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Ron Larson, Calculus, 8th Edition. Boston, Houghton Mifflin, 2006.

Section P.2

In Exercises 7 and 8, sketch the lines through the point with the indicated slopes. Make the sketches on the same set of coordinate axes.
7. (2, 3); (a) 1     (b) -2

In Exercises 9–14, plot the pair of points and find the slope of the line passing through them.
9. (3, -4), (5, 2)

In Exercises 15–18, use the point on the line and the slope of the line to find three additional points that the line passes through.
15. (2, 1); m = 0

19. Conveyor Design. A moving conveyor is built to rise 1 meter for each 3 meters of horizontal change. (a) Find the slope of the conveyor. (b) Suppose the conveyor runs between two floors in a factory. Find the length of the conveyor if the vertical distance between floors is 10 feet.

20. Rate of Change. Each of the following is the slope of a line representing daily revenue y in terms of time x in days. Use the slope to interpret any change in daily revenue for a one day increase in time. a) m=800 b) m=250 c) m=0 

21. Modeling Data The table shows the populations (in millions) of the United States for 1996–2001. The variable  represents the time in years, with  corresponding to 1996. (a) Plot the data by hand and connect adjacent points with a line segment. (b) Use the slope of each line segment to determine the year when the population increased least rapidly.

22. Modeling Data. The table shows the rate (in miles per hour) that a vehicle is traveling after seconds. (a) Plot the data by hand and connect adjacent points with a line segment. (b) Use the slope of each line segment to determine the interval when the vehicle’s rate changed most rapidly. How did the
rate change?

23. Find the slope and the y-intercept (if possible) of the line
x + 5y = 20

27. Find an equation of the line that passes through the point and has the indicated slope. Sketch the line.
(0, 3); m = ¾

33. Find an equation of the line that passes through the points, and sketch the line.
(0, 0), (2, 6).

43. Find an equation of the vertical line with x-intercept 3.

44. Show that the line with intercepts and has the following equation x/a + y/b = 1

In Exercises 59– 64, write an equation of the line through the point (a) parallel to the given line and (b) perpendicular to the given line.
59. (2, 1); 4x – 2y = 3

Rate of Change. In Exercises 65– 68, you are given the dollar value of a product in 2004 and the rate at which the value of the product is expected to change during the next 5 years. Write a linear equation that gives the dollar value of the product in terms of the year (Let t = 0 represent 2000.)
65. $2540       $125 increase per year

In Exercises 71 and 72, determine whether the points are collinear. (Three points are collinear if they lie on the same line.)
71. (-2, 1), (-1, 0), (2, -2)

In Exercises 73 –75, find the coordinates of the point of intersection of the given segments. Explain your reasoning.
73. Perpendicular bisectors

76. Show that the points of intersection in Exercises 73, 74, and 75 are collinear.

77. Temperature conversion. Find a linear equation that expresses the relationship between the temperature in degrees Celsius C and the temperature in degrees Fahrenheit F. Use the fact that water freezes at 0 °C (32 °F) and boils at 100 °C (212 °F). Use the equation to convert 72 °F to degrees Celsius.

78. Reimbursed Expenses. A company reimburses its sales representatives at $150 per day for lodging  and meals plus 34¢ per mile driven. Write a linear function that gives the daily cost C in terms of x, the number of miles driven. How much does it cost the company if a sales representative drives 137 miles on a given day?

79. Career Choice. An employee has two options for positions in a large corporation. One position pays $12.50 per hour plus and additional unit rate of $0.75 per unit produced. The other pays $9.20 per hour plus a unit rate of $1.30. a) Find linear equations for the hourly wages W in terms of x, the number of units produced per hour, for each option. b) Use a graphing utility to graph both equations and find the point of intersection. c) Interpret the meaning of the point of intersection of the graph in part (b). How would you use this information to select the correct option if the goal were to obtain the highest hourly wage?

80. Straight-Line Depreciation. A small business purchases a piece of equipment for $875. After 5 years, the equipment is outdated, having no value. (a) Write a linear equation giving the value y of the equipment in terms of the time x, 0 ≤ x ≤ 5. (b) Find the value of the equipment when x = 2. (c) Estimate (to two-decimal-place accuracy) the time when the value of the equipment is $200.

81. Apartment Rental A real estate office handles an apartment complex with 50 units. When the rent is $580 per month, all 50 units are occupied. However, when the rent is $625, the average number of occupied units drops to 47. Assume that the relationship between the monthly rent p and the demand x is linear. (Note: The term demand refers to the number of occupied units.) (a) Write a linear equation giving the demand x in terms of the rent p.  (b) Linear extrapolation Use a graphing utility to graph the demand equation and use the trace feature to predict the number of units occupied if the rent is raised to $655. (c) Linear interpolation Predict the number of units occupied if the rent is lowered to $595. Verify graphically.

82. Modeling Data An instructor gives regular 20-point quizzes and 100-point exams in a mathematics course. Average scores for six students, given as ordered pairs (x, y) where x is the average quiz score and y is the average test score, are (18, 87), (10, 55), (19, 96), (16, 79), (13, 76) and (15, 82) (a) Use the regression capabilities of a graphing utility to find the least squares regression line for the data. (b) Use a graphing utility to plot the points and graph the regression line in the same viewing window. (c) Use the regression line to predict the average exam score for a student with an average quiz score of 17. (d) Interpret the meaning of the slope of the regression line. (e) The instructor adds 4 points to the average test score of everyone in the class. Describe the changes in the positions of the plotted points and the change in the equation of the line.

83. Tangent Line. Find an equation of the line tangent to the circle x2 + y2 = 169 at the point (5, 12).

84. Tangent Line Find an equation of the line tangent to the circle (x – 1)2 + (y – 1)2 = 25 at the point (4, -3).

Distance. In Exercises 85–90, find the distance between the point  and line, or between the lines, using the formula for the distance between the point (x1, y1) and the line Ax + By + C = 0
Distance = |Ax1 + By1 + C|/√(A2 + B2)

85. Point:  (0, 0)      Line: 4x + 3y = 10

91. Show that the distance between the point (x1, y1) and the line Ax + By + C = 0 is
Distance = |Ax1 + By1 + C|/√(A2 + B2)

92. Write the distance d between the point (3, 1) and the line in terms of m. Use a graphing utility to graph the equation. When is the distance 0? Explain the result geometrically.

93. Prove that the diagonals of a rhombus intersect at right angles. (A rhombus is a quadrilateral with sides of equal lengths.)

94. Prove that the figure formed by connecting consecutive midpoints of the sides of any quadrilateral is a parallelogram.

