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Jan Stewart, Calculus: Early Transcendetals, 6 Edition

Section 4.3

61. A graph of a population of yeast cells in a new laboratory culture as a function of time is shown. (a) Describe how the rate of population increase varies. (b) When is this rate highest? (c) On what intervals is the population function concave upward or downward? (d) Estimate the coordinates of the inflection point.

65. A drug response curvedescribes the level of medication in the bloodstream after a drug is administered. A surge function S(t) = Atpe-kt is often used to model the response curve, reflecting an initial surge in the drug level and then a more gradual decline. If, for a particular drug, A = 0.01, p = 4, k = 0.07, and is measured in minutes, estimate the times corresponding to the inflection points and explain their significance. If you have a graphing device, use it to graph the drug response curve.

67. Find a cubic function f(x) = ax3 + bx2 + cx + d that has a local maximum value of 3 at -2 and a local minimum value of 0 at 1.

68. For what values of the numbers a and b does the function f(x) = axebx^2 have the maximum value f(2) = 1?

69. Show that the curve y = (1 + x)/(1 + x2) has three points of inflection and they all lie on one straight line.

70. Show that the curves y = e-x and y = -e-x touch the curve y = e-x sin x at its inflection points.

 

Section 4.4

73. If the initial amount Ao of money is invested at an interest rate r compounded n times a years is A = A0(1 + r/n)nt. If we let n →∞, we refer to the continuous compounding of interest. Use l’Hopital’s Rule to show that if interest is compounded continuously, then the amount after t years A = A0ert.

74. If a metal ball with mass m is projected in water and the force of resistance is proportional to the square of the velocity, then the distance the ball travels in time t is S(t) = m/c ln cosh√(gc/mt) where c is a positive constant. Find lim(c → 0+) s(t).

75. If an electrostatic field E acts on a liquid or a gaseous polar dielectric, the net dipole moment P per unit volume is P(E) = (eE + e-E)/ (eE - e-E) – 1/E. Show that lim (E → 0+) P(E) = 0.

76. A metal cable has radius r and is covered by insulation, so that the distance from the center of the cable to the exterior of the insulation is R. The velocity v of an electrical impulse in the cable is v = -c(r/R)2 ln(r/R) where c is a positive constant. Find the following limits and interpret your answers. (a) lim(R → r+) v   (b) lim(r → 0+) v

78. The figure shows a sector of a circle with central angle θ. Let A(θ) be the area of the segment between the chord PR and the arc PR. Let B(θ) be the area of the triangle PQR. lim(θ → 0+) A(θ)/B(θ).


Section 4.5


Use the guidelines of this section to sketch the curve
1. y = x3 + x

57–60. Find an equation of the slant asymptote. Do not sketch the curve.
y = (x2 + 1)/(x + 1)

67. Show that the curve y = x – tan-1 x has two slant asymptotes: y = x + π/2 and y = x - π/2. Use this fact to help sketch the curve.


Section 4.6

1-8. Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f ′ and f ″ to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points. F(x) = 4x4 – 32x3 + 89x2 – 95x + 29

 

Section 4.7

1. Consider the following problem: Find two numbers whose sum is 23 and whose product is a maximum. (a) Make a table of values, like the following one, so that the sum of the numbers in the first two columns is always 23. On the basis of the evidence in your table, estimate the answer to the problem. (b) Use calculus to solve the problem and compare with your answer to part (a).

2. Find two numbers whose difference is 100 and whose product is a minimum.

3. Find two positive numbers whose product is 100 and whose sum is a minimum.

4. Find a positive number such that the sum of the number and its reciprocal is as small as possible.

5. Find the dimensions of a rectangle with perimeter 100 m whose area is as large as possible.

6. Find the dimensions of a rectangle with area 1000 m3 whose perimeter is as small as possible.

7. A model used for the yield Yof an agricultural crop as a function of the nitrogen level N in the soil (measured in appropriate units) is Y= kN/(1 + N2) where k is a positive constant. What nitrogen level gives the best yield?

8. The rate (in mg/m3/h) at which photosynthesis takes place for a species of phytoplankton is modeled by the function P = 100I/(I2 + I + 4) where I is the light intensity (measured in thousands of foot-candles). For what light intensity is P a maximum?

9. Consider the following problem: A farmer with 750 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible total area of the four pens? (a) Draw several diagrams illustrating the situation, some with shallow, wide pens and some with deep, narrow pens. Find the total areas of these configurations. Does it appear that there is a maximum area? If so, estimate it. (b) Draw a diagram illustrating the general situation. Introduce notation and label the diagram with your symbols. (c) Write an expression for the total area. (d) Use the given information to write an equation that relates the variables. (e) Use part (d) to write the total area as a function of one variable. (f) Finish solving the problem and compare the answer with your estimate in part (a).

10. Consider the following problem: A box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have. (a) Draw several diagrams to illustrate the situation, some short boxes with large bases and some tall boxes with small bases. Find the volumes of several such boxes. Does it appear that there is a maximum volume? If so, estimate it. (b) Draw a diagram illustrating the general situation. Introduce notation and label the diagram with your symbols. (c) Write an expression for the volume. (d) Use the given information to write an equation that relates the variables. (e) Use part (d) to write the volume as a function of one variable. (f) Finish solving the problem and compare the answer with your estimate in part (a).

11. A farmer wants to fence an area of 1.5 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. How can he do this so as to minimize the cost of the fence?

12. A box with a square base and open top must have a volume of 32,000 cm3. Find the dimensions of the box that minimize the amount of material used.

