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All Textbook Solutions for Calculus: Early Transcendentals (3rd Edition)
Letf(x)=x2/3. Show that there is no value of c in the interval (1, 8) for which f(c)=f(8)f(1)8(1) and explain why this does not violate the Mean Value Theorem.Rolles Theorem Determine whether Rolles Theorem applies to the following functions on the given interval. If so, find the point(s) that are guaranteed to exist by Rolles Theorem. 7. f(x) = x(x 1)2; [0, 1]Rolles Theorem Determine whether Rolles Theorem applies to the following functions on the given interval. If so, find the point(s) that are guaranteed to exist by Rolles Theorem. 8. f(x) = sin 2x; [0, /2]Rolles Theorem Determine whether Rolles Theorem applies to the following functions on the given interval. If so, find the point(s) that are guaranteed to exist by Rolles Theorem. 9. f(x) = cos 4x; [/8, 3/8]Rolles Theorem Determine whether Rolles Theorem applies to the following functions on the given interval. If so, find the point(s) that are guaranteed to exist by Rolles Theorem. 10. f(x) = 1 |x|; [1, 1]Rolles Theorem Determine whether Rolles Theorem applies to the following functions on the given interval. If so, find the point(s) that are guaranteed to exist by Rolles Theorem. 11. f(x) = 1 x2/3; [1, 1]Rolles Theorem Determine whether Rolles Theorem applies to the following functions on the given interval. If so, find the point(s) that are guaranteed to exist by Rolles Theorem. 12. f(x) = x3 2x2 8x; [2, 4]Rolles Theorem Determine whether Rolles Theorem applies to the following functions on the given interval. If so, find the point(s) that are guaranteed to exist by Rolles Theorem. 13. g(x) = x3 x2 5x 3; 1, 3Rolles Theorem Determine whether Rolles Theorem applies to the following functions on the given interval. If so, find the point(s) that are guaranteed to exist by Rolles Theorem. 14. h(x)=ex2; [a, a], where a 0Lapse rates in the atmosphere Concurrent measurements indicate that at an elevation of 6.1 km, the temperature is 10.3C, and at an elevation of 3.2 km, the temperature is 8.0C. Based on the Mean Value Theorem, can you conclude that the lapse rate exceeds the threshold value of 7C/km at some intermediate elevation? Explain.Drag racer acceleration The fastest drag racers can reach a speed of 330 mi/hr over a quarter-mile strip in 4.45 seconds (from a standing start). Complete the following sentence about such a drag racer: At some point during the race, the maximum acceleration of the drag racer is at least _____ mi/hr/s.Mean Value Theorem Consider the following functions on the given interval [a, b]. a.Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b]. b.If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem. 21.f(x)=7x2;[1,2]Mean Value Theorem Consider the following functions on the given interval [a, b]. a.Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b]. b.If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem. 22.f(x)=x32x2;[0,1]Mean Value Theorem Consider the following functions on the given interval [a, b]. a.Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b]. b.If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem. 23.f(x)={2xifx0xifx0;[1,1]Mean Value Theorem Consider the following functions on the given interval [a, b]. a.Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b]. b.If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem. 24.f(x)=1(x1)2;[0,2]Mean Value Theorem Consider the following functions on the given interval [a, b]. a.Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b]. b.If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem. 25.f(x)=ex;[0,1]Mean Value Theorem Consider the following functions on the given interval [a, b]. a.Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b]. b.If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem. 26.f(x)=ln2x;[1,e]Mean Value Theorem Consider the following functions on the given interval [a, b]. a.Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b]. b.If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem. 27.f(x)=sinx;[2,2]Mean Value Theorem Consider the following functions on the given interval [a, b]. a.Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b]. b.If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem. 28.f(x)=tanx;[0,4]Mean Value Theorem Consider the following functions on the given interval [a, b]. a.Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b]. b.If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem. 29.f(x)=sin1x;[0,12]Mean Value Theorem Consider the following functions on the given interval [a, b]. a.Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b]. b.If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem. 30.f(x)=x+1x;[1,3]Mean Value Theorem Consider the following functions on the given interval [a, b]. a.Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b]. b.If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem. 31.f(x)=2x1/3;[8,8]Mean Value Theorem Consider the following functions on the given interval [a, b]. a.Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b]. b.If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem. 32.f(x)=xx+2;[1,2]Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The continuous function f(x) = 1 |x| satisfies the conditions of the Mean Value Theorem on the interval [1, 1]. b. Two differentiable functions that differ by a constant always have the same derivative. c. If f(x) = 0, then f(x) = 10.34EAnother inverse tangent identity a.Use derivatives to show that tan12x2 and tan1(x+1)tan1(x1) differ by a constant. b.Prove that tan12x2=tan1(x+1)tan1(x1), for x0.Questions about derivatives 26. Without evaluating derivatives, which of the functions f(x) = ln x, g(x) = ln 2x, h(x) = ln x2, and p(x) = ln 10x2 have the same derivative?Questions about derivatives 27. Without evaluating derivatives, which of the functions g(x) = 2x10, h(x) = x10 + 2, and p(x) = x10 ln 2 have the same derivative as f(x) = x10?Questions about derivatives 28. Find all functions f whose derivative is f(x) = x + 1.Mean Value Theorem and graphs By visual inspection, locate all points on the interval (4, 4) at which the slope of the tangent line equals the average rate of change of the function on the interval [4, 4].Mean Value Theorem and graphs Find all points on the interval (1, 3) at which the slope of the tangent line equals the average rate of change of f on [1, 3]. Reconcile your results with the Mean Value Theorem. 