96. Prove that if the slopes of two nonvertical lines are negative reciprocals of each other, then the lines are perpendicular.

True or False? In Exercises 97 and 98, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

97. The lines represented by ax + by = c1 and bx – ay = c2 are perpendicular. Assume and a ≠ 0 and b ≠ 0.

98. It is possible for two lines with positive slopes to be perpendicular to each other.

 

Section P.3

In Exercises 29–36, sketch a graph of the function and find its domain and range. Use a graphing utility to verify your graph.
29. f(x) = 4 – x

In exercises 43-46, determine whether y is a function of x.
43. x2 + y2 = 4

In exercises 59-62, find the composite functions (f ◦ g) and (g ◦ f). What is the domain of each composite function? Are the two composite functions equal?

59. f(x) = x2, g(x) = √x

64. Ripples. A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius (in feet) of the outer ripple is given by r(t) = 0.6t, where is the time in seconds after the pebble strikes the water. The area of the circle is given by the function A(r) = πr2. Find and interpret
(A ◦ r)(t).

In Exercises, find the coordinates of a second point on the graph of a function f if the given point is on the graph and the function is (a) even and (b) odd.

72. (4, 9)

73. The graphs of f, g and h are shown in the figure. Decide whether each function is even, odd, or neither.

74. The domain of the function shown in the figure is -6 ≤ x ≤ 6. (a) Complete the graph of given that is even. (b) Complete the graph of given that is odd.

In Exercises 75–78, write an equation for a function that has the given graph.

75. Line segment connecting (-4, 3) and (0, -5).

77. The bottom half of the parabola x + y2 = 0

78. The bottom half of the circle x2 + y2 = 4

84. Water runs into a vase of height 30 centimeters at a constant rate. The vase is full after 5 seconds. Use this information and the shape of the vase shown to answer the questions if d is the depth of the water in centimeters and t is the time in seconds (see figure). (a) Explain why d is a function of t.  (b) Determine the domain and range of the function. (c) Sketch a possible graph of the function.

86. Automobile Aerodynamics. The horsepower H required to overcome wind drag on a certain automobile is approximated by H(x) = 0.002x2 + 0.005x – 0.029 where x is the speed of the car in miles per hour. (a) Use a graphing utility to graph H.  (b) Rewrite the power function so that x represents the speed in kilometers per hour. 

87. Write the function f(x) = |x| + |x – 2| without using absolute value signs.

89. Prove that the function is odd
f(x) = a2n+1x2n+1 + … + a3x3 + a1x

90. Prove that the function is even
f(x) = a2nx2n + a2n-2x2n-2 +… + a2x2 + a0

91. Prove that the product of two even (or two odd) functions is even.

92. Prove that the product of an odd function and an even function is odd.

93. Volume. An open box of maximum volume is to be made from a square piece of material 24 centimeters on a side by cutting equal squares from the corners and turning up the sides (see figure). (a) Write the volume V as a function of x, the length of the corner squares. What is the domain of the function? (b) Use a graphing utility to graph the volume function and approximate the dimensions of the box that yield a maximum volume. (c) Use the table feature of a graphing utility to verify your answer in part (b). (The first two rows of the table are shown.)

94. Length A right triangle is formed in the first quadrant by the x- and y-axes and a line through the point (3,2) (see figure). Write the length L of the hypotenuse as a function of x.

Section P.4

10. Brinell Hardness. The data in the table show the Brinell hardness of 0.35 carbon steel when hardened and tempered at temperature (degrees Fahrenheit). (Source: Standard Handbook for Mechanical Engineers) (a) Use the regression capabilities of a graphing utility to find a linear model for the data. (b) Use a graphing utility to plot the data and graph the model. How well does the model fit the data?  Explain your reasoning. (c) Use the model to estimate the hardness when is 500 F.

11. Automobile Costs. The data in the table show the variable costs for operating an automobile in the United States for several recent years. The functions y1, y2 and y3 represent the costs in cents per mile for gas and oil, maintenance, and tires, respectively. (Source: American Automobile Manufacturers Association) (a) Use the regression capabilities of a graphing utility to find a cubic model for y1 and linear models for y2 and y3 (b) Use a graphing utility to graph y1, y2, y3, and y1 + y2 + y3 in the same viewing window. Use the model to estimate the total variable cost per mile in year.

12. Beam Strength. Students in a lab measured the breaking strength (in pounds) of wood 2 inches thick, inches high, and 12 inches long. The results are shown in the table. (a) Use the regression capabilities of a graphing utility to fit a quadratic model to the data. (b) Use a graphing utility to plot the data and graph the model. (c) Use the model to approximate the breaking strength when

13. Health Maintenance Organizations. The bar graph shows the numbers of people (in millions) receiving care in HMOs for the years 1990 through 2002. (Source: Centers for Disease Control) (a) Let be the time in years, with corresponding to 1990. Use the regression capabilities of a graphing utility to find linear and cubic models for the data. (b) Use a graphing utility to graph the data and the linear and cubic models. (c) Use the graphs in part (b) to determine which is the better model. (d) Use a graphing utility to find and graph a quadratic model for the data. (e) Use the linear and cubic models to estimate the number of people receiving care in HMOs in the year 2004. (f) Use a graphing  utility  to find other models for the data. Which models do you think best represent the data? Explain.

14. Car Performance. The time  (in seconds) required to attain a speed  of  miles  per  hour  from  a  standing  start  for  a  Dodge Avenger is shown in the table. (Source: Road & Track) (a) Use the regression capabilities of a graphing utility to find a quadratic model for the data. (b) Use a graphing utility to plot the data and graph the model. (c) Use the graph in part (b) to state why the model is not appropriate  for  determining the times required  to  attain speeds less than 20 miles per hour. (d) Because the test began from a standing start, add the point to the data. Fit a quadratic model to the revised data and graph the new model. (e) Does the quadratic model more accurately model the behavior of the car for low speeds? Explain.

Review Exercises for Chapter P

In Exercises 1–4, find the intercepts (if any).
1. y = 2x – 3

In Exercises 5 and 6, check for symmetry with respect to both axes and to the origin.
5. x2y – x2 + 4y = 0

20. Think about it. For what value of y = kx3 does the graph of pass through the point?
(a) (1, 4)     (b) (-2, 1)

In Exercises 25–28, find an equation of the line that passes through the point with the indicated slope. Sketch the line.
25. (0, -5), m = 3/2

29. Find equations of the lines passing through (-2, 4) and having the following characteristics. (a) Slope of 7/16, (b) Parallel to the line 5x – 3y = 3, (c) Passing through the origin, (d) Parallel to the y-axis

30. Find equations of the lines passing through (1, 3) and having the following characteristics.
(a)  Slope of -2/3   (b) Perpendicular to the line x + y = 0 (c) Passing through the point (2, 4)     (d) Parallel to the  x-axis

31. Rate of Change The purchase price of a new machine is $12,500, and its value will decrease by $850 per year. Use this information to write a linear equation that gives the value V of the machine t years after it is purchased. Find its value at the end of 3 years.

