Name: Class: Date: Calculus 437 Semester 1 Review Chapters 1, 2, and 3 January 2016 Short Answer 1. Decide whether the following problem can be solved using precalculus, or whether calculus is required. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, use a graphical or numerical approach to estimate the solution. Find the distance traveled in 16 seconds by an object traveling at a constant velocity of 20 feet per second. 2. Complete the table and use the result to estimate the limit. lim x 3 x 3 x 2 16x + 39 x 2.9 2.99 2.999 3.001 3.01 3.1 f(x) 3. Determine the following limit. (Hint: Use the graph to calculate the limit.) lim ( 5 x) x 1 1
4. Let f x Ï ( ) = x2 + 5, Ì Ô x 1 ÓÔ 1, x = 1. Determine the following limit. (Hint: Use the graph to calculate the limit.) lim f( x) x 1 5. Find the limit L. lim (x + 2) x 7 6. Find the limit. lim x 4 9x 2 + 36x 7. Find the limit. lim x 6 x x 2 + 8 8. Let f(x) = 3 + 2x 2 and g ( x) = x + 3. Find the limit. lim x 2 g f( x) ˆ 2
9. Find the following limit (if it exists). Write a simpler function that agrees with the given function at all but one point. 8x 2 + 40x + 32 lim x 4 x + 4 10. Find the limit (if it exists). lim x 8 x + 8 x 2 64 11. Use the graph as shown to determine the following limits, and discuss the continuity of the function at x = 3. (i) lim f(x) x 3 + (ii) lim f(x) (iii) lim f(x) x 3 x 3 3
12. Use the graph to determine the following limits, and discuss the continuity of the function at x = 3. (i) lim f(x) x 3 + (ii) lim f(x) (iii) lim f(x) x 3 x 3 13. Find the limit (if it exists). lim x 11 + 11 x x 2 121 14. Find the limit (if it exists). x 6 lim x 36 x 36 4
15. Discuss the continuity of the function f( x) = x 2 4 x 2. 16. Find the x-values (if any) at which f( x) = x x 2 2x is not continuous. 17. Find the slope m of the line tangent to the graph of the function f( x) = 2 7x at the point 1,9 ˆ. 18. Find the slope m of the line tangent to the graph of the function g ( x) = 9 x 2 at the point 4, 7 ˆ. 19. Find the derivative of the function g ( x) = 2 by the limit process. 20. Find the derivative of the following function f(x) = 3x 2 + 6x 8 using the limiting process. 21. Find the derivative of the following function using the limiting process. f( x) = 3x 3 9x 2 8 22. Find the derivative of the following function using the limiting process. f( x) = 1 x 4 23. Find the derivative of the function f( x) = 20 x by the limit process. 5
24. Find an equation of the line that is tangent to the graph of f and parallel to the given line. f( x) = 3x 3, 9x y + 9 = 0 25. Identify the graph which has the following characteristics. f( 0) = 2 f ʹ ( x) = 2, < x < Graph 1 Graph 2 Graph 3 Graph 4 26. Use the alternative form of the derivative to find the derivative of the function f( x) = x 2 9 at x = 5. 27. Use the alternative form of the derivative to find the derivative of the function f( x) = 3 at x = 2. 2 x 6
28. Describe the x-values at which the graph of the function f( x) = x 2 x 2 9 given below is differentiable. 29. Find the derivative of the function. f( x) = x 4 30. Find the derivative of the function. f( x) = 1 x 8 31. Find the derivative of the function f( x) = 7x 3 + 4x 2 + 1. 32. Find the derivative of the function f( x) = 5x 3 2 sin ( x). 33. Find the slope of the graph of the function at the given value. f( x) = 2x 2 + 6 x when x = 5 2 34. Find the slope of the graph of the function f( x) = x 2x 3 ˆ + 7 at x = 4. 35. Find the slope of the graph of the function f( x) = ( 4x 7) 2 at x = 4. 7
