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All Textbook Solutions for Calculus: Early Transcendentals (3rd Edition)

Describe the set of antiderivatives of f(x) = 1.Why do two different antiderivatives of a function differ by a constant?Give the antiderivatives of xp. For what values of p does your answer apply?Give the antiderivatives of a/1x2, where a is a constant.Give the antiderivatives of 1/x.Evaluate acosxdxand asinxdx, where a is a constant.If F(x) = x2 3x + C and F(1) = 4, what is the value of C?For a given function f, explain the steps used to solve the initial value problem F(t) = f(t), F(0) = 10.Finding antiderivatives Find all the antiderivatives of the following functions. Check your work by taking derivatives. 11. f(x) = 5x4 Finding antiderivatives Find all the antiderivatives of the following functions. Check your work by taking derivatives. 12. g(x) = 11x10 Finding antiderivatives Find all the antiderivatives of the following functions. Check your work by taking derivatives. 13. f(x)=2sinx+1 Finding antiderivatives Find all the antiderivatives of the following functions. Check your work by taking derivatives. 14. g(x)=4cosxx Finding antiderivatives Find all the antiderivatives of the following functions. Check your work by taking derivatives. 15. p(x)=3sec2x Finding antiderivatives Find all the antiderivatives of the following functions. Check your work by taking derivatives. 16. q(s)=csc2s Finding antiderivatives Find all the antiderivatives of the following functions. Check your work by taking derivatives. 17. f(y) = 2/y3 Finding antiderivatives Find all the antiderivatives of the following functions. Check your work by taking derivatives. 18. H(z) = 6z7 Finding antiderivatives Find all the antiderivatives of the following functions. Check your work by taking derivatives. 19. f(x) = ex Finding antiderivatives Find all the antiderivatives of the following functions. Check your work by taking derivatives. 20. h(y) = y1 21E Finding antiderivatives Find all the antiderivatives of the following functions. Check your work by taking derivatives. 22. (l)=Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 23. (3x55x9)dx Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 24. (3u24u2+1)du Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 25. (4x4x)dx Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 26. (5t2+4t2)dt Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 27. (5s+3)2ds Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 28. 5m(12m310m)dm Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 29. (3x1/3+4x1/3+6)dx Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 30. 6x3dx Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 31. (3x+1)(4x)dx Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 32. (4z1/3z1/3)dz Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 33. (3x4+23x2)dx Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 34. r25dr Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 35. 4x46x2xdx Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 36. 12t8tt3/2dt Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 37. x236x6dx Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 38. y39y2+20yy4dy Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 39. (csc2+223)d Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 40. (csc2+1)d Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 41. 2+3cosysin2ydy Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 42. sint(4csctcott)dt Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 43. (sec2x1)dx Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 44. sec3vsec2vsecv1dv Indefinite integrals involving trigonometric functions Determine the following indefinite integrals. Check your work by differentiation. 41. (sec2+sectan)d Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 44. sin1cos2d Indefinite integrals involving trigonometric functions Determine the following indefinite integrals. Check your work by differentiation. 43. (3t2+sec22t)dt Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 48. cscx(cotxcscxc)dx Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 49. sec(tan+sec+cos)d Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 50. csc3x+1cscxdx Other indefinite integrate Determine the following indefinite integrals. Check your work by differentiation. 47. 12ydy Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 52. e2t1et1dt Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 53. 644x2dx Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 54. v3+v+11+v2dv Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 55. 4xx21dx Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 56. 225z2+25dz Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 57. 1x36x236dx Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 58. (4949x2)1/2dx Other indefinite integrate Determine the following indefinite integrals. Check your work by differentiation. 