Section 2.1

1. A ball is dropped from a state of rest at time t = 0. The distance traveled after t seconds is s(t) = 16t² ft.
(a) How far does the ball travel during the time interval [2, 2.5]?
(b) Compute the average velocity over [2, 2.5].
(c) Compute the average velocity over time intervals [2, 2.01], [2, 2.005], [2, 2.001], [2, 2.00001]. Use this to estimate the object’s instantaneous velocity at t = 2.

2.  A wrench is released from a state of rest at time t = 0. Estimate the wrench’s instantaneous velocity at t = 1, assuming that the distance traveled after t seconds is s(t) = 16t².

3.  Let v = 20√T as in Example 2. Estimate the instantaneous ROC of v with respect to T when T = 300 K.

4.  Compute Dy /Dx for the interval [2, 5], where y = 4x − 9. What is the instantaneous ROC of y with respect to x at x = 2?

In Exercises 5–6, a stone is tossed in the air from ground level with an initial velocity of 15 m/s. Its height at time t is h(t) = 15t − 4.9t² m.

5. Compute the stone’s average velocity over the time interval [0.5, 2.5] and indicate the corresponding secant line on a sketch of the graph of h(t).

6. Compute the stone’s average velocity over the time intervals [1, 1.01], [1, 1.001], [1, 1.0001] and [0.99, 1], [ 0.999, 1] , [0.9999, 1] .Use this to estimate the instantaneous velocity at t = 1.

 7.  With an initial deposit of $100, the balance in a bank account after t years is f(t) = 100(1.08)^t dollars.
(a) What are the units of the ROC of f(t)?
(b) Find the average ROC over [0, 0.5] and [0, 1].
(c) Estimate the instantaneous rate of change at t = 0.5 by computing the average ROC over intervals to the left and right of t = 0.5.

8. The distance traveled by a particle at time t is s(t) = t³ + t. Compute the average velocity over the time interval [1, 4] and estimate the instantaneous velocity at t = 1.

In Exercises 9–16, estimate the instantaneous rate of change at the point indicated.

9. P(x) = 4x² − 3; x = 2

17. The atmospheric temperature T (in ◦F) above a certain point on earth is T = 59 − 0.00356h, where h is the altitude in feet (valid for h ≤ 37,000). What are the average and instantaneous rates of change of T with respect to h? Why are they the same? Sketch the graph of T for h ≤ 37,000.

18. The height (in feet) at time t (in seconds) of a small weight oscillating at the end of a spring is h(t) = 0.5 cos (8t).
(a) Calculate the weight’s average velocity over the time intervals [0, 1] and [3, 5].
(b) Estimate its instantaneous velocity at t = 3.

19. The number P(t) of E. coli cells at time t (hours) in a petri dish is plotted in Figure 9.
(a)  Calculate the average ROC of  P ( t ) over the time interval [ 1 , 3 ] and draw the corresponding secant line.
(b)  Estimate the slope m of the line in Figure 9. What does m represent?

24.  An epidemiologist finds that the percentage N(t) of susceptible children who were infected on day t during the first three weeks of a measles outbreak is given, to a reasonable approximation, by the formula

A graph of N ( t ) appears in Figure 13.
(a)  Draw the secant line whose slope is the average rate of increase in infected children over the intervals between days 4 and 6 and between days 12 and 14. Then compute these average rates (in units of percent per day).
(b)  Estimate the ROC of N ( t ) on day 12.

25.  The fraction of a city’s population infected by a flu virus is plotted as a function of time (in weeks) in Figure 14.
(a)  Which quantities are represented by the slopes of lines A and B? Estimate these slopes.
(b)  Is the flu spreading more rapidly at t = 1, 2, or 3?
(c)  Is the flu spreading more rapidly at t = 4, 5, or 6?

26.  The fungus fusarium exosporium infects a field of flax plants through the roots and causes the plants to wilt. Eventually, the entire field is infected. The percentage f(t) of infected plants as a function of time t (in days) since planting is shown in Figure 15.
(a) What are the units of the rate of change of f(t) with respect to t? What does this rate measure?
(b) Use the graph to rank (from smallest to largest) the average infection rates over the intervals [0, 12], [20, 32], and [40, 52].
(c) Use the following table to compute the average rates of infection over the intervals [30, 40], [40, 50] ,[30, 50]
(d) Draw the tangent line at t = 40 and estimate its slope. Choose any two points on the tangent line for the computation.

27. Let v = 20√T as in Example 2. Is the ROC of v with respect to T greater at low temperatures or high temperatures? Explain in terms of the graph.

28. If an object moving in a straight line (but with changing velocity) covers Ds  feet in Dt  seconds, then its average velocity is v₀ = Ds /Dt  ft/s. Show that it would cover the same distance if it traveled at constant velocity v₀ over the same time interval of Dt seconds. This is a justification for calling Ds /Dt the average velocity.

29. Sketch the graph of  f(x) = x(1 − x) over [0, 1] . Refer to the graph and, without making any computations, find:
(a) The average ROC over [0, 1]
(b) The (instantaneous) ROC at x = 1/2
(c) The values of x at which the ROC is positive

30. Which graph in Figure 16 has the following property: For all x, the average ROC over [ 0, x] is greater than the instantaneous ROC at x? Explain.

31. The height of a projectile fired in the air vertically with initial velocity 64 ft / s is h (t) = 64t − 16t² ft.
(a) Compute h(1). Show that h(t) − h(1) can be factored with (t − 1) as a factor.
(b) Using part (a), show that the average velocity over the interval [1, t] is −16(t − 3).
(c) Use this formula to find the average velocity over several intervals [1, t] with t close to 1. Then estimate the instantaneous velocity at time t = 1.

32. Let Q(t) = t². As in the previous exercise, find a formula for the average ROC of Q over the interval [1, t] and use it to estimate the instantaneous ROC at t = 1. Repeat for the interval [2 , t] and estimate the ROC at t = 2. 

33.  Show that the average ROC of f(x) = x³ over [1, x] is equal to x² + x + 1. Use this to estimate the instantaneous ROC of f (x) at x = 1.

34.  Find a formula for the average ROC of f(x) = x³ over [2, x] and use it to estimate the instantaneous ROC at x = 2.

Section 2.3

58. Investigate numerically for several values of n and then guess the value in general.

59.  Show numerically that for b = 3, 5 appears to equal ln 3, ln 5, where ln x is the natural logarithm. Then make a conjecture (guess) for the value in general and test your conjecture for two additional values of b.

60.  Investigate  for (m , n) equal to (2, 1), (1, 2), (2, 3), and (3, 2). Then guess the value of the limit in general and check your guess for at least three additional pairs.

Section 2.4

2. Find the points of discontinuity of  f(x) and state whether  f (x) is left- or right-continuous (or neither) at these points.

3. At which point c does f(x) have a removable discontinuity? What value should be assigned to f(c) to make f  continuous at x = c?

4. Find the point c₁ at which f (x) has a jump discontinuity but is left-continuous. What value should be assigned to f (c₁) to make f right-continuous at x = c₁?

5. (a)  For the function shown in Figure 16, determine the one-sided limits at the points of discontinuity.
(b)  Which of these discontinuities is removable and how should f be redefined to make it continuous at this point?

In Exercises 37–50, determine the domain of the function and prove that it is continuous on its domain using the Laws of Continuity and the facts quoted in this section.

37.

51.  Suppose that  f (x) = 2 for x > 0 and  f (x) = − 4 for x < 0. What is  f (0) if  f  is left-continuous at x = 0? What is  f (0) if  f  is right-continuous at x = 0?

52.  Sawtooth Function  Draw the graph of  f (x) = x − [x]. At which points is f discontinuous? Is it left- or right-continuous at those points?