36. Find the derivative of the function f( x) = x 5 9. x 4 37. Determine all values of x, (if any), at which the graph of the function has a horizontal tangent. y( x) = x 3 + 12x 2 + 8 38. Determine all values of x, (if any), at which the graph of the function has a horizontal tangent. y( x) = x 4 4x + 4 39. Suppose the position function for a free-falling object on a certain planet is given by s( t) = 16t 2 + v 0 t + s 0. A silver coin is dropped from the top of a building that is 1372 feet tall. È Determine the average velocity of the coin over the time interval ÎÍ 3, 4. 40. Suppose the position function for a free-falling object on a certain planet is given by s( t) = 12t 2 + v 0 t + s 0. A silver coin is dropped from the top of a building that is 1372 feet tall. Find the instantaneous velocity of the coin when t = 4. 41. A ball is thrown straight down from the top of a 300-ft building with an initial velocity of 12 ft per second. The position function is s( t) = 16t 2 + v 0 t + s 0. What is the velocity of the ball after 4 seconds? 42. Find the derivative of the algebraic function H( v) = v 5 ˆ 3 v3 ˆ + 3. 43. Use the Product Rule to differentiate f( u) = u 5 u 6 ˆ Ë Á. 44. Use the Product Rule to differentiate f( s) = s 5 cos s. 45. Use the Quotient Rule to differentiate the function f( x) = 8x x 5. + 3 46. Use the Quotient Rule to differentiate the function f ʹ ( x) = 4 + x x 2. + 9 8
47. Use the Quotient Rule to differentiate the function f( x) = sin x x 2. + 3 48. Find the derivative of the algebraic function f( x) = x 3 4 ˆ Ë Á x + 6. 49. Find the derivative of the function. f( s) = 9s sin s + 5cos. s 50. Find the derivative of the function. f( t) = 2t 3 sin t + 5t 6 cos t 51. Determine all values of x, (if any), at which the graph of the function has a horizontal tangent. y( x) = 6x ( x 9 ) 2 5 9 52. Find the second derivative of the function f( x) = 8 x. 53. Find the second derivative of the function f( x) = 3 x2 + 5 x 4 x 54. Suppose that an automobile's velocity starting from rest is v( t) = 240 t where v is measured in feet per 5 t + 13 second. Find the acceleration at 9 seconds. Round your answer to one decimal place. 55. Find the derivative of the algebraic function f( x) = x 6 ˆ + 4 56. Find the derivative of the function. 5.. f( t) = ( 1 + 8t) 5 9 9
57. Find the derivative of the function. g ( x) = x + 6 x 2 + 7 ˆ 4 58. Find the derivative of the function y = 8 sin 5x. 59. Find the derivative of the function. f( t) = 5 sec 2 ( 7πt 5) 60. Evaluate the derivative of the function at the given point. f( t) = 7 t 1, 5, 7 4 ˆ 61. Find an equation to the tangent line to the graph of the function f( x) = tan 8 x at the point 4π 5,0.078 ˆ The coefficients below are given to two decimal places.. 62. Find the second derivative of the function f( x) = sin 5x 6. 63. Find dy dx by implicit differentiation. x 2 + y 2 = 25 64. Find dy dx by implicit differentiation. x 2 + 5x + 9xy y 2 = 4 65. Find dy dx by implicit differentiation given that 2xy = 9. 66. Find dy dx by implicit differentiation. x 3 + 8x + x 13 y y 6 = 4 10
67. Find dy dx by implicit differentiation. sin x + 7 cos14y = 2 68. Find the slope of the tangent line ( 16 x)y 2 = x 3 at the given point 8,8 ˆ. Round your answer to two decimal places. 69. Find an equation of the tangent line to the graph of the function given below at the given point. 7x 2 2xy + 4y 2 36 = 0, 2, 1 ˆ (The coefficients below are given to two decimal places.) 70. Assume that x and y are both differentiable functions of t. Find dy when x = 49 and dx dt dt equation y = x. = 17 for the 71. A point is moving along the graph of the function y = Find dy dt when x = 2. 1 9x 2 + 4 such that dx dt = 2 centimeters per second. 72. The radius, r, of a circle is decreasing at a rate of 5 centimeters per minute. Find the rate of change of area, A, when the radius is 6. 73. The radius r of a sphere is increasing at a rate of 6 inches per minute. Find the rate of change of the volume when r = 11 inches. 