55. t+1tdt Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 60. t2e2tt+etdt Other indefinite integrate Determine the following indefinite integrals. Check your work by differentiation. 57. ex+2dx Other indefinite integrate Determine the following indefinite integrals. Check your work by differentiation. 58. 10t53tdt Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 63. e2w5ew+4ew1dw Miscellaneous indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 102. (x23+x3)dx Miscellaneous indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 107. 1+xxdx Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 66. 16cos2w81sin2w4cosw9sinwdw Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 67. x(2x643x)dx Miscellaneous indefinite integrals Determine the following indefinite integrals. Check your work by differentiation. 108. 2+x21+x2dx Particular antiderivatives For the following functions f, find the antiderivative F that satisfies the given condition. 69. f(x)=x52x2+1;F(0)=1 Particular antiderivatives For the following functions f, find the antiderivative F that satisfies the given condition. 70. f(x)=4x+6;F(1)=8 Particular antiderivatives For the following functions f, find the antiderivative F that satisfies the given condition. 71. f(x)=8x3+sinx;F(0)=2 Particular antiderivatives For the following functions f, find the antiderivative F that satisfies the given condition. 72. f(t)=sec2t;F(/4)=1,/2t/2 Particular antiderivatives For the following functions f, find the antiderivative F that satisfies the given condition. 73. f(v)=secvtanv;F(0)=0,/2v/2 Particular antiderivatives For the following functions f, find the antiderivative F that satisfies the given condition. 64. f(u) = 2eu + 3; F(0) = 8 Particular antiderivatives For the following functions f, find the antiderivative F that satisfies the given condition. 75. f(y)=3y3+5y;F(1)=3,y0 Particular antiderivatives For the following functions f, find the antiderivative F that satisfies the given condition. 66. f()=2sin24cos4;F(4)=2 Solving initial value problems Find the solution of the following initial value problems. 67. f(x) = 2x 3; f(0) = 4 Solving initial value problems Find the solution of the following initial value problems. 68. g(x) = 7x6 4x3 + 12; g(1) = 24 Solving initial value problems Find the solution of the following initial value problems. 69. g(x)=7x(x617);g(1)=2 Solving initial value problems Find the solution of the following initial value problems. 80. h(t)=1+6sint;h(3)=3 Solving initial value problems Find the solution of the following initial value problems. 81. f(u)=4(cosusinu);f(2)=0 Solving initial value problems Find the solution of the following initial value problems. 82. p(t)=10et+70;p(0)=100 Solving initial value problems Find the solution of the following initial value problems. 83. y(t)=3t+6;y(1)=8,t0 Solving initial value problems Find the solution of the following initial value problems. 84. u(x)=xe2x+4exxex;u(1)=0,x0 Solving initial value problems Find the solution of the following initial value problems. 85. y()=2cos3+1cos3;y(4)=3,/2/2 Solving initial value problems Find the solution of the following initial value problems. 86. v(x)=4x1/3+2x1/3;v(8)=40,x0 Graphing general solutions Graph several functions that satisfy the following differential equations. Then find and graph the particular function that satisfies the given initial condition. 77. f(x) = 2x 5; f(0) = 4 88E Graphing general solutions Graph several functions that satisfy each of the following differential equations. Then find and graph the particular function that satisfies the given initial condition. 89. f(x)=3x+sinx;f(0)=3 Graphing general solutions Graph several functions that satisfy each of the following differential equations. Then find and graph the particular function that satisfies the given initial condition. 90. f(x)=cosxsinx+2;f(0)=1 Velocity to position Given the following velocity functions of an object moving along a line, find the position function with the given initial position. Then graph both the velocity and position functions. 83. v(t) = 2t + 4; s(0) = 0 Velocity to position Given the following velocity functions of an object moving along a line, find the position function with the given initial position. 92. v(t)=et+4;s(0)=2 Velocity to position Given the following velocity functions of an object moving along a line, find the position function with the given initial position. 93. v(t)=2t;s(0)=1 Velocity to position Given the following velocity functions of an object moving along a line, find the position function with the given initial position. Then graph both the velocity and position functions. 86. v(t) = 2 cos t; s(0) = 0 Velocity to position Given the following velocity functions of an object moving along a line, find the position function with the given initial position. Then graph both the velocity and position functions. 