30.Mean Value Theorem and graphs Find all points on the interval (1, 3) at which the slope of the tangent line equals the average rate of change of f on [1, 3]. Reconcile your results with the Mean Value Theorem. 31.Avalanche forecasting Avalanche forecasters measure the temperature gradient dT/dh, which is the rate at which the temperature in a snowpack T changes with respect to its depth h. A large temperature gradient may lead to a weak layer in the snowpack. When these weak layers collapse, avalanches occur. Avalanche forecasters use the following rule of thumb: If dT/dh exceeds l0C/m anywhere in the snowpack, conditions are favorable for weak-layer formation, and the risk of avalanche increases. Assume the temperature function is continuous and differentiable. a. An avalanche forecaster digs a snow pit and takes two temperature measurements. At the surface (h = 0), the temperature is 16C. At a depth of 1.1 m, the temperature is 2C, Using the Mean Value Theorem, what can he conclude about the temperature gradient? Is the formation of a weak layer likely? b. One mile away, a skier finds that the temperature at a depth of 1.4 m is 1C, and at the surface, it is 12C. What can be concluded about the temperature gradient? Is the formation of a weak layer in her location likely? c. Because snow is an excellent insulator, the temperature of snow-covered ground is often near 0C. furthermore, the surface temperature of snow in a particular area does not vary much from one location to the next. Explain why a weak layer is more likely to form in places where the snowpack is not too deep. d. The term isothermal is used to describe the situation where all layers of the snowpack are at the same temperature (typically near the freezing point). Is a weak layer likely to form in isothermal snow? Explain.Mean Value Theorem and the police A state patrol officer saw a car start from rest at a highway on-ramp. She radioed ahead to a patrol officer 30 mi along the highway. When the car reached the location of the second officer 28 min later, it was clocked going 60 mi/hr. The driver of the car was given a ticket for exceeding the 60-mi/hr speed limit. Why can the officer conclude that the driver exceeded the speed limit?Mean Value Theorem and the police again Compare carefully to Exercise 43. A state patrol officer saw a car start from rest at a highway on-ramp. She radioed ahead to another officer 30 mi along the highway. When the car reached the location of the second officer 30 min later, it was clocked going 60 mi/hr. Can the patrol officer conclude that the driver exceeded the speed limit? Mean Value Theorem and the police A state patrol officer saw a car start from rest at a highway on-ramp. She radioed ahead to a patrol officer 30 mi along the highway. When the car reached the location of the second officer 28 min later, it was clocked going 60 mi/hr. The driver of the car was given a ticket for exceeding the 60-mi/hr speed limit. Why can the officer conclude that the driver exceeded the speed limit?Running pace Explain why if a runner completes a 6.2-mi (10-km) race in 32 min, then he must have been running at exactly 11 mi/hr at least twice in the race. Assume the runners speed at the finish line is zero.Mean Value Theorem for linear functions Interpret the Mean Value Theorem when it is applied to any linear function.Mean Value Theorem for quadratic functions Consider the quadratic function f(x) = Ax2 + Bx + C, where A, B, and C are real numbers with A 0. Show that when the Mean Value Theorem is applied to f on the interval [a, b], the number c guaranteed by the theorem is the midpoint of the interval.Means a. Show that the point c guaranteed to exist by the Mean Value Theorem for f(x) = x2 on [a, b] is the arithmetic mean of a and b; that is, c = (a + b)/2. b. Show that the point c guaranteed to exist by the Mean Value Theorem for f(x) = 1/x on [a, b], where 0 a b, is the geometric mean of a and b; that is, c=ab.Equal derivatives Verify that the functions f(x) = tan2 x and g(x) = sec2 x have the same derivative. What can you say about the difference f g? Explain.100-m speed The Jamaican sprinter Usain Bolt set a world record of 9.58 s in the 100-m dash in the summer of 2009. Did his speed ever exceed 37 km/hr during the race? Explain.Verify the identity sec1x=cos1(1/x),forx0.52ESuppose f(x)2, for allx2, and f(2) = 7. Show that f(4) 11.Suppose f(x)1, for all x 0, and f(0) = 0. Show that f(x) x, for all x 0.Use the Mean Value Theorem to prove that 1+a21+a for a 0. (Hint: For a given value of a 0. let f(x)=1+x on [0, a] and use the fact that 1+c1, for c 0.)Prove the following statements. a.|sinasinb||ab|, for any real numbers a and b b.|sina||a|, for any real number aGeneralized Mean Value Theorem Suppose the functions f and g are continuous on a, b and differentiable on (a, b), where g(a) g(b). Then there is a point c in (a, b) at which f(b)f(a)g(b)g(a)=f(c)g(c). This result is known as the Generalized (or Cauchys) Mean Value Theorem. a. If g(x) = x, then show that the Generalized Mean Value Theorem reduces to the Mean Value Theorem. b. Suppose f(x) = x2 l, g(x) = 4x + 2, and [a, b] = [0, 1]. Find a value of c satisfying the Generalized Mean Value Theorem.58EExplain why a positive derivative on an interval implies that the function is increasing on the interval.Sketch a function f that is differentiable on (−∞, ∞) with the following properties: (i) x = 0 and x = 2 are critical points; (ii) f is increasing on (−∞, 2); (iii) f is decreasing on (2, ∞).
Explain how the First Derivative Test determines whether f(x)=x2 has a local maximum or local minimum at the critical point x=0.
Verify that the function f(x) = x4 is concave up for x>0 and for x<0. Is x=0 an inflection point? Explain.