 In Exercises 53–56, draw the graph of a function on [0, 5] with the given properties.

53. f(x) is not continuous at x = 1, but and  exist and are equal.

54. f(x) is left-continuous but not continuous at x = 2 and right-continuous but not continuous at x = 3.

55. f(x) has a removable discontinuity at x = 1, a jump discontinuity at x = 2, and , .

56. f (x) is right- but not left-continuous at x = 1, left- but not right-continuous at x = 2, and neither left- nor right-continuous at x = 3.

57. Each of the following statements is false. For each statement, sketch the graph of a function that provides a counterexample.

(a) If exists, then f (x) is continuous at x = a.

(b) If f(x) has a jump discontinuity at x = a, then f(a) is equal to either  or .

(c) If f (x) has a discontinuity at  x = a,  then  and  exist but are not equal.

(d)  The one-sided limits   and  always exist, even if does not exist.

 

Section 2.7

1. Use the IVT to show that  f (x) = x³ + x takes on the value 9 for some x in [1, 2].

2. Show that takes on the value 0.499 for some t in [0, 1].

3. Show that g(t) = t² tan t  takes on the value ½ for some t in [0 , π/4].

4. Show that takes on the value 0.4.

5.  Show that cos x = x  has a solution in the interval [0, 1].

6.  Use the IVT to find an interval of length ½ containing a root of f(x) = x³ + 2x + 1.

In Exercises 7–16, use the IVT to prove each of the following statements.

7.  for some number c.

8. For all integers n, sin nx = cos x for some x Î [0 ,π] .

9. √2 exists. Hint: Consider f(x) = x²

10. A positive number c has an nth root for all positive integers n. (This fact is usually taken for granted, but it requires proof.)

11. For all positive integers k, there exists x such that cos x = x^k.

12. 2^x = bx has a solution if b > 2.

13. 2^x = b has a solution for all b > 0 (treat b ≥ 1 first).

14. tan x = x has infinitely many solutions.

15. The equation e^x + ln x = 0 has a solution in (0, 1).

16.  tan⁻¹x = cos⁻¹x has a solution.

17. Carry out three steps of the Bisection Method for f(x) = 2^x − x³ as follows:
(a) Show that f(x) has a zero in [1, 1.5].
(b) Show that f(x) has a zero in [1.25, 1.5].
(c) Determine whether [1.25, 1.375] or [1.375, 1.5] contains a zero.

18. Figure 4 shows that f(x) = x³ − 8x − 1 has a root in the interval [2.75, 3]. Apply the Bisection Method twice to find an interval of length 1/16 containing this root.

19. Find an interval of length ¼ in [0, 1] containing a root of x⁵ − 5x + 1 = 0.

20. Show that tan³θ − 8tan²θ + 17tan θ − 8 = 0 has a root in [0.5, 0.6]. Apply the Bisection Method twice to find an interval of length 0.025 containing this root.

In Exercises 21–24, draw the graph of a function f (x) on [0, 4] with the given property.

21. Jump discontinuity at x = 2 and does not satisfy the conclusion of the IVT.

22. Jump discontinuity at x = 2, yet does satisfy the conclusion of the IVT on [0, 4].

23. Infinite one-sided limits at x = 2 and does not satisfy the conclusion of the IVT.

24. Infinite one-sided limits at x = 2, yet does satisfy the conclusion of the IVT on [0, 4].

26. Take any map (e.g., of the United States) and draw a circle on it anywhere. Prove that at any moment in time there exists a pair of diametrically opposite points on that circle corresponding to locations where the temperatures at that moment are equal. Hint: Let θ be an angular coordinate along the circle and let f (θ) be the difference in temperatures at the locations corresponding to θ and θ + π.

27. Assume that f(x) is continuous and that 0 ≤ f (x) ≤ 1 for 0 ≤ x ≤ 1 (see Figure 5). Show that f(c) = c for some c in [0, 1].

28. Use the IVT to show that if f (x) is continuous and one-to-one on an interval [a, b], then f(x) is either an increasing or a decreasing function.

Section 2.7

2. Let f (x) = 2x² − 3x − 5. Show that the slope of the secant line through (2 , f (2)) and (2 + h , f (2 + h)) is 2h + 5. Then use this formula to compute the slope of:
(a) The secant line through (2, f(2)) and (3, f(3))
(b) The tangent line at x = 2 (by taking a limit)

22. First find the slope and then an equation of the tangent line to the graph of  f (x) = √x at x = 4. 

In Exercises 23–40, compute the derivative at x = a using the limit definition and find an equation of the tangent line.

23. f(x) = 3x² + 2x,    a = 2

41. What is an equation of the tangent line at x = 3, assuming that f (3) = 5 and  f ' ( 3 ) = 2?

42. Suppose that y = 5x + 2 is an equation of the tangent line to the graph of y = f(x) at a = 3. What is f(3)? What is f ' (3)?

43. Consider the “curve” y = 2x + 8. What is the tangent line at the point (1, 10) ? Describe the tangent line at an arbitrary point.

44. Suppose that f (x) is a function such that f (2 + h) − f (2) = 3h² + 5h.
(a) What is f ' (2)?
(b) What is the slope of the secant line through (2, f (2)) and (6, f(6))?

49. The vapor pressure of water is defined as the atmospheric pressure  P  at which no net evaporation takes place. The following table and Figure 13 give P (in atmospheres) as a function of temperature T in kelvins.
(a) Which is larger: P ' (300) or P ' (350) ? Answer by referring to the graph.
(b) Estimate P ' ( T ) for T = 303, 313, 323, 333, 343 using the table and the average of the difference quotients for h = 10 and − 10:

In Exercises 50–51, traffic speed S along a certain road (in mph) varies as a function of traffic density q (number of cars per mile on the road). Use the following data to answer the questions:

50. Estimate S ' (q) when q = 120 cars per mile using the average of difference quotients at h and − h as in Exercise 48.

51. The quantity V = q S is called traffic volume. Explain why V  is equal to the number of cars passing a particular point per hour. Use the data to compute values of V as a function of q and estimate V ' (q) when q = 120.

52. For the graph in Figure 14, determine the intervals along the x-axis on which the derivative is positive.

59. Sketch the graph of  f (x) = sin x on [ 0 ,π ] and guess the value of f ' (π/2). Then calculate the slope of the secant line between x = π/2 and x = π/2 + h for at least three small positive and negative values of h. Are these calculations consistent with your guess?

60. Figure 15(A) shows the graph of f (x) =√x. The close-up in (B) shows that the graph is nearly a straight line near x = 16. Estimate the slope of this line and take it as an estimate for f ' (16). Then compute f ' (16) and compare with your estimate.

65. Apply the method of Example 6 to f (x) = sin x to determine f ' (p/4) accurately to four decimal places.

66. Apply the method of Example 6 to f(x) = cos x to determine f ' (π/5) accurately to four decimal places. Use a graph of f(x) to explain how the method works in this case.

In Exercises 70–72, i ( t ) is the current (in amperes) at time t (seconds) flowing in the circuit shown in Figure

69. According to Kirchhoff’s law, i(t) = Cv ' ( t ) + R⁻¹v(t) , where v(t) is the voltage (in volts) at time t, C the capacitance (in farads), and R the resistance (in ohms). 

70.  Calculate the current at t = 3 if v( t ) = 0.5t + 4 V, C = 0.01 F, and R = 100 W.

71.  Use the following table to estimate v ¢(10). For a better estimate, take the average of the difference  quotients for h and −h as described in Exercise 48. Then estimate i(10), assuming C = 0.03 and R = 1000.

 72. Assume that R = 200 W but C is unknown. Use the following data to estimate v ' (4) as in Exercise 71 and deduce an approximate value for the capacitance C.

