then h( x ) A) f ( x ) f ( x )C)B) f ( x ) f ( x ) 2D) f ( x ) f ( x ) 2 f ( x ) f ( x )2E) 2 f ( x ) f ( x )47. If u, v, and w are nonzero differentiable functions, then the derivative ofuv u vw u v w uv w B)w2uv w uv w u v ww2u v w uv w uv w D)w2A)C)E)uviswuv w u v w uv w w248. If f ( x ) 2x 1 , then the fourth derivative of f ( x ) at x 0 is4A) 0B) 24C) 48D) 240E) 384d2 y x 49. If y 2cos , thendx 2 2 x A) 8cos 2 x B) 2cos 2 1 x E) cos 2 2 x C) sin 2 x D) cos 2 50. If f ( x ) x 2 2x 1 3 , then f (0) is2A)43B) 0C) 23D) 43E) 251. If f ( x ) sin e x , then f ( x ) A) cos e x B) cos e x e xC) cos e x e xD) e x cos e x 8E) e x cos e x

AP Calculus BCAP Exam ProblemsChapters 1 – 3 52. If f ( x ) tan 2x , then f 6 A)B) 2 33C) 4D) 4 3E) 853. If f and g are twice differentiable functions and if h( x ) f g( x ) , then h ( x ) A) f g( x ) g ( x ) f g( x ) g ( x )D) f g( x ) g ( x )B) f g( x ) g ( x ) f g( x ) g ( x )E) f g( x ) 2C) f g( x ) g ( x ) 254. Findd x2 dx B) 2x 1 xA) 2x 155. FindA)C) 2x ln2D) 2x 1 ln2 E)C) 1 xD) x 12xln2d 1 ln dx 1 x 11 xB)1x 1E) 1 x 56. If f ( x ) x ln x 2 , then f ( x ) A) ln x 2 157. If f ( x ) lnA) 2x2B) ln x 2 2C) ln x 2 1xD)11E)2xx x , then f (x ) B) 12x 2C) 12xD) 12x32E)2x258. If f ( x ) e x , then ln f (2) A) 2B) 0C)1e2D) 2e9E) e22

AP Calculus BCAP Exam ProblemsChapters 1 – 359. Findd ln cos dx x x 2 cos x B) tan x 1 cos x D) tan x x A)C)E) tan x x 260. If f ( x ) ln e2 x , then f ( x ) A) 1B) 2D) e 2xC) 2xE) 2e 2x261. If f ( x ) e tan x , then f ( x ) 2A) e tan2C) tan2 x e tanx2B) sec2 x e tan2x 1E) 2tan x e tan2xD) 2tan x sec2 x e tanxx x 1 , then dy 62. If y 102dx x 1 A) ln10 102C) x 1 B) 2x 1063. If y A)2 x 2 1 10 x 1 2E) x 2 ln10 102 x 1 2D) 2x ln10 102ln xdy, then xdx1x64. Find xB)1x2C)ln x 1x2D)1 ln xx2E) d ln xx dxA) x ln xB) ln x x 2 ln x x ln xxD) ln x x ln x 1C)10 E) 2 ln x x ln x1 ln xx2

AP Calculus BCAP Exam ProblemsChapters 1 – 365. If f ( x ) eA) eB) 3 ln x 2 , then f (x) 3ln x 2 C) 6 ln x e3 3 ln x 2 ex266. For 0 x 2 3ln x 2E) 6x 5D) 5x 4, if y sin x , thenxdy dxA) x ln sin x C) x sin x B) sin x cot xD) sin x x cos x sin x xx 1 cos x E) sin x x cot x ln sin x xx67. If e f ( x ) 1 x 2 , then f ( x ) 11 x22xB)1 x2C) 2x 1 x 2 A) D) 2x e1 x2E) 2x ln 1 x 2 68. If x 2 xy y3 0 , then, in terms of x and y,A) 2x yx 3 y2C) 2x1 3 y2B) x 3 y22x yD) 2 xx 3 y269. If x 2xy y2 2 , then at the point (1, 1),A)32B)12C) 0dy dxE) 2x yx 3 y2 1dy dxD) 1132E) nonexistent