5QCExplain how the first derivative of a function determines where the function is increasing and decreasing.Explain how to apply the First Derivative Test.Suppose the derivative of f isf(x)=x3. a.Find the critical points of f. b.On what intervals is f increasing and on what intervals is f decreasing?Suppose the derivative of f isf(x)=(x1)(x2). a.Find the critical points of f. b.On what intervals is f increasing and on what intervals is f decreasing?Sketch the graph of a function that has neither a local maximum nor a local minimum at a point where f(x) = 0.The following graph of the derivative g' has exactly two roots. a.Find the critical points of g. b.For what values of x in (0, 3) is g increasing? Decreasing? c.For what values of x in (0, 3) does g have a local maximum? A local minimum?Functions from derivatives The following figures give the graph of the derivative of a continuous function f that passes through the origin. Sketch a possible graph of f on the same set of axes. 15.Functions from derivatives The following figures give the graph of the derivative of a continuous function f that passes through the origin. Sketch a possible graph of f on the same set of axes. 16.Sketches from properties Sketch a graph of a function that is continuous on (, ) and has the following properties. Use a sign graph to summarize information about the function.f(x) 0 on (, 2); f(x) 0 on (2, 5); f(x) 0 on (5, ) 1114. Sketches from properties Sketch a graph of a function that is continuous on (, ) and has the following properties. Use a sign graph to summarize information about the function. 12. f(1) is undefined: f(x) 0 on (, 1); f(x) 0 on (1, )Sketches from properties Sketch a graph of a function that is continuous on (, ) and has the following properties. Use a sign graph to summarize information about the function. 13. f(0) = f(4) = f(0) = f(2) = f(4) = 0; f(x) 0 on (, )Sketches from properties Sketch a graph of a function that is continuous on (, ) and has the following properties. Use a sign graph to summarize information about the function. 14. f(2) = f(2) = f(6) = 0; f(x) 0 on (, )Supposeg(x)=2x. a.On what intervals is g concave up and on what intervals is g concave down? b.State the inflection points of g.The following graph of g has exactly three x-intercepts. a.For what values of x in (-4, 3) is the graph of g concave up? Concave down? b.State the inflection points of g that lie in (4, 3).Is it possible for a function to satisfy f(x) 0, f(x) 0, and f(x) 0 on an interval? Explain.Sketch a function that changes from concave up to concave down as x increases. Describe how the second derivative of this function changes.Give a function that does not have an inflection point at a point where f(x) = 0.Suppose f is continuous on an interval containing a critical point c and f(c) = 0. How do you determine whether f has a local extreme value at x = c?Increasing and decreasing functions Find the intervals on which f is increasing and the intervals on which it is decreasing. 19.f(x)=4x2Increasing and decreasing functions Find the intervals on which f is increasing and the intervals on which it is decreasing. 20.f(x)=x216Increasing and decreasing functions Find the intervals on which f is increasing and the intervals on which it is decreasing. 21.f(x)=(x1)2Increasing and decreasing functions Find the intervals on which f is increasing and the intervals on which it is decreasing. 22.f(x)=x3+4xIncreasing and decreasing functions Find the intervals on which f is increasing and the intervals on which it is decreasing. 23.f(x)=x335x22+4xIncreasing and decreasing functions Find the intervals on which f is increasing and the intervals on which it is decreasing. 24.f(x)=x33+x22+2xIncreasing and decreasing functions Find the intervals on which f is increasing and the intervals on which it is decreasing. 25.f(x)=12+xx2Increasing and decreasing functions Find the intervals on which f is increasing and the intervals on which it is decreasing. 26.f(x)=x44x3+4x2Increasing and decreasing functions Find the intervals on which f is increasing and the intervals on which it is decreasing. 27.f(x)=x44+x3x2Increasing and decreasing functions Find the intervals on which f is increasing and the intervals on which it is decreasing. 28.f(x)=2x515x44+5x33Increasing and decreasing functions Find the intervals on which f is increasing and the intervals on which it is decreasing. 29.f(x)=x2lnx2+1Increasing and decreasing functions Find the intervals on which f is increasing and the intervals on which it is decreasing. 30.f(x)=exe2x+1Increasing and decreasing functions Find the intervals on which f is increasing and the intervals on which it is decreasing. 31.f(x)=2cosxxon[0,2]Increasing and decreasing functions Find the intervals on which f is increasing and the intervals on which it is decreasing. 32.f(x)=2sinxxon[0,2]Increasing and decreasing functions Find the intervals on which f is increasing and decreasing. 27. f(x) = 3 cos 3x on [, ]Increasing and decreasing functions Find the intervals on which f is increasing and decreasing. 28. f(x) = cos2 x on [, ]Increasing and decreasing functions Find the intervals on which f is increasing and decreasing. 29. f(x) = x2/3(x2 4)Increasing and decreasing functions Find the intervals on which f is increasing and decreasing. 30. f(x)=x29x2 on (3, 3)Increasing and decreasing functions Find the intervals on which f is increasing and the intervals on which it is decreasing. 37.f(x)=9x2+sin1(x/3)Increasing and decreasing functions Find the intervals on which f is increasing and the intervals on which it is decreasing. 38.f(x)=xlnx2x+3on(0,)Increasing and decreasing functions Find the intervals on which f is increasing and decreasing. 33. f(x) = 12x5 + 75x4 80x3Increasing and decreasing functions Find the intervals on which f is increasing and decreasing. 34. f(x) = x2 2 ln xIncreasing and decreasing functions Find the intervals on which f is increasing and decreasing. 35. f(x) = 2x4 + x2 + 10Increasing and decreasing functions Find the intervals on which f is increasing and decreasing. 36. f(x)=x448x33+15x22+8Increasing and decreasing functions Find the intervals on which f is increasing and decreasing. 37. f(x)=xex2/2Increasing and decreasing functions Find the intervals on which f is increasing and decreasing. 38. f(x)=tan1(xx2+2)First Derivative Test a. Locale the critical points of f. b. Use the First Derivative Test to locale the local maximum and minimum values. c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist). 39. f(x) = x2 + 3 on [3, 2]First Derivative Test a. Locale the critical points of f. b. Use the First Derivative Test to locale the local maximum and minimum values. c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist). 40. f(x) = x2 x + 2 on [4, 4]First Derivative Test a. Locale the critical points of f. b. Use the First Derivative Test to locale the local maximum and minimum values. c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist). 41. f(x)=x4x2 on [2, 2]First Derivative Test a. Locale the critical points of f. b. Use the First Derivative Test to locale the local maximum and minimum values. c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist). 42. f(x) = 2x3 + 3x2 12x + 1 on [2, 4]First Derivative Test a. Locale the critical points of f. b. Use the First Derivative Test to locale the local maximum and minimum values. c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist). 43. f(x) = x3 + 9x on [4, 3]First Derivative Test a. Locale the critical points of f. b. Use the First Derivative Test to locale the local maximum and minimum values. c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist). 44. f(x) = 2x5 5x4 10x3 + 4 on [2, 4]First Derivative Test a. Locale the critical points of f. b. Use the First Derivative Test to locale the local maximum and minimum values. c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist). 45. f(x) = x2/3 (x 5) on [5, 5]First Derivative Test a. Locale the critical points of f. b. Use the First Derivative Test to locale the local maximum and minimum values. c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist). 46. f(x)=x2x21 on [4, 4]First Derivative Test a. Locale the critical points of f. b. Use the First Derivative Test to locale the local maximum and minimum values. c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist). 47. f(x)=xlnx on [0, ]First Derivative Test a.Locate the critical points of f. b.Use the First Derivative Test to locate the local maximum and minimum values. c.Identify the absolute maximum and minimum values of the function on the given interval (when they exist). 54.f(x)=x2tan1xon[3,3]Absolute extreme values Verify that the following functions satisfy the conditions of Theorem 4.5 on their domains. Then find the location and value of the absolute extremum guaranteed by the theorem. 49. f(x) = xexAbsolute extreme values Verify that the following functions satisfy the conditions of Theorem 4.5 on their domains. Then find the location and value of the absolute extremum guaranteed by the theorem. 