 Section 3.2

52. Sketch the graph of  f(x) = x − 3x² and find the values of x for which the tangent line is horizontal.

53. Find the points on the curve y = x² + 3x − 7 at which the slope of the tangent line is equal to 4.

54. Sketch the graphs of f(x) = x² − 5x + 4 and g(x) = −2x + 3. Find the value of x at which the graphs have parallel tangent lines.

55. Find all values of x where the tangent lines to y = x³ and y = x⁴ are parallel.

56. Show that there is a unique point on the graph of the function f(x) = ax² + bx + c where the tangent line is horizontal (assume a > 0). Explain graphically.

57. Determine coefficients a and b such that  p(x) = x² + ax + b satisfies p(1) = 0 and p ' (1) = 4.

58. Find all values of x such that the tangent line to the graph of y = 4x² + 11x + 2 is steeper than the tangent line to y = x³

59. Let f(x) = x³ − 3x + 1. Show that f ' (x) ≥ −3 for all x, and that for every m > −3, there are precisely two points where f ' (x) = m. Indicate the position of these points and the corresponding tangent lines for one value of m in a sketch of the graph of f (x).

60. Show that if the tangent lines to the graph of at x = a and at x = b are parallel, then either a = b or a + b = 2.

61. Compute the derivative of f (x) = x⁻² using the limit definition.

63. Find an approximation to m4 using the limit definition and estimate the slope of the tangent line to y = 4^x at x = 0 and x = 2.

64. Let f(x) = xe^x. Use the limit definition to compute f ¢(0) and find the equation of the tangent line at x = 0.

65. The average speed (in meters per second) of a gas molecule is , where T is the temperature (in kelvin),  M is the molar mass (kg /mol) and  R = 8.31. Calculate dv_avg /dT at T = 300 K for oxygen, which has a molar mass of 0.032 kg/mol.

66. Biologists  have  observed  that  the  pulse  rate  P  (in  beats  per minute) in animals is related to body mass (in kilograms) by the approximate formula P = 200m^−1/4. This is one of many allometric scaling laws prevalent in biology. Is the absolute value |dP/dm| increasing or decreasing as m increases? Find an equation of the tangent line at the points on the graph in Figure 17 that represent goat (m = 33) and man (m = 68).

67. Some studies suggest that kidney mass K in mammals (in kilograms) is related to body mass m (in kilograms) by the approximate formula K = 0.007m^0.85. Calculate dK/dm at m = 68. Then calculate the derivative with respect to m of the relative kidney-to-mass ratio K/m at m = 68.

68. The relation between the vapor pressure P (in atmospheres) of water and the temperature T (in kelvin) is given by the Clausius–Clapeyron law:

where k is a constant. Use the table below and the approximation to estimate dP/dT  for T = 303, 313, 323, 333, 343. Do your estimates seem to confirm the Clausius–Clapeyron law? What is the approximate value of k? What are the units of k?

69. Let L be a tangent line to the hyperbola xy = 1 at x = a, where a > 0. Show that the area of the triangle bounded by L and the coordinate axes does not depend on a.

70. In the notation of Exercise 69, show that the point of tangency is the midpoint of the segment of L lying in the first quadrant.

71. Match the functions (A)–(C) with their derivatives (I)–(III) in Figure 18.

94. Two small arches have the shape of parabolas. The first is given by f(x) = 1 − x² for −1 ≤ x ≤ 1 and the second by g(x) = 4 − (x − 4)² for 2 ≤ x ≤ 6. A board is placed on top of these arches so it rests on both (Figure 22). What is the slope of the board?

95. A vase is formed by rotating y = x² around the y-axis. If we drop in a marble, it will either touch the bottom point of the vase or be suspended above the bottom by touching the sides (Figure 23). How small must the marble be to touch the bottom?

98. Verify the Power Rule for the exponent 1/n, where n is a positive integer, using the following trick: Rewrite the difference quotient for y = x^1/n at x = b in terms of u = (b + h)^1/n and a = b^1/n.

99.  Infinitely Rapid Oscillations    Define

Show that  f(x) is continuous at x = 0 but f ' (0) does not exist (see Figure 12).

100.  Prove that  f(x) = e^x is not a polynomial function. Hint: Differentiation lowers the degree of a polynomial by 1.

101.  Consider the equation e^x = λx, where λ is a constant.
(a)  For which λ does it have a unique solution? For intuition, draw a graph of y = e^x and the line y = λx.
(b)  For which λ does it have at least one solution?

Section 3.4

1. Find the ROC of the area of a square with respect to the length of its side s when s = 3 and s = 5.

2. Find the ROC of the volume of a cube with respect to the length of its side s when s = 3 and s = 5.

3. Find the ROC of y = x⁻¹ with respect to x for x = 1, 10.

4.  At what rate is the cube root changing with respect to x when x = 1, 8, 27?

In Exercises 5–8, calculate the ROC.

5. dV/dr, where V is the volume of a cylinder whose height is equal to its radius (the volume of a cylinder of height h and radius r is πr²h)

6.  ROC of the volume V of a cube with respect to its surface area A

7.  ROC of the volume V of a sphere with respect to its radius.

8. , where A is the surface area of a sphere of diameter D (the surface area of a sphere of radius r is 4πr²)

9. (a) Estimate the average velocity over [0.5, 1].
(b) Is average velocity greater over [1, 2] or [2, 3]?
(c) At what time is velocity at a maximum?

10. Match the description with the interval (a)–(d).
(i) Velocity increasing
(ii) Velocity decreasing
(iii) Velocity negative
(iv) Average velocity of 50 mph

(a) [0, 0.5]
(b) [0, 1]
(c) [1.5, 2]
(d) [2.5, 3] 

11. Figure 11 displays the voltage across a capacitor as a function of time while the capacitor is being charged. Estimate the ROC of voltage at t = 20 s. Indicate the values in your calculation and include proper units. Does voltage change more quickly or more slowly as time goes on? Explain in terms of tangent lines.

12. Use Figure 12 to estimate dT/dh at h = 30 and 70, where T is atmospheric temperature (in degrees Celsius) and h is altitude (in kilometers). Where is dT/dh equal to zero?

13. A stone is tossed vertically upward with an  initial  velocity  of 25 ft /s from the top of a 30-ft building.
(a)  What is the height of the stone after 0.25 s?
(b)  Find the velocity of the stone after 1 s.
(c)  When does the stone hit the ground?

14. The height (in feet) of a skydiver at time t (in seconds) after opening his parachute is h(t) = 2000 − 15t ft. Find the skydiver’s velocity after the parachute opens.

15.  The temperature of an object (in degrees Fahrenheit) as a function of time (in minutes) is for 0 ≤ t ≤ 20. At what rate does the object cool after 10 min (give correct units)?

 16. The velocity (in centimeters per second) of a blood molecule flowing through a capillary of radius 0.008 cm is given by the formula v = 6.4 × 10⁻⁸ − 0.001r², where r is the distance from the molecule to the center of the capillary. Find the ROC of velocity as a function of distance when r = 0.004 cm.

 17.  The earth exerts a gravitational force of (in Newtons) on an object with a mass of 75 kg, where r is the distance (in meters) from the center of the earth. Find the ROC of force with respect to distance at the surface of the earth, assuming the radius of the earth is 6.77 × 10⁶ m

18. The escape velocity at a distance r meters from the center of the earth is v_esc = (2.82 × 10⁷)r^–1/2 m/s. Calculate the rate at which v_esc changes with respect to distance at the surface of the earth.

19.  The power delivered by a battery to an apparatus of resistance R (in ohms) is W. Find the rate of change of power with respect to resistance for R = 3 and R = 5 W.

