BC Exam 2 - Part I 28 questions No Calculator Allowed. C. 1 x n D. e x n E. 0

Transcription

1 1. If f x ( ) = ln e A. n x x n BC Exam - Part I 8 questions No Calculator Allowed, and n is a constant, then f ( x) = B. x n e C. 1 x n D. e x n E.. Let f be the function defined below. Which of the following statements about f are NOT true? ( ) = f x x 3 1,x 1 x 1 3x,x =1 I. f has a limit at x =1. II. f is continuous at x = 1. III. f is differentiable at x = 1. IV. The derivative of f is continuous at x =1. A. IV only B. III and IV only C. II, III, and IV only D. I, II, III, and IV E. All statements are true 3. x cos4x dx = A. x sin4x + cos4x + C B. x sin4x cos4x + C C. 8x sin4x + 3cos4x + C D. 8x sin4x 3cos4x + C E. x sin4x cos4x + C BC Exam Illegal to post on Internet

2 4. Find lim ( x 1)ln x 1 x 1 + ( ) A. B. 1 C. 1 D. -1 E. nonexistent ( ) ( 3x 1) x 4 5. Find lim x ( x +1) ( x 1) A. 3 B. 3 C. 3 4 D. 1 E. 6. The graph of the function f is shown to the right for - x. Four calculations are made: LS Left Riemann sum approximation of subintervals of equal length RS Right Riemann sum approximation of subintervals of equal length TS Trapezoidal sum approximation of DI f ( x) dx f ( x) dx with 4 f ( x) dx with 4 Arrange the results of the calculations from highest to lowest. f ( x) dx with 4 subintervals of equal length A. DI RS TS - LS B. RS TS DI - LS C. TS DI LS - RS D. LS DI TS - RS E. RS DI TS - LS BC Exam Illegal to post on Internet

3 7. What is the equation of the line normal to the graph of y = sin x cos x sin x at x = 3π? A. y = x 3π +1 B. y = x + 3π +1 C. y = x 3π 4 +1 D. y = x + 3π 4 +1 E. y = x + 3π Let f x x ( ) = g( t) dt where g has the graph shown to the right. Which of a the following could be the graph of f? A. B. C. D. E. 9. The rate of change of the volume V of the oil in a tank with respect to time t is inversely proportional to the square root of V. If the tank is empty at time t = and V = 9 when t =, find the equation which describes the relationship between t and V. A. V = 9 4 t B. V = t 36 C. V 3 = 4 9 t D. V 3 = 7 t E. V = 3t BC Exam Illegal to post on Internet

4 1. The three regions A, B, and C in the figure to the right are bounded by the graph of the function f and the x-axis. If the areas of A, B, and C are 5,, and 1 respectively, what is the value of 1 [ f ( x ) + ] dx? 4 A. 8 B. 1 C. 6 D. -4 E If f ( x) = sinx, which of the following is the Taylor series for f about x =? x A. 1 x C. x 4x 6 E. x 8x 6 3! + x 4 5! x 6 7! 3! + 16x1 5! 3! + 3x1 5! 64x14 7! +... B. x x 3 18x14 7! +... D. x 4x ! + x 5 5! x 7 3! + 16x 9 5! 7! x13 7! BC Exam Illegal to post on Internet

5 1. The derivative of a function f is defined by f ( x) = g ( x ), 1 x 1. 1 e x +1, 3 x < 1 The graph of the continuous function f is shown to the right with the graph of g being a semicircle. Which of the following statements are true? I. f has a relative maximum at x = 1 ln II. f has an inflection point at x = 1 III. f is increasing on (1,3) A. I only B. II only C. I and II only D. II and III only E. I, II and III 13. Which of the following series converge? I. 3 II. n +1 n =1 n e III. π n = n =1 n n +1 A. I only B. II only C. III only D. II and III only E. None f ( x) f ( ) 14. If f ( x) = tan 1 ( x), find lim x x. A. 17 B C. 15 D E. nonexistent BC Exam Illegal to post on Internet