50. f(x)=4x+1/xAbsolute extreme values Verify that the following functions satisfy the conditions of Theorem 4.5 on their domains. Then find the location and value of the absolute extremum guaranteed by the theorem. 51. A(r) = 24/r + 2r2, r 0Absolute extreme values Verify that the following functions satisfy the conditions of Theorem 4.5 on their domains. Then find the location and value of the absolute extremum guaranteed by the theorem. 52. f(x)=x3x59E60E61E62EConcavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points. 57. f(x) = x4 2x3 + 1Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points. 58. f(x) = x4 2x3 + 12x3Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points. 59. f(x) = 5x4 20x3 + 10Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points. 60. f(x)=11+x2Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points. 61. f(x) = ex(x 3)Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points. 62. f(x) = 2x2 ln x 5x2Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points. 63. g(t) = ln (3t2 + 1)Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points. 64. g(x)=x43Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points. 65. f(x)=ex2/2Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points. 72.p(x)=x4ex+xConcavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points. 67. f(x)=xlnxConcavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points. 74.h(t)=2+cos2ton[0,]Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points. 69. g(t) = 3t5 30t4 + 80t3 + 100Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points. 70. f(x) = 2x4 + 8x3 + 12x2 x 2Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima. 71. f(x) = x3 3x2Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima. 72. f(x) = 6x2 x3Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima. 73. f(x) = 4 x2Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima. 80.f(x)=x332x236xSecond Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima. 75. f(x) = ex(x 7)82ESecond Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima. 77. f(x) = 2x3 3x2 + 12Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima. 84.f(x)=exx+1Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima. 79. f(x) = x2exSecond Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima. 80. g(x)=x4212x2Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima. 81. f(x) = 2x2 ln x 11x2Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima. 82. f(x)=x(127x34x2)Second Derivative Test Locate the critical points of the following functions and use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima. 96. p(t) = 2t3 + 3t2 36tSecond Derivative Test Locate the critical points of the following functions and use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima. 97. f(x)=x445x334x2+48xSecond Derivative Test Locate the critical points of the following functions and use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima. 98. h(x) = (x + a)4; a constantSecond Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima. 92.f(x)=x313x29xSecond Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima. 93.f(x)=6x4lnx27x4Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima. 94.f(x)=2x3x2Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. If f(x) 0 and f(x) 0 on an interval, then f is increasing at a decreasing rate on the interval. b. If f(c) 0 and f(c) = 0, then f has a local maximum at c. c. Two functions that differ by an additive constant both increase and decrease on the same intervals. d. If f and g increase on an interval, then the product fg also increases on that interval. e. There exists a function f that is continuous on (, ) with exactly three critical points, all of which correspond to local maxima.Is it possible? Determine whether the following properties can be satisfied by a function that is continuous on (, ). If such a function is possible, provide an example or a sketch of the function. If such a function is not possible, explain why. a. A function f is concave down and positive everywhere. b. A function f is increasing and concave down everywhere. c. A function f has exactly two local extrema and three inflection points. d. A function f has exactly four zeros and two local extrema.Matching derivatives and functions The following figures show the graphs of three functions (graphs a-c). Match each function with its first derivative (graphs A-C) and its second derivative (graphs i-iii).
100E101EDesigner functions Sketch the graph of a function that is continuous on (, ) and satisfies the following sets of conditions. 90. f(x) 0 on (, 2); f(2) = 0; f(1) = f(1) = 0; f(2) = 0; f(3) = 0; f(x) 0 on (4, )103EDesigner functions Sketch the graph of a function that is continuous on (, ) and satisfies the following sets of conditions. 92. f(x) 0, for all x in the domain of f; f(2) and f(1) do not exist; f(0) = 0Designer functions Sketch the graph of a function that is continuous on (, ) and satisfies the following sets of conditions. 93. f(x) 0 on (, 2); f(x) 0 on (2, 1); f(x) 0 on (1, 3); f(x) 0 on (3, )Graph carefully Graph the function f(x) = 60x5 901x3 + 27x in the window [4, 4] [10000, 10000]. How many extreme values do you see? Locate all the extreme values by analyzing f.Interpreting the derivative The graph of f on the interval [3, 2] is shown in the figure. a. On what interval(s) is f increasing? Decreasing? b. Find the critical points of f. Which critical points correspond to local maxima? Local minima? Neither? c. At what point(s) does f have an inflection point? d. On what interval(s) is f concave up? Concave down? e. Sketch the graph of f. f. Sketch one possible graph of f.108E109E110EPopulation models The population of a species is given by the function P(t)=Kt2t2+b, where t 0 is measured in years and K and b are positive real numbers. a. With K = 300 and b = 30, what is limtP(t), the carrying capacity of the population? b. With K = 300 and b = 30, when does the maximum growth rate occur? c. For arbitrary positive values of K and b, when does the maximum growth rate occur (in terms of K and b)?Tangent lines and concavity Give an argument to support the claim that if a function is concave up at a point, then the tangent line at that point lies below the curve near that point.General quartic Show that the general quartic (fourth-degree) polynomial f(x) = x4 + ax3 + bx2 + cx + d has either zero or two inflection points, and that the latter case occurs provided that b 3a2/8.Properties of cubics Consider the general cubic polynomial f(x) = x3 + ax2 + bx + c, where a, b, and c are real numbers. a. Prove that f has exactly one local maximum and one local minimum provided that a2 3b. b. Prove that f has no extreme values if a2 3b.Concavity of parabolas Consider the general parabola described by the function f(x) = ax2 + bx + c. For what values of a, b, and c is f concave up? For what values of a, b, and c is f concave down?Graph f(x) = x3/3 - 400x using various windows on a graphing calculator. Find a window that gives a better graph of f than those in Figure 4.44. Figure 4.44Explain why the functions f and f + C, where C is a constant, have the same derivative properties.3QCWhy is it important to determine the domain of f before graphing f?2E3EWhere are the vertical asymptotes of a rational function located?How do you find the absolute maximum and minimum values of a function that is continuous on a closed interval?Describe the possible end behavior of a polynomial.Shape of the curve Sketch a curve with the following properties. 7. f 0 and f 0, for x 3 f 0 and f 0, for x 3Shape of the curve Sketch a curve with the following properties. 8. f 0 and f 0, for x 1 f 0 and f 0, for 1 x 2 f 0 and f 0, for 2 x 8 f 0 and f 0, for 8 x 10 f 0 and f 0, for x 10Designer functions Sketch a continuous function f on some interval that has the properties described. 57. The function f has one inflection point but no local extrema.Designer functions Sketch a continuous function f on some interval that has the properties described. 58. The function f has three real zeros and exactly two local minima.Designer functions Sketch a continuous function f on some interval that has the properties described. 59. The function f satisfies f(2) = 2, f(0) =0, f(1) = 3, and f(4) = 1.Designer functions Sketch a continuous function f on some interval that has the properties described. 60. The function f has the same finite limit as x and has exactly one absolute minimum and one absolute maximum.Let f(x)=(x3)(x+3)2. a.Verify that f(x)=3(x1)(x+3) and f(x)=6(x+1). b.Find the critical points and possible inflection points of f. c.Find the intervals on which f is increasing or decreasing. d.Determine the intervals on which f is concave up or concave down. e.Identify the local extreme values and inflection points of f. f.State the x- and y-intercepts of the graph of f. g.Use your work in parts (a) through (f) to sketch a graph of f.If , it can be shown that and . Use these functions to complete the following steps.