32. Break-Even Analysis A contractor purchases a piece of equipment for $36,500 that costs an average of $9.25 per hour for fuel and maintenance. The equipment operator is paid $13.50 per hour, and customers are charged $30 per hour. (a) Write an equation for the cost C of operating this equipment for t hours. (b) Write an equation for the revenue R derived from t hours of use. (c) Find the break-even point for this equipment by finding the time at which R = C.

45. Area. A wire 24 inches long is to be cut into four pieces to form a rectangle whose shortest side has a length of x (a) Write the area of the rectangle as a function of x (b) Determine the domain of the function and use a graphing utility to graph the function over that domain. (c) Use the graph of the function to approximate the maximum area of the rectangle. Make a conjecture about the dimensions that yield a maximum area.


P.S. Problem solving


1. Consider the circle x2 + y2 – 6x – 8y = 0 as shown in the figure. (a) Find the center and radius of the circle. (b) Find an equation of the tangent line to the circle at the point (0, 0) (c) Find an equation of the tangent line to the circle at the point (6, 0) (d) Where do the two tangent lines intersect?

2. There are two tangent lines from the point (0, 1) to the circle x2 + (y + 1)2 = 1. Find equations of these two lines by using the fact that each tangent line intersects the circle in exactly one point.

5. A rancher plans to fence a rectangular pasture adjacent to a river. The rancher has 100 meters of fence, and no fencing is needed along the river (see figure). (a) Write the area A of the pasture as a function of x, the length of the side parallel to the river. What is the domain of A? (b) Graph the area function A(x) and estimate the dimensions that yield the maximum amount of area for the pasture. (c) Find the dimensions that yield the maximum amount of area for the pasture by completing the square.

6. A rancher has 300 feet of fence to enclose two adjacent pastures.  (a) Write the total area A of the two pastures as a function of x (see figure). What is the domain of A? (b) Graph the area function and estimate the dimensions that yield the maximum amount of area for the pastures.  (c) Find the dimensions that yield the maximum amount of area for the pastures by completing the square.

7. You are in a boat 2 miles from the nearest point on the coast. You are to go to a point Q located 3 miles down the coast and 1 mile inland (see figure). You can row at 2 miles per hour and walk at 4 miles per hour. Write the total time T of the trip as a function of x.

8. You drive to the beach at a rate of 120 kilometers per hour. On the return trip, you drive at a rate of 60 kilometers per hour. What is your average speed for the entire trip? Explain your reasoning.

Section 1.1

1. Find the distance traveled in 15 seconds by an object traveling at a constant velocity of 20 feet per second.

2. Find the distance traveled in 15 seconds by an object moving with a velocity of v(t) = 20 + 7cos t feet per second.

3. A bicyclist is riding on a path modeled by the function f(x) = 0.04(8x – x2), where x and f(x) are measured in miles. Find the rate of change of elevation when x = 2.

4. A bicyclist is riding on a path modeled by the function f(x)=0.08x, where x and f(x) are measured in miles. Find the rate of change of elevation when x = 2.

5. Find the area of the shaded region.

7. Secant Lines Consider the function f(x)= 4x – x2  and the point P(1,3) on the graph of f. (a) Graph f and the secant lines passing through P(1,3) and Q(x,f(x) for x-values of 2, 1.5, and 0.5. (b) Find the slope of each secant line. (c) Use the results of part (b) to estimate the slope of the tangent line of at P(1,3). Describe how to improve your approximation of the slope.

9. (a) Use the rectangles in each graph to approximate the area of the region bounded by y=5/x, y = 0, x=1, and x=5. (b) Describe how you could continue this process to obtain a more accurate approximation of the area.

10. (a) Use the rectangles in each graph to approximate the area of the region bounded by y = sin x, y = 0, x = 0 and x = π. (b) Describe how you could continue this process to obtain a more accurate approximation of the area.


Section 1.2

Complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result.

57. Jewelry. A jeweler resizes a ring so that its inner circumference is 6 centimeters. (a) What is the radius of the ring? (b) If the ring’s inner circumference can vary between 5.5 centimeters and 6.5 centimeters, how can the radius vary? (c) Use the ε-δ definition of limit to describe this situation. Identify ε and δ.

58. Sports. A sporting goods manufacturer  designs a golf ball having a volume of 2.48 cubic inches. (a) What is the radius of the golf ball? (b) If the ball’s volume can vary between 2.45 cubic inches and 2.51 cubic inches, how can the radius vary? (c) Use the ε-δ definition of limit to describe this situation. Identify ε and δ.

59. Consider the function f(x) = (1 +x)1/x. Estimate the limit lim(x →0)(1 + x) by evaluating f at x-values near 0. Sketch the graph of f.


Section 1.3


In Exercises 1–4, use a graphing utility to graph the function and visually estimate the limits.

1. h(x) = x2 – 5x

In Exercises 23–26, find the limits.

In Exercises 27– 36, find the limit of the trigonometric function.

In Exercises 45–48, find the limit of the function (if it exists). Write a simpler function that agrees with the given function at all but one point. Use a graphing utility to confirm your result.

In  Exercises  87  and  88, use  the  Squeeze  Theorem  to  find lim(x →c)f(x)

101. If a construction worker drops a wrench from a height of 1000 feet, how fast will the wrench be falling after 5 seconds?

102.  If a construction worker drops a wrench from a height of 1000 feet, when will the wrench hit the ground? At what velocity will the wrench impact the ground?

Free-Falling Object. In Exercises 103 and 104, use the position function s(t) = -4.9t2 + 150 which gives the height (in meters) of an object that has fallen from a height of 150 meters. The velocity at time  t = a seconds is given by lim(x→a)(s(a) – s(t))/(a – t).

103. Find the velocity of the object when t = 3.

104. At what velocity will the object impact the ground?

 

Section 1.4

Find the x-values (if any) at which f is not continuous. Which of the discontinuities are removable
33. f(x) = x2 – 2x + 1

Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of c guaranteed by the theorem.
103. f(x) = x2  + x – 1, [0,5], f(c) = 11

95. Swimming Pool. Every day you dissolve 28 ounces of chlorine in a swimming pool. The graph shows the amount of chlorine f(t) in the pool after t days. Estimate and interpret lim(t →4-)f(t) and lim(t →4+)f(t).