AP Calculus BCAP Exam ProblemsChapters 1 – 370. If x 3 3xy 2 y3 17 , then, in terms of x and y,A) x2 yx 2 y2C) x2 yx 2yB) x2 yx y2D) x2 y2 y271. If x 2 xy 10 , then when x 2,A) 72B) 2C)dy dxE) x21 2 y2E)72dy dx27D)3272. If xy2 2xy 8 , then at the point (1, 2), y A) 52B) 4373. If y2 2xy 16 , thenA)xy xB)C) 1D) 12E) 0dy dxyx yC)yy xD)y2y xE)2yx yE)12d2 ydy2 74. If 1 y , thendx 2dxA) 2yB) yC) y1 y2D) yTangents and Normals75. The slope of the line tangent to the curve y 2 xy 1 0 at 2, 1 is3A) 32B) 34C) 0D)34E)3276. The slope of the line tangent to the graph of ln xy x at the point where x 1 isA) 0B) 1C) eD) e212E) 1 e

AP Calculus BCAP Exam ProblemsChapters 1 – 3 x 77. The slope of the line tangent to the graph of y ln at x 4 is 2 A)18B)14C)12D) 1E) 478. The slope of the line normal to the graph of y 2ln sec x at x A) 2B) 12C)12D) 2 is4E) nonexistent79. An equation of the line tangent to the graph of f ( x ) x 1 2x at the point 1, 1 is3A) y 7x 6B) y 6x 5C) y 2x 1D) y 2x 380. An equation of the line tangent to the graph of y A) 13x y 8B) 13x y 18E) y 7x 82x 3at the point (1, 5) is3x 2C) x 13 y 64D) x 13 y 66E) 2x 3 y 1381. An equation of the line tangent to the graph of y x cos x at the point (0, 1) isA) y 2x 1B) y x 1C) y xD) y x 1E) y 082. Let f be the function given by f ( x ) 3e2x and let g be the function given by g( x ) 6x 3 . At whatvalue of x do the graphs of f and g have parallel tangent lines? (Calculator)A) 0.701B) 0.567C) 0.391D) 0.302E) 0.25883. Which of the following is an equation of the line tangent to the graph of f ( x ) x 4 2x 2 at thepoint where f ( x ) 1 ? (Calculator)A) y 8x 5B) y x 7C) y x 0.763D) y x 0.12213E) y x 2.146

AP Calculus BCAP Exam ProblemsChapters 1 – 384. An equation of the line normal to the graph of y x 3 3x 2 7x 1 at the point where x 1 isA) 4x y 10B) x 4 y 23C) 4x y 2D) x 4 y 25E) x 4 y 25Position/Velocity/Acceleration85. A particle moves along the x-axis so that at any time t 0 its position is given byx(t ) t 3 3t 2 9t 1 . For what values of t is the particle at rest?A) NoneB) 1 onlyC) 3 onlyD) 5 onlyE) 1 and 386. The position of a particle moving along a straight line at any time t is given by s(t ) t 2 4t 4 .What is the acceleration of the particle when t 4?A) 0B) 2C) 4D) 8E) 1287. A particle moves along a line so that at time t, where 0 t , its position is given bys(t ) 4cos t t2 10 . What is the velocity of the particle when its acceleration is zero?2(Calculator)A) 5.19B) 0.74C) 1.32D) 2.55E) 8.1388. A particle moves along the x-axis so that its position at time t is given by x(t ) t 2 6t 5 . Forwhat value of t is the velocity of the particle zero?A) 1B) 2C) 3D) 4E) 589. The position of a particle moving along the x-axis is x(t ) sin 2t cos 3t for time t 0 .When t , the acceleration of the particle isA) 9B)19C) 0D) 1419E) 9

AP Calculus BCAP Exam ProblemsChapters 1 – 3Free Response Questions1. Let f be the function given by f ( x ) (a)(b)(c)(d)2.Find the domain of f.Write an equation for each vertical and horizontal asymptote for the graph of f.Find f ( x ) .Write an equation for the line tangent to the graph of f at the point 0, f (0) .Let f be the function given by f ( x ) x 4 16x 2 .(a)(b)(c)(d)3.2x 5.x2 4Find the domain of f.Describe the symmetry, if any, of the graph of f.Find f ( x ) .Find the slope of the line normal to the graph of f at x 5.A particle moves on the x-axis so that its position at time t 0 is given by x(t ) 2te t .(a)(b)(c)Find the acceleration of the particle at t 0.Find the velocity of the particle when its acceleration is 0.Find the total distance traveled by the particle from t 0 to t 5.15

AP Calculus BCAP Exam ProblemsChapters 1 – 316

AP Exam Problems Chapters 1 – 3 1 Precalculus Review 1. If f is a continuous function defined for all real numbers x and if the maximum value of f(x) is 5 and the minimum value of f(x) is 7, then which of the following must be true? I. The maximum value of fx is 5. II. The maximum value of fx() is 7. III. The minimum value of fx is 0.