87. v(t) = 6t2 + 4t 10; s(0) = 0 Velocity to position Given the following velocity functions of an object moving along a line, find the position function with the given initial position. 96. v(t) = 4t + sin t; s(0) = 0 Acceleration to position Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position. 89. a(t) = 32; v(0) = 20, s(0) = 0 Acceleration to position Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position. 90. a(t) = 4; v(0) = 3, s(0) = 2 Acceleration to position Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position. 91. a(t) = 0.2 t; v(0) = 0, s(0) = 1 Acceleration to position Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position. 92. a(t) = 2 cos t; v(0) = 1, s(0) = 0 Acceleration to position Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position. 101. a(t)=2+3sint;v(0)=1,s(0)=10 102E A car starting at rest accelerates at 16 ft/s2-for 5 seconds on a straight road. How far does it travel during this time?104E Races The velocity function and initial position of Runners A and B are given. Analyze the race that results by graphing the position functions of the runners and finding the lime and positions (if any) at which they first pass each other. 95. A: v(t) = sin t, s(0) = 0; B: V(t) = cos t, S(0) = 0 106E Motion with gravity Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation a(t) = v(t) = -g, where g = 9.8 m/s2. a. Find the velocity of the object for all relevant times. b. Find the position of the object for all relevant times. c. Find the time when the object reaches its highest point. What is the height? d. Find the time when the object strikes the ground. 107. A softball is popped up vertically (from the ground) with a velocity of 30 m/s.Motion with gravity Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation a(t) = v (t) = g, where g = 9.8 m/s2. a. Find the velocity of the object for all relevant times. b. Find the position of the object for all relevant times. c. Find the time when the object reaches its highest point. What is the height? d. Find the time when the object strikes the ground. 108. A stone is thrown vertically upward with a velocity of 30 m/s from the edge of a cliff 200 m above a river.Motion with gravity Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation a(t) = v (t) = g, where g = 9.8 m/s2. a. Find the velocity of the object for all relevant times. b. Find the position of the object for all relevant times. c. Find the time when the object reaches its highest point. What is the height? d. Find the time when the object strikes the ground. 109. A payload is released at an elevation of 400 m from a hot-air balloon that is rising at a rate of 10 m/s.Motion with gravity Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation a(t) = v (t) = g, where g = 9.8 m/s2. a. Find the velocity of the object for all relevant times. b. Find the position of the object for all relevant times. c. Find the time when the object reaches its highest point. What is the height? d. Find the time when the object strikes the ground. 110. A payload is dropped at an elevation of 400 m from a hot-air balloon that is descending at a rate of 10 m/s.Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. F(x) = x3 4x + 100 and G(x) = x3 4x 100 are anti-derivatives of the same function. b. If F(x) = f(x), then f is an antiderivative of F. c. If F(x) = f(x), then f(x)dx=F(x)+C. d. f(x) = x3 + 3 and g(x) = x3 4 are derivatives of the same function. e. If F(x) = G(x), then F(x) = G(x).112E Functions from higher derivatives Find the function F that satisfies the following differential equations and initial conditions. 111. F(x) = cos x, F(0) = 3, F() = 4 Functions from higher derivatives Find the function F that satisfies the following differential equations and initial conditions. 112. F(x) = 4x, F(0) = 0, F(0) = 1, F(0) = 3 115E 116E How rate A large tank is filled with water when an outflow valve is opened at t = 0. Water flows out at a rate, in gal/min. given by Q(t) = 0.1(100 t2), for 0 t 10. a. Find the amount of water Q(t) that has flowed out of the tank after t minutes, given the initial condition Q(0) = 0. b. Graph the flow function Q, for 0 t 10. c. How much water flows out of the tank in 10 min?118E Verifying indefinite integrals Verify the following indefinite integrals by differentiation. These integrals are derived in later chapters. 118. cosxxdx=2sinx+C 120E 121E 122E Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. If f(c) = 0, then f has a local maximum or minimum at c. b. If f(c) = 0, then f has an inflection point at c. c. F(x) = x2 + 10 and G(x) = x2 100 are antiderivatives of the same function. d. Between two local minima of a function continuous on (, ), there must be a local maximum. e. The Linear approximation to f(x) = sin x at x = 0 is L(x) = x. f. If limxf(x)= and limxg(x)=, then limx(f(x)g(x))=0.Locating extrema Consider the graph of a function f on the interval [3, 3]. a. Give the approximate coordinates of the local maxima and minima of f. b. Give the approximate coordinates of the absolute maximum and minimum of f (if they exist). c. Give the approximate coordinates of the inflection point(s) of f. d. Give the approximate coordinates of the zero(s) of f. e. On what intervals (approximately) is f concave up? f. On what intervals (approximately) is f concave down?Designer functions Sketch the graph of a function continuous on the given interval that satisfies the following conditions. 