20. The position of a particle moving in a straight line during a 5-s trip is s(t) = t² − t + 10 cm.
(a) What is the average velocity for the entire trip?
(b) Is there a time at which the instantaneous velocity is equal to this average velocity? If so, find it.

21.  By Faraday’s Law, if a conducting wire of length l meters moves at velocity v m/s perpendicular to a magnetic field of strength B (in teslas), a voltage of size V = − Blv is induced in the wire. Assume that B = 2 and l = 0.5.
(a) Find the rate of change dV/dv.
(b) Find the rate of change of V with respect to time t if v = 4t + 9.

22. The height (in  feet) of a helicopter at  time  t  (in  minutes)  is s(t) = −3t + 400t for 0 ≤ t ≤ 10.
(a)  Plot the graphs of height s(t) and velocity v(t).
(b)  Find the velocity at t = 6 and t = 7.
(c)  Find the maximum height of the helicopter.

23. The population P(t) of a city (in millions) is given by the formula P(t) = 0.00005t² + 0.01t + 1, where t denotes the number of years since 1990.
(a) How large is the population in 1996 and how fast is it growing?
(b) When does the population grow at a rate of 12,000 people per year?

24. According to Ohm’s Law, the voltage V, current I, and resistance R in a circuit are related by the equation V = IR, where the units are volts, amperes, and ohms. Assume that voltage is constant with V = 12 V. Calculate (specifying the units):
(a) The average ROC of  I  with respect to  R  for the interval from R = 8 to R = 8.1
(b)  The ROC of I with respect to R when R = 8
(c)  The ROC of R with respect to I when I = 1.5

25. Ethan finds that with h hours of tutoring, he is able to answer correctly S (h) percent of the problems on a math exam. What is the meaning of the derivative S ¢(h)? Which would you expect to be larger: S ' (3) or S ' (30) ? Explain.

26. Suppose θ(t) measures the angle between a clock’s minute and hour hands. What is θ ' (t) at 3 o’clock? 

27. Table 2 gives the total U.S. population during each month of 1999 as determined by the U.S. Department of Commerce.
(a)  Estimate P ¢(t) for each of the months January–November.
(b)  Plot these data points for P ¢(t) and connect the points by a smooth curve.
(c)  Write a newspaper headline describing the information contained in this plot.

28. The tangent lines to the graph of f(x) = x² grow steeper as x increases. At what rate do the slopes of the tangent lines increase?

29. According to a formula widely used by doctors to determine drug dosages, a person’s body surface area (BSA) (in meters squared) is given by the formula BSA = /60, where is the height in centimeters and w the weight in kilograms. Calculate the ROC of BSA with respect to weight for a person of constant height h = 180. What is this ROC for w = 70 and w = 80? Express your result in the correct units. Does BSA increase more rapidly with respect to weight at lower or higher body weights?

30. A slingshot is used to shoot a pebble in the air vertically from ground level with an initial velocity 200 m/s. Find the pebble’s maximum velocity and height.

31. What is the velocity of an object dropped from a height of 300 m when it hits the ground?

32. It takes a stone 3 s to hit the ground when dropped from the top of a building. How high is the building and what is the stone’s velocity upon impact?

33. A ball is tossed up vertically from ground level and returns to earth 4 s later. What was the initial velocity of the stone and how high did it go?

34. An object is tossed up vertically from ground level and hits the ground T s later. Show that its maximum height was reached after T/2 s.

35. A man on the tenth floor of a building sees a bucket (dropped by a window washer) pass his window and notes that it hits the ground 1.5 s later. Assuming a floor is 16 ft high (and neglecting air friction), from which floor was the bucket dropped?

36. Which of the following statements is true for an object falling under the influence of gravity near the surface of the earth? Explain.
(a) The object covers equal distance in equal time intervals.
(b) Velocity increases by equal amounts in equal time intervals.
(c) The derivative of velocity increases with time.

37. Show that for an object rising and falling according to Galileo’s formula in Eq. (3), the average velocity over any time interval [t₁, t₂] is equal to the average of the instantaneous velocities at t₁ and t₂.

38. A weight oscillates up and down at the end of a spring. Figure 13 shows the height y of the weight through one cycle of the oscillation. Make a rough sketch of the graph of the velocity as a function of time.

In Exercises 39–46, use Eq. (2) to estimate the unit change.

39.  Estimate  and . Compare your estimates with the actual values.

40. Suppose that f(x) is a function with f ¢(x) = 2^−x. Estimate f (7) − f (6). Then estimate f(5), assuming that  f (4) = 7.

41. Let F(s) = 1.1s + 0.03s² be the stopping distance as in Example 3. Calculate F(65) and estimate the increase in stopping distance if speed is increased from 65 to 66 mph. Compare your estimate with the actual increase.

42. According to Kleiber’s Law, the metabolic rate P (in kilocalories per day) and body mass m (in kilograms) of an animal are related by a three-quarter power law  P = 73.3m^3/4. Estimate the increase in metabolic rate when body mass increases from 60 to 61 kg.

43. The dollar cost of producing x bagels is C(x) = 300 + 0.25x − 0.5(x/1000)³. Determine the cost of producing 2,000 bagels and estimate the cost of the 2001st bagel. Compare your estimate with the actual cost of the 2001st bagel.

44. Suppose the dollar cost of producing x video cameras is C(x) = 500x − 0.003x² + 10⁻⁸x³.
(a) Estimate the marginal cost at production level x = 5000 and compare it with the actual cost C(5001) − C(5000).
(b) Compare the marginal cost at x = 5000 with the average cost per camera, defined as C(x)/x.

45. The demand for a commodity generally decreases as the price is raised. Suppose that the demand for oil (per capita per year) is D(p) = 900/p barrels, where p is the price per barrel in dollars. Find the demand when p = $40. Estimate the decrease in demand if p rises to $41 and the increase if p is decreased to $39.

46. The reproduction rate of the fruit fly Drosophila melanogaster, grown in bottles in a laboratory, decreases as the bottle becomes more crowded. A researcher has found that when a bottle contains p flies, the number of offspring per female per day is f(p) = (34 − 0.612p)p^−0.658
(a) Calculate  f(15) and  f ' (15).
(b) Estimate the decrease in daily offspring per female when p is increased from 15 to 16. Is this estimate larger or smaller than the actual value f(16) − f(15)?
(c) Plot f (p) for 5 ≤ p ≤ 25 and verify that f(p) is a decreasing function of p. Do you expect f ' (p) to be positive or negative? Plot f ' (p) and confirm your expectation.

47. Let A = s². Show that the estimate of A(s + 1) − A(s) provided by Eq. (2) has error exactly equal to 1. Explain this result using Figure 14.

48. According to Steven’s Law in psychology, the perceived magnitude of a stimulus (how strong a person feels the stimulus to be) is proportional to a power of the actual intensity I  of the stimulus. Although not an exact law, experiments show that the perceived brightness B of a light satisfies B = kI^2/3, where I  is the light intensity, whereas the perceived heaviness  H  of a weight W  satisfies H = kW^3/2 (k is a constant that is different in the two cases). Compute dB/dI and dH/dW and state whether they are increasing or decreasing functions. Use this to justify the statements:
(a) A one-unit increase in light intensity is felt more strongly when I is small than when I is large.
(b) Adding another pound to a load W is felt more strongly when Wis large than when W is small.

49. Let M(t) be the mass (in kilograms) of a plant as a function of time (in years). Recent studies by Niklas and Enquist have suggested that for a remarkably wide range of plants (from algae and grass to palm trees), the growth rate during the life span of the organism satisfies a three-quarter power law, that is, dM/dt = CM^3/4for some constant C.
(a) If a tree has a growth rate of 6 kg/year when M = 100 kg, what is its growth rate when M = 125 kg?
(b) If M = 0.5 kg, how much more mass must the plant acquire to double its growth rate?