13. If 1200 cm2 of material is available to make a box with a square base and an open top, find the largest possible volume of the box.

14. A rectangular storage container with an open top is to have a volume of 10 m . The length of its base is twice the width. Material for the base costs $10 per square meter. Material for the sides costs $6 per square meter. Find the cost of materials for the cheapest such container.

15. Do Exercise 14 assuming the container has a lid that is made from the same material as the sides.

16. (a) Show that of all the rectangles with a given area, the one with smallest perimeter is a square. (b) Show that of all the rectangles with a given perimeter, the one with greatest area is a square.

17. Find the point on the line y = 4x + 7 that is closest to the origin.

18. Find the point on the line 6x + y = 9 that is closest to the point (-3, 1).

19. Find the points on the ellipse 4x2 + y2 = 4 that are farthest away from the point (1, 0).

20. Find, correct to two decimal places, the coordinates of the point on the curve y=tan x that is closest to the point (1,1).

21. Find the dimensions of the rectangle of largest area that can be inscribed in a circle of radius r.

22. Find the area of the largest rectangle that can be inscribed in the ellipse x2/a2 + y2/b2 = 1.

23. Find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle of side L if one side of the rectangle lies on the base of the triangle.

24. Find the dimensions of the rectangle of largest area that has its base on the x-axis and its other two vertices above the x-axis and lying on the parabola y=8 - x2.

25. Find the dimensions of the isosceles triangle of largest area that can be inscribed in a circle of radius r.

26. Find the area of the largest rectangle that can be inscribed in a right triangle with legs of lengths 3 cm and 4 cm if two sides of the rectangle lie along the legs.

27. A right circular cylinder is inscribed in a sphere of radius r. Find the largest possible volume of such a cylinder.

28. A right circular cylinder is inscribed in a cone with height h and base radius r. Find the largest possible volume of such a cylinder.

29. A right circular cylinder is inscribed in a sphere of radius r. Find the largest possible surface area of such a cylinder.

30. A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus the diameter of the semicircle is equal to the width of the rectangle. See Exercise 56 on page 23.) If the perimeter of the window is 30 ft, find the dimensions of the window so that the greatest possible amount of light is admitted.

31. The top and bottom margins of a poster are each 6 cm and the side margins are each 4 cm. If the area of printed material on the poster is fixed at 384 cm2, find the dimensions of the poster with the smallest area.

32. A poster is to have an area of 180 in2 with 1-inch margins at the bottom and sides and a 2-inch margin at the top. What dimensions will give the largest printed area?

33. A piece of wire 10 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. How should the wire be cut so that the total area enclosed is (a) a maximum? (b) A minimum?

35. A cylindrical can without a top is made to contain V cm3 of liquid. Find the dimensions that will minimize the cost of the metal to make the can.

36. A fence 8 ft tall runs parallel to a tall building at a distance of 4 ft from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building?

37. A cone-shaped drinking cup is made from a circular piece of paper of radius R by cutting out a sector and joining the edges CA and CB. Find the maximum capacity of such a cup.

38. A cone-shaped paper drinking cup is to be made to hold 27 cm3 of water. Find the height and radius of the cup that will use the smallest amount of paper.

39. A cone with height is inscribed in a larger cone with height H so that its vertex is at the center of the base of the larger cone. Show that the inner cone has maximum volume when h = (1/3)H

40. An object with weight W is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle with a plane, then the magnitude of the force is F = μW/(μ sin θ + cos θ ) where μ is a constant called the coefficient of friction. For what value of θ is F smallest?

41. If a resistor of R ohms is connected across a battery of E volts with internal resistance r ohms, then the power (in watts) in the external resistor is P = E2R/(R + r)2. If E and r are fixed but R varies, what is the maximum value of the power?

42. For a fish swimming at a speed relative to the water, the energy expenditure per unit time is proportional to v3. It is believed that migrating fish try to minimize the total energy required to swim a fixed distance. If the fish are swimming against a current u (u < v), then the time required to swim a distance L is L/(v – u) and the total energy required to swim the distance is given by E(v) = av3L/(v – u) where a is the proportionality constant. (a) Determine the value of v that minimizes E. (b) Sketch the graph of E.

43. In a beehive, each cell is a regular hexagonal prism, open at one end with a trihedral angle at the other end as in the figure. It is believed that bees form their cells in such a way as to minimize the surface area for a given volume, thus using the least amount of wax in cell construction. Examination of these cells has shown that the measure of the apex angle q is amazingly consistent. Based on the geometry of the cell, it can be shown that the surface area S is given by S = 6sh – (3/2)s2 cot θ + (3s2√3/2) csc θ where s, the length of the sides of the hexagon, and h, the height, are constants. (a) Calculate dS/dq. (b) What angle should the bees prefer? (c) Determine the minimum surface area of the cell (in terms of s and h).

44. A boat leaves a dock at 2:00 PM and travels due south at a speed of 20 km/h. Another boat has been heading due east at 15 km/h and reaches the same dock at 3:00 PM. At what time were the two boats closest together?

46. A woman at a point A on the shore of a circular lake with radius 2 mi wants to arrive at the point C diametrically opposite A on the other side of the lake in the shortest possible time. She can walk at the rate of 4 mi/h and row a boat at 2 mi/h. How should she proceed?

47. An oil refinery is located on the north bank of a straight river that is 2 km wide. A pipeline is to be constructed from the refinery to storage tanks located on the south bank of the river 6 km east of the refinery. The cost of laying pipe is $400,000/km over land to a point P on the north bank and $800,000/km under the river to the tanks. To minimize the cost of the pipeline, where P should be located?