97. Telephone Charges. A dial-direct long distance call between two cities costs $1.04 for the first 2 minutes and $0.36 for each additional minute or fraction thereof. Use the greatest integer function to write the cost C of a call in terms of time t (in minutes). Sketch the graph of this function and discuss its continuity.

98. Inventory management. The number of units in inventory in a small company is given by N(t) = 25(2[[(t+2)/2]] – t) where t is the time in months. Sketch the graph of this function and discuss its continuity. How often must this company replenish its inventory?

99. At 8:00 A.M. on Saturday a man begins running up the side of a mountain to his weekend campsite (see figure). On Sunday morning at 8:00 A.M. he runs back down the mountain. It takes him 20 minutes to run up, but only 10 minutes to run down. At some point on the way down, he realizes that he passed the same place at exactly the same time on Saturday. Prove that he is correct.

100. Volume. Use the Intermediate Value Theorem to show that for all spheres with radii in the interval [1, 5] there is one with a volume of 275 cubic centimeters.

105. Modeling Data. After an object falls for t seconds, the speed S (in feet per second) of the object is recorded in the table. (a) Create a line graph of the data. (b) Does there appear to be a limiting speed of the object? If there is a limiting speed, identify a possible cause.

106. Creating Models. A swimmer crosses a pool of width b by swimming in a straight line from (0, 0) to (2b, b). (a) Let f be a function defined as the y-coordinate of the point on the long side of the pool that is nearest the swimmer at any given time during the swimmer’s path across the pool. Determine the function f and sketch its graph. Is it continuous? Explain. (b) Let g be the minimum distance between the swimmer and the long sides of the pool. Determine the function g and sketch its graph. Is it continuous? Explain.

107. Find all values of c such that f is continuous on (-∞, ∞)
f(x) = { 1 – x2, x ≤ c
            x,        x > c

 

Section 1.5

 

In Exercises 9–28, find the vertical asymptotes (if any) of the graph of the function.
9. f(x) = 1/x2

29. Determine whether the graph of the function has a vertical asymptote or a removable discontinuity at x = -1. Graph the function using a graphing utility to confirm your answer.
f(x) = (x2 – 1)/(x + 1)

58. Boyle’s Law. For a quantity of gas at a constant temperature, the pressure P is inversely proportional to the volume V. Find the limit of P as V →0+.

59. A patrol car is parked 50 feet from a long warehouse. The revolving light on top of the car turns at a rate of 1/2 revolution per second. The rate at which the light beam moves along the wall is r = 50π sec2 θ ft/sec. (a) Find the rate r when θ = π/6. (b) Find the rate r when θ = π/3. (c) Find the limit of r as θ → π/2-.

60. Illegal Drugs. The cost in millions of dollars for a governmental agency to seize x% of an illegal drug is C = 528x/ (100 – x), 0 ≤ x < 100. (a) Find the cost of seizing 25% of the drug. (b) Find the cost of seizing 50% of the drug. (c) Find the cost of seizing 75% of the drug. (d) Find the limit of C as x → 100- and interpret its meaning.

61. Relativity. According to the theory of relativity, the mass of a particle depends on its velocity v. That is m = m0/√(1 – v2/c2) where m0 is the mass when the particle is at rest and c is the speed of light. Find the limit of the mass as approaches c-.

62. Rate of Change. A 25-foot ladder is leaning against a house (see figure). If the base of the ladder is pulled away from the house at a rate of 2 feet per second, the top will move down the wall at a rate of r = 2x/√(625 – x2) ft/s where  is the distance between the base of the ladder and the house. (a) Find the rate r when x is 7 feet. (b) Find the rate r when x is 15 feet. (c) Find the limit of r as x → 25-.

63. Average Speed. On a trip of d miles to another city, a truck driver’s average speed was x miles per hour. On the return trip the average speed was y miles per hour. The average speed for the round trip was 50 miles per hour. (a) Verify that y = 25x/(x – 25). What is the domain? (b) Complete the table. Are the values of y different than you expected? Explain. (c) Find the limit of y as x → 25+ and interpret its meaning.

64. Numerical and Graphical Analysis Use a graphing utility to complete the table for each function and graph each function to estimate the limit. What is the value of the limit when the power on x in the denominator is greater than 3?

65. Numerical and Graphical Analysis. Consider  the  shaded region  outside  the  sector  of  a  circle  of  radius  10  meters  and inside a right triangle (see figure). (a) Write the area A = f(θ) of the region as a function of θ. Determine the domain of the function. (b) Use a graphing utility to complete the table and graph the function over the appropriate domain. (c) Find the limit of A as θ → (π/2)+.

66. Numerical and Graphical Reasoning. A crossed belt connects a 20-centimeter pulley (10-cm radius) on an electric motor with a 40-centimeter pulley (20-cm radius) on a saw arbor (see figure). The electric motor runs at 1700 revolutions per minute. (a) Determine the number of revolutions per minute of the saw. (b) How does crossing the belt affect the saw in relation to the motor? (c) Let L be the total length of the belt. Write L as a function of φ where φ is measured in radians. What is the domain of the function? (Hint: Add the lengths of the straight sections of the belt and the length of the belt around each pulley.) (d) Use a graphing utility to complete the table. (e) Use a graphing utility to graph the function over the appropriate domain. (f) Find lim(φ → (π/2)-) L. Use a geometric argument as the basis of a second method of finding this limit. (g) Find lim(φ → 0+) L.

 
Review Exercises for Chapter 1

47. Determine the value of c such that the function is continuous on the entire real line.
f(x) = { x + 3,             x ≤ 2
           cx + 6            x > 2

49. Use the Intermediate Value Theorem to show that f(x) = 2x3 – 3 has a zero in the interval [1, 2].

50. Delivery Charges. The cost of sending an overnight package from New York to Atlanta is $9.80 for the first pound and $2.50 for each additional pound or fraction thereof. Use the greatest integer function to create a model for the cost C of overnight delivery of a package weighing x pounds. Use a graphing utility to graph the function and discuss its continuity.

Section 2.1

In Exercises 5 –10, find the slope of the tangent line to the graph of the function at the given point.
5. f(x) = 3 – 2x, (-1, 5)

In Exercises 11–24, find the derivative by the limit process.
11. f(x) = 3

41. The tangent line to the graph of y=g(x) at the point (5,2) passes through the point (9,0). Find g(5) and g ′(5).

Section 2.2

In Exercises 57–62, determine the point(s) (if any) at which the graph of the function has a horizontal tangent line.
57. y = x4 – 8x2 + 2

73. Sketch the graphs of y = x2 and y = -x2 + 6x – 5, and sketch the two lines that are tangent to both graphs.  Find equations of these lines.

