Virginia Tech Math 16 : Past CTE problems 1. It requires 1 in-pounds of work to stretch a spring from its natural length of 1 in to a length of 1 in. How much additional work (in inch-pounds) is done in stretching it one inch further? (a) 15 7 (b) 845 7 (c) 5 (d) 45. The region in Quadrant I bounded by the graph of y = x, the y-axis, and the circle x + y = 4 is revolved about the line x = 4. Which integral expresses the volume of the solid? (a) π (4 y) dy (b) π (c) π (d) π 4 x (4 x) dx (16 (4 y) ) dy ( 4 x x)(4 x) dx 1
3. e x x dx (a) x3 3 ex + C (b) x e x xe x + e x + C (c) x e x xe x + C (d) x e x xe x e x + C 4. Evaluate (a) -1 (b) π cos 3 x dx (c) 3 (d)
5. Which integral equals the area of the region bounded by the graphs of y = x 1 and y = x + 1? (a) (b) (c) (d) 3 3 1 1 [(x + 1) (x 1)] dx [(x 1) (x + 1)] dx [(x + 1) (x 1)] dx [(x 1) (x 1)] dx 6. The region bounded by the x-axis and the graph y = sin x, x π is revolved around y = 1. Which integral expresses the volume of the solid? π (a) π (sin x + 1) dx (b) π (c) π (d) π π π π (sin x + sin x) dx x sin x dx x(sin x + 1) dx 3
7. The region bounded by the graph of y = e x, the y-axis and y = is revolved around x = 1. What is the volume of the solid? (a) π (ln y) dy (b) π (c) π (d) π ln ln 1 (x + 1)( e x ) dx x(e x ) dx (1 + ln y) dy 8. x cos x dx (a) x sin x + x cos x sin x + C (b) x3 3 sin x + C (c) x sin x x cos x + sin x + C (d) x sin x + x cos x + sin x + C 4
9. The region bounded by the graphs of y = x + 3 and y = x + 1 is revolved about the x-axis. What is the volume of the solid? 5 (a) π (b) π (c) π (d) π 1 5 1 1 1 y y 1 dy y( y 1 y + 3) dx [(x + 1) (x + 3) ] dx [(x + 3) (x + 1) ] dx 1. Find the volume of the solid generated by rotating the region bounded by y = x and y = 1 about the line y =. 1 (a) π [( x ) 1] dx (b) π (c) π (d) π 1 1 1 1 1 1 1 [(x ) 1] dx [( x ) 1] dx [x 1] dx 5
11. Evaluate the integral ln x, dx (a) x ln x x + C (b) x ln x x + C (c) ln x + C (d) ln x x + C 1. Evaluate the integral x sec (3x), dx (a) 1 x sec 3 (3x) + C (b) x tan(3x) 1 tan (3x) + C (c) x tan(3x) + ln sec(3x) + C (d) 1 3 x tan(3x) 1 ln sec(3x) + C 9 6
13. sin 3 (x), dx (a) 1 6 cos3 (x) 1 cos(x) + C (b) 1 6 cos3 (x) + 1 cos(x) + C (c) 1 3 cos3 (x) + cos(x) + C (d) 1 8 sin4 (x) + C 14. Set up the integral to find the volume of the solid generated by rotating the region bounded by x = 8 y and x = y + 6 about the line x = 3. 1 (a) π [(8 y ) (y + 6) ] dy (b) π (c) π (d) π 1 1 1 [(3 + (8 y )) (3 + (y + 6)) ] dy [(3 (8 y )) (3 (y + 6)) ] dy (y + 3)[(8 y ) (y + 6)] dy 7
15. The tank shown in the figure is initially full of water weighing 6.5 pounds per cubic foot. How much work is done in pumping all of the water to a point feet above the top of the tank? 1 (a) 4π(6.5) (b) 4π(6.5) (c) 4π(6.5) (d) 4π(6.5) 1 1 1 y dy (1 y) dy y dy (1 y) dy 16. Find the area between y = x and y = x on the interval [, 3] (a) 1 (b) 16 (c) 44 3 (d) 4 3 8
