BC Exam 1 - Part I 8 questions No Calculator Allowed - Solutions 6x 5 8x 3 1. Find lim x 0 9x 3 6x 5 A. 3 B. 8 9 C. 4 3 D. 8 3 E. nonexistent ( ) f ( 4) f x. Let f be a function such that lim x 4 x 4 I. f is continuous at x = 4. II. f is differentiable at x = 4. III. The derivative of f is continuous at x = 4. =. Which of the following must be true? A. I only B. II only C. I and II only D. I and III only E. I, II and III 3. If f ( x) = ( x +1) x 3 A. x 3 ( ) 4, then ( ) = f x ( ) 3 ( x + 4 x 1) B. 4( x +1) ( x 3) C. 8x( x +1) ( x 3) ( ) 3 ( 3x + x 3) E. ( x 3) ( 9x + 4x 3) D. x 3 011 www.mastermathmentor.com - 1 - BC Exam 1 Illegal to post on Internet
4. ( x 4 x) dx = A. x 9 D. 9 + x 3 3 + C B. x 9 ( x 4 x ) 3 + C E. 3 9 x 6 6 + x 3 3 + C C. x 9 9 x 6 3 + x 3 3 + C ( x 4 x ) 3 3 4x 3 x ( ) + C e x + cos x x 5. Find lim x 0 x 4 x 3 A. 1 30 B. 1 4 C. 1 6 D. 0 E. nonexistent 6. For what values of p does the infinite series n converge? n p 1 + n =1 A. p > 0 B. p >1 C. p 1 D. p > 1 E. p > 3 011 www.mastermathmentor.com - - BC Exam 1 Illegal to post on Internet
7. The table below gives selected values of v( t), of a particle moving along the x-axis. At time t = 0, the particle is at the origin. Which of the following could be the graph of the position, x( t), of the particle for 0 t 4? t 0 1 3 4 v( t) 3 0-1 1 3 A. B. C. D. E. 8. The graph of a twice-differentiable function f is shown in the figure to the right. Which of the following is true? ( ) < f ( ) < f ( ) ( ) < f ( ) < f ( ) ( ) < f ( ) < f ( ( ) < f ( ) < f ( ) ( ) < f ( ) < f ( ) A. f B. f C. f D. f E. f 011 www.mastermathmentor.com - 3 - BC Exam 1 Illegal to post on Internet
9. x 3 x 1 dx = A. x x + ln x +1 + ln x x 1 1 + C B. + C C. x ln x 1 + C D. x + ( ln x 1 + ln x +1) + C E. ln x + 3 5 + C x +1 10. The region bounded by the graph of y = 4, the line x = 4 and the x - axis is rotated about the x-axis. x Find the volume of the solid. A. π B. π C. 4π D. 16π E. infinite 11. If the length of a curve from x = to x = 8 is given by 1 + 81x 4 dx, and the curve passes through the point (-1, 4), which of the following could be the equation for the curve? A. y =13 9x B. y = 4 3x 3 C. y = 7 + 3x 3 D. y = 1 3x 3 E. y = 9x 5 8 011 www.mastermathmentor.com - 4 - BC Exam 1 Illegal to post on Internet
1. Let f ( x) be the power series for sin x, centered at x = 0. Which of the following is a power series? I. f ( x ). II. f ( x ) III. f ( e x ) A. I only B. II only C. III only D. I and II only E. I, II and III 13. Find the y-intercept of the tangent line to 4 x + y = x + y + 3 at the point (9, 4). A. - B. 10 C. 5 D. 15 E. 4 9 14. The function f is continuous and non-linear for 3 x 7 and f 3 value c, where 3 < c < 7, for which f c A. For some k, where 3 < k < 7, f k B. For some k, where 3 < k < 7, f k C. For some k, where 3 < k < 7, f k D. For 3 < k < 7, f ( k) exists. E. For some k, where 3 < k < 7, f k ( ) = 5 and f ( 7) = 5. If there is no ( ) = 1, which of the following statements must be true? ( ) < 1. ( ) > 1. ( ) = 0. ( ) does not exist. 011 www.mastermathmentor.com - 5 - BC Exam 1 Illegal to post on Internet
15. Let f be a function having derivatives for all orders of real numbers. The first three derivatives of f at x = 0 are given in the table below. Use the third-degree Taylor polynomial at x = 0 to approximate 1 f. x f x f x f x f x ( ) ( ) ( ) ( ) 0 5 8 4 A. 1 B. 11 C. 7 D. 17 3 E. 3 16. A large block of ice in the shape of a cube is melting. All sides of the cube melt at the same rate. At the time that the block is s feet on each side, its surface area is decreasing at the rate of 4 ft hr. At what rate is the volume of the block decreasing at that time? A. 1s ft 3 hr B. 6s ft 3 hr C. 4s ft 3 hr D. s ft 3 hr E. s ft 3 hr 17. The position of an object moving in the xy-plane with position function r( t) = 1 + sint,t + cost, t 0. What is the maximum speed attained by the object? A. 1 B. C. D. 4 E. 011 www.mastermathmentor.com - 6 - BC Exam 1 Illegal to post on Internet