50. As an epidemic spreads through a population, the percentage p of infected individuals at time t (in days) satisfies the equation (called a differential equation) dp/dt = 4p − 0.06p², 0 ≤ p ≤ 100
(a) How fast is the epidemic spreading when p = 10% and when p = 70%?
(b) For which p is the epidemic neither spreading nor diminishing?
(c) Plot dp/dt as a function of p.
(d) What is the maximum possible rate of increase and for which p does this occur?

51. The size of a certain animal population P(t) at time t (in months) satisfies dP/dt = 0.2(300 − P).
(a) Is P growing or shrinking when P = 250? when P = 350?
(b) Sketch the graph of dP/dt as a function of P for 0 ≤ P ≤ 300.
(c) Which of the graphs in Figure 15 is the graph of P(t) if P(0) = 200?

In Exercises 53–54, the average cost per unit at production level x is defined as C_avg (x) = C(x)/x, where C(x) is the cost function. Average cost is a measure of the efficiency of the production process.

53.  Show that C_avg(x) is equal to the slope of the line through the origin and the point (x, C(x)) on the graph of C(x). Using this interpretation, determine whether average cost or marginal cost is greater at points A, B, C, D in Figure 16.

54. The cost in dollars of producing alarm clocks is C(x) = 50x³ − 750x² + 3740x + 3750 where x is in units of 1,000.
(a) Calculate the average cost at x = 4, 6, 8, and 10.
(b) Use the graphical interpretation of average cost to find the production level x₀ at which average cost is lowest. What is the relation between average cost and marginal cost at x₀ (see Figure 17)?

Section 3.5

37. (a) Find the acceleration at time t = 5 min of a helicopter whose height (in feet) is h(t) = − 3t³ + 400t.
(b) Plot the acceleration h ²(t) for 0 ≤ t ≤ 6. How does this graph show that the helicopter is slowing down during this time interval?

38. Find an equation of the tangent to the graph of y = f ¢(x) at x = 3, where f(x) = x⁴. 

39. Figure 5 shows f , f ' , and  f ''. Determine which is which. 

40. The second derivative f '' is shown in Figure 6. Determine which graph, (A) or (B), is f and which is f ''. 

41. Figure 7 shows the graph of the position of an object as a function of time. Determine the intervals on which the acceleration is positive.

42. Find the second derivative of the volume of a cube with respect to the length of a side.

43. Find a polynomial f(x) satisfying the equation xf '' (x) + f (x) = x². 

44. Find a value of n such that y = xⁿe^x satisfies  the  equation xy ' = (x − 3)y. 

45. Which of the following descriptions could not apply to Figure 8? Explain.
(a) Graph of acceleration when velocity is constant
(b) Graph of velocity when acceleration is constant
(c) Graph of position when acceleration is zero 

46. A servomotor controls the vertical movement of a drill bit that will drill a pattern of holes in sheet metal. The maximum vertical speed of the drill bit is 4 in ./s, and while drilling the hole, it must move no more than 2.6 in ./s to avoid warping the metal. During a cycle, the bit begins and ends at rest, quickly approaches the sheet metal, and quickly returns to its initial position after the hole is drilled. Sketch possible graphs of the drill bit’s vertical velocity and acceleration. Label the point where the bit enters the sheet metal. 

52. Find the 100th derivative of p(x) = (x + x⁵ + x⁷)¹⁰(1 + x²)¹¹(x³ + x⁵ + x⁷) 

54. Use the Product Rule twice to find a formula for (f g)'' in terms of the first and second derivative of f and g. 

55. Use the Product Rule to find a formula for (f g)'' and compare your result with the expansion of (a + b)³. Then try to guess the general formula for (f g)⁽ⁿ⁾ 

Section 3.6 

42.  Find the values of x between 0 and 2 π where the tangent line to the graph of y = sin x cos x is horizontal. 

43.  Calculate the first five derivatives of f(x) = cos x. Then determine f⁽⁸⁾ and f⁽³⁷⁾.

44.  Find y⁽¹⁵⁷⁾, where y = sin x. 

48. Show that no tangent line to the graph of f(x) = tan x has zero slope. What is the least slope of a tangent line? Justify your response by sketching the graph of (tan x) '. 

49. The height at time t (s) of a weight, oscillating up and down at the end of a spring, is s(t) = 300 + 40 sin t cm. Find the velocity and acceleration at t = π.

50. The horizontal range R of a projectile launched from ground level at an angle θ and initial velocity v₀ m/s is . Calculate dR/dθ. If θ = 7π /24, will the range increase or decrease if the angle is increased slightly? Base your answer on the sign of the derivative.

51. If you stand 1 m from a wall and mark off points on the wall at equal increments δ of angular elevation (Figure 4), then these points grow increasingly far apart. Explain how this illustrates the fact that the derivative of tan θ is increasing.

52.  Use the limit definition of the derivative and the addition law for the cosine to prove that (cos x) ' = −sin x.

53.  Show that a nonzero polynomial function y = f (x) cannot satisfy the equation y ² = − y. Use this to prove that neither sin x nor cos x is a polynomial.

56.  Show that if π / 2 < θ  < π , then the distance along the x-axis between θ and the point where the tangent line intersects the x-axis is equal to |tan θ| (Figure 5).

Section 3.7

74.  The average molecular velocity v of a gas in a certain container is given by v = 29√T m/s, where T is the temperature in kelvins. The temperature is related to the pressure (in atmospheres) by T = 200P.

Find

76. Assume that f(0) = 2 and  f ' (0) = 3. Find the derivatives of (f (x))³ and f (7x ) at x = 0. 

77. Compute the derivative of h(sin x) at x = π/6, assuming that h ' (0.5) = 10. 

78. Let F(x) = f(g(x)), where the graphs of f  and g are shown in Figure 1. Estimate g ' (2) and f ' (g(2)) from the graph and compute F ' (2). 

88. Use the Chain Rule to express the second derivative of  f ◦ g in terms of the first and second derivatives of f  and g.

89. Compute the second derivative of sin (g(x)) at x = 2, assuming that g (2) = π/4, g ' (2) = 5, and g '' (2) = 3. 

90. An expanding sphere has radius r = 0.4t cm at time t (in seconds). Let V be the sphere’s volume. Find dV/dt when (a) r = 3 and (b) t = 3.

91. The power P in a circuit is P = Ri², where R is resistance and i the current. Find dP/dt at t = 2 if R = 1000 W and i varies according to i = sin(4πt) (time in seconds).

92. The price (in dollars) of a computer component is P = 2C − 18C⁻¹, where C is the manufacturer’s cost to produce it. Assume that cost at time t (in years) is C = 9 + 3t⁻¹ and determine the ROC of price with respect to time at t = 3.

93. The force F (in Newtons) between two charged objects is F = 100/r², where r is the distance (in meters) between them. Find dF/dt at t = 10 if the distance at time t (in seconds) is r = 1 + 0.4t².

94.  According to the U.S. standard atmospheric model, developed by the National Oceanic and Atmospheric Administration for use in aircraft and rocket design, atmospheric temperature T (in degrees Celsius), pressure P (kPa = 1000 Pascals), and altitude h (meters) are related by the formulas (valid in the troposphere h ≤ 11000):

T = 15.04 – 0.000649h,    

Calculate dP/dh. Then estimate the change in P (in Pascals, Pa) per additional meter of altitude when h = 3000.

Section 3.8

35. Find the points on the graph of  y² = x³ − 3x + 1 (Figure 6) where the tangent line is horizontal.
(a) First show that 2yy ¢ = 3x² − 3, where y ¢= dy/dx.
(b) Do not solve for y ¢. Rather, set y ¢ = 0 and solve for x. This gives two possible values of x where the slope may be zero.
(c) Show that the positive value of x does not correspond to a point on the graph.
(d) The negative value corresponds to the two points on the graph where the tangent line is horizontal. Find the coordinates of these two points.