17. A tank is in the shape of an inverted cone with height 14 feet and radius on the ground 7 feet. If the tank is filled to within feet of the top with water weighing 6.4 lb/ft 3, how much work does it take to empty the tank through an outlet at the top of the tank? 14 (a) π(6.4) (7 1 y) (1 y) dy (b) π(6.4) (c) π(6.4) (d) π(6.4) 1 1 14 (7 1 y) (14 y) dy (7) (14 y) dy (7) (1 y) dy 18. Evaluate (a) π (b) 1 π x cos(3x) dx. (c) 1 9 π 6 (d) 1 3 π 6 9
19. Assuming the function T (t) = 1 + 3t + 1 t approximiates the temperature t hours past noon on a typical August day in Las Vegas, then the average temperature between noon and 6pm would be: (a) 14.5 (b) 1 (c) 115 (d) 13. A 5 lb ball is suspended from a 4 foot cable that weighs lbs ft. Find the total work required to wind the cable up 5 feet. (a) 5 ft-lbs (b) 687 ft-lbs (c) 875 ft-lbs (d) 9 ft-lbs 1
1. Find the average value of f(x) = sin(x) cos(x) over the interval [, π ]. (a) 1 π (b) 1 π (c) 1 (d) 1. Evaluate x sin(4x) dx (a) x sin(4x) + 4x cos(4x) + C (b) 1 4 x cos(4x) + 1 8 x sin(4x) + 1 3 cos(4x) + C (c) x3 sin(4x) + x cos(4x) + C (d) 1 4 x cos(4x) 1 8 x sin(4x) 1 3 cos(4x) + C 11
3. Suppose f(x) = e x + c where c is a constant. Suppose also that the average value of f 1 over [, ln ] is ln. Then, (a) c = 1 ln (b) c = 1 (ln ) 1 ln (c) c = 1 (ln ) ln (d) c = 4. A tank with the shape of an inverted right circular cone (the vertex is down) is filled with milk (weighing 64.5 lb/ft 3 ) to a depth of one half its height. If the height is ft and its diameter is 8 ft, find the work done by pumping all of the milk to the top of the tank. Assume the origin is at the vertex. 1 (a) 64.5π (5y) ( y) dy (b) 64.5π (c) 64.5π (d) 64.5π 1 1 1 ( ) y ( y) dy 5 ( ) y ( y) dy 5 ( y 5 ) (y 1) dy 1
dx 5. The integral 1 x (a) 1 (b) 3 (c) 7 4 has value (d) No value. Integral Diverges. 6. Evaluate 3x 1 (x + 1)(x + 1) dx (a) ln(x + 1) + tan 1 x ln x + 1 + C (b) tan 1 x ln x + 1 + C (c) 3 ln(x + 1) + 3 ln x + 1 + C (d) ln(x + 1) + ln x + 1 + C 13
7. Evaluate lim x +(1 + tan x)3/x (a) e 6 (b) 6 (c) 1 (d) Does not exist 8. Which of the following is the Simpson s Rule approximation to ln x dx with 6 subintervals? (a) ln 1 + ln + ln 3 +... + ln 7 (b) 1 [ ln 1 + 4 ln + ln 3 + + 4 ln 6 + ln 7] 3 (c) 1 [ln 1 + ln + ln 3 + + ln 6 + ln 7] (d) 1 [ln 1 + 4 ln + ln 3 + + 4 ln 6 + ln 7] 3 7 1 14
9. Let R be the region bounded by the graphs of y = x + and y = x. The moment of the region about the y-axis and the y-coordinate of the centroid are: (a) M y = (b) M y = (c) M y = 1 (d) M y = 1 1 1 x(x + x ) dx; ȳ = M y Area x(x + x ) dx; ȳ = M x Area 1 1 [(x + ) x 4 )] dx; ȳ = M y Area [(x + ) x 4 )] dx; ȳ = M x Area 3. Evaluate lim x +(1 + sin 3x)/x (a) (b) e (c) 6 (d) e 6 15