74. Show that the graphs of the two equations y = x and y = 1/x have tangent lines that are perpendicular to each other at their point of intersection.

75. Show that the graph of the function f(x) = 3x + sin(x) + 2 does not have a horizontal tangent line.

76. Show that the graph of the function f(x) = x5 + 3x3 + 5x does not have a tangent line with a slope of √3.

79. Linear Approximation. Use a graphing utility, with a square window setting, to zoom in on the graph of f(x) = 4 – 1/2x2 to approximate f ′(1). Use the derivative to find f ′(1).

80. Linear Approximation. Use a graphing utility, with a square window setting, to zoom in on the graph of f(x) = 4√x + 1 to approximate f ′(4). Use the derivative to find f ′(4).

93. A silver dollar is dropped from the top of a building that is 1362 feet tall. (a) Determine the position and velocity functions for the coin. (b) Determine the average velocity on the interval [1, 2]. (c) Find the instantaneous velocities when t =1 and t=2. (d) Find the time required for the coin to reach ground level. (e) Find the velocity of the coin at impact.

94. A ball is thrown straight down from the top of a 220-foot building with an initial velocity of –22 feet per second. What is its velocity after 3 seconds? What is its velocity after falling 108 feet?

95. A projectile is shot upward from the surface of Earth with an initial velocity of 120 meters per second. What is its velocity after 5 seconds? After 10 seconds?

96. To estimate the height of a building, a stone is dropped from the top of the building into a pool of water at ground level. How high is the building if the splash is seen 6.8 seconds after the stone is dropped?

101. Modeling Data. The stopping distance of an automobile, on dry, level  pavement, traveling  at  a  speed v (kilometers  per hour)  is  the  distance  (meters) the car travels during the reaction time of the driver plus the distance  (meters) the car travels after the brakes are applied (see  figure). The table shows the results of an experiment. (a) Use the regression capabilities of a graphing utility to find a linear model for reaction time distance. (b) Use the regression capabilities of a graphing utility to find a quadratic model for braking distance. (c) Determine the polynomial giving the total stopping distance T. (d) Use a graphing utility to graph the functions R, B, and T in the same viewing window. (e) Find the derivative of T and the rates of change of the total stopping distance for v = 40, v = 80, and v = 100. (f) Use the results of this exercise to draw conclusions about the total stopping distance as speed increases.

102. Fuel Cost. A car is driven 15,000 miles a year and gets miles per gallon. Assume that the average fuel cost is $1.55 per gallon. Find the annual cost of fuel C as a function of x and use this function to complete the table. Who would benefit more from a one-mile-per-gallon increase in fuel efficiency—the driver of a car that gets 15 miles per gallon or the driver of a car that gets 35 miles per gallon? Explain.

103. Volume. The volume of a cube with sides of length s is given by V = s3. Find the rate of change of the volume with respect to s when s = 4 centimeters.

104. Area. The area of a square with sides of length s is given by A = s2. Find the rate of change of the area with respect to s when s = 4 centimeters.

105. Velocity. Verify that the average velocity over the time interval [t0 – Δt, t0 + Δt] is the same as the instantaneous velocity at t = t0 for the position function s(t) = -(1/2)at2 + c.

106. Inventory Management. The annual inventory cost C for a manufacturer is C = 1008000/Q + 6.3Q where Q is the order size when the inventory is replenished. Find the change in annual cost when Q is increased from 350 to 351, and compare this with the instantaneous rate of change when Q = 350.

107. Writing. The number of gallons N of regular unleaded gasoline sold by a gasoline station at a price of p dollars per gallon is given by N = f(p). (a) Describe the meaning of f ′(1.479). (b) Is f ′(1.479) usually positive or negative? Explain.

108. Newton’s Law of Cooling. This law states that the rate of change of the temperature of an object is proportional to the difference between the object’s temperature T and the temperature Ta of the surrounding medium. Write an equation for this law.

109. Find an equation of the parabola y = ax2 + bx + c that passes through (0, 1) and is tangent to the line y = x – 1 at (1, 0).

110. Let (a, b) be an arbitrary point on the graph of y = 1/x, x > 0. Prove that the area of the triangle formed by the tangent line through (a, b) and the coordinate axes is 2.

111. Find the tangent line(s) to the curve y = x3 – 9x through the point (1, -9).


Section 2.3


In Exercises 1–6, use the Product Rule to differentiate the function.
1. g(x) = (x2 + 1)(x2 – 2x)

77. Tangent Lines. Find equations of the tangent lines to the graph of f(x) = (x + 1)/(x – 1) that are parallel to the line 2y + x = 6. Then graph the function and the tangent lines.

78. Tangent Lines. Find equations of the tangent lines to the graph of f(x) = x/(x – 1) that pass through the point (-1, 5). Then graph the function and the tangent lines.

83. Area. The length of a rectangle is given by 2t + 1 and its height is √t where is time in seconds and the dimensions are in centimeters. Find the rate of change of the area with respect to time.

84. Volume. The radius of a right circular cylinder is given by √(t + 2) and its height is 1/2√t, where t is time in seconds and the dimensions are in inches. Find the rate of change of the volume with respect to time.

85. Inventory Replenishment. The ordering and transportation cost C for the components used in manufacturing a product is C = 100(200/x2 + x/(x + 30)) where C is measured in thousands of dollars and x is the order size in hundreds. Find the rate of change of C with respect to x when (a) x = 10, (b) x = 15, and (c) x = 20.  What do these rates of change imply about increasing order size?

86. Boyle’s Law. This law states that if the temperature of a gas remains constant, its pressure is inversely proportional to its volume. Use the derivative to show that the rate of change of the pressure is inversely proportional to the square of the volume.

87. Population Growth. A population of 500 bacteria is introduced into a culture and grows in number according to the equation P(t) = 500(1 + 4t/(50 + t2)) where t measured in hours. Find the rate at which the population is growing when t = 2.

88. Gravitational Force. Newton’s Law of Universal Gravitation states that the force F between two masses, m1 and m2, is F = Gm1m2/d2 where G is a constant and d is the distance between the masses. Find an equation that gives an instantaneous rate of change of F with respect to d.

90. Rate of Change Determine whether there exist any values of x in the interval [0, 2π) such that the rate of change of f(x) = sec x and the rate of change of g(x) = csc x are equal.