18. The graph of f ( x), the derivative of f, is shown to the right. Which of the following statements is not true? A. f is increasing on x 3. B. f has a local minimum at x = 1. C. f has a local maximum at x = 0. D. f is differentiable at x = 3. E. f is concave down on - x 1. 1 19. A power series is used to approximate e x3 dx with a maximum error of 0.01. What is the minimum number of terms needed to obtain this approximation? 0 A. B. 3 C. 4 D. 5 E. 6 0. The function f is continuous on the closed interval [0, 8] and has the values given in the table below. The trapezoidal approximation for 8 f ( x ) dx found with 3 subintervals is 0k. What is the value of k? 0 x 0 3 5 8 ( ) 5 k 7 10 f x A. 4 B. ±4 C. 8 D. -8 E. No values of k 011 www.mastermathmentor.com - 7 - BC Exam 1 Illegal to post on Internet
1. The Maclaurin series for a certain function f converges to f ( x) for all x in the interval of convergence. The n th derivative of f at x = -1 is given by ( ) = 1 for n ( 1 n) f ( n ) 1 ( +1 ( )n n +1)! If the graph of f has a horizontal tangent at (-1, -4), describe the behavior of the graph of f at x = -1. A. relative maximum B. relative minimum C. cusp point D. inflection point E. none of these. The line x + y = k, where k is a constant, is tangent to the graph of y = x 3 9x x +1. What are the only possible values of k? A. 1 only B. 0 and - 9 C. 1 and -9 D. 0 and 3 E. 1 and -6 3. The shaded region between the graph of y = tan 1 x and the x-axis for 0 x 1 as shown in the figure is the base of a solid whose cross-sections perpendicular to the x-axis are squares. Find the volume of the solid. A. π + ln 1 B. π + e ln 4 C. π ln D. π 4 ln E. π ln 011 www.mastermathmentor.com - 8 - BC Exam 1 Illegal to post on Internet
4. The average value of sin x cos x on the interval π, 3π is A. B. C. 0 D. -1 E. 1 3π 3π ( ( )) = x for all x. If 5. The functions f and g are differentiable and f g x f ( 4) = 8, g( 4) = 8, f ( 8) =, what is the value of g ( 4)? A. 1 8 B. 1 C. D. 4 E. Insufficient data 6. A particle moves along the x-axis so that its velocity v( t) =1te t t +1. At t = 0, the particle is at position x = 0.5. What is the total distance that the particle traveled from t = 0 to t = 3? A. 1.448 B. 1.948 C..911 D. 4.181 E. 4.681 011 www.mastermathmentor.com - 9 - BC Exam 1 Illegal to post on Internet
7. The graph of f ( x) = x + 0.0001 0.01 is shown in the graph to the right. Which of the following statements are true? ( ) = 0. I. lim f x x 0 II. f is continuous at x = 0. III. f is differentiable at x = 0. A. I only B. II only C. I and II only D. I, II, and III E. None are true 8. The hyperbolic sine function is defined as sinh x = 1 ( ex e x ). Give the general term for the Maclaurin series for sinh x. A. n +1 x ( n +1)! B. ( 1) n x ( n +1)! n +1 C. x n ( n)! D. ( 1) n +1 x n ( n)! E. ( 1) n x n ( n)! 011 www.mastermathmentor.com - 10 - BC Exam 1 Illegal to post on Internet
BC Exam 1 - Part II 17 questions Calculators Allowed - Solutions 9. The slope field for the equation in the figure to the right could be A. dy dx = x + y D. dy dx = x + y B. dy dx = x y dy E. dx = x y C. dy dx = xy = cos x with initial condition f ( 0) = 0. Find the difference 30. Consider the differential equation dy dx π π between the exact value of f and an Euler approximation of f using two equal steps. A. 0 B. 0.30 C. 0.341 D. 0.555 E. 0.707 011 www.mastermathmentor.com - 11 - BC Exam 1 Illegal to post on Internet