3. f is continuous on the interval [4, 4]; f(x) = 0 for x = 2, 0, and 3; f has an absolute minimum at x = 3; f has a local minimum at x = 2; f has a local maximum at x = 0; f has an absolute maximum at x = 4.Designer functions Sketch the graph of a function continuous on the given interval that satisfies the following conditions. 4. f is continuous on (, ); f(x) 0 and f(x) 0 on (, 0); f(x) 0 and f(x) 0 on (0, ).Use the graphs of f and f to complete the following steps. a.Find the critical points of f and determine where f is increasing and where it is decreasing. b.Determine the locations of the inflection points of f and the intervals on which f is concave up or concave down. c.Determine where f has local maxima and minima. d.Plot a possible graph of f.Critical points Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist). 6. f(x)=x36x2on[1,5]Critical points Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist). 7. f(x)=3x46x2+9on[2,2]Critical points Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist). 8. g(x)=x450x2on[1,5]Critical points Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist). 9. f(x)=2x33x236x+12on(,)Critical points Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist). 10. f(x)=x3lnxon(0,)Critical points Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist). 11. f(x)=ln(x22x+2)on[0,2]Critical points Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist). 12. f(x)=sin2x+3on[,]Critical points Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist). 13. g(x)=12sinx+12sinxcosxon[0,2]Critical points Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist). 14. f(x)=4x1/2x5/2on[0,4]Critical points Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist). 15. f(x)=2xlnx+10on(0,4)Critical points Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist). 16. g(x)=xsin1xon[1,1]Absolute values Consider the function f(x) = |x 2| + |x + 3| on [4, 4]. Graph f, identify the critical points, and give the coordinates of the local and absolute extreme values.Use f and f to complete parts (a) and (b). a. Find the intervals on which f is increasing and the intervals on which it is decreasing. b. Find the intervals on which f is concave up and the intervals on which it is concave down. 18. f(x)=ex1+ex Use f and f to complete parts (a) and (b). a.Find the intervals on which f is increasing and the intervals on which it is decreasing. b.Find the intervals on which f is concave up and the intervals on which it is concave down. 19. f(x)=x99+3x516x Use f and f to complete parts (a) and (b). a.Find the intervals on which f is increasing and the intervals on which it is decreasing. b.Find the intervals on which f is concave up and the intervals on which it is concave down. 20. f(x)=xx+9 Inflection points Does f(x) = 2x5 10x4 + 20x3 + x + 1 have any inflection points? If so, identify them.Does f(x)=x62+5x4415x2 have any inflection points? If so, identify them.Identify the critical points and the inflection points of f(x) = (x a)(x + a)3, for a 0.Curve sketching Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work. 24. f(x)=x(x+4)3 Curve sketching Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work. 25. f(x)=x423x2+4x+1 Curve sketching Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work. 26. f(x)=3xx2+3 Curve sketching Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work. 27. f(x)=4cos((x1))on[0,2]Curve sketching Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work. 28. f(x)=ln(x2+9)Curve sketching Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work. 29. f(x)=x2+3x1 Curve sketching Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work. 30. f(x)=xex/2 Curve sketching Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work. 31. f(x)=10x2x2+3 32RE Curve sketching Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work. 33. f(x)=x3x+2 34RE Optimal popcorn box A small popcorn box is created from a 12 12 sheet of paperboard by first cutting out four shaded rectangles, each of length x and width x/2 (see figure). The remaining paperboard is folded along the solid lines to form a box. What dimensions of the box maximize the volume of the box?Minimizing time Hannah is standing on the edge of an island, 1 mile from a straight shoreline (see figure) She wants to return to her beach house that is 2 miles from the point P on shore that is closest to where she is standing. Given that she can swim at 2 mi/hr and jog at 6 mi/hr, find the point at which she should come ashore to minimize the total time of her trip. If she