92. Satellites. When satellites observe Earth, they can scan only part of Earth’s surface. Some satellites have sensors that can measure the angle θ shown in the figure. Let h represent the satellite’s distance from Earth’s surface and let r represent Earth’s radius. (a) Show that h = r(csc θ – 1). (b) Find the rate at which h is changing with respect to θ when θ = 30°.

115. Acceleration. The velocity of an object in meters per second is v(t) = 36 – t2, 0 ≤ t ≤ 6. Find the velocity and acceleration of the object when t = 3. What can be said about the speed of the object when the velocity and acceleration have opposite signs?

116. Acceleration. An automobile’s velocity starting from rest is v(t) = 100t/(2t + 15) where t is measured in feet per second. Find the acceleration at (a) 5 seconds, (b) 10 seconds, and (c) 20 seconds.

117. Stopping Distance. A car is traveling at a rate of 66 feet per second (45 miles per hour) when the brakes are applied. The position function for the car is s(t) = -8.25t2 + 66t, where s is  measured  in  feet  and t is measured in seconds.  Use this function to complete the table, and find the average velocity during each time interval.

135. Find a second-degree polynomial f(x) = ax2 + bx + c such that its graph has a tangent line with slope 10 at the point (2, 7) and an x-intercept at (1, 0).


Section 2.4

In Exercises 7–32, find the derivative of the function.
7. y = (2x – 7)3

In  Exercises 75–78, (a)  use a graphing utility to find the derivative of the function at the given point, (b) find an equation of  the tangent line to the graph of the function at the given point, and (c) use  the  utility to graph the function and its tangent line in the same viewing window.

81. Horizontal Tangent Line. Determine the point(s) in the interval (0, 2π) at which the graph of f(x) = 2cos x + sin2x has a horizontal tangent.

In Exercises 83–86, find the second derivative of the function.
83. f(x) = 2(x2 – 1)3

101. Doppler Effect. The frequency F of a fire truck siren heard by a stationary observer is F = 132400/(331 ± v) where ±v represents the velocity of the accelerating fire truck in meters per second (see figure). Find the rate of change of F with respect to v when (a) the fire truck is approaching at a velocity of 30 meters per second (use –v). (b) the fire truck is moving away at a velocity of 30 meters per second (use v).

102. Harmonic Motion. The displacement from equilibrium of an object in harmonic motion on the end of a spring is y = 1/3 cos 12t – 1/4 sin 12t
where y is measured in feet and t is the time in seconds. Determine the position and velocity of the object when t = π/8.

103. Pendulum. A 15-centimeter pendulum moves according to the equation θ = 0.2cos 8t, where θ is the angular displacement from the vertical in radians and t is the time in seconds. Determine the maximum angular displacement and the rate of change of θ when t = 3 seconds.

104. Wave Motion. A buoy oscillates in simple harmonic motion y = A cos wt as waves move past it. The buoy moves a total of 3.5 feet (vertically) from its low point to its high point. It returns to its high point every 10 seconds. (a) Write an equation describing the motion of the buoy if it is at its high point at t = 0.
(b) Determine the velocity of the buoy as a function of t.

105. Circulatory System. The speed S of blood that is r centimeters from the center of an artery is S = C(R2 – r2) where C is a constant, R is the radius of the artery, and S is measured in centimeters per second. Suppose a drug is administered and the artery begins to dilate at a rate of dR/dt. At a constant distance r find the rate at which S changes with respect to t for C =1.76×105, R =1.2×10–2 and dR/dt = 10 –5

107. Modeling Data. The cost of producing x units of a product is C = 60x + 1350. For one week management determined the number of units produced at the end of t hours during an eight-hour shift  The average values of x for the week are shown in the table. (a) Use a graphing utility to fit a cubic model to the data. (b) Use the Chain Rule to find dC/dt. (c) Explain  why  the  cost  function  is  not  increasing  at  a constant rate during the 8-hour shift.


Section 2.5


41. (a) Use implicit differentiation to find an equation of the tangent line to the ellipse x2/2 + y2/8 =1 at (1,2)

57. Find the points at which the graph of the equation has a vertical or horizontal tangent line.
25x2 + 16y2 + 200x – 160y + 400 = 0

72. Weather Map. The weather map shows several isobars— curves that represent areas of constant air pressure. Three high pressures H and one low pressure L are shown on the map. Given that wind speed is greatest along the orthogonal trajectories of the isobars, use the map to determine the areas having high wind speed.

76. Slope. Find all points on the circle x2 + y2 = 25 where the slope is ¾.

77. Horizontal Tangent. Determine the point(s) at which the graph of y4 = y2 – x2 has a horizontal tangent.

78. Tangent Lines. Find equations of both tangent lines to the ellipse x2/4 + y2/9 =1 that passes through the point (4,0).

79. Normals to a Parabola. The graph shows the normal lines from the point (2,0)to the graph of the parabola x = y2. How many normal lines are there from the point (xo, 0) to the graph of the parabola if (a) x0 = ¼; (b) x0 = ½; and (c) x0 = 1? For what value of x0 are two of the normal lines perpendicular to each other?


Section 2.6


13. Find the rate of change of the distance between the origin and a moving point on the graph of y = x2 + 1 if dx/dt = 2 centimeters per second.

14. Find the rate of change of the distance between the origin and a moving point on the graph of y = sin x if dx/dt = 2centimeters per second.

15. Area. The radius r of a circle is increasing at a rate of 3 centimeters per minute. Find the rates of change of the area when (a) r = 6 centimeters and (b) r = 24 centimeters.

16. Area. Let A be the area of a circle of radius r that is changing with respect to time. If dr/dt is constant, is  dA/dt constant? Explain.

17. Area. The included angle of the two sides of constant equal length s of an isosceles triangle is q. (a) Show that the area of the triangle is given by A = ½ s2sin θ. (b) If q is increasing at the rate of ½ radian per minute, find the rates of change of the area when θ = π/6 and (c) θ = π/3. Explain why the rate of change of the area of the triangle is not constant even though dθ/dt is constant.

19. Volume. A spherical balloon is inflated with gas at the rate of 800 cubic centimeters per minute. How fast is the radius of the balloon increasing at the instant the radius is (a) 30 centimeters and (b) 60 centimeters?

20. Volume. All edges of a cube are expanding at a rate of 3 centimeters per second. How fast is the volume changing when each edge is (a) 1 centimeter and (b) 10 centimeters?

21. Surface Area. The conditions are the same as in Exercise 20. Determine how fast the surface area is changing when each edge is (a) 1 centimeter and (b) 10 centimeters.

22. Volume. The formula for the volume of a cone is V = 1/3 pr2h. Find the rate of change of the volume if dr/dt is 2 inches per minute and h = 3r when (a) r = 6 inches and (b) r = 24 inches.