6 15. Which of the following is equal to the area of the region inside the polar curve r = 3sinθ and outside the polar curve r = sinθ? π A. 5 sin θ dθ B. 5 sin θ dθ C. 1 sin θ dθ π D. 5 sin θ dθ E. 1 sin θ dθ π π π 16. If f ( x) = k sin x + ln x where k is a constant has a critical point at x = π, determine which statement below is true. A. f has a relative minimum at x = π B. f has a relative maximum at x = π C. f has a point of inflection at x = π D. f has no concavity at x = π E. None of the above 17. How many horizontal (H) and vertical (V) tangent lines does the graph of x = 4 3 y 3 y 4 have? A. H, V B. 1 H, V C. 1 H, 3V D. H, 1V E. H, V BC Exam Illegal to post on Internet

7 18. Consider the differential equation dy dx = y x the exact value of f 8 with initial condition f ( ) = 4. Find the difference between ( ) and an Euler approximation of f ( 8) using a step of.5. A. B. 1 C. D. 5 E A lighthouse is km from a point P along a straight shoreline and its light makes 1 revolution per minute. How fast, in km/min, is the beam of light moving along the shoreline when it is 1 km from point P? A..5 B. 1π C. π D. 5π E. 1. The ratio test is applied to the series inequalities results? e A. lim n n < 1 e D. lim n n 1 < 1 e n to show convergence. Which of the following n =1 ( n 1)! B. lim n E. lim n e n ( n 1)! e n! < 1 e < 1 C. lim n n < BC Exam Illegal to post on Internet

8 1. A position of a particle moving in the xy-plane is given by x = t 3 6t + 9t +1 and y = t 3 9t. For what values of t is the particle at rest? A. only B. 1 only C. 3 only D. and 1 only E., 1 and 3. Find the 6 th degree term of the Taylor series for sin( x ) cos( x 3 ) about x =. A. x 6 3 B. x 6 3 C. x 6 6 D. x 6 3 E. x 6 3. Consider the differential equation dy dx = ky ( L y ). Let y = f ( x) be the particular solution to the differential equation with f ( ) =. If x, find the range of f ( x). A. (,L) B. (,) C. ( L,] D. [,L) E. [,kl) BC Exam Illegal to post on Internet

9 4. A lake shaped in a right triangle is located next to a park as shown in the figure to the right. The depth of the lake at any point along a strip x miles from the park s edge is f ( x) feet. Which of the following expressions gives the total volume of the lake? 3 A x f x 4 ( ) dx B x 4 4 D. f ( x) dx E. 3 f ( x) dx f x ( ) dx C. f ( x) 3 dx ( ) for t 5. ( ) is shown to the right with the positive areas 5. A particle moves along the y-axis with velocity v t The graph of v t indicated by P, Q, and R. What is the difference between the distance that the particle travels and its displacement for t 5? A. P - R B. (P + Q + R) C. Q D. R E. (Q + R) 6. x 1 lnt t dt = A. x 1 B. x C. ( ln x) D. ( ln x) 1 E. ln x BC Exam Illegal to post on Internet

10 7. e x + dx = 8x +16 A. e x + + C B. e x + + C C. e x + 8 D. e x + + C E. e x + + C + C 8. In the 5 th degree Taylor polynomial for f x ( x 1) 5. ( ) = ln x centered at x = 1, write the coefficient for A. 1 5! B. 1 5! C. ( 49! ) D. 1 5 E BC Exam Illegal to post on Internet

11 BC Exam - Part II 17 Questions Calculators Allowed 9. An ant walks around the first quadrant region R bounded by the y-axis, the line y = x and the curve f x to the right. Find the distance the ant walked. ( ) = 6 4x 3 as shown in the figure A B C D E A particle moving along the x-axis such that at any time t its velocity is given by v( t) = t 3 is the acceleration of the particle at t = e? A. 6e e B. 6e 3 C. e 3 D. 4e E. 6e 4e 5 ln t. What 31. If cos( xy) = x + y, find dy dx. A. 1 sin( xy) B. ( ) ( ) 1 + y sin xy D. 1 x sin xy ( ) x sin( xy) + y sin xy E. 1 + y 1 x ( ) 1 sin( xy) 1 + y sin xy C BC Exam Illegal to post on Internet