Find the critical points and possible inflection points of f.
Find the intervals on which f is increasing or decreasing.
Determine the intervals on which f is concave up or concave down.
Identify the local extreme values and inflection points of f.
State the x- and y-intercepts of the graph of f.
Find the asymptotes of f.
Use your work in parts (a) through (f) to sketch a graph of f.
Graphing functions Use the guidelines of this section to make a complete graph of f. 15.f(x)=x26xGraphing functions Use the guidelines of this section to make a complete graph of f. 16.f(x)=xx2Graphing functions Use the guidelines of this section to make a complete graph of f. 17.f(x)=x36x2+9xGraphing functions Use the guidelines of this section to make a complete graph of f. 18.f(x)=3xx3Graphing functions Use the guidelines of this section to make a complete graph of f. 19.f(x)=x46x2Graphing functions Use the guidelines of this section to make a complete graph of f. 20.f(x)=x4+4x3Graphing functions Use the guidelines of this section to make a complete graph of f. 21.f(x)=(x6)(x+6)2Graphing functions Use the guidelines of this section to make a complete graph of f. 22.f(x)=27(x2)2(x+2)Graphing functions Use the guidelines of this section to make a complete graph of f. 23.f(x)=x36x2135xGraphing functions Use the guidelines of this section to make a complete graph of f.
f(x) = x4 + 8x3 − 270x2 + 1
Graphing functions Use the guidelines of this section to make a complete graph of f. 25.f(x)=x33x2144x140Graphing functions Use the guidelines of this section to make a complete graph of f. 26.f(x)=x3147x+286Graphing functions Use the guidelines of this section to make a complete graph of f.
f(x) = x − 2
Graphing functions Use the guidelines of this section to make a complete graph of f. 28.f(x)=3xx3/2Graphing rational functions Use the guidelines of this section to make a complete graph of f. 17. f(x)=3xx21Graphing rational functions Use the guidelines of this section to make a complete graph of f. 18. f(x)=2x32x8Graphing rational functions Use the guidelines of this section to make a complete graph of f. 15. f(x)=x2x2Graphing rational functions Use the guidelines of this section to make a complete graph of f. 16. f(x)=x2x24Graphing rational functions Use the guidelines of this section to make a complete graph of f. 19. f(x)=x2+122x+1Graphing functions Use the guidelines of this section to make a complete graph of f. 34.f(x)=4xx2+3Graphing functions Use the guidelines of this section to make a complete graph of f.
f(x) = tan−1
Graphing functions Use the guidelines of this section to make a complete graph of f. 36.f(x)=ln(x2+1)Graphing functions Use the guidelines of this section to make a complete graph of f.
f(x) = x + 2 cos x on [−2π, 2π]
38EGraphing functions Use the guidelines of this section to make a complete graph of f. 39.f(x)=x3x1/340EGraphing functions Use the guidelines of this section to make a complete graph of f. 41.f(x)=sinxxon[0,2]Graphing functions Use the guidelines of this section to make a complete graph of f. 42.f(x)=xx+3Graphing functions Use the guidelines of this section to make a complete graph of f. 43.f(x)=exsinxon,Graphing functions Use the guidelines of this section to make a complete graph of f. 44.f(x)=1ex1Graphing functions Use the guidelines of this section to make a complete graph of f. 45.f(x)=x+tanxon(2,2)Graphing functions Use the guidelines of this section to make a complete graph of f. 46.f(x)=ex2/2Functions from graphs Use the graphs of f and f to find the critical points and inflection points of f, the intervals on which f is increasing and decreasing, and the intervals of concavity. Then graph f assuming f(0) = 0. 49.Functions from graphs Use the graphs of f and f to find the critical points and inflection points of f, the intervals on which f is increasing and decreasing, and the intervals of concavity. Then graph f assuming f(0) = 0. 48.Graphing with technology Make a complete graph of the following functions. A graphing utility is useful in locating intercepts, local extreme values, and inflection points. 37. f(x)=13x32x25x+2Graphing with technology Make a complete graph of the following functions. A graphing utility is useful in locating intercepts, local extreme values, and inflection points. 38. f(x)=115x3x+1Graphing with technology Make a complete graph of the following functions. A graphing utility is useful in locating intercepts, local extreme values, and inflection points. 39. f(x) = 3x4 + 4x3 12x2Graphing with technology Make a complete graph of the following functions. A graphing utility is useful in locating intercepts, local extreme values, and inflection points. 40. f(x) = x3 33x2 + 216x 2Graphing with technology Make a complete graph of the following functions. A graphing utility is useful in locating intercepts, local extreme values, and inflection points. 41. f(x)=3x5x21Graphing with technology Make a complete graph of the following functions. A graphing utility is useful in locating intercepts, local extreme values, and inflection points. 42. f(x) = x1/3(x 2)2Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. If the zeros of f are 3, 1, and 4, then the local extrema of f are located at these points. b. If the zeros of f are 2 and 4, then the inflection points of f are located at these points. c. If the zeros of the denominator of f are 3 and 4, then f has vertical asymptotes at these points. d. If a rational function has a finite limit as x , then it must have a finite limit as xFunctions from derivatives Use the derivative f to determine the x-coordinates of the local maxima and minima of f, and the intervals of increase and decrease. Sketch a possible graph of f (f is not unique). 44. f(x) = (x 1)(x + 2)(x + 4)Functions from derivatives Use the derivative f to determine the x-coordinates of the local maxima and minima of f, and the intervals of increase and decrease. Sketch a possible graph of f (f is not unique). 45. f(x) = 10 sin 2x on [2, 2]Functions from derivatives Use the derivative f to determine the x-coordinates of the local maxima and minima of f, and the intervals of increase and decrease. Sketch a possible graph of f (f is not unique). 46. f(x)=16(x+1)(x2)2(x3)Functions from derivatives Use the derivative f to determine the x-coordinates of the local maxima and minima of f, and the intervals of increase and decrease. Sketch a possible graph of f (f is not unique). 