starts swimming at noon, can she make it home before 1 p.m.?Minimizing sound intensity Two sound speakers are 100 m apart and one speaker is three times as loud as the other speaker. At what point on a line segment between the speakers is the sound intensity the weakest? (Hint: Sound intensity is directly proportional to the sound level and inversely proportional to the square of the distance from the sound source.)Hockey problem A hockey player skates on a line that is perpendicular to the goal line. If the line on which he is skating is 10 feet to the left of the center of the hockey goal and if the hockey goal is 6 ft wide, then where (what value of x) should he shoot the puck to maximize the angle on goal (see figure)? (Hint: = .)Optimization A right triangle has legs of length h and r, and a hypotenuse of length 4 (see figure). It is revolved about the leg of length h to sweep out a right circular cone. What values of h and r maximize the volume of the cone? (Volume of a cone = r2h/3.)T 22. Rectangles beneath a curve A rectangle is constructed with one side on the positive x-axis, one side on the positive y-axis, and the vertex opposite the origin on the curve y = cos x, for 0 x /2. Approximate the dimensions of the rectangle that maximize the area of the rectangle. What is the area?Maximum printable area A rectangular page in a textbook (with width x and length y) has an area of 98 in2, top and bottom margins set at 1 in, and left and right margins set at 12in. The printable area of the page is the rectangle that lies within the margins. What are the dimensions of the page that maximize the printable area?Nearest point What point on the graph of f(x)=52x2is closest to the origin? (Hint: You can minimize the square of the distance.)Maximum area A line segment of length 10 joins the points (0, p) and (q, 0) to form a triangle in the first quadrant. Find the values of p and q that maximize the area of the triangle.Minimum painting surface A metal cistern in the shape of a right circular cylinder with volume V = 50 m3 needs to be painted each year to reduce corrosion. The paint is applied only to surfaces exposed to the elements (the outside cylinder wall and the circular top). Find the dimensions r and h of the cylinder that minimize the area of the painted surfaces.Linear approximation a. Find the linear approximation to f at the given point a. b. Use your answer from part (a) to estimate the given function value. Does your approximation underestimate or overestimate the exact function value? 27. f(x) = x2/3; a = 27; f(29)Linear approximation a. Find the linear approximation to f at the given point a. b. Use your answer from part (a) to estimate the given function value. Does your approximation underestimate or overestimate the exact function value? 28. f(x) = sin1 x; a = 1/2; f(0.48)Estimations with linear approximation Use linear approximation to estimate the following quantities. Choose a value of a to produce a small error. 29. 1/4.22 Estimations with linear approximation Use linear approximation to estimate the following quantities. Choose a value of a to produce a small error. 30. tan1 1.05 Change in elevation The elevation h (in feet above the ground) of a stone dropped from a height of 1000 ft is modeled by the equation h(t) = 1000 16t2, where t is measured in seconds and air resistance is neglected. Approximate the change in elevation over the interval 5 t 5.7 (recall that h h(a)t).Change in energy The energy E (in joules) released by an earthquake of magnitude M is modeled by the equation E(M) = 25, 000 101.5M. Approximate the change in energy released when the magnitude changes from 7.0 to 7.5 (recall that E E(a)M).Mean Value Theorem For the function f(x)=10x and the interval [a, b] = [0, 10], use the graph to make a conjecture about the value of c for which f(b)f(a)ba=f(c). Then verify your conjectured value by solving the equation f(b)f(a)ba=f(c) for c.Mean Value Theorem Explain why the Mean Value Theorem does not apply to the function f(x) = |x| on [a, b] = [1, 2].Mean Value Theorem The population of a culture of cells grows according to the function P(t)=100tt+1, where t 0 is measured in weeks. a. What is the average rate of change in the population over the interval [0, 8]? b. At what point of the interval [0, 8] is the instantaneous rate of change equal to the average rate of change?Growth rate of bamboo Bamboo belongs to the grass family and is one of the fastest-growing plants in the world. a. A bamboo shoot was 500 cm tall at 10.00 A.M. and 515 cm at 3:00 P.M. Compute the average growth rate of the bamboo shoot in cm/hr over the period of time from 10:00 A.M. to 3:00 P.M. b. Based on the Mean Value Theorem, what can you conclude about the instantaneous growth rate of bamboo measured in millimeters per second between 10:00 A.M. and 3:00 P.M.?Newtons method Use Newtons method to approximate the roots of f(x) = 3x3 4x2 + 1 to six digits.56RE Newtons method Use Newtons method to approximate the x-coordinate of the inflection points of f(x) = 2x5 6x3 4x + 2 to six digits.Two methods Evaluate the following limits in two different ways: with and without lHpitals Rule. 58. limx4x4x2x4+x1 Two methods Evaluate the following limits in two different ways: with and without lHpitals Rule. 59. limx2x5x+15x6+x Limits Evaluate the following limits. Use lHpitals Rule when needed. 