74. A spherical balloon is inflated with gas at the rate of 300 cubic centimeters per minute. How fast is the radius of the balloon increasing at the instant the radius is 70 centimeters? 75. All edges of a cube are expanding at a rate of 9 centimeters per second. How fast is the volume changing when each edge is 2 centimeters? 76. A conical tank (with vertex down) is 12 feet across the top and 18 feet deep. If water is flowing into the tank at a rate of 18 cubic feet per minute, find the rate of change of the depth of the water when the water is 10 feet deep. 11
77. A ladder 20 feet long is leaning against the wall of a house (see figure). The base of the ladder is pulled away from the wall at a rate of 2 feet per second. How fast is the top of the ladder moving down the wall when its base is 13 feet from the wall? Round your answer to two decimal places. 78. A ladder 25 feet long is leaning against the wall of a house (see figure). The base of the ladder is pulled away from the wall at a rate of 2 feet per second. Consider the triangle formed by the side of the house, the ladder, and the ground. Find the rate at which the area of the triangle is changed when the base of the ladder is 11 feet from the wall. Round your answer to two decimal places. 12
79. A man 6 feet tall walks at a rate of 10 feet per second away from a light that is 15 feet above the ground (see figure). When he is 13 feet from the base of the light, at what rate is the tip of his shadow moving? 80. A man 6 feet tall walks at a rate of 13 feet per second away from a light that is 15 feet above the ground (see figure). When he is 5 feet from the base of the light, at what rate is the length of his shadow changing? 13
81. A man 6 feet tall walks at a rate of 2 ft per second away from a light that is 16 ft above the ground (see figure). When he is 8 ft from the base of the light, find the rate.at which the tip of his shadow is moving. 82. Find the value of the derivative (if it exists) of the function f( x) = 0,0 ˆ. x 2 x 2 + 49 at the extremum point 14
83. Find the value of the derivative (if it exists) of the function f( x) = 15 x at the extremum point 0, 15 ˆ. 84. Find all critical numbers of the function g ( x) = x 4 4x 2. 85. Find any critical numbers of the function g ( t) = t 2 t, t < 2. 86. Find all critical numbers of the function f( x) = sin 2 6x + cos6x, 0 < x < π 3. 87. Locate the absolute extrema of the function g ( x) = 4x + 5 5 È on the closed interval ÎÍ 0, 5. 88. Locate the absolute extrema of the function g ( t) = t 2 t 2 + 2 È on the closed interval ÎÍ 3, 3. È 89. Locate the absolute extrema of the function f(x) = sinπx on the closed interval 0, 1 ÎÍ 3. 90. The formula for the power output of battery is P = VI RI 2 where V is the electromotive force in volts, R is the resistance, and I is the current. Find the current (measured in amperes) that corresponds to a maximum value of P in a battery for which V = 12 volts and R = 0.8 ohm. Assume that a 10-ampere fuse bounds the output in the interval 0 I 10. Round your answer to two decimal places. 91. Determine whether Rolle's Theorem can be applied to f( x) = x 2 È + 10x on the closed interval ÎÍ 0,10. If Rolle's Theorem can be applied, find all values of c in the open interval 0, 10 ˆ such that f ʹ ( c) = 0. 15