47. f(x) = x2(x + 2)(x 1)e e Prove that e e by first finding the maximum value of f(x) = ln x/x.Oscillations Consider the function f(x) = cos (ln x), for x 0. Use analytical techniques and a graphing utility. a. Locate all local extrema on the interval (0, 4], b. Identify the inflection points on the interval (0, 4]. c. Locate the three smallest zeros of f on the interval (0.1, ). d. Sketch the graph of f.Local max/min of x1/x Use analytical methods to find all local extrema of the function f(x) = x1/x, for x 0. Verify your work using a graphing utility.Local max/min of xx Use analytical methods to find all local extrema of the function f(x) = xx, for x 0. Verify your work using a graphing utility.Derivative information Suppose a continuous function f is concave up on (, 0) and (0, ). Assume f has a local maximum at x = 0. What, if anything, do you know about f(0)? Explain with an illustration.66ECombining technology with analytical methods Use a graphing utility together with analytical methods to create a complete graph of the following functions. Be sure to find and label the intercepts, local extrema, inflection points, and asymptotes, and find the intervals on which the function is increasing or decreasing, and the intervals on which the function is concave up or concave down. 68.f(x)=3x4x2Combining technology with analytical methods Use a graphing utility together with analytical methods to create a complete graph of the following functions. Be sure to find and label the intercepts, local extrema, inflection points, and asymptotes, and find the intervals on which the function is increasing or decreasing, and the intervals on which the function is concave up or concave down. 69.f(x)=3x444x3+60x2 (Hint: Two different graphing windows may be needed.)Combining technology with analytical methods Use a graphing utility together with analytical methods to create a complete graph of the following functions. Be sure to find and label the intercepts, local extrema, inflection points, and asymptotes, and find the intervals on which the function is increasing or decreasing, and the intervals on which the function is concave up or concave down. 70.f(x)=sinx1+sinxon[0,2]Special curves The following classical curves have been studied by generations of mathematicians. Use analytical methods (including implicit differentiation) and a graphing utility to graph the curves. Include as much detail as possible. 75. x2/3 + y2/3 = 1 (Astroid or hypocycloid with four cusps)Special curves The following classical curves have been studied by generations of mathematicians. Use analytical methods (including implicit differentiation) and a graphing utility to graph the curves. Include as much detail as possible. 76. y=8x2+4(WitchofAgnesi)Special curves The following classical curves have been studied by generations of mathematicians. Use analytical methods (including implicit differentiation) and a graphing utility to graph the curves. Include as much detail as possible. 77. x3 + y3 = 3xy (Folium of Descartes)Special curves The following classical curves have been studied by generations of mathematicians. Use analytical methods (including implicit differentiation) and a graphing utility to graph the curves. Include as much detail as possible. 80. y2 = x3(1 x) (Pear curve)Special curves The following classical curves have been studied by generations of mathematicians. Use analytical methods (including implicit differentiation) and a graphing utility to graph the curves. Include as much detail as possible. 81. x4 x2 + y2 = 0 (Figure-8 curve)78E79EVerify that in the example to the right, the same result is obtained if the constraint x + y = 20 is used to eliminate x rather than y. To introduce the ideas behind optimization problems, think about pairs of nonnegative real numbers x and y between 0 and 20 with the property that their sum is 20, that is, x + y = 20. Of all possible pairs, which has the greatest product?Find the objective function in Example 1 (in terms of x) (i) if there is no interior fence and (ii) if there is one interior fence that forms a right angle with the barn, as in Figure 4.58. Example 1 Ranchers Dilemma A rancher has 400ft of fence for constructing a rectangular corral. One side of the corral will be formed by a barn and requires no fence. Three exterior fences and two interior fences partition the corral into three rectangular regions as shown in Figure 4.58. What are the dimensions of the corral that maximize the enclosed area? What is the area of that corral? Figure 4.58Find the objective function in Example 2 (in terms of w) if the constraint is that the sum of length and width and height cannot exceed 108 in. Example 2 Airline Regulations Suppose an airline policy states that all baggage must be box-shaped with a sum of length, width, and height not exceeding 64 in. What are the dimensions and volume of a square-based box with the greatest volume under these conditions?Fill in the blanks: The goal of an optimization problem is to find the maximum or minimum value of the _____ function subject to the _____.2ESuppose the objective function is Q = x2y and you know that x + y = 10. Write the objective function first in terms of and then in terms of y.Suppose you wish to minimize a continuous objective function on a closed interval, but you find that it has only a single local maximum. Where should you look for the solution to the problem?Suppose the objective function P = xy is subject to the constraint 10x + y = 100, where x and y are real numbers. a.Eliminate the variable y from the objective function so that P is expressed as a function of one variable x. b.Find the absolute maximum value of P subject to the given constraint.Suppose S=x+2y is an objective function subject to the constraint xy = 50, for x 0 and y 0 a.Eliminate the variable y from the objective function so that S is expressed as a function of one variable x. b.Find the absolute minimum value of S subject to the given constraint.Maximum product What two nonnegative real numbers with a sum of 23 have the largest possible product?Sum of squares What two nonnegative real numbers a and b whose sum is 23 maximize a2 + b2? Minimize a2 + b2?Minimum sum What two positive real numbers whose product is 50 have the smallest possible sum?Maximum product Find numbers x and y satisfying the equation 3x + y = 12 such that the product of x and y is as large as possible.Maximum area rectangles Of all rectangles with a perimeter of 10, which one has the maximum area? (Give the dimensions.)Maximum area rectangles Of all rectangles with a fixed perimeter of P, which one has the maximum area? (Give the dimensions in terms of P.)Minimum perimeter rectangles Of all rectangles of area 100, which one has the minimum perimeter?Minimum perimeter rectangles Of all rectangles with a fixed area A, which one has the minimum perimeter? (Give the dimensions in terms of A.)Minimum sum Find positive numbers x and y satisfying the equation xy = 12 such that the sum 2x + y is as small as possible.Pen problems a. A rectangular pen is built with one side against a barn. Two hundred meters of fencing are used for