38. limt2t3t22tt24 Limits Evaluate the following limits. Use lHpitals Rule when needed. 39.limt01cos6t2t Limits Evaluate the following limits. Use lHpitals Rule when needed. 40. limx5x2+2x5x41 Limits Evaluate the following limits. Use lHpitals Rule when needed. 41. lim03sin222 Limits Evaluate the following limits. Use lHpitals Rule when needed. 42. limx(x2+x+1x2x)Limits Evaluate the following limits. Use lHpitals Rule when needed. 43. lim02cot3 Limits Evaluate the following limits. Use lHpitals Rule when needed. 44. limx0e2x1+2xx2 Limits Evaluate the following limits. Use lHpitals Rule when needed. 45. limy0+ln10yy 68RE 69RE Limits Evaluate the following limits. Use lHpitals Rule when needed. 48. limxlnx100x Limits Evaluate the following limits. Use lHpitals Rule when needed. 49. limx0cscxsin1x Limits Evaluate the following limits. Use lHpitals Rule when needed. 50. limxln3xx 73RE Limits Evaluate the following limits. Use lHpitals Rule when needed. 74. limx0+(1+x)lnx Limits Evaluate the following limits. Use lHpitals Rule when needed. 75. limx/2(sinx)tanx Limits Evaluate the following limits. Use lHpitals Rule when needed. 76. limx(x+1)1/x Limits Evaluate the following limits. Use lHpitals Rule when needed. 77. limx0+|lnx|x 78RE 79RE 80RE Limits Evaluate the following limits. Use lHpitals Rule when needed. 81. limx1(x1)sinx Comparing growth rates Determine which of the two functions grows faster or state that they have comparable growth rates. 60. x100 and 1.1x 83RE 84RE Comparing growth rates Determine which of the two functions grows faster or state that they have comparable growth rates. 63. x and ln10 x Comparing growth rates Determine which of the two functions grows faster or state that they have comparable growth rates. 64. 10x and ln x2 87RE Comparing growth rates Determine which of the two functions grows faster or state that they have comparable growth rates. 66. x6+10 and x3 Comparing growth rates Determine which of the two functions grows faster or state that they have comparable growth rates. 67. 2x and 44x/2 Indefinite integrals Determine the following indefinite integrals. 68. (x83x3+1)dx Indefinite integrals Determine the following indefinite integrals. 69. (2x+1)2dx Indefinite integrals Determine the following indefinite integrals. 92. x53xdx Indefinite integrals Determine the following indefinite integrals. 71. (1x22x5/2)dx Indefinite integrals Determine the following indefinite integrals. 72. x42x+2x2dx Indefinite integrals Determine the following indefinite integrals. 95. (1+3cos)d Indefinite integrals Determine the following indefinite integrals. 74. 2sec2d 97RE Indefinite integrals Determine the following indefinite integrals. 98. 2e2x+exexdx 99RE Indefinite integrals Determine the following indefinite integrals. 78. dx1x2 101RE 102RE 103RE 104RE 105RE Functions from derivatives Find the function f with the following properties. 106. f(t)=t2+t2;f(1)=1,fort0 Functions from derivatives Find the function f with the following properties. 107. f(x)=x421+x2;f(1)=23 108RE 109RE Distance traveled A car starting at rest accelerates at 20 ft/s2 for 5 seconds on a straight road. How far does it travel during this time?111RE Logs of logs Compare the growth rates of ln x, ln (ln x), and ln (ln x), and ln (ln (ln x)).113RE 114RE 115RE 116RE 117RE A family of super-exponential functions Let f(x) = (a + x)x, where a 0. a. What is the domain of f (in terms of a)? b. Describe the end behavior of f (near the left boundary of its domain and as x ). c. Compute f. Then graph f and f, for a = 0.5, 1, 2, and 3. d. Show that f has a single local minimum at the point z that satisfies (z + a) ln (z + a) + z = 0. e. Describe how z (found in part (d)) varies as a increases. Describe how f(z) varies as a increases.What is the displacement of an object that travels at a constant velocity of 10 mi/hr for a half hour, 20 mi/hr for the next half hour, and 30 mi/hr for the next hour?In Example 1, if we used n = 32 subintervals of equal length, what would be the length of each subinterval? Find the midpoint of the first and last subinterval.If the interval [1, 9] is partitioned into 4 subintervals of equal length, what is x? List the grid points x0, x1, x2, x3, and x4.If the function in Example 2 is replaced with f(x) = 1/x, does the left Riemann sum of the right Riemann sum overestimate the area under the curve?Suppose an object moves along a line at 15 m/s, for 0 t 2, and at 25 m/s, for 2 t 5, where t is measured in seconds. Sketch the graph of the