23. Volume. At a sand and gravel plant, sand is falling off a conveyor and onto a conical pile at a rate of 10 cubic feet per minute. The diameter of the base of the cone is approximately three times the altitude. At what rate is the height of the pile changing when the pile is 15 feet high?

24. Depth. A conical tank (with vertex down) is 10 feet across the top and 12 feet deep. If water is flowing into the tank at a rate of 10 cubic feet per minute, find the rate of change of the depth of the water when the water is 8 feet deep.

25. Depth. swimming pool is 12 meters long, 6 meters wide, 1 meter deep at the shallow end, and 3 meters deep at the deep end (see figure). Water is being pumped into the pool at ¼ cubic meter per minute, and there is 1 meter of water at the deep end. (a) What percent of the pool is filled? (b) At what rate is the water level rising?

26. Depth. A trough is 12 feet long and 3 feet across the top (see figure). Its ends are isosceles triangles with altitudes of 3 feet. (a) If water is being pumped into the trough at 2 cubic feet per minute, how fast is the water level rising when h is 1 foot deep? (b) If the water is rising at a rate of 3/8 inch per minute when h = 2 determine the rate at which water is being pumped into the trough.

27. Moving Ladder. ladder 25 feet long is leaning against the wall of a house (see figure). The base of the ladder is pulled away from the wall at a rate of 2 feet per second. (a) How fast is the top of the ladder moving down the wall when its base is 7 feet, 15 feet, and 24 feet from the wall? (b) Consider the triangle formed by the side of the house, the ladder, and the ground. Find the rate at which the area of the triangle is changing when the base of the ladder is 7 feet from the wall. (c) Find the rate at which the angle between the ladder and the wall of the house is changing when the base of the ladder is 7 feet from the wall.

28. Construction. A construction worker pulls a five-meter plank up the side of a building under construction by means of a rope tied to one end of the plank (see figure). Assume the opposite end of the plank follows a path perpendicular to the wall of the building and the worker pulls the rope at a rate of 0.15 meter per second. How fast is the end of the plank sliding along the ground when it is 2.5 meters from the wall of the building?

29. Construction. A winch at the top of a 12-meter building pulls a pipe of the same length to a vertical position, as shown in the figure. The winch pulls in rope at a rate of –0.2 meter per second. Find the rate of vertical change and the rate of horizontal change at the end of the pipe when y=6.

30. Boating. A boat is pulled into a dock by means of a winch 12 feet above the deck of the boat (see figure). (a) The winch pulls in rope at a rate of 4 feet per second. Determine the speed of the boat when there is 13 feet of rope out. What happens to the speed of the boat as it gets closer to the dock? (b) Suppose the boat is moving at a constant rate of 4 feet per second. Determine the speed at which the winch pulls in rope when there is a total of 13 feet of rope out. What happens to the speed at which the winch pulls in rope as the boat gets closer to the dock?

31. Air Traffic Control. An air traffic controller spots two planes at the same altitude converging on a point as they fly at right angles to each other (see figure). One plane is 150 miles from the point moving at 450 miles per hour. The other plane is 200 miles from the point moving at 600 miles per hour. (a) At what rate is the distance between the planes decreasing? (b) How much time does the air traffic controller have to get one of the planes on a different flight path?

32. Air Traffic Control. An airplane is flying at an altitude of 5 miles and passes directly over a radar antenna (see figure on previous page). When the plane is 10 miles away (s=10), the radar detects that the distance s is changing at a rate of 240 miles per hour. What is the speed of the plane?

33. Sports. A baseball diamond has the shape of a square with sides 90 feet long (see figure). A player running from second base to third base at a speed of 28 feet per second is 30 feet from third base. At what rate is the player’s distance s from home plate changing?

34. Sports. For the baseball diamond in Exercise 33, suppose the player is running from first to second at a speed of 28 feet per second. Find the rate at which the distance from home plate is changing when the player is 30 feet from second base.

35. Shadow Length. A man 6 feet tall walks at a rate of 5 feet per second away from a light that is 15 feet above the ground (see figure). When he is 10 feet from the base of the light, (a) at what rate is the tip of his shadow moving? (b) at what rate is the length of his shadow changing?

36. Shadow Length. Repeat Exercise 35 for a man 6 feet tall walking at a rate of 5 feet per second toward a light that is 20 feet above the ground (see figure).

37. Machine Design. The endpoints of a movable rod of length 1 meter have coordinates (x,0) and (0,y) (see figure). The position of the end on the x-axis is x(t) = ½(sin(pt/6) where t is the time in seconds. (a) Find the time of one complete cycle of the rod. (b) What is the lowest point reached by the end of the rod on the y-axis? (c) Find the speed of the y-axis endpoint when the axis endpoint is (1/4,0).

39. Evaporation. As a spherical raindrop falls, it reaches a layer of dry air and begins to evaporate at a rate that is proportional to its surface area (S=4pr2). Show that the radius of the raindrop decreases at a constant rate.

40. Electricity. The combined electrical resistance R of R1 and R2 connected in parallel, is given by 1/R = 1/R1 + 1/R2 where R, R1 and R2 are measured in ohms. R1 and R2 are increasing at rates of 1 and 1.5 ohms per second, respectively. At what rate R is changing when R1 = 50 ohms and R2 = 50 ohms?
41. Adiabatic Expansion. When a certain polyatomic gas undergoes adiabatic expansion, its pressure p and volume V satisfy the equation pV1.3=k, where k is a constant. Find the relationship between the related rates dp/dt and dV/dt.

42. Roadway Design. Cars on a certain roadway travel on a circular arc of radius r. In order not to rely on friction alone to overcome the centrifugal force, the road is banked at an angle of magnitude q from the horizontal (see figure). The banking angle must satisfy the equation rg tan q = v0 where v is the velocity of the cars and g = 32 feet per second per second is the acceleration due to gravity. Find the relationship between the related rates dv/dt and dq/dt.

43. Angle of Elevation. A balloon rises at a rate of 3 meters per second from a point on the ground 30 meters from an observer. Find the rate of change of the angle of elevation of the balloon from the observer when the balloon is 30 meters above the ground.

44. Angle of Elevation. A fish is reeled in at a rate of 1 foot per second from a point 10 feet above the water (see figure). At what rate is the angle between the line and the water changing when there is a total of 25 feet of line out?

45. Angle of Elevation. An airplane flies at an altitude of 5 miles toward a point directly over an observer (see figure). The speed of the plane is 600 miles per hour. Find the rates at which the angle of elevation q is changing when the angle is (a) θ = 30° (b) θ = 60°  and (c) θ = 75°.