the other three sides of the pen. What dimensions maximize the area of the pen? b. A rancher plans to make four identical and adjacent rectangular pens against a barn, each with an area of 100 m2 (see figure). What are the dimensions of each pen that minimize the amount fence that must be used?Rectangles beneath a semicircle A rectangle is constructed with its base on the diameter of a semicircle with radius 5 and its two other vertices on the semicircle. What are the dimensions of the rectangle with maximum area?Rectangles beneath a parabola A rectangle is constructed with its base on the x-axis and two of its vertices on the parabola y = 48 x2. What are the dimensions of the rectangle with the maximum area? What is the area?Minimum-surface-area box Of all boxes with a square base and a volume of 8 m3, which one has the minimum surface area? (Give its dimensions.)Maximum-volume box Suppose an airline policy states that all baggage must be box-shaped with a sum of length, width, and height not exceeding 108 in. What are the dimensions and volume of a square-based box with the greatest volume under these conditions?Shipping crates A square-based, box-shaped shipping crate is designed to have a volume of l6 ft3. The material used to make the base costs twice as much (per square foot) as the material in the sides, and the material used to make the top costs half as much (per square foot) as the material in the sides. What are the dimensions of the crate that minimize the cost of materials?Closest point on a line What point on the line y = 3x + 4 is closest to the origin?Closest point on a curve What point on the parabola y=1x2 is closest to the point (1.1)?Minimum distance Find the point P on the curve y = x2 that is closest to the point (18, 0). What is the least distance between P and (18, 0)? (Hint: Use synthetic division.)Minimum distance Find the point P on the line y = 3x that is closest to the point (50, 0). What is the least distance between P and (50, 0)?Walking and rowing A boat on the ocean is 4 mi from the nearest point on a straight shoreline; that point is 6 mi from a restaurant on the shore. A woman plans to row the boat straight to a point on the shore and then walk along the shore to the restaurant. a. If she walks at 3 mi/hr and rows at 2 mi/hr, at which point on the shore should she land to minimize the total travel time? b. If she walks at 3 mi/hr, what is the minimum speed at which she must row so that the quickest way to the restaurant is to row directly (with no walking)?Laying cable An island is 3.5 mi from the nearest point on a straight shoreline; that point is 8 mi from a power station (see figure). A utility company plans to lay electrical cable underwater from the island to the shore and then underground along the shore to the power station. Assume that it costs 2400/mi to lay underwater cable and 1200/mi to lay underground cable. At what point should the underwater cable meet the shore to minimize the cost of the project?Laying cable again Solve the problem in Exercise 28, but this time minimize the cost with respect to the smaller angle between the underwater cable and the shore. (You should get the same answer.) 28.Laying cable An island is 3.5 mi from the nearest point on a straight shoreline; that point is 8 mi from a power station (see figure). A utility company plans to lay electrical cable underwater from the island to the shore and then underground along the shore to the power station. Assume it costs 2400/mi to lay underwater cable and 1200/mi to lay underground cable. At what point should the underwater cable meet the shore in order to minimize the cost of the project?Shortest ladder A 10-ft-tall fence runs parallel to the wall of a house at a distance of 4 ft. Find the length of the shortest ladder that extends from the ground to the house without touching the fence. Assume the vertical wall of the house and the horizontal ground have infinite extent.Shortest laddermore realistic An 8-ft-tall fence runs parallel to the wall of a house at a distance of 5 ft. Find the length of the shortest ladder that extends from the ground to the house without touching the fence. Assume that the vertical wall of the house is 20 ft high and the horizontal ground extends 20 ft from the fence.Circle and square A piece of wire of length 60 is cut, and the resulting two pieces are formed to make a circle and a square. Where should the wire be cut to (a) minimize and (b) maximize the combined area of the circle and the square?Maximum-volume cone A cone is constructed by cutting a sector from a circular sheet of metal with radius 20. The cut sheet is then folded up and welded (see figure). Find the radius and height of the cone with maximum volume that can be formed in this way.34EOptimal soda can a. Classical problem Find the radius and height of a cylindrical soda can with a volume of 354 cm3 that minimize the surface area. b. Real problem Compare your answer in part (a) to a real soda can, which has a volume of 354 cm3, a radius of 3.1 cm. and a height of 12.0 cm, to conclude that real soda cans do not seem to have an optimal design. Then use the fact that real soda cans have a double thickness in their top and bottom surfaces to find the radius and height that minimizes the surface area of a real can (the surface areas of the top and bottom are now twice their values in part (a)). Are these dimensions closer to the dimensions of a real soda can?Covering a marble Imagine a flat-bottomed cylindrical pot with a circular cross section of radius 4. A marble with radius 0 r 4 is placed in the bottom of the pot. What is the radius of the marble that requires the most water to cover it completely?Optimal garden A rectangular flower garden with an area of 30 m2 is surrounded by a grass border 1 m wide on two sides and 2 m wide on the other two sides (see figure). What dimensions of the garden minimize the combined area of the garden and borders?Rectangles beneath a line a. 