31. A lamina with constant density ρ = 1 covers the region in Quadrant I bounded by the graphs of y = x and y = x 3. The x-coordinate of the centroid is 1 x(x x 3 ) dx (a) Mass 1 1 (x x 6 ) dx (b) Mass 1 y(y 1/3 y) dy (c) Mass (d) None of the above 3. 4 x 1 dx (a) ln x 1 x + 1 + C (b) 4 tan 1 x + C (c) ln x 1 + C (d) 4 ln x 1 + C 16
33. lim x e x 1 x x e x 1 x + 3x (a) 1 (b) 3 (c) 3 7 (d) None of the above 34. The integral (a) (b) 1 dx e x has the value (c) 1 e (d) No Value. Integral Diverges. 17
35. x 4 + x 3 + 1 x 4 + 1 (a) 16 17 (b) 33 17 (c) + 1 ln 17 4 (d) + 1 ln 17 dx equals: 36. Find the moment with respect to the y-axis, M y, for the region of density δ = 1 bounded by x = 8 y and x = y + 6. (a) (b) (c) (d) 1 1 1 1 1 [(8 y ) (y + 6) ] dy 1 [(8 y ) (y + 6) ] dy y[(8 y ) (y + 6)] dy y[(8 y ) (y + 6)] dy [(8 y ) (y + 6)] dy 18
37. Decide whether the following integral converges or diverges. If it converges, find the value. x (x + ) dx (a) 1 4 (b) 1 (c) 1 (d) diverges 38. Evaluate: x 4 x dx (a) 4 3 sin3 x + C (b) sin 1 x x + C (c) sin 1 x x 4 x + C (d) sin 1 x 4 x + C 19
39. Suppose an individual calculated the velocity v of a vehicle at time t and found the following: t (seconds) 4 8 1 16 v (feet/second) 1 3 5 8 Use Simpson s rule to estimate the distance traveled in feet. (a) 195 (b) 134 3 (c) 5 (d) 536 * Assume the function, f(t), can be used to model the data above. If it is known that 3 f (4) (t) for all t, estimate the error involved in the approximation above. ln ln x 4. Use L Hospital s Rule to calculate the limit: lim x ln x (a) e (b) (c) 1 (d) Cannot be calculated from L Hospital s Rule
41. 5x + 4 x + x dx = ( ) x 1 (a) 5 ln x + x + 4 tan 1 + C x + (b) 3 ln x 1 + ln x + + C (c) 5 ln x + x + C (d) 3 ln x 1 ln x + + C 4. Evaluate the integral x + x (x + 1) dx = (a) x tan 1 (x) + C (b) x 1 ln(x + 1) + C (c) x tan 1 (x) + C (d) x ln(x + 1) + C 1
43. Determine the behavior of the limits (a) Both converge (b) A converges and B diverges (c) A diverges and B converges (d) Both diverge ln x (x 1) A = lim B = lim x 1 (x 1) x 1 ln x 44. Let R be the region bounded by the graphs of y = (x 1) and y = x + 1. Set up an integral which expresses the y-coordinate of the centroid of R. (a) (b) (c) (d) 4 4 3 3 3 3 3 3 1 ((x + 1) (x 1) 4 ) dx ((x + 1) (x 1) ) dx 1 ((x + 1) (x 1) 4 ) dx ((x + 1) (x 1) ) dx 1 ((x + 1) (x 1) ) dx ((x + 1) (x 1) ) dx x((x + 1) (x 1) ) dx ((x + 1) (x 1) ) dx
4 1 45. Which of the following represents the Simpson s Rule approximation of dx with 1 x n = 6? (a) 1 [ 1 + 8 6 9 + 1 + 8 5 + 9 + 8 49 + 1 ] 16 (b) 1 [ 1 + 1 3 4 + 1 9 + 1 ] 16 (c) 1 [1 + + 3 + 4] 6 (d) 1 [ 1 + 16 6 9 + 1 + 16 5 + 9 + 16 49 + 1 ] 16 46. NOT ON SPRING 15 SYLLABUS After using the table formula u a du = u u a a ln u + u a + C, the integral x x 4 4 dx becomes ( x (a) x4 4 4 ln x + ) x 4 4 + C ( x (b) x4 4 4 ln x + x 4 4 ) + C (c) 1 ( x x4 4 4 ln x + ) x 4 4 + C (d) 1 ( x x4 4 4 ln x + x 4 4 ) + C 3