92. Determine whether Rolle's Theorem can be applied to the function f( x) = x 2 2x 3 on the closed interval [ 1,3]. If Rolle's Theorem can be applied, find all values of c in the open interval ( 1,3) such that f ʹ ( c) = 0. 93. Determine whether Rolle's Theorem can be applied to f( x) = x 2 13 È on the closed interval 13,13 x ÎÍ. If Rolle's Theorem can be applied, find all values of c in the open interval 13,13 ˆ such that f ʹ ( c) = 0. 94. Determine whether the Mean Value Theorem can be applied to the function f( x) = x 2 on the closed interval [3,9]. If the Mean Value Theorem can be applied, find all numbers c in the open interval (3,9) such that f ʹ ( c) = f( 9) f( 3). 9 ( 3) 95. Determine whether the Mean Value Theorem can be applied to the function f( x) = x 3 on the closed interval [0,16]. If the Mean Value Theorem can be applied, find all numbers c in the open interval (0,16) such that f ʹ ( c) = f( 16) f( 0). 16 0 96. The height of an object t seconds after it is dropped from a height of 550 meters is s( t) = 4.9t 2 + 550. Find the average velocity of the object during the first 7 seconds. 97. The height of an object t seconds after it is dropped from a height of 250 meters is s( t) = 4.9t 2 + 250. Find the time during the first 8 seconds of fall at which the instantaneous velocity equals the average velocity. 98. A company introduces a new product for which the number of units sold S is S ( t) = 300 7 10 ˆ 5 + t where t is the time in months since the product was introduced. Find the average value of S ( t) during the first year. 99. A plane begins its takeoff at 2:00 P.M. on a 2200-mile flight. After 12.5 hours, the plane arrives at its destination. Explain why there are at least two times during the flight when the speed of the plane is 100 miles per hour. 100. Which of the following functions passes through the point 0,10 ˆ and satisfies f ʹ ( x) = 12? 101. Find a function f that has derivative f ʹ ( x) = 12x 6 and with graph passing through the point (5,6). 16
102. Use the graph of the function y = x3 3x given below to estimate the open intervals on which the 4 function is increasing or decreasing. 103. Identify the open intervals where the function f( x) = 6x 2 6x + 4 is increasing or decreasing. 104. Identify the open intervals where the function f( x) = x 30 x 2 is increasing or decreasing. 105. Find the critical number of the function f( x) = 6x 2 + 108x + 7. 106. Find the open interval(s) on which f( x) = 2x 2 + 12x + 8 is increasing or decreasing. 107. Find the relative extremum of f( x) = 9x 2 + 54x + 2 by applying the First Derivative Test. 108. For the function f( x) = 4x 3 48x 2 + 6: (a) Find the critical numbers of f (if any); (b) Find the open intervals where the function is increasing or decreasing; and (c) Apply the First Derivative Test to identify all relative extrema. Then use a graphing utility to confirm your results. 17
109. The graph of f is shown in the figure. Sketch a graph of the derivative of f. 110. The graph of f is shown in the figure. Sketch a graph of the derivative of f. 18
111. The graph of f is shown in the figure. Sketch a graph of the derivative of f. 112. The graph of f is shown in the figure. Sketch a graph of the derivative of f. 113. Determine the open intervals on which the graph of f( x) = 3x 2 + 7x 3 is concave downward or concave upward. 114. Determine the open intervals on which the graph of y = 6x 3 + 8x 2 + 6x 5 is concave downward or concave upward. 115. Determine the open intervals on which the graph of the function y = 3x tan x, π 2, π ˆ is concave 2 upward or concave downward. 19
116. Find all points of inflection on the graph of the function f( x) = 1 2 x 4 + 2x 3. 117. Find the points of inflection and discuss the concavity of the function. f( x) = 4x 3 5x 2 + 5x 7 118. Find the points of inflection and discuss the concavity of the function f( x) = x x + 16. 119. Find all points of inflection, if any exist, of the graph of the function f x ( ) = x x + 3. Round your answers to two decimal places. 20