49. The illumination of an object by a light source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. If two light sources, one three times as strong as the other, are placed 10 ft apart, where should an object be placed on the line between the sources so as to receive the least illumination?

50. Find an equation of the line through the point (3,5) that cuts off the least area from the first quadrant.

51. Let a and b be positive numbers. Find the length of the shortest line segment that is cut off by the first quadrant and passes through the point (a,b).

53. (a) If C(x) is the cost of producing units of a commodity, then the average cost per unit is (x)=C(x)/x. Show that if the average cost is a minimum, then the marginal cost equals the average cost. (b) If C(x) = 16000 + 200x + 4x3/2, in dollars, find (i) the cost, average cost, and marginal cost at a production level of 1000 units; (ii) the production level that will minimize the average cost; and (iii) the minimum average cost.

54. (a) Show that if the profit P(x) is a maximum, then the marginal revenue equals the marginal cost. (b) If C(x)=16000 +500x - 1.6x2 + 0.004x3 is the cost function and p(x)=1700-7x is the demand function, find the production level that will maximize profit.

55. A baseball team plays in a stadium that holds 55,000 spectators. With ticket prices at $10, the average attendance had been 27,000. When ticket prices were lowered to $8, the average attendance rose to 33,000. (a) Find the demand function, assuming that it is linear. (b) How should ticket prices be set to maximize revenue?

56. During the summer months Terry makes and sells necklaces on the beach. Last summer he sold the necklaces for $10 each and his sales averaged 20 per day. When he increased the price by $1, he found that the average decreased by two sales per day. (a) Find the demand function, assuming that it is linear.
(b) If the material for each necklace costs Terry $6, what should the selling price be to maximize his profit?

57. A manufacturer has been selling 1000 television sets a week at $450 each. A market survey indicates that for each $10 rebate offered to the buyer, the number of sets sold will increase by 100 per week. (a) Find the demand function. (b) How large a rebate should the company offer the buyer in order to maximize its revenue? (c) If its weekly cost function is C(x)=68,000+150x, how should the manufacturer set the size of the rebate in order to maximize its profit?

58. The manager of a 100-unit apartment complex knows from experience that all units will be occupied if the rent is $800 per month. A market survey suggests that, on average, one additional unit will remain vacant for each $10 increase in rent. What rent should the manager charge to maximize revenue?

59. Show that of all the isosceles triangles with a given perimeter, the one with the greatest area is equilateral.

60. The frame for a kite is to be made from six pieces of wood. The four exterior pieces have been cut with the lengths indicated in the figure. To maximize the area of the kite, how long should the diagonal pieces be?

61. A point P needs to be located somewhere on the line AD so that the total length L of cables linking P to the points A, B, and C is minimized (see the figure). Express L as a function of x=|AP| and use the graphs of L and dL/dx to estimate the minimum value.

62. The graph shows the fuel consumption c of a car (measured in gallons per hour) as a function of the speed v of the car. At very low speeds the engine runs inefficiently, so initially c decreases as the speed increases. But at high speeds the fuel consumption increases. You can see that c(v) is minimized for this car when v»30 mi/h. However, for fuel efficiency, what must be minimized is not the consumption in gallons per hour but rather the fuel consumption in gallons per mile. Let’s call this consumption G. Using the graph, estimate the speed at which G has its minimum value.

63. Let v1 be the velocity of light in air and v2 the velocity of light in water. According to Fermat’s Principle, a ray of light will travel from a point A in the air to a point B in the water by a path ACB that minimizes the time taken. Show that sin θ1/ sin θ2 = v1/v2 where θ1 (the angle of incidence) and θ2 (the angle of refraction) are as shown. This equation is known as Snell’s Law.

64. Two vertical poles PQ and ST are secured by a rope PRS going from the top of the first pole to a point R on the ground between the poles and then to the top of the second pole as in the figure. Show that the shortest length of such a rope occurs when θ12.

65. The upper right-hand corner of a piece of paper, 12 in. by 8 in., as in the figure, is folded over to the bottom edge. How would you fold it so as to minimize the length of the fold? In other words, how would you choose x to minimize y?

66. A steel pipe is being carried down a hallway 9 ft wide. At the end of the hall there is a right-angled turn into a narrower hallway 6 ft wide. What is the length of the longest pipe that can be carried horizontally around the corner?

67. An observer stands at a point P, one unit away from a track. Two runners start at the point S in the figure and run along the track. One runner runs three times as fast as the other. Find the maximum value of the observer’s angle q of sight between the runners.

68. A rain gutter is to be constructed from a metal sheet of width 30 cm by bending up one-third of the sheet on each side through an angle q. How should q be chosen so that the gutter will carry the maximum amount of water?

69. Where should the point P be chosen on the line segment AB so as to maximize the angle θ?

70. A painting in an art gallery has height h and is hung so that its lower edge is a distance d above the eye of an observer (as in the figure). How far from the wall should the observer stand to get the best view? (In other words, where should the observer stand so as to maximize the angle q subtended at his eye by the painting?)

