BC Exam 1 - Part I 28 questions No Calculator Allowed - Solutions C = 2. Which of the following must be true?

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1 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 BC Exam 1 Illegal to post on Internet

2 4. ( x 4 x) dx = A. x 9 D. 9 + x C B. x 9 ( x 4 x ) 3 + C E. 3 9 x x C C. x 9 9 x x 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 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 > BC Exam 1 Illegal to post on Internet

3 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 v( t) 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 BC Exam 1 Illegal to post on Internet

4 9. x 3 x 1 dx = A. x x + ln x +1 + ln x x C B. + C C. x ln x 1 + C D. x + ( ln x 1 + ln x +1) + C E. ln x C x 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 x 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 BC Exam 1 Illegal to post on Internet

5 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 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 BC Exam 1 Illegal to post on Internet

6 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 ( ) ( ) ( ) ( ) A. 1 B. 11 C. 7 D E 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 BC Exam 1 Illegal to post on Internet

7 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 A power series is used to approximate e x3 dx with a maximum error of What is the minimum number of terms needed to obtain this approximation? 0 A. B. 3 C. 4 D. 5 E 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 ( ) 5 k 7 10 f x A. 4 B. ±4 C. 8 D. -8 E. No values of k BC Exam 1 Illegal to post on Internet

8 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 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 BC Exam 1 Illegal to post on Internet

9 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 B C..911 D E BC Exam 1 Illegal to post on Internet

10 7. The graph of f ( x) = x 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)! BC Exam 1 Illegal to post on Internet

11 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 C D E BC Exam 1 Illegal to post on Internet

12 ( ) =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, ) B. (-3.683, 1.718) C. (.317, ) D. (4.47, 6.053) E. (-1.573, ) BC Exam 1 Illegal to post on Internet

13 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 B C D E BC Exam 1 Illegal to post on Internet

14 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 B C D E 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 B C D E 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 ( ) BC Exam 1 Illegal to post on Internet

15 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 B C D E 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 E BC Exam 1 Illegal to post on Internet

16 4. The expression 1 76 ln ln + ln A. ln x dx B. ln x 75 dx C ln is a Riemann sum approximation for 150 ln x dx D. ln x dx E 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) BC Exam 1 Illegal to post on Internet

17 45. The rate of change of people waiting in line to buy tickets to a concert is given by w t ( ) for 0 t 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? A. w ( t) dt B. w( t) dt C. 800 w t D. w ( t) dt E. w t ( ) dt ( ) dt BC Exam 1 Illegal to post on Internet