ID: A Calculus 437 Semester 1 Review Chapters 1, 2, and 3 January 2016 Answer Section SHORT ANSWER 1. precalculus, 320 ft 2. 0.100000 3. 4 4. 6 5. L = 9 6. 0 3 7. 22 8. 14 9. 24 10. 1 16 11. 1,1,1, not continuous 12. 0, 1, does not exist, not continuous 13. 1 22 1 14. 12 15. f( x) is discontinuous at x = 2. 16. f( x) is not continuous at x = 0,2 and f( x) has a removable discontinuity at x = 0. 17. m = 7 18. m = 8 19. g ʹ ( x) = 0 20. f ʹ ( x) = 6x + 6 21. f ʹ ( x) = 9x 2 18x 22. f ʹ ( x) = 4 x 5 23. f ʹ ( x) = 10 24. y = 9x 6 and y = 9x + 6 x 25. Graph 3 26. f ʹ 5 ˆ = 10 x 27. f ʹ ( 2) = 3 4 28. f( x) is differentiable everywhere except at x = ±3. 29. f ʹ ( x) = 4x 3 1
ID: A 30. f ʹ ( x) = 8 x 9 31. f ʹ ( x) = 21x 2 + 8x 32. f ʹ ( x) = 15x 2 2 cos( x) 33. f ʹ ( 5) = 2512 125 34. f ʹ ( 4) = 519 35. f ʹ ( 4) = 72 36. f ʹ ( x) = 1 + 36 x 5 37. x = 0 and x = 8 38. x = 1 39. 112 ft/sec 40. 96 ft/sec 41. The velocity after 4 seconds is 140 ft per second. 42. H ʹ ( s) = 8v 7 + 15v 4 9v 2 43. f ʹ ( u) = 6u 11 2 + 5 u 6 2 u 44. f ʹ ( s) = s 5 sin s + 5s 4 cos s 8 3 + 4x 5 ˆ 45. f ʹ Ë Á ( x) = x 5 ˆ2 + 3 Ë Á 46. f ʹ ( x) = 47. f ʹ ( x) = 2 ˆ 9 8x x Ë Á x 2 ˆ2 + 9 Ë Á 2 ˆ 3 + x Ë Á cos x 2x sin x x 2 ˆ2 + 3 Ë Á 2
ID: A 48. f ʹ ( x) = 49. f ʹ s 50. f ʹ t 51. x = 9 84 + 36x + 3x2 ( x + 6) 2 ( ) = 9s cos s + 4 sin s ( ) = 6t 2 5t 6 ˆ Ë Á sin t + 2t3 + 30t 5 ˆ Ë Á 52. f ʺ ( x) = 160 81 x 13 9 53. f ʺ ( s) = 8 x 3 54. 0.9 ft/sec 2 55. f ʹ ( x) = 30x 5 x 6 ˆ + 4 56. f ʹ ( t) = 40 ( 1 + 8t) 4 9 9 4 7 12x x 2 57. g ʹ ( x) = 58. yʹ = 40 cos5x 4 7 + x 2 ˆ ˆ ( 6 + x) 3 59. f ʹ ( t) = 70πsec 2 ( 7πt 5) tan ( 7πt 5) 60. f ʹ ( 5) = 7 16 61. y = 1.31x + 3.36 62. f ʺ ( x) = 150x 4 cos5x 6 900x 10 sin 5x 6 dy 63. dx = x y 64. 65. 66. 67. dy dx dy = 2x + 5 + 9y 2y 9x dx = y x dy dx = 3x 2 + 8 + 13x 12 y 6y 5 x 13 dy dx = 68. 2.00 cosx 98 sin 14y 5 cos t 3
ID: A 69. y = 2.50x 6.00 70. 71. 72. 73. 74. dy dt = 17 14 dy dt = 9 200 da = 60π sq cm/min dt dv = 2904π in 3 / min dt dr 3 = dt 196π cm/min 75. 108 cm 3 / sec 81 76. 50π ft/min 77. 1.71 ft/sec 78. 20.06ft 2 /sec 50 79. 3 ft/sec 26 80. 3 ft/sec 16 81. ft per minute 5 82. 0 83. does not exist 84. critical numbers: x = 0, x = 2, x = 2 4 85. 3 π 86. 18, π 6, 5π 18 87. absolute maximum: 5, 5 ˆ absolute minimum: 0, 1 ˆ 9 88. The absolute maximum is, and it occurs at either endpoint x = ±3. 11 The absolute minimum is 0, and it occurs at the critical number x = 0. 89. The absolute min imum is 0, and it occurs at the left endpoint x = 0. 3 The absolute max imum is 2, and it occurs at the right endpoint x = 1 3. 90. 45.00 amperes 91. Rolle's Theorem applies; c = 5 4
ID: A 92. Rolle's Theorem applies; c = 1 93. Rolle's Theorem does not apply 94. MVT applies; c = 6 95. MVT applies; 16 3 3 96. 34.30 m/sec 97. 4 seconds 600 98. 17 99. By the Mean Value Theorem, there is a time when the speed of the plane must equal the average speed of 88 mi/hr. The speed was 100 mi/hr when the plane was accelerating to 88 mi/hr and decelerating from 88 mi/hr. 100. f( x) = 12x + 10 101. f( x) = 6x 2 6x 114 102. increasing on Ë Á, 2 ˆ and Ë Á2, ˆ ; decreasing on 2,2 ˆ 103. decreasin g:, 1 ˆ 2 ; increasin g: 1 2, ˆ ˆ ˆ ˆ 104. increasing: 15, 15 ; decreasing: 30, 15 15, 30 105. x = 9 106. increasing on Ë Á,3 ˆ ; decreasing on 3, ˆ 107. relative maximum: 3,83 ˆ 108. (a) x = 0, 8 (b) increasing: Ë Á,0 ˆ Ë Á8, ˆ ; decreasing: 0,8 ˆ (c) relative max: f( 0) = 6 ; relative min: f( 8) = 1018 109. 5
ID: A 110. 111. 112. 113. concave upward on, ˆ 114. concave upward on, 4 ˆ 9 ; concave downward on 4 9, ˆ 115. concave upward: π 2,0 ˆ ; concave downward: 0, π ˆ 2 6
ID: A 116. 0,0 ˆ 2, 8 ˆ 117. inflection point at x = 5 12 ; concave upward on, 5 ˆ 12 ; concave downward on 5 12, ˆ 118. no inflection points; concave up on 16, ˆ 119. no points of inflection 7