velocity function and find the displacement of the object for 0 t 5.Given the graph of the positive velocity of an object moving along a line, what is the geometrical representation of its displacement over a time interval [a, b]?3E The velocity in ft/s or an object moving along a line is given by v f(t) on the interval 0 t 6 (see figure), where t is measured in seconds. a.Divide the interval [0, 6] into n 3 subintervals, [0, 2], [2, 4], and [2, 6]. On each subinterval, assume the object moves at a constant velocity equal to the value of v evaluated at the left endpoint of the subinterval, and use these approximations to estimate the displacement of the object on [0, 6] (see part (a) of the figure). b.Repeat part (a) for n = 6 subintervals (see part (b) of the figure).The velocity in ft/s of an object moving along a line is given by v f(t) on the interval 0 t 6 (see figure), where t is measured in seconds. a.Divide the interval [0, 6] into n = 3 subintervals [0, 2], [2, 4] and [4, 6]. On each subinterval, assume the object moves at a constant velocity equal to the value of v evaluated at the right endpoint of the subinterval, and use these approximations to estimate the displacement of the object on [0, 6] (see part (a) of the figure). b.Repeat part (a) n = 6 subintervals (see part (b) of the figure).The velocity in ft/s of an object moving along a line is given by v f(t) on the interval 0 t 10 (see figure), where t is measured in seconds. a.Divide the interval [0, 10] into n 5 subintervals. On each subinterval, assume the object moves at a constant velocity equal to the value of v evaluated at the left endpoint of the subinterval, and use these approximations to estimate the displacement of the object on [0, 10]. b.Repeat part (a) using the right endpoints to estimate the displacement on [0, 10]. c.Repeat part (a) using the midpoints of each subinterval to estimate the displacement on [0, 10].7E Explain how Riemann sum approximations to the area of a region under a curve change as the number of subintervals increases.9E 10E Suppose the interval [1, 3] is partitioned into n = 4 subintervals. What is the subinterval length x? List the grid points x0, x1, x2, x3, and x4. Which points are used for the left, right, and midpoint Riemann sums?12E Does a right Riemann sum underestimate or overestimate the area of the region under the graph of a function that is positive and decreasing on an interval [a, b]? Explain.Does a left Riemann sum underestimate or overestimate the area of the region under the graph of a function that is positive and increasing on an interval [a, b]? Explain.Approximating displacement The velocity in ft/s of an object moving along a line is given by v = 3t2 + 1 on the interval 0 t 4. a. Divide the interval [0, 4] into n = 4 subintervals, [0, 1], [1, 2], [2, 3], and [3, 4]. On each subinterval, assume the object moves at a constant velocity equal to v evaluated at the midpoint of the subinterval and use these approximations to estimate the displacement of the object on [0, 4] (see part (a) of the figure). b. Repeat part (a) for n = 8 subintervals (see part (b) of the figure).Approximating displacement The velocity in ft/s of an object moving along a line is given by v=10t on the interval 1 t 7. a. Divide the time interval [1, 7] into n = 3 subintervals. [1, 3], [3, 5], and [5, 7]. On each subinterval, assume the object moves at a constant velocity equal to v evaluated at the midpoint of the subinterval and use these approximations to estimate the displacement of the object on [1, 7] (see part (a) of the figure). b. Repeat part (a) for n = 6 subintervals (see part (b) of the figure).Approximating displacement The velocity of an object is given by the following functions on a specified interval. Approximate the displacement of the object on this interval by subdividing the interval into n subintervals. Use the left endpoint of each subinterval to compute the height of the rectangles. 11. v = 2t + 1 (m/s), for 0 t 8; n = 2 Approximating displacement The velocity of an object is given by the following functions on a specified interval. Approximate the displacement of the object on this interval by subdividing the interval into n subintervals. Use the left endpoint of each subinterval to compute the height of the rectangles. 12. v = et (m/s), for 0 t 3; n = 3 Approximating displacement The velocity of an object is given by the following functions on a specified interval. Approximate the displacement of the object on this interval by subdividing the interval into n subintervals. Use the left endpoint of each subinterval to compute the height of the rectangles. 13. v=12t+1(m/s), for 0 t 8; n = 4 Approximating displacement The velocity of an object is given by the following functions on a specified interval. Approximate the displacement of the object on this interval by subdividing the interval into n subintervals. Use the left endpoint of each subinterval to compute the height of the rectangles. 