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 line y = 10 2x. What dimensions maximize the area of the rectangle? What is the maximum area? b. Is it possible to construct a rectangle with a greater area than that found in part (a) by placing one side of the rectangle on the line y = 10 2x and the two vertices not on that line on the positive x- and y-axes? Find the dimensions of the rectangle of maximum area that can be constructed in this way.Designing a box Two squares of length x are cut out of adjacent corners of an 18 18 piece of cardboard and two rectangles of length 9 and width x are cut out of the other two corners of the cardboard (see figure). The resulting piece of cardboard is then folded along the dashed lines to form an enclosed box. Find the dimensions and volume of the largest box that can be formed in this way.Folded boxes a. Squares with sides of length x are cut out of each corner of a rectangular piece of cardboard measuring 5 ft by 8 ft. The resulting piece of cardboard is then folded into a box without a lid. Find the volume of the largest box that can be formed in this way. b. Squares with sides of length x are cut out of each corner of a square piece of cardboard with sides of length . Find the volume of the largest open box that can be formed in this way.A window consists of rectangular pane of glass surmounted by a semicircular pane of glass (see figure). If the perimeter of the window is 20 feet, determine the dimensions of the window that maximize the area of the window.Light transmission A window consists of a rectangular pane of clear glass surmounted by a semicircular pane of tinted glass. The clear glass transmits twice as much light per unit of surface area as the tinted glass. Of all such windows with a fixed perimeter P, what are the dimensions of the window that transmits the most light?Keplers wine barrel Several mathematical stories originated with the second wedding of the mathematician and astronomer Johannes Kepler. Here is one: While shopping for wine for his wedding, Kepler noticed that the price of a barrel of wine (here assumed to be a cylinder) was determined solely by the length d of a dipstick that was inserted diagonally through a centered hole in the top of the barrel to the edge of the base of the barrel (see figure). Kepler realized that this measurement does not determine the volume of the barrel and that for a fixed value of d, the volume varies with the radius r and height h of the barrel. For a fixed value of d, what is the ratio r/h that maximizes the volume of the barrel?Blood testing Suppose a blood test for a disease is given to a population of N people, where N is large. At most, N individual blood tests must be done. The following strategy reduces the number of tests. Suppose 100 people are selected from the population and their blood samples are pooled. One test determines whether any of the 100 people test positive. If that test is positive, those 100 people are tested individually, making 101 tests necessary. However, if the pooled sample tests negative, then 100 people have been tested with one test. This procedure is then repeated. Probability theory shows that if the group size is x (for example, x = 100, as described here), then the average number of blood tests required to test N people is N(1 qx + 1/x), where q is the probability that any one person tests negative. What group size x minimizes the average number of tests in the case that N = 10, 000 and q = 0.95? Assume x is a real number between 1 and 10,000.Maximum-volume cylinder in a sphere Find the dimensions of the right circular cylinder of maximum volume that can be placed inside a sphere of radius R.Maximizing profit Suppose you own a tour bus and you book groups of 20 to 70 people for a day tour. The cost per person is 30 minus 0.25 for every ticket sold. If gas and other miscellaneous costs are 200, how many tickets should you sell to maximize your profit? Treat the number of tickets as a nonnegative real number.Cone in a cone A right circular cone is inscribed inside a larger right circular cone with a volume of 150 cm3. The axes of the cones coincide and the vertex of the inner cone touches the center of the base of the outer cone. Find the ratio of the heights of the cones that maximizes the volume of the inner cone.Cylinder in a sphere Find the height h, radius r, and volume of a right circular cylinder with maximum volume that is inscribed in a sphere of radius R.
Travel costs A simple model for travel costs involves the cost of gasoline and the cost of a driver. Specifically, assume that gasoline costs p/gallon and the vehicle gets g miles per gallon. Also assume that the driver earns w/hour. a. A plausible function to describe how gas mileage (in mi/gal) varies with speed v is g(v) = v(85 v)/60. Evaluate g(0), g(40), and g(60) and explain why these values are reasonable. b. At what speed does the gas mileage function have its maximum? c. Explain why the formula C(v) = Lp/g(v) + Lw/v gives the cost of the trip in dollars, where L is the length of the trip and v is the constant speed. Show that the dimensions are consistent. d. Let L = 400 mi, p = 4/gal, and w = 20/hr. At what (constant) speed should the vehicle be driven to minimize the cost of the trip? e. Should the optimal speed be increased or decreased (compared with part (d)) if L is increased from 400 mi to 500 mi? Explain. f. Should the optimal speed be increased or decreased (compared with part (d)) if p is increased from 4/gal to 4.20/gal? Explain. g. Should the optimal speed be increased or decreased (compared with part (d)) if w is decreased from 20/hr to 15/hr? Explain.Do dogs know calculus? A mathematician stands on a beach with his dog at point A. He throws a tennis ball so that it hits the water at point B. The dog, wanting to get to the tennis ball as quickly as possible, runs along the straight beach line to point D and then swims from point D to point B to retrieve his ball. Assume C is the point on the edge of the beach closest to the tennis ball (see figure). a. Assume the dog runs at speed r and swims at speed s, where r s and both are measured in meters/second. Also assume the lengths of BC, CD, and AC are x, y, and z, respectively. Find a function T(y) representing the total time it takes the dog to get to the ball. b. Verify that the value of y that minimizes the time it takes to retrieve the ball isy=xr/s+1r/s1. c. If the dog runs at 8 m/s and swims at 1 m/s, what ratio y/x produces the fastest retrieving time? d. A dog named Elvis who runs at 6.4 m/s and swims at 0.910 m/s was found to use an average ratio y/x of 0.144 to retrieve his ball. Does Elvis appear to know calculus? (Source: Do Dogs Know Calculus? T. Pennings. The College Mathematics Journal, 34, 3, May 2003)Viewing angles An auditorium with a flat floor has a large screen on one wall. The lower edge of the screen is 3 ft above eye level, and the upper edge of the screen is 10 ft above eye level (see figure). How far from the screen should you stand to maximize your viewing angle ?Suspension system A load must be suspended 6 m below a high ceiling using cables attached to two supports that are 2 m apart (see figure). How far below the ceiling (x in the figure) should the cables be joined to minimize the total length of cable used?Light sources The intensity of a light source at a distance is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. Two light sources, one twice as strong as the other, are 12 m apart. At what point on the line segment joining the sources is the intensity the weakest?