36. Find all points on the graph of 3x² + 4y² + 3xy = 24 where the tangent line is horizontal (Figure 7).
(a) By differentiating the equation of the curve implicitly and setting y ¢ = 0, show that if the tangent line is horizontal at (x , y) , then y = −2x.
(b) Solve for x by substituting y = −2x in the equation of the curve.  

37. Show that no point on the graph of x² − 3xy + y² = 1 has a horizontal tangent line.

38. Figure 1 shows the graph of y⁴ + xy = x³ − x + 2. Find dy/dx at the two points on the graph with x-coordinate 0 and find an equation of the tangent line at (1, 1).

39. If the derivative dx/dy exists at a point and dx/dy = 0, then the tangent line is vertical. Calculate dx/dy for the equation y⁴ + 1 = y² + x² and find the points on the graph where the tangent line is vertical.

40. Differentiate the equation xy = 1 with respect to the variable t and derive the relation .

44. The volumeV and pressure P of gas in a piston (which vary in time t) satisfy PV^3/2 = C, where C is a constant. Prove that
The ratio of the derivatives is negative. Could you have predicted this from the relation PV^3/2 = C?

46. Find all points on the folium x³ + y³ = 3xy at which the tangent line is horizontal.

58.  Show that if P lies on the intersection of the two curves x² − y² = c and xy = d (c, d constants), then the tangents to the curves at P are perpendicular.

Section 3.10

79. The energy E (in joules) radiated as seismic waves from an earthquake of Richter magnitude  M  is given by the formula log₁₀E = 4.8 + 1.5M.
(a) Express E as a function of M.
(b) Show that when M increases by 1, the energy increases by a factor of approximately 31.
(c) Calculate dE/dM

Section 3.11

1. How fast is the water level rising if water is filling the tub at a rate of 0.7 ft³/min? 

2. At what rate is water pouring into the tub if the water level rises at a rate of 0.8 ft/min?

3. The radius of a circular oil slick expands at a rate of 2 m/min.
(a) How fast is the area of the oil slick increasing when the radius is 25 m?
(b) If the radius is 0 at time t = 0, how fast is the area increasing after 3 min? 

4. At what rate is the diagonal of a cube increasing if its edges are increasing at a rate of 2 cm/s?

In Exercises 5–8, assume that the radius r of a sphere is expanding at a rate of 14 in./min. The volume of a sphere is .

5. Determine the rate at which the volume is changing with respect to time when r = 8 in.

6.  Determine the rate at which the volume is changing with respect to time at t = 2 min, assuming that r = 0 at t = 0.

7. Determine the rate at which the surface area is changing when the radius is r = 8 in.

8. Determine the rate at which the surface area is changing with respect to time at t = 2 min, assuming that r = 3 at t = 0.

9. A road perpendicular to a highway leads to a farmhouse located 1 mile away (Figure 9). An automobile travels past the farmhouse at a speed of 60 mph. How fast is the distance between the automobile and the farmhouse increasing when the automobile is 3 miles past the intersection of the highway and the road?

10. A conical tank has height 3 m and radius 2 m at the top. Water flows in at a rate of 2 m³/min. How fast is the water level rising when it is 2 m?

11. Follow the same set-up as Exercise 10, but assume that the water level is rising at a rate of 0 . 3 m / min when it is 2 m. At what rate is water flowing in?

12. Sonya and Isaac are in motorboats located at the center of a lake. At time t = 0, Sonya begins traveling south at a speed of 32 mph. At the same time, Isaac takes off, heading east at a speed of 27 mph.
(a) How far have Sonya and Isaac each traveled after 12 min?
(b) At  what  rate  is  the  distance  between  them  increasing  at  t = 12 min? 

13. Answer (a) and (b) in Exercise 12 assuming that Sonya begins moving 1 minute after Isaac takes off.

14. A 6-ft man walks away from a 15-ft lamppost at a speed of 3 ft /s (Figure 10). Find the rate at which his shadow is increasing in length.

15. At a given moment, a plane passes directly above a radar station at an altitude of 6 miles.
(a) If the plane’s speed is 500 mph, how fast is the distance between the plane and the station changing half an hour later?
(b) How fast is the distance between the plane and the station changing when the plane passes directly above the station? 

16. In the setting of Exercise 15, suppose that the line through the radar station and the plane makes an angle θ with the horizontal. How fast is θ changing 10 min after the plane passes over the radar station?

17. A hot air balloon rising vertically is tracked by an observer located 2 miles from the lift-off point. At a certain moment, the angle between the observer’s line-of-sight and the horizontal is π/5, and it is changing at a rate of 0.2 rad/min. How fast is the balloon rising at this moment?

18. As a man walks away from a 12-ft lamppost, the tip of his shadow moves twice as fast as he does. What is the man’s height?

In Exercises 19–23, refer to a 16-ft ladder sliding down a wall, as in Figures 1 and 2. The variable h is the height of the ladder’s top at time t, and x is the distance from the wall to the ladder’s bottom.

19. Assume the bottom slides away from the wall at a rate of 3 ft/s. Find the velocity of the top of the ladder at t = 2 if the bottom is 5 ft from the wall at t = 0.

20. Suppose that the top is sliding down the wall at a rate of 4 ft/s. Calculate dx/dt when h = 12.

21. Suppose that h(0) = 12 and the top slides down the wall at a rate of 4 ft/s. Calculate x and dx/dt at t = 2 s.

22. What is the relation between h and x at the moment when the top and bottom of the ladder move at the same speed?

23. Show that the velocity dh / dt approaches infinity as the ladder slides down to the ground (assuming dx / dt is constant). This suggests that our mathematical description is unrealistic, at least for small values of h. What would, in fact, happen as the top of the ladder approaches the ground?

24. The radius r of a right circular cone of fixed height h = 20 cm is increasing at a rate of 2 cm/s. How fast is the volume increasing when r = 10?

25. Suppose that both the radius r and height h of a circular cone change at a rate of 2 cm/s. How fast is the volume of the cone increasing when r = 10 and h = 20?

26. A particle moves counterclockwise around the ellipse 9x² + 16y² = 25 (Figure 11).
(a) In which of the four quadrants is the derivative dx / dt positive? Explain your answer.
(b) Find a relation between dx/dt and dy/dt.
(c) At what rate is the x-coordinate changing when the particle passes the point (1, 1) if its y-coordinate is increasing at a rate of 6 ft /s?
(d) What is dy/dt when the particle is at the top and bottom of the ellipse?

27. A searchlight rotates at a rate of 3 revolutions per minute. The beam hits a wall located 10 miles away and produces a dot of light that moves horizontally along the wall. How fast is this dot moving when the angle θ between the beam and the line through the searchlight perpendicular to the wall is π/6?

28. A rocket travels vertically at a speed of 800 mph. The rocket is tracked through a telescope by an observer located 10 miles from the launching pad. Find the rate at which the angle between the telescope and the ground is increasing 3 min after lift-off.

29. A plane traveling at an altitude of 20,000 ft passes directly overhead at time t = 0. One minute later you observe that the angle between the vertical and your line of sight to the plane is 1.14 rad and that this angle is changing at a rate of 0.38 rad/min. Calculate the velocity of the airplane.

30. Calculate the rate (in cm²/s) at which area is swept out by the second hand of a circular clock as a function of the clock’s radius.

31. A jogger runs around a circular track of radius 60 ft. Let (x, y) be her coordinates, where the origin is at the center of the track. When the jogger’s coordinates are (36, 48), her x-coordinate is changing at a rate of 14 ft/s. Find dy/dt.