51. Find the acceleration of the top of the ladder described in Exercise 27 when the base of the ladder is 7 feet from the wall.

52. Find the acceleration of the boat in Exercise 30(a) when there is a total of 13 feet of rope out.

54. Moving Shadow. A ball is dropped from a height of 20 meters, 12 meters away from the top of a 20-meter lamppost (see figure). The ball’s shadow, caused by the light at the top of the lamppost, is moving along the level ground. How fast is the shadow moving 1 second after the ball is released?


Review exercises for Chapter 2

33. Vibrating String. When a guitar string is plucked, it vibrates with a frequency of F = 200√T, where F is measured in vibrations per second and the tension T is measured in pounds. Find the rates of change of F when (a) T = 4 and (b) T = 9.

34. Vertical Motion. A ball is dropped from a height of 100 feet. One second later, another ball is dropped from a height of 75 feet. Which ball hits the ground first?

35. Vertical Motion. To estimate the height of a building, a weight is dropped from the top of the building into a pool at ground level. How high is the building if the splash is seen 9.2 seconds after the weight is dropped?

36. Vertical Motion. A bomb is dropped from an airplane at an altitude of 14,400 feet. How long will it take for the bomb to reach the ground? (Because of the motion of the plane, the fall will not be vertical, but the time will be the same as that for a vertical fall.) The plane is moving at 600 miles per hour. How far will the bomb move horizontally after it is released from the plane?

37. Projectile Motion A ball thrown follows a path described by y = x – 0.02x2. (a) Sketch a graph of the path. (b) Find the total horizontal distance the ball is thrown. (c) At what x-value does the ball reach its maximum height? (Use the symmetry of the path.) (d) Find an equation that gives the instantaneous rate of change of  the  height  of  the  ball  with  respect  to  the  horizontal change. Evaluate the equation at x = 10, 25, 30, and 50. (e) What is the instantaneous rate of change of the height when the ball reaches its maximum height?

38. Projectile Motion. The path of a projectile thrown at an angle of 45° with level ground is y = x – 32x2/v02 where the initial velocity is v0 feet per second. (a) Find the x-coordinate of the point where the projectile strikes the ground. Use the symmetry of the path of the projectile to locate the x-coordinate of the point where the projectile reaches its maximum height. (b) What is the instantaneous rate of change of the height when the projectile is at its maximum height? (c) Show that doubling the initial velocity of the projectile multiplies both the maximum height and the range by a factor of 4. (d) Find the maximum height and range of a projectile thrown with an initial velocity of 70 feet per second. Use a graphing utility to graph the path of the projectile.

39. Horizontal Motion. The position function of a particle moving along the x-axis is x(t) = t2 – 3t + 2 for -∞ ≤ t ≤ ∞. (a) Find the velocity of the particle. (b) Find the open t-interval(s) in which the particle is moving to the left. (c) Find the position of the particle when the velocity is 0. (d) Find the speed of the particle when the position is 0.

40. Modeling Data. The speed of a car in miles per hour and the stopping distance in feet are recorded in the table. (a) Use the regression capabilities of a graphing utility to find a quadratic model for the data. (b) Use a graphing utility to plot the data and graph the model. (c) Use a graphing utility to graph dy/dx. (d) Use the model to approximate the stopping distance at a speed of 65 miles per hour. (e) Use the graphs in parts (b) and (c) to explain the change in stopping distance as the speed increases.

59. Acceleration. The velocity of an object in meters per second is v(t) = 36 – t2, 0 ≤ t ≤ 6. Find the velocity and acceleration of the object when t = 4.

60. Acceleration. An automobile’s velocity starting from rest is v(t) = 90t/(4t +10) where v is measured in feet per second. Find the vehicle’s velocity and acceleration at each of the following times. (a) 1 second (b) 5 seconds (c) 10 seconds.

99. Refrigeration. The temperature T of food put in a freezer is T = 700/(t2 + 4t + 10) where t is the time in hours. Find the rate of change of T with respect to t at each of the following times. (a) t = 1 (b) t = 3   (c) = 5    (d) = 10

100. Fluid Flow. The emergent velocity of a liquid flowing from a hole in the bottom of a tank is given by v = √(2gh), where g is the acceleration due to gravity (32 feet per second per second) and h is the depth of the liquid in the tank. Find the rate of change of v with respect to h when (a) h = 9 and (b) h = 4.

109. A point moves along the curve y = √x in such a way that the y-value is increasing at a rate of 2 units per second. At what rate is x changing for each of the following values? (a) x = ½  (b) x = 1 (c) x = 4.

110. Surface Area. The edges of a cube are expanding at a rate of 5 centimeters per second. How fast is the surface area changing when each edge is 4.5 centimeters?

111. Depth. The cross section of a five-meter trough is an isosceles trapezoid with a two-meter lower base, a three-meter upper base, and an altitude of 2 meters. Water is running into the trough at a rate of 1 cubic meter per minute. How fast is the water level rising when the water is 1 meter deep?

112. Linear and Angular Velocity. A rotating beacon is located 1 kilometer off a straight shoreline (see figure). If the beacon rotates at a rate of 3 revolutions per minute, how fast (in kilometers per hour) does the beam of light appear to be moving to a viewer who is ½ kilometer down the shoreline?

113. Moving Shadow. A sandbag is dropped from a balloon at a height of 60 meters when the angle of elevation to the sun is 30° (see figure). Find the rate at which the shadow of the sandbag is traveling along the ground when the sandbag is at a height of 35 meters.


P.S. Problem Solving

9. A man 6 feet tall walks at a rate of 5 feet per second toward a streetlight that is 30 feet high (see figure). The man’s 3-foot-tall child follows at the same speed, but 10 feet behind the man. At times, the shadow behind the child is caused by the man, and at other times, by the child. (a) Suppose the man is 90 feet from the streetlight. Show that the man’s shadow extends beyond the child’s shadow. (b) Suppose the man is 60 feet from the streetlight. Show that the child’s shadow extends beyond the man’s shadow. (c) Determine the distance d from the man to the streetlight at which the tips of the two shadows are exactly the same distance from the streetlight. (d) Determine how fast the tip of the shadow is moving as a function of x, the distance between the man and the street light. Discuss the continuity of this shadow speed function.

10. A particle is moving along the graph of y = 3√x (see figure). When x = 8, the y-component of its position is increasing at the rate of 1 centimeter per second. (a) How fast is the x-component changing at this moment? (b) How fast is the distance from the origin changing at this moment? (c) How  fast  is  the angle of inclination θ changing at this moment?

 

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