71. Find the maximum area of a rectangle that can be circumscribed about a given rectangle with length L and width W.

72. The blood vascular system consists of blood vessels (arteries, arterioles, capillaries, and veins) that convey blood from the heart to the organs and back to the heart. This system should work so as to minimize the energy expended by the heart in pumping the blood. In particular, this energy is reduced when the resistance of the blood is lowered. One of Poiseuille’s Laws gives the resistance R of the blood as R = CL/r4 where L is the length of the blood vessel, r is the radius, and C is a positive constant determined by the viscosity of the blood. (Poiseuille established this law experimentally, but it also follows from Equation 8.4.2.) The figure shows a main blood vessel with radius r1 branching at an angle θ into a smaller vessel with radius r2. (a) Use Poiseuille’s Law to show that the total resistance of the blood along the path ABC is R = C((a – b cot θ)/r14 + b csc θ/r24) where a and b are the distances shown in the figure. (b) Prove that this resistance is minimized when cos θ = r24/r14 (c) Find the optimal branching angle (correct to the nearest degree) when the radius of the smaller blood vessel is two-thirds the radius of the larger vessel.

Section 4.8

9. Use Newton’s method with initial approximation x1=-1 to find x2, the second approximation to the root of the equation x3 + x + 3 = 0. Explain how the method works by first graphing the function and its tangent line at (-1, 1).

39. Use Newton’s method to find the coordinates, correct to six decimal places, of the point on the parabola y=(x-1)2 that is closest to the origin.


Section 4.9

63. A stone is dropped from the upper observation deck (the Space Deck) of the CN Tower, 450 m above the ground. (a) Find the distance of the stone above ground level at time t. (b) How long does it take the stone to reach the ground? (c) With what velocity does it strike the ground? (d) If the stone is thrown downward with a speed of 5 m/s, how long does it take to reach the ground?

66. Two balls are thrown upward from the edge of the cliff in Example 7. The first is thrown with a speed of 48 ft/s and the other is thrown a second later with a speed of 24 ft/s. Do the balls ever pass each other?

67. A stone was dropped off a cliff and hit the ground with a speed of 120 ft/s. What is the height of the cliff?

69. A company estimates that the marginal cost (in dollars per item) of producing x items is 1.92-0.002x. If the cost of producing one item is $562, find the cost of producing 100 items.

70. The linear density of a rod of length 1 m is given by ρ(x) = 1/√x, in grams per centimeter, where x is measured in centimeters from one end of the rod. Find the mass of the rod.

72. A car is traveling at 50 mi/h when the brakes are fully applied, producing a constant deceleration of 22 ft/s2. What is the distance traveled before the car comes to a stop?

73. What constant acceleration is required to increase the speed of a car from 30 mi/h to 50 mi/h in 5 s?

74. A car braked with a constant deceleration of 16 ft/s2, producing skid marks measuring 200 ft before coming to a stop. How fast was the car traveling when the brakes were first applied?

75. A car is traveling at 100 km/h when the driver sees an accident 80 m ahead and slams on the brakes. What constant deceleration is required to stop the car in time to avoid a pileup?


Chapter 4 Review

50. Find two positive integers such that the sum of the first number and four times the second number is 1000 and the product of the numbers is as large as possible.

52. Find the point on the hyperbola xy=8 that is closest to the point (3, 0).

53. Find the smallest possible area of an isosceles triangle that is circumscribed about a circle of radius r.

54. Find the volume of the largest circular cone that can be inscribed in a sphere of radius r.

58. A metal storage tank with volume V is to be constructed in the shape of a right circular cylinder surmounted by a hemisphere. What dimensions will require the least amount of metal?

59. A hockey team plays in an arena with a seating capacity of 15,000 spectators. With the ticket price set at $12, average attendance at a game has been 11,000. A market survey indicates that for each dollar the ticket price is lowered, average attendance will increase by 1000. How should the owners of the team set the ticket price to maximize their revenue from ticket sales?

60. A manufacturer determines that the cost of making units of a commodity is C(x)=1800 + 25x - 0.2x2 + 0.001x3 and the demand function is p(x)=48.2 - 0.03x. (a) Graph the cost and revenue functions and use the graphs to estimate the production level for maximum profit. (b) Use calculus to find the production level for maximum profit. (c) Estimate the production level that minimizes the average cost.

77. A canister is dropped from a helicopter 500 m above the ground. Its parachute does not open, but the canister has been designed to withstand an impact velocity of 100 m/s. Will it burst?

78. In an automobile race along a straight road, car A passed car B twice. Prove that at some time during the race their accelerations were equal. State the assumptions that you make.

Section 5.1

13. Oil leaked from a tank at a rate of r(t) liters per hour. The rate decreased as time passed and values of the rate at two hour time intervals are shown in the table. Find lower and upper estimates for the total amount of oil that leaked out.

 

Section 5.1

1. Evaluate the Riemann sum for f(x) = 3 – 1/2x, 2 ≤ x ≤ 14, with six subintervals, taking the sample points to be left endpoints. Explain, with the aid of a diagram, what the Riemann sum represents.

3. If f(x)=ex - 2, 0 ≤ x ≤ 2 , find the Riemann sum with n = 4 correct to six decimal places, taking the sample points to be midpoints. What does the Riemann sum represent? Illustrate with a diagram.

 

Section 5.4
51. If oil leaks from a tank at a rate of r(t) gallons per minute at time t, what does represent?
52. A honeybee population starts with 100 bees and increases at a rate n′(t) of bees per week. What does 100 + represent?
61. The linear density of a rod of length 4 m is given by ρ(x) = 9 + 2√x measured in kilograms per meter, where x is measured in meters from one end of the rod. Find the total mass of the rod.

62. Water flows from the bottom of a storage tank at a rate r(t)=200-4t of liters per minute, where 0 ≤ t ≤ 50 . Find the amount of water that flows from the tank during the first 10 minutes.

Section 5.5

77. An oil storage tank ruptures at time t=0 and oil leaks from the tank at a rate of r(t)=100e-0.01t liters per minute. How much oil leaks out during the first hour?