32. A car travels down a highway at 55 mph. An observer is standing 500 ft from the highway.
(a) How fast is the distance between the observer and the car increasing at the moment the car passes in front of the observer? Can you justify your answer without relying on any calculations?
(b)  How fast is the distance between the observer and the car increasing 1 min later? 

In Exercises 33–34, assume that the pressure P (in kilopascals) and volume V  (in cubic centimeters) of an expanding gas are related by PV^b = C, where b and C are constants (this holds in adiabatic expansion, without heat gain or loss).

33. Find  dP/dt  if  b = 1.2,  P = 8 kPa, V = 100 cm²,  and dV/dt = 20 cm³/min.

34. Find b if P = 25 kPa, dP/dt = 12 kPa/min, V = 100 cm², and dV/dt = 20 cm³/min.

35.  A point moves along the parabola y = x² + 1. Let l(t) be the distance between the point and the origin. Calculate l ¢(t) , assuming that the x-coordinate of the point is increasing at a rate of 9 ft/s.

36. The base x of the right triangle in Figure 12 increases at a rate of 5 cm/s, while the height remains constant at h = 20. How fast is the angle θ changing when x = 20?

37. A water tank in the shape of a right circular cone of radius 300 cm and height 500 cm leaks water from the vertex at a rate of 10 cm³/min. Find the rate at which the water level is decreasing when it is 200 cm.

38. Two parallel paths 50 ft apart run through the woods. Shirley jogs east on one path at 6 mph, while Jamail walks west on the other at 4 mph. If they pass each other at time t = 0, how far apart are they 3 s later, and how fast is the distance between them changing at that moment?

39. Henry is pulling on a rope that passes through a pulley on a 10-ft pole and is attached to a wagon (Figure 13). Assume that the rope is attached to a loop on the wagon 2 ft off the ground. Let x be the distance between the loop and the pole.
(a) Find a formula for the speed of the wagon in terms of x and the rate at which Henry pulls the rope.
(b) Find the speed of the wagon when it is 12 ft from the pole, assuming that Henry pulls the rope at a rate of 1 . 5 ft/s.

40. A roller coaster has the shape of the graph in Figure 14. Show that when the roller coaster passes the point ( x, f (x)), the vertical velocity of the roller coaster is equal to  f ¢(x) times its horizontal velocity.

41. Using a telescope, you track a rocket that was launched 2 miles away, recording the angle θ between the telescope and the ground at half-second intervals. Estimate the velocity of the rocket if θ(10) = 0 .205 and θ(10.5) = 0.225.

42. Two trains leave a station at t = 0 and travel with constant velocity v along straight tracks that make an angle θ.
(a) Show  that  the  trains  are  separating  from  each  other  at  a  rate .
(b) What does this formula give for θ = π?

43. A baseball player runs from home plate toward first base at 20 ft/s. How fast is the player’s distance from second base changing when the player is halfway to first base? See Figure 15.

44. As the wheel of radius r cm in Figure 16 rotates, the rod of length L attached at the point P drives a piston back and forth in a straight line. Let x be the distance from the origin to the point Q at the end of the rod as in the figure.
(a) Use the Pythagorean Theorem to show that
(b) Differentiate Eq. (8) with respect to t to prove that

(c) Calculate the speed of the piston when θ = π/2, assuming that r = 10 cm, L = 30 cm, and the wheel rotates at 4 revolutions per minute.

45. A spectator seated 300 m away from the center of a circular track of radius 100 m watches an athlete run laps at a speed of 5 m/s. How fast is the distance between the spectator and athlete changing when the runner is approaching the spectator and the distance between them is 250 m?

46. A cylindrical tank of radius R and length L lying horizontally as in Figure 17 is filled with oil to height h.
(a) Show that the volume V(h) of oil in the tank as a function of height h is

(b)  Show that
(c)  Suppose that R = 4 ft and L = 30 ft, and that the tank is filled at a constant rate of 10 ft³/min. How fast is the height h increasing when h = 5?

CHAPTER REVIEW EXERCISES 

113. Water pours into the tank in Figure 7 at a rate of 20 m³/min. How fast is the water level rising when the water level is h = 4 m?

114. The minute hand of a clock is 4 in. long and the hour hand is 3 in. long. How fast is the distance between the tips of the hands changing at 3 o’clock?

115. A light moving at 3 ft / s approaches a 6-ft man standing 12 ft from a wall (Figure 8). The light is 3 ft above the ground. How fast is the tip P of the man’s shadow moving when the light is 24 ft from the wall?

116. A bead slides down the curve xy = 10. Find the bead’s horizontal velocity if its height at time t seconds is y = 80 − 16t² cm.

117. (a)  Side x of the triangle in Figure 9 is increasing at 2 cm/s and side y is increasing at 3 cm/s. Assume that θ decreases in such a way that the area of the triangle has the constant value 4 cm². How fast is θ decreasing when x = 4, y = 4?
(b) How fast is the distance between P and Q changing when x = 2, y = 3? 

Section 4.1

In exercises 1-6 use the linear approximation to estimate Df = f(3.02) – f(3) for the given function
1. f(x) = x²

25. The cube root of 27 is 3. How much larger is the cube root of 27.2? Estimate using the Linear Approximation. 

26. Which is larger: √2.1 − √2 or √9.1 − √9? Explain using the Linear Approximation.

27. Estimate sin 61◦ − sin 60◦ using the Linear Approximation.

28. A thin silver wire has length L = 18 cm when the temperature is T = 30◦C. Estimate the length when T = 25◦C if the coefficient of thermal expansion is k = 1.9 × 10⁻⁵ ◦C⁻¹.

29.  The atmospheric pressure P (in kilopascals) at altitudes h (in kilometers) for 11 ≤ h ≤ 25 is approximately P(h) = 128e^−0.157h.
(a) Use the Linear Approximation to estimate the change in pressure at h = 20 when Dh = 0.5.
(b) Compute the actual change and compute the percentage error in the Linear Approximation.

30. The resistance R of a copper wire at temperature T = 20◦C is R = 15 W. Estimate the resistance at T = 22◦C, assuming that

31. The side s of a square carpet is measured at 6 ft. Estimate the maximum error in the area A of the carpet if s is accurate to within half an inch.

32. A spherical balloon has a radius of 6 in. Estimate the change in volume and surface area if the radius increases by 0.3 in.

33. A stone tossed vertically in the  air with initial velocity v ft/s reaches a maximum height of h = v²/64 ft.
(a) Estimate Dh if v is increased from 25 to 26 ft / s.
(b) Estimate Dh if v is increased from 30 to 31 ft / s.
(c) In general, does a 1 ft / s increase in initial velocity cause a greater change in maximum height at low or high initial velocities? Explain.

 

34. If the price of a bus pass from Albuquerque to Los Alamos is set at x dollars, a bus company takes in a monthly revenue of R(x) = 1.5x − 0.01x² (in thousands of dollars).
(a) Estimate the change in revenue if the price rises from $50 to $53.
(b) Suppose that x = 80. How will revenue be affected by a small increase in price? Explain using the Linear Approximation.

35. The stopping  distance  for  an  automobile  (after  applying  the brakes) is approximately  F(s) = 1.1s + 0.054s² ft, where s is the speed in mph. Use the Linear Approximation to estimate the change in stopping distance per additional mph when s = 35 and when s = 55.

36. Juan measures the circumference C of a spherical ball at 40 cm and computes the ball’s volume V . Estimate the maximum possible error in V if the error in C is at most 2 cm. Recall that C = 2πr and , where r is the ball’s radius.

37. Estimate the weight loss per mile of altitude gained for a 130-lb pilot. At which altitude would she weigh 129.5 lb? See Example 4.