14. v = t2/2 + 4 (ft/s), for 0 t 12; n = 6 Approximating displacement The velocity of an object is given by the following functions on a specified interval. Approximate the displacement of the object on this interval by subdividing the interval into n subintervals. Use the left endpoint of each subinterval to compute the height of the rectangles. 15. v=4t+1(mi/hr), for 0 t 15; n = 5 Approximating displacement The velocity of an object is given by the following functions on a specified interval. Approximate the displacement of the object on this interval by subdividing the interval into n subintervals. Use the left endpoint of each subinterval to compute the height of the rectangles. 16. v=t+36(m/s), for 0 t 4; n = 4 23E 24E 25E 26E 27E 28E 29E 30E 31E 32E A midpoint Riemann sum Approximate the area of the region bounded by the graph of f(x) = 100 x2 and the x-axis on [0, 10] with n = 5 subintervals. Use the midpoint of each subinterval to determine the height of each rectangle (see figure).34E Free fall On October 14, 2012, Felix Baumgartner stepped off a balloor capsule at an attitude of a most 39 km above Earths surface and began his free fall. His velocity in m/s during the fall is given in the figure. It is claimed that Felix reached the speed of sound 34 seconds into his fall and that he continued to fall at supersonic speed for 30 seconds. (Source: http://www.redbullstratos.com) a.Divide the interval [34, 64] into n 5 subintervals with the gridpoints x0 34, x1 40, x2 46, x3 52, x4 58, and x3 = 64. Use left and right Riemann sums to estimate how far Felix fall while travelling at supersonic speed. b.It is claimed that the actual distance that Felix fell at supersonic speed was approximately 10.495 m. Which estimate in part (a) produced the more accurate estimate? c.How could you obtain more accurate estimates of the total distance fallen than those found in part(a)?Free fall Use geometry and the figure given in Exercise 35 to estimate how far Felix fell in the first 20 seconds of his free fall. Free fall On October 14, 2012, Felix Baumgartner stepped off a balloor capsule at an attitude of a most 39 km above Earths surface and began his free fall. His velocity in m/s during the fall is given in the figure. It is claimed that Felix reached the speed of sound 34 seconds into his fall and that he continued to fall at supersonic speed for 30 seconds. (Source: http://www.redbullstratos.com) a.Divide the interval [34, 64] into n 5 subintervals with the gridpoints x0 34, x1 40, x2 46, x3 52, x4 58, and x3 = 64. Use left and right Riemann sums to estimate how far Felix fall while travelling at supersonic speed. b.It is claimed that the actual distance that Felix fell at supersonic speed was approximately 10.495 m. Which estimate in part (a) produced the more accurate estimate? c.How could you obtain more accurate estimates of the total distance fallen than those found in part(a)?37E Midpoint Riemann sums Complete the following steps for the given function, interval, and value of n. a. Sketch the graph of the function on the given interval. b. Calculate x and the grid points x0, x1, , xn. c. Illustrate the midpoint Riemann sum by sketching the appropriate rectangles. d. Calculate the midpoint Riemann sum. 30. f(x) = 2 cos1 x on [0, 1]; n = 5 39E 40E 41E 42E Riemann sums from tables Evaluate the left and right Riemann sums for f over the given interval for the given value of n. 35. n = 4; [0, 2]44E Displacement from a table of velocities The velocities (in mi/hr) of an automobile moving along a straight highway over a two-hour period are given in the following table. a. Sketch a smooth curve passing through the data points. b. Find the midpoint Riemann sum approximation to the displacement on [0, 2] with n = 2 and n = 4.Displacement from a table of velocities The velocities (in m/s) of an automobile moving along a straight freeway over a four-second period are given in the following table. a. Sketch a smooth curve passing through the data points. b. Find the midpoint Riemann sum approximation to the displacement on [0, 4] with n = 2 and n = 4 subintervals.Sigma notation Express the following sums using sigma notation. (Answers are not unique.) a. 1 + 2 + 3 + 4 + 5 b. 4 + 5 + 6 + 7 + 8 + 9 c. 12 + 22 + 32 + 42 d. 1+12+13+14 Sigma notation Express the following sums using sigma notation. (Answers are not unique.) a. 1+3+5+7++99 b. 4+9+14++44 c. 3+8+13++63 d. 112+123+134++14950 Sigma notation Evaluate the following expressions. a. k=110k b. k=16(2k+1) c. k=14k2 d. n=15(1+n2) e. m=132m+23 f. j=13(3j4) g. p=15(2p+p2) h. n=04sinn2 Evaluating sums Evaluate the following expressions by two methods. (i) Use Theorem 5.1. (ii) Use a calculator. a. k=145k b. k=145(5k1) c. k=1752k2 d. n=150(1+n2) e. m=1752m+23 f. j=120(3j4) g. p=135(2p+p2) h. n=040(n2+3n1)51E 52E 53E 54E 55E 56E 57E 58E

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