Chaper 5 Review

 

56. A particle moves along a line with velocity function v(t)=t2 - t, where v is measured in meters per second. Find (a) the displacement and (b) the distance traveled by the particle during the time interval [0,5].


Section 6.1

42. The widths (in meters) of a kidney-shaped swimming pool were measured at 2-meter intervals as indicated in the figure. Use the Midpoint Rule to estimate the area of the pool.

43. A cross-section of an airplane wing is shown. Measurements of the height of the wing, in centimeters, at 20-centimeter intervals are 5.8, 20.3, 26.7, 29.0, 27.7, 27.3, 23.8, 20.5, 15.1, 8.7, and 2.8. Use the Midpoint Rule to estimate the area of the wing’s cross-section.

45. Two cars, A and B, start side by side and accelerate from rest. The figure shows the graphs of their velocity functions. (a) Which car is ahead after one minute? Explain. (b) What is the meaning of the area of the shaded region? (c) Which car is ahead after two minutes? Explain. (d) Estimate the time at which the cars are again side by side.

48. Find the area of the region bounded by the parabola y=x2, the tangent line to this parabola at (1,1), and the x-axis.

49. Find the number b such that the line y=b divides the region bounded by the curves y=x2 and y=4 into two regions with equal area.

51. Find the values of c such that the area of the region bounded by the parabolas y=x2 – c2 and y=c2 – x2 is 576.

 

Section 6.2

 

1. Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

31. Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.

49. Find the volume of the described solid. A right circular cone with height h and base radius r.

50. Find the volume of the described solid. A frustum of a right circular cone with height h, lower base radius R, and top radius r.

51. Find the volume of the described solid. A cap of a sphere with radius r and height h.

52. Find the volume of the described solid. A frustum of a pyramid with square base of side b, square top of side a, and height h.

53. Find the volume of the described solid. A pyramid with height h and rectangular base with dimensions b and 2b.

54. Find the volume of the described solid. A pyramid with height h and base an equilateral triangle with side a (a tetrahedron).

56. The base of S is a circular disk with radius r. Parallel cross sections perpendicular to the base are squares.

57. The base of S is an elliptical region with boundary curve 9x2+4y2 = 36. Cross-sections perpendicular to the x-axis are isosceles right triangles with hypotenuse in the base.

58. The base of S is the triangular region with vertices (0,0), (1,0), and (0,1). Cross-sections perpendicular to the y–axis are equilateral triangles.

59. The base of S is the region enclosed by the parabola y=1-x2 and the x-axis. Cross-sections perpendicular to the x-axis are squares.

62. The base of is a circular disk with radius r. Parallel cross-sections perpendicular to the base are isosceles triangles with height h and unequal side in the base. (a) Set up an integral for the volume of S. (b) By interpreting the integral as an area, find the volume of S.

66. Find the volume common to two circular cylinders, each with radius r, if the axes of the cylinders intersect at right angles.

67. Find the volume common to two spheres, each with radius r, if the center of each sphere lies on the surface of the other sphere.

68. A bowl is shaped like a hemisphere with diameter 30 cm. A ball with diameter 10 cm is placed in the bowl and water is poured into the bowl to a depth of h centimeters. Find the volume of water in the bowl.

69. A hole of radius r is bored through a cylinder of radius R>r at right angles to the axis of the cylinder. Set up, but do not evaluate, an integral for the volume cut out.

70. A hole of radius r is bored through the center of a sphere of radius R>r. Find the volume of the remaining portion of the sphere.


Section 6.3

 

3. Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis. Sketch the region and a typical shell.

8. Let V be the volume of the solid obtained by rotating about the y-axis the region bounded by y = √x and y=x2. Find V both by slicing and by cylindrical shells. In both cases draw a diagram to explain your method.

9. Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. Sketch the region and a typical shell.

15. Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis. Sketch the region and a typical shell.


Section 6.4

 

1. How much work is done in lifting a 40-kg sandbag to a height of 1.5 m?

2. Find the work done if a constant force of 100 lb is used to pull a cart a distance of 200 ft.

3. A particle is moved along the x-axis by a force that measures 10/(1+x)2 pounds at a point x feet from the origin. Find the work done in moving the particle from the origin to a distance of 9 ft.

7. A force of 10 lb is required to hold a spring stretched 4 in. beyond its natural length. How much work is done in stretching it from its natural length to 6 in. beyond its natural length?

8. A spring has a natural length of 20 cm. If a 25-N force is required to keep it stretched to a length of 30 cm, how much work is required to stretch it from 20 cm to 25 cm?

9. Suppose that 2 J of work is needed to stretch a spring from its natural length of 30 cm to a length of 42 cm. (a) How much work is needed to stretch the spring from 35 cm to 40 cm? (b) How far beyond its natural length will a force of 30 N keep the spring stretched?

10. If the work required to stretch a spring 1 ft beyond its natural length is 12 ft-lb, how much work is needed to stretch it 9 in. beyond its natural length?

11. A spring has natural length 20 cm. Compare the work W1 done in stretching the spring from 20 cm to 30 cm with the work W2 done in stretching it from 30 cm to 40 cm. How are W2 and W1 related?

12. If 6 J of work is needed to stretch a spring from 10 cm to 12 cm and another 10 J is needed to stretch it from 12 cm to 14 cm, what is the natural length of the spring?

13. A heavy rope, 50 ft long, weighs 0.5 lb/ft and hangs over the edge of a building 120 ft high. (a) How much work is done in pulling the rope to the top of the building? (b) How much work is done in pulling half the rope to the top of the building?