47. You are planning to use Simpson s Rule to estimate the value of the integral 1 f(x) dx with an error magnitude less than 1 5. You have determined that f (4) 3 throughout the interval of integration. How many subintervals should you use to assure the required accuracy? (a) 6 (b) 8 (c) 1 (d) None of these 48. According to the error-bound formula for the Trapezoidal Rule, how many subintervals should you use to be sure of estimating the value of ln 3 = 3 1 1 x dx by the Trapezoidal Rule with an error of no more than 1 4? (a) 15 (b) 8 (c) 116 (d) None of these 4
49. The length of time spent waiting in line at a certain bank is modeled by an exponential density function with mean 8 minutes. What is the probability that a customer has to wait more than 1 minutes? (a).313 (b).687 (c).87 (d).713 5. Suppose f(x) = k for x 1 and f(x) = for x < 1. Find the value of k for which x3/ f(x) is a probability density function. (a) 1/ (b) 1 (c) (d) 3/ 5
51. Consider the sequence: {a n } = { ln(5n+6) n } + n 1 3n+1, n = 1,, 3,.... This sequence: (a) converges to 3 (b) converges to (c) converges to 5 3 (d) diverges 5. Consider the sequence a n = n sin ( ) 1 n for n = 1,, 3,... (a) The sequence diverges (b) The sequence converges and a n (c) The sequence converges and a n 1 (d) The sequence converges and a n 1 6
53. Evaluate the following infinite series ( 1) n n n= 3 n (a) 4 15 (b) 4 3 (c) 5 (d) n 54. Suppose that the series a n has partial sums s n = a k satisfying k=1 Then the series a n s n = (a) diverges because for every n, s n 1. (b) converges to 1. ( 1 + 1 ) n n (c) converges to e. (d) There is not enough information to determine whether the series or diverges. a n converges 7
55. Let S n = correct? n k=1 a k. If S n = n + 3 n + 1 for each n 1, which of the following statements is (a) lim n a n = (b) lim n a n = (c) a n is convergent and its sum is 3 (d) a n is divergent. 56. The sum of the series 5( ) n 3 n+1 equals (if it converges) (a) 3 (b) 1 (c) 5 (d) The series diverges 8
57. The radius of convergence of the power series (a) 1 (b) 1 (c) (d) n n+1 (x + 1)n is 58. Find the open interval of convergence for the following power series (a) ( 1, ) 3 n (x 1)n n! (b) (, ) (c) ( 1, 3) (d) (, ) 9
59. Determine the interval of convergence of the series (a) [ 1, 5) (b) [ 5, 1) (c) [ 3, 3) (d) [1, 3) (x ) n n3 n. 6. Consider the series ( x) n n. This series: (a) converges at x = 1 and at x = 1 (b) converges at x = 1 but diverges at x = 1 (c) diverges at x = 1 but converges at x = 1 (d) diverges at x = 1 and at x = 1 3