12 3. The graph shows the polar curve r = + cos( 4θ) for θ π. What is the total area enclosed by the graph? A B C D E x ( x +1) dx = A. π 4 B. π C. π D. π E. infinite 34. A cup of coffee is heated to boiling ( 1 F ) and taken out of a microwave and placed in a 7 F room at time t = minutes. The coffee cools at the rate of 16e.11t degrees Fahrenheit per minute. To the nearest degree, what is the temperature of the coffee at time t = 5 minutes? A. 15 F B. 133 F C. 166 F D. 151 F E. 3 F 35. The function f is twice-differentiable and its derivatives are continuous. The table below gives the value of f, f and f for x = and x =. Find the value of x f ( x ) dx. x ( ) f ( x) f ( x) f x A. -1 B. -8 C. 6 D. 14 E BC Exam Illegal to post on Internet

13 36. A particle moves on a plane curve such that at any time t >, its x-coordinate is t t + t 3 while its y-coordinate is t ( ). Find the magnitude of the particle s acceleration at t = 1. A. 4 B. C. D. 3 E Let f be the function given by f ( t) = 4 4t + 4t 4 4t = ( 1) n 4t n and F be the function given by F x x ( ) = f ( t) dt. Find the interval of convergence of the power series for F ( x) about t =. A. ( 1,1 ) B. ( 1,1 ] C. [ 1,1 ] D. [ 4,4] E. ( 4,4] n = 38. Let f be a function having derivatives for all orders of real numbers. The function and its first three derivatives of f at x = are given in the table below. The fourth-derivative of f satisfies the inequality f ( 4) ( x) 6 for all x in the interval [,1]. Find the maximum value of f (.5). x ( ) f ( x) f ( x) f ( x) f x A..34 B C D E BC Exam Illegal to post on Internet

14 39. Find the derivative of f 1 ( x) for f ( x) = x 3 3x x +1 at x =. A..64 B..5 C D..5 E A continuous function f is defined on the closed interval -4 x 4. The graph of the function, shown in the figure to the right consists of a line and two curves. There is a value a, -4 a < 4, for which the Mean Value Theorem, applied to the interval [a, 4] guarantees a value c, a c < 4 at which f c I. -4 II. III. ( ) = 3. What are possible values of a? A I only B. II only C. III only D. II and III only E. I, II, and III 41. Let f be the function with the first derivative defined as f x ( ) for x. Let c be ( ) = sin x 3 4x + the value of x where f attains its minimum value in the interval x and let d be the value of x where f attains its maximum value in the interval x. Find c d. A..539 B C D E BC Exam Illegal to post on Internet

15 4. Matthew is visiting Gregory at his home on North Street. Shortly after Matthew leaves, Gregory realizes that Matthew left his wallet and begins to chase him. When Gregory is 3 miles from the 9 intersection along North Street traveling at 4 mph towards the intersection, Matthew is 4 miles along East street traveling away from the intersection at 5 mph. At that time, how fast is the distance between the two men changing? A. getting closer at 4 mph B. getting further away at 44 mph C. getting closer at 44 mph D. Getting closer at 5 mph E. getting closer at 7 mph 43. Region R is defined as the region between the graph 9 of y =, x = and the x-axis as shown in the x + x figure to the right. Find the area of region R. A. ln1 B ln4 C. 3ln 1 4 D. 6ln E. infinite BC Exam Illegal to post on Internet

16 ( ) = 3x 6x. For how many positive values of b does the average value of f ( x) on the interval ( ) on the interval [, b]? 44. Let f x [, b] equal the average rate of change of f x A. 4 B. 3 C. D. 1 E. none 45. The graph of the polar curve r = 4 cos3θ is shown at the right. Which of the following does not represent the area of the shaded region? π 6 I. 48 cos 3θ dθ II. 4 cos 3θ dθ III. 8 cos 3θ dθ π 6 π 6 π A. I only B. II only C. III only D. I and II only E. I, II and III BC Exam Illegal to post on Internet