14. A chain lying on the ground is 10 m long and its mass is 80 kg. How much work is required to raise one end of the chain to a height of 6 m?

15. A cable that weighs 2 b/ft is used to lift 800 lb of coal up a mine shaft 500 ft deep. Find the work done.

16. A bucket that weighs 4 lb and a rope of negligible weight are used to draw water from a well that is 80 ft deep. The bucket is filled with 40 lb of water and is pulled up at a rate of 2 ft/s, but water leaks out of a hole in the bucket at a rate of 0.2 lb/s. Find the work done in pulling the bucket to the top of the well.

17. A leaky 10-kg bucket is lifted from the ground to a height of 12 m at a constant speed with a rope that weighs 0.8 kg/m. Initially the bucket contains 36 kg of water, but the water leaks at a constant rate and finishes draining just as the bucket reaches the 12 m level. How much work is done?

18. A 10-ft chain weighs 25 lb and hangs from a ceiling. Find the work done in lifting the lower end of the chain to the ceiling so that it’s level with the upper end.

19. An aquarium 2 m long, 1 m wide, and 1 m deep is full of water. Find the work needed to pump half of the water out of the aquarium. (Use the fact that the density of water is 1000kg/m3.)

20. A circular swimming pool has a diameter of 24 ft, the sides are 5 ft high, and the depth of the water is 4 ft. How much work is required to pump all of the water out over the side? (Use the fact that water weighs 62.5 lb/ft3.)

21. A tank is full of water. Find the work required to pump the water out of the spout. In Exercises 23 and 24 use the fact that water weighs 62.5 lb/ft3.

25. Suppose that for the tank in Exercise 21 the pump breaks down after 4.7×105 J of work has been done. What is the depth of the water remaining in the tank?


Section 6.5

18. (a) A cup of coffee has temperature 95 °C and takes 30 minutes to cool to 61° C in a room with temperature 20 °C. Use Newton’s Law of Cooling (Section 3.8) to show that the temperature of the coffee after minutes is T(t) = 20 + 75e-kt where k»0.02. (b) What is the average temperature of the coffee during the first half hour?

Section 7.1

63. A particle that moves along a straight line has velocity v(t)=t2e-t meters per second after t seconds. How far will it travel during the first t seconds?

 

Section 7.3

41. Find the area of the crescent-shaped region (called a lune) bounded by arcs of circles with radii r and R.

42. A water storage tank has the shape of a cylinder with diameter 10 ft. It is mounted so that the circular cross-sections are vertical. If the depth of the water is 7 ft, what percentage of the total capacity is being used?

 

Section 8.1

19.Find the length of the arc of the curve from point P to point Q.

35. Find the arc length function for the curve y = sin-1 x + √(1 – x2) with starting point (0,1).

36. A steady wind blows a kite due west. The kite’s height above ground from horizontal position x=0 to x=80 ft is given by y = 150 – (1/40)(x – 50)2. Find the distance traveled by the kite.

37. A hawk flying at 15 m/s at an altitude of 180 m accidentally drops its prey. The parabolic trajectory of the falling prey is described by the equation y = 180 – x2/45 until it hits the ground, where y is its height above the ground and x is the horizontal distance traveled in meters. Calculate the distance traveled by the prey from the time it is dropped until the time it hits the ground. Express your answer correct to the nearest tenth of a meter.

Section 8.2

5.Find the area of the surface obtained by rotating the curve about the x-axis. y = x3, 0 ≤ x ≤ 2.

17. Use Simpson’s Rule with n=10 to approximate the area of the surface obtained by rotating the curve about the x-axis. Compare your answer with the value of the integral produced by your calculator. y=ln x, 1 ≤ x ≤ 3.

26. If the infinite curve y=e-x, x≥0, is rotated about the x-axis, find the area of the resulting surface.

33. Find the area of the surface obtained by rotating the circle x2 + y2 = r2 about the line y=r.

Section 8.3

12.A large tank is designed with ends in the shape of the region between the curves y = (1/2)x2 and y = 12, measured in feet. Find the hydrostatic force on one end of the tank if it is filled to a depth of 8 ft with gasoline. (Assume the gasoline’s density is 42.0 lb/ft3.)

13. A trough is filled with a liquid of density 840 kg/m3. The ends of the trough are equilateral triangles with sides 8 m long and vertex at the bottom. Find the hydrostatic force on one end of the trough.

14. A vertical dam has a semicircular gate as shown in the figure. Find the hydrostatic force against the gate.

15. A cube with 20-cm-long sides is sitting on the bottom of an aquarium in which the water is one meter deep. Estimate the hydrostatic force on (a) the top of the cube and (b) one of the sides of the cube.

16. A dam is inclined at an angle 30° of from the vertical and has the shape of an isosceles trapezoid 100 ft wide at the top and 50 ft wide at the bottom and with a slant height of 70 ft. Find the hydrostatic force on the dam when it is full of water.

29. Find the centroid of the region bounded by the given curves. y=x2, x=y2.

Section 8.4

1.The marginal cost function C ′(x) was defined to be the derivative of the cost function. (See Sections 3.7 and 4.7.) If the marginal cost of manufacturing x meters of a fabric is C ′(x) = 5 – 0.008x + 0.000009x2 (measured in dollars per meter) and the fixed start-up cost is C(0)=$20,000, use the Net Change Theorem to find the cost of producing the first 2000 units.

2. The marginal revenue from the sale of units of a product is 12 - 0.0004x. If the revenue from the sale of the first 1000 units is $12,400, find the revenue from the sale of the first 5000 units.