61. Find the sum of the series ( 1 + n ) n= 3 n+1 (a) 5 (b) 9 (c) 3 (d) 1 6. Let a n = π 3 n Which one of the following statements is true? (a) The sequence {a n } is convergent and the series a n is convergent. (b) The sequence {a n } is divergent and the series (c) The sequence {a n } is divergent and the series (d) The sequence {a n } is convergent and the series a n is divergent. a n is convergent. a n is divergent. 31
63. Find the interval of convergence for the series (x + 5) n n (a) ( 6, 4) (b) [ 3, ) (c) ( 6, 4] (d) ( 3, ] 64. Find the sum of the series if it converges. (a) 38 3 (b) 9 3 (c) 17 3 ( ) 5 4 + n+1 n 3 n (d) The series diverges. 3
65. Let A = then we can say the following: (a) Both A and B diverge. (b) A converges, but B diverges. (c) Both A and B converge. (d) A diverges, but B converges. ( 1 + 5 n) 1/n, B = n= cos(nπ) n 66. What is the sum of the series below if it converges? (a) The series diverges. (b) 1 (c) 5 5( 3) n 1 4 n (d) 5 7 33
67. The interval of convergence of the series ( 1) n (x + ) n 3 n n is (a) [ 3, 3) (b) [ 3, 3] (c) [ 5, 1) (d) ( 5, 1] 68. Let A be a series whose partial sum is S n = + ( 1) n sin ( ) 1 n. Then (a) A converges to 1. (b) A converges to. (c) A converges to 3. (d) A diverges. 34
69. Let A and B denote the infinite series A = 7 n + 4 n+1, B = ( 1) n+1 n n 3 (a) A diverges and B converges (b) A converges and B diverges (c) Both series converge (d) Both series diverge 7. The open interval of convergence of (a) ( 1, 1 ) (x + 1) n n! equals (b) ( 1, 1) (c) ( 1, ) (d) (, ) 35
71. Which of the following series converges? (a) (b) (c) (d) ( 1) n+1 n ( 1 n (1 1 n ) 1 + sin n n ) 1 n + 1 7. The open interval of convergence of the series (a) (b) ( 13 ) 3, 11 3 ( 11 3, 13 ) 3 3 n (x + 4) n n is (c) { 5, 3} (d) ( 7, 1) 36
73. The first three nonzero terms of the Maclaurin series for f(x) = x cos(x) are (a) x x 3 + 3 x5 (b) x 4x 3 + 16x 5 (c) x 8x 4 + 3x 6 (d) x 4 3 x4 + 4 15 x6 74. With the help of sin x = ( 1) n x n+1 1/ n= (n + 1)! the integral sin x dx can be written as x the following power series (a) (b) (c) (d) n= n= n= n= ( 1) n (1/) 4n (n + 1)! ( 1) n (1/) 4n (n + 1)! 4 ( 1) n (1/) 4n (n + 1)! 4n ( 1) n (1/) 4n (n + 1)! 4(4n + ) 37
75. The power series representation of f(x) = x3 x 1 is (a) (b) (c) (d) x n+3 n= x n+3 n= x n+ n= x n+ n= 76. The first four terms of the Taylor series generated by f(x) = 1 x at x = 1 are (a) 1 (x 1) + (x 1) 6(x 1) 3 (b) 1 x (c) 1 x (x 1) + x x (x 3 1) 6 (x 1)3 x4 (x 1) + 1 x x (x 3 1) 1 (x 1)3 x4 (d) 1 (x 1) + (x 1) (x 1) 3 38
77. The following is the sum of the first four terms of a Taylor series of a function f at a = 1. 1 + 3(x 1) + 5(x 1) + (x 1) 3 Which one of the following statements must be true? (a) f (1) = 5 (b) f (1) + f (1) = 4 (c) f() = (d) f (1) + f (1) = 16 78. The first three nonzero terms of the MacLaurin series for f(x) = 1 x cos(πx) are: (a) π x π4 4 x4 + π6 7 x6 (b) 1 x + 1 6 x3 (c) x 1 x3 + 1 4 x5 (d) 1 x + π x3 39