( ) =1 + ( x 7) + ( x 7) +... + ( x 7) n +... for all real 31. The function f is defined by the power series f x numbers x for which the series converges. What is the range of f ( x)within the interval of convergence? A. 1, B. 1, C. 0, [ ) D. ( 0, ) E. (, ) 3. An object moving along a curve in the xy-plane has position x( t),y t dx dt = t dy + 3t +1 and dt = et 1 for t 0. ( ( )) at time t with At time t = 0, the object is at position (-6, -7). Find the position of the object at t =. A. (4.667, 13.053) B. (-3.683, 1.718) C. (.317, 19.718) D. (4.47, 6.053) E. (-1.573, -0.947) 011 www.mastermathmentor.com - 1 - BC Exam 1 Illegal to post on Internet
33. At which points is the tangent line to the curve 8x + y = 6xy +14 vertical? I. (-, -3) II (3, 8) III. (4, 6) A. I only B. II only C. III only D. I and II only E. I and III only ( ) for all x in the interval of convergence. 34. The Maclaurin series for a certain function f converges to f x The nth derivative of f at x = 0 is given by f ( n ) ( 0) = ( n +1)! for n 0. Find the radius of ( 0.5) n 3 n convergence for f, if it exists, about x = 0. A. 1 B. 1 C. D. All reals E. does not converge 35. Let F ( x) be an antiderivative of x 3 + x +1. If F ( 1) =.15, then F ( 4) = A. -15.879 B. -11.69 C. 7.74 D. 15.879 E. 11.69 011 www.mastermathmentor.com - 13 - BC Exam 1 Illegal to post on Internet
36. What is the area of the region in the first quadrant enclosed by the graph of y = cos x, y = x, and the x-axis? A. 0.816 B. 1.184 C. 1.59 D. 1.794 E. 1.999 37. The graph of the polar curve r = 4sinθ is a limaçon with two loops as shown in the figure to the right. Find the area between the two loops. A. 5.688 B. 35.187 C. 35.55 D. 37.361 E. 37.699 38. Find the Lagrange error in calculating f 0.1 about x = 0. ( ) for the third degree Taylor polynomial for f ( x) = xe x A. 1 4 B. ( 0.1) 3 6 C. ( 0.1) 3 4 D. ( 0.1) 4 6 E. 0.1 ( ) 4 4 011 www.mastermathmentor.com - 14 - BC Exam 1 Illegal to post on Internet
39. A particle moves along a straight line with velocity given by v( t) = t.5 cost. What is the acceleration of the particle at t =? A. 0.168 B. 0.38 C. 0.451 D. 0.584 E. 1.450 40. A particle moving along the polar curve r =1 sinθ has position x( t),y t θ = 0 when t = 0. The particle moves along the curve such that dr dt = particle at t = π 6. I. Getting closer to the x-axis II. Getting closer to the y-axis III. Getting closer to the origin ( ( )) at time with dr. Describe the motion of the dθ A. I only B. II only C. I and II only D. II and III only E. I, II and III 41. The rate at which the gasoline is changing in the tank of a hybrid car is modeled by f t 6-hour trip was the gasoline in the tank going down most rapidly? ( ) = t +.5sint.5 gallons per hour, t hours after a 6-hour trip starts. At what time during the A. 0 B..9 C. 3.8 D. 4.03 E. 6 011 www.mastermathmentor.com - 15 - BC Exam 1 Illegal to post on Internet
4. The expression 1 76 ln 75 75 77 78 + ln + ln 75 75 A. ln x dx B. ln x 75 dx C. 1 75 75 1 150 76 +... + ln is a Riemann sum approximation for 150 ln x dx D. ln x dx E. 1 75 76 1 1 ln x dx 43. Let f be the function given by f x ( ) = cos t + t percentage of values of x for x is f x x 0 ( ) dt for x. Approximately, for what ( ) decreasing? A. 30% B. 6% C. 44% D. 50% E. 59% 44. A curve C is defined by the parametric equations x = t t 4 and y = t 3 7t. Which of the following is the equation of the line tangent to the graph of C at the point (, 4)? A. y = 6 x B. x 4y +14 = 0 C. 5x 3y + = 0 D. y = 4x 4 E. No tangent line at (, 4) 011 www.mastermathmentor.com - 16 - BC Exam 1 Illegal to post on Internet
45. The rate of change of people waiting in line to buy tickets to a concert is given by w t ( ) for 0 t 4. 800 people are already waiting in line when the box office ( ) =100 t 3 4t t + 7 opens at t = 0. Which of the following expressions give the change in people waiting in line when the line is getting shorter? 3.773 A. w ( t) dt B. w( t) dt C. 800 w t 1.480.786 3.773 1.480 D. w ( t) dt E. w t 0.786 0 ( ) dt 3.773 1.480 ( ) dt 011 www.mastermathmentor.com - 17 - BC Exam 1 Illegal to post on Internet