3. The marginal cost of producing units of a certain product is 74 + 1.1x - 0.002x2 + 0.00004x3 (in dollars per unit). Find the increase in cost if the production level is raised from 1200 units to 1600 units.

4. The demand function for a certain commodity is p =20 - 0.05x. Find the consumer surplus when the sales level is 300. Illustrate by drawing the demand curve and identifying the consumer surplus as an area.

5. A demand curve is given by p=450/(x+8). Find the consumer surplus when the selling price is $10.

7. If a supply curve is modeled by the equation p = 200 + 0.2x3/2, find the producer surplus when the selling price is $400.

10. A movie theater has been charging $7.50 per person and selling about 400 tickets on a typical weeknight. After surveying their customers, the theater estimates that for every 50 cents that they lower the price, the number of moviegoers will increase by 35 per night. Find the demand function and calculate the consumer surplus when the tickets are priced at $6.00.


Section 9.1


14. Suppose you have just poured a cup of freshly brewed coffee with temperature 95 °C in a room where the temperature is 20 °C. (a) When do you think the coffee cools most quickly? What happens to the rate of cooling as time goes by? Explain. (b) Newton’s Law of Cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings, provided that this difference is not too large. Write a differential equation that expresses Newton’s Law of Cooling for this particular situation. What is the initial condition? In view of your answer to part (a), do you think this differential equation is an appropriate model for cooling?


Section 9.3


19.
Find an equation of the curve that passes through the point (0, 1) and whose slope at (x, y) is xy.

38. A sphere with radius 1m has temperature 15 °C. It lies inside a concentric sphere with radius 2 m and temperature 25 °C. The temperature T(r) at a distance r from the common center of the spheres satisfies the differential equation d2T/dr2 + (2/r)dT/dr = 0. If we let S = dT/dr, then satisfies a first-order differential equation. Solve it to find an expression for the temperature T(r) between the spheres.

39. A glucose solution is administered intravenously into the bloodstream at a constant rate r. As the glucose is added, it is converted into other substances and removed from the bloodstream at a rate that is proportional to the concentration at that time. Thus a model for the concentration C=C(t) of the glucose solution in the bloodstream is dC/dt = r – kC where k is a positive constant. (a) Suppose that the concentration at time t=0 is Co. Determine the concentration at any time by solving the differential equation. (b) Assuming that Co<r/k, find lim(t →∞) C(t) and interpret your answer.

41. A tank contains 1000 L of brine with 15 kg of dissolved salt. Pure water enters the tank at a rate of 10 L/min. The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt is in the tank (a) after t minutes and (b) after 20 minutes?

42. The air in a room with volume 180 m3 contains 0.15% carbon dioxide initially. Fresher air with only 0.05% carbon dioxide flows into the room at a rate of 2 m3/min and the mixed air flows out at the same rate. Find the percentage of carbon dioxide in the room as a function of time. What happens in the long run?

43. A vat with 500 gallons of beer contains 4% alcohol (by volume). Beer with 6% alcohol is pumped into the vat at a rate of 5 gal/min and the mixture is pumped out at the same rate. What is the percentage of alcohol after an hour?

44. A tank contains 1000 L of pure water. Brine that contains 0.05 kg of salt per liter of water enters the tank at a rate of 5 L/min. Brine that contains 0.04 kg of salt per liter of water enters the tank at a rate of 10 L/min. The solution is kept thoroughly mixed and drains from the tank at a rate of 15 L/min. How much salt is in the tank (a) after t minutes and (b) after one hour?


Section 9.4

1. Suppose that a population develops according to the logistic equation dP/dt = 0.05P – 0.0005P2 where is measured in weeks. (a) What is the carrying capacity? What is the value of k? (b) A direction field for this equation is shown. Where are the slopes close to 0? Where are they largest? Which solutions are increasing? Which solutions are decreasing?

5. The population of the world was about 5.3 billion in 1990. Birth rates in the 1990s ranged from 35 to 40 million per year and death rates ranged from 15 to 20 million per year. Let’s assume that the carrying capacity for world population is 100 billion. (a) Write the logistic differential equation for these data. (Because the initial population is small compared to the carrying capacity, you can take k to be an estimate of the initial relative growth rate.) (b) Use the logistic model to estimate the world population in the year 2000 and compare with the actual population of 6.1 billion. (c) Use the logistic model to predict the world population in the years 2100 and 2500. (d) What are your predictions if the carrying capacity is 50 billion?

7. One model for the spread of a rumor is that the rate of spread is proportional to the product of the fraction y of the population who have heard the rumor and the fraction who have not heard the rumor. (a) Write a differential equation that is satisfied by y. (b) Solve the differential equation. (c) A small town has 1000 inhabitants. At 8 AM, 80 people have heard a rumor. By noon half the town has heard it. At what time will 90% of the population have heard the rumor?

8. Biologists stocked a lake with 400 fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be 10,000. The number of fish tripled in the first year. (a) Assuming that the size of the fish population satisfies the logistic equation, find an expression for the size of the population after t years. (b) How long will it take for the population to increase to 5000?

 

Section 9.5

27. In the circuit shown in Figure 4, a battery supplies a constant voltage of 40 V, the inductance is 2 H, the resistance is 10W, and I(0) = 0. (a) Find I(t). (b) Find the current after 0.1 s.

32. Two new workers were hired for an assembly line. Jim processed 25 units during the first hour and 45 units during the second hour. Mark processed 35 units during the first hour and 50 units the second hour. Using the model of Exercise 31 and assuming that P(0)=0, estimate the maximum number of units per hour that each worker is capable of processing.