79. Which of the following represents the graph of the parametric equations x = 5t + 1 and y = 4t 1 on the interval t? (a) (b) 5 1 1 1 5 5 1 5 1 (c) (d) 6 4 5 1 1 6 4 4 6 4 5 6 4
8. Which of the following is the graph of the parametric equations x = 3 sin(3t) y = 3 cos(3t), t π 5 (a) 3 (b) 3 1 1 3 1 1 3 3 1 1 3 1 1 (c) 3 3 (d) 3 3 1 1 3 1 1 3 3 1 1 3 1 1 3 3 41
81. Which of the following represents the Cartesian equation of the parametric equations: x = sin t + 3 y = 4 sin 4 t (a) x + 3 (b) 3 x /8 (c) (x 3) (d) x 3 8. Describe the motion of the parametric equations x = cos t y = sin t, 3π 4 t 5π 4 (a) The motion starts at (, 1/ ) and goes Clockwise (b) The motion starts at (, 1/ ) and goes Counterclockwise (c) The motion starts at (, 1/ ) and goes Clockwise (d) The motion starts at (, 1/ ) and goes Counterclockwise 4
83. Which of the following is the graph of the parametric equations x = sin t y = cos(t), t π (a) 1. (b) 1..5.5 1..5.5 1..5.5.5.5 (c) 1. 1. (d) 1. 1..5.5 1..5.5 1..5.5.5.5 1. 1. 43
1 x 5 y 4 84. 5.5 1. 1.5..5 3. t.5 1. 1.5..5 3. t 1 4 Given the graph of x = f(t) and the graph of y = g(t), shown above, which graph represents the parametric curve x = f(t), y = g(t)? (a) y 4 (b) y 4 1 5 5 1 x 1 5 5 1 x (c) 4 y 4 (d) 4 y 4 1 5 5 1 1 5 5 1 x 4 4 44
85. Which of the following is the Cartesian equation of the parametric equations x = cos (sin t)) y = 3 sin (sin t)) (a) x + y = (b) x + y 3 = (c) x + 4y 9 = 4 (d) x + y 9 = 4 86. The Cartesian coordinates for the polar point (r, θ) = ( 3, 7π 6 ) are (a) (b) (c) (d) ( 3 3 4, 3 ) 4 ( 3 ) 3 4, 3 4 ( ) 3 3, 3 ( 3 ) 3, 3 45
87. The polar equation for the curve given by x + (y 3) = 9 is (a) r = 3 (b) r = 3 sin θ (c) r = 6 sin θ (d) r = 3 cos θ 88. The polar equation for the curve given by (x 4) + y = 16 is (a) r = 8 cos θ (b) r = 4 sin θ (c) r = 4 (d) r = 4 cos θ 46
89. Which of the following graphs represents r = sin(3θ) for π 6 θ 5π 6? (a) 1. (b) 1..5.5 1..5.5 1. 1..5.5 1..5.5 1. 1. (c) 1. (d) 1..5.5 1..5.5 1. 1..5.5 1..5.5 1. 1. 47
9. Which of the following graphs represents the polar equation r = 5 cos θ + 1? (a) 6 (b) 4 4 6 4 4 6 4 6 4 4 6 4 (c) 6 (d) 6 3 4 1 6 4 4 6 4 6 1 3 1 3 4 5 6 48
91. Which of the following graphs represents r 1, π 3 θ 7π 6? (a) (b) (c) (d) 49
9. Which Cartesian equation is equivalent to the polar equation r = (a) y = x 8 (b) y = x 8 + (c) y = 4 x 4 (d) y = x 4 + 4 8 sin θ +? 93. How many of the following polar points represent the point shown in the graph below? ( 1, 5π 6 ), ( 1, π ), 6 ( 1, 7π 6 ), ( 1, 11π 6 ), ( 1, π ) 6 1 1 1 1 (a) 1 of the answers (b) of the answers (c) 3 of the answers (d) 4 of the answers (e) 5 of the answers 5