34. A tank with a capacity of 400 L is full of a mixture of water and chlorine with a concentration of 0.05 g of chlorine per liter. In order to reduce the concentration of chlorine, fresh water is pumped into the tank at a rate of 4 L/s. The mixture is kept stirred and is pumped out at a rate of 10 L/s. Find the amount of chlorine in the tank as a function of time.


Section 10.1

1. (a) Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases. (b) Eliminate the parameter to find a Cartesian equation of the curve. x = 3t – 5, y = 2t + 1

24. Match the graphs of the parametric equations x=f(t) and y=g(t) in (a)–(d) with the parametric curves labeled I–IV. Give reasons for your choices.

28. Match the parametric equations with the graphs labeled I-VI. Give reasons for your choices. (Do not use a graphing device.)

33. Find parametric equations for the path of a particle that moves along the circle x2 + (y – 1)2 = 4 in the manner described. (a) Once around clockwise, starting at (2, 1). (b) Three times around counterclockwise, starting at (2, 1). (c) Halfway around counterclockwise, starting at (0, 3).

41. If a and b are fixed numbers, find parametric equations for the curve that consists of all possible positions of the point P in the figure, using the angle as q the parameter. Then eliminate the parameter and identify the curve.


Section 10.2


1. Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter. x = t4 + 1, y = t3 + t, t = –1

7. Find an equation of the tangent to the curve at the given point by two methods: (a) without eliminating the parameter and (b) by first eliminating the parameter. x = 1 + ln t, y = t2 + 2; (2, 3)

17. Find the points on the curve where the tangent is horizontal or vertical. If you have a graphing device, graph the curve to check your work. x = 10 – t2, y = t3 – 12t

25. Show that the curve x= cos t, y= sin t cos t has two tangents at (0,0) and find their equations. Sketch the curve.

30. Find equations of the tangents to the curve x = 3t2 + 1, y = 2t3 + 1 that pass through the point (4, 3).

32. Find the area enclosed by the curve x = t2 – 2t, y = √t and the y-axis.

33. Find the area enclosed by the x-axis and the curve x = 1 + et, y = t – t2.

41. Find the exact length of the curve. x = 1 + 3t2, y = 4 + 2t3, 0 ≤ t ≤ 1

59. Find the exact area of the surface obtained by rotating the given curve about the x-axis. x = t3, y = t2, 0 ≤ t ≤ 1

65. Find the surface area generated by rotating the given curve about the y-axis. x = 3t2, y =2t3, 0 ≤ t ≤ 5.

73. A string is wound around a circle and then unwound while being held taut. The curve traced by the point P at the end of the string is called the involute of the circle. If the circle has radius r and center O and the initial position of P is (r,0), and if the parameter q is chosen as in the figure, show that parametric  equations of the involute are x = r(cos θ + θ sin θ), y = r(sin θ - θ cos θ).

74. A cow is tied to a silo with radius r by a rope just long enough to reach the opposite side of the silo. Find the area available for grazing by the cow.


Section 10.3



3. Plot the point whose polar coordinates are given. Then find the Cartesian coordinates of the point.
(a) (1, π)        (b) (2, –2π/3)       (c) (–2, 3π/4)

7. Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. 1 ≤ r ≤ 2

13. Find the distance between the points with polar coordinates (2, π/3) and (4, 2π/3).

14. Find a formula for the distance between the points with polar coordinates (r1, θ1) and (r2, θ2).

21. Find a polar equation for the curve represented by the given Cartesian equation. x = 3.

29. Sketch the curve with the given polar equation. θ = –π/6.

56. Match the polar equations with the graphs labeled I–VI. Give reasons for your choices. (Don’t use a graphing device.)

57. Find the slope of the tangent line to the given polar curve at the point specified by the value of θ.
r = 2sin θ, θ = π/6.

63. Find the points on the given curve where the tangent line is horizontal or vertical. r = 3 cos θ


Section 10.4


1.
Find the area of the region that is bounded by the given curve and lies in the specified sector. r = θ2, 0 ≤ θ ≤ π/4.

5. Find the area of the shaded region.

6. Sketch the curve and find the area that it encloses. r = 3 cos θ.

17. Find the area of the region enclosed by one loop of the curve. r = sin2θ

23. Find the area of the region that lies inside the first curve and outside the second curve. r = 2 cosθ, r = 1.

45. Find the exact length of the polar curve. r = 3 sin θ, 0 ≤ θ ≤ π/3.

 

Section 10.5

 

1. Find the vertex, focus, and directrix of the parabola and sketch its graph. x = 2y2

11. Find the vertices and foci of the ellipse and sketch its graph. x2/9 + y2/5 = 1

19. Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph. x2/144 - y2/25 = 1

25. Identify the type of conic section whose equation is given and find the vertices and foci.  x2 = y + 1


Section 10.6

1.Write a polar equation of a conic with the focus at the origin and the given data. Parabola, eccentricity 7/4, directrix y = 6.

9. (a) Find the eccentricity, (b) identify the conic, (c) give an equation of the directrix, and (d) sketch the conic. r = 1/(1 + sin θ)


Chapter 10 review

49. Find an equation of the ellipse with foci (0, ±4) and vertices (±5, 0).

50. Find an equation of the parabola with focus (2, 1) and directrix x = –4.

51. Find an equation of the hyperbola with foci (0, ±4) and asymptotes y = ±3x.

52. Find an equation of the ellipse with foci (3, ±2) and major axis with length 8.

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