38. How much would a 160-lb astronaut weigh in a satellite orbiting the earth at an altitude of 2,000 miles? Estimate the astronaut’s weight loss per additional mile of altitude beyond 2,000.

39. The volume of a certain gas (in liters) is related to pressure P (in atmospheres) by the formula PV = 24. Suppose that V = 5 with a possible error of ± 0.5 L.
(a) Compute P and estimate the possible error.
(b) Estimate the maximum allowable error in V  if  P  must have an error of at most 0.5 atm.

40. The dosage D of diphenhydramine for a dog of body mass w kg is D = k w^2/3 mg, where k is a constant. A cocker spaniel has mass w = 10 kg according to a veterinarian’s scale. Estimate the maximum allowable error in w if the percentage error in the dosage D must be less than 5%.

In Exercises 41–50, find the linearization at x = a.

41. y = cos x sin x,  a = 0

51. Estimate √16.2 using the linearization L(x) of  f(x) = √x at a = 16. Plot f(x) and L(x) on the same set of axes and determine if the estimate is too large or too small.

52. Estimate 1/√15 using a suitable linearization of  f(x) = 1/√x. Plot f(x) and L(x) on the same set of axes and determine if the estimate is too large or too small. Use a calculator to compute the percentage error.

In Exercises 53–61, approximate using linearization and use a calculator to compute the percentage error.

53.√17

62. Plot f(x) = tan x and its linearization L(x) at a = π/4 on the same set of axes.
(a) Does the linearization overestimate or underestimate f(x)?
(b) Show, by graphing y = f(x) − L(x) and y = 0.1 on the same set of  axes, that the error |f(x) − L(x)| is at most 0.1 for 0.55 ≤ x ≤ 0.95.
(c) Find an interval of x-values on which the error is at most 0.05.

63. Compute the linearization L(x) of f(x) = x² − x^3/2 at a = 2. Then plot f(x) − L(x) and find an interval around a = 1 such that |f(x) − L(x)| ≤ 0.1.

In Exercises 64–65, use the following fact derived from Newton’s Laws: An object released at an angle θ with initial velocity v ft/s travels a total distance

64. A player located 18.1 ft from a basket launches a successful jump shot from a height of 10 ft (level with the rim of the basket), at an angle θ = 34^o and initial velocity of v = 25 ft/s.
(a) Show that the distance s of the shot changes by approximately 0.255Dθ ft if the angle changes by an amount Dθ. Remember to convert the angles to radians in the Linear Approximation.
(b) Is it likely that the shot would have been successful if the angle were off by 2◦?

65. Estimate the change in the distance s of  the shot if  the angle changes from 50◦ to 51◦ for v = 25 ft / s and v = 30 ft / s. Is the shot more sensitive to the angle when the velocity is large or small? Explain.

66. Compute the linearization of f (x) = 3x − 4 at a = 0 and a = 2. Prove more generally that a linear function coincides with its linearization at x = a for all a.

67. According to (3), the error in the Linear Approximation is of “order two” in h. Show that the Linear Approximation to f(x) =√x at x = 9 yields the estimate. Then compute the error E for h = 10⁻ⁿ, 1 ≤ n ≤ 4, and verify numerically that E ≤ 0.006h².

68. Show that the Linear Approximation to f(x) = tan x at x = π/4 yields the estimate tan (π/4 + h) – 1 » 2h. Compute the error E  for h = 10⁻ⁿ, 1 ≤ n ≤ 4, and verify that E satisfies the Error Bound (3) with K = 6.2.

69. Show that for any real number k, (1 + x) k ≈ 1 + kx for small x. Estimate (1.02)^0.7 and (1.02)^−0.3.

70. (a) Show that  f(x) = sin x and g(x) = tan x have the same linearization at a = 0.
(b) Which function is approximated more accurately? Explain using a graph over [0, π/6].
(c) Calculate the error in these linearizations at x = π/6. Does the answer confirm your conclusion in (b)?

71. Let Df = f(5 + h) − f(5) , where f(x) = x², and let E = |Df − f ¢(5)h| be the error in the Linear Approximation. Verify directly that E satisfies (3) with K = 2 (thus E is of order two in h).

Section 4.2

1. The following questions refer to Figure 15.
(a) How many critical points does  f(x) have?
(b) What is the maximum value of  f(x) on [0, 8]?
(c) What are the local maximum values of f(x)?
(d) Find a closed interval on which both the minimum and maximum values of f(x) occur at critical points.
(e) Find an interval on which the minimum value occurs at an endpoint.

In Exercises 3–14, find all critical points of the function.

3.   f(x) = x² − 2x + 4

15. Let f(x) = x² − 4x + 1.
(a) Find the critical point c of f(x) and compute f(c).
(b) Compute the value of f(x) at the endpoints of the interval [0, 4].
(c) Determine the min and max of f(x) on [0, 4].
(d) Find the extreme values of f(x) on [0, 1]. 

16. Find the extreme values of 2x³ − 9x² + 12x on [0, 3] and [0, 2]. 

17. Find the minimum value of y = tan⁻¹(x² − x). 

18. Find the critical points of f(x) = sin x + cos x and determine the extreme values on [0, π/2]. 

19. Compute the critical points of h(t) = (t² − 1)^1/3. Check that your answer is consistent with Figure 17. Then find the extreme values of h(t) on [0, 1] and [0, 2]. 

20. Plot f(x) = 2√x − x on [0, 4] and determine the maximum value graphically. Then verify your answer using calculus. 

21. Plot f (x) = ln x − 5 sin x on [0, 2π] (choose an appropriate viewing rectangle) and approximate both the critical points and extreme values. 

22. Approximate the critical points of g(x) = x arccos x and estimate the maximum value of g(x). 

In Exercises 23–56, find the maximum and minimum values of the function on the given interval.
23.  y = 2x² − 4x + 2, [0, 3].

63.  Let f(x) = 3x − x³. Check that f (−2) = f(1). What may we conclude from Rolle’s Theorem? Verify this conclusion.

In Exercises 64–67, verify Rolle’s Theorem for the given interval.

64. f(x) = x + x⁻¹, [1/2, 2] 

68. Use Rolle’s Theorem to prove that f(x) = x⁵ + 2x³ + 4x − 12 has at most one real root.

69. Use Rolle’s Theorem to prove that f (x) = x³/6 + x²/2 + x + 1 has at most one real root.

70. The concentration C ( t ) (in mg/cm³) of a drug in a patient’s bloodstream after t hours is . Find the maximum concentration and the time at which it occurs.

84. Show, by considering its minimum, that f(x) = x² − 2x + 3 takes on only positive values. More generally, find the conditions on r and s under which the quadratic function f(x) = x² + rx + s takes on only positive values. Give examples of r and s for which f takes on both positive and negative values.

85. Show that if the quadratic polynomial f(x) = x² + rx + s takes on both positive and negative values, then its minimum value occurs at the midpoint between the two roots.

88. Find the minimum and maximum values of  f(x) = x^p(1 − x)^q on [0, 1], where p and q are positive numbers.

Section 4.3

In Exercises 1–10, find a point c satisfying the conclusion of the MVT for the given function and interval.

1. y = x⁻¹, [1, 4]

11. Let f(x) = x⁵ + x². Check that the secant line between x = 0 and x = 1 has slope 2. By the MVT, f ' (c) = 2 for some c Îin the interval (0,1). Estimate c graphically as follows. Plot f(x) and the secant line on the same axes. Then plot the lines y = 2x + b for different values of b until you find a value of b for which it is tangent to y = f(x). Zoom in on the point of tangency to find its x-coordinate.

12. Determine the intervals on which f (x) is positive and negative, assuming that Figure 12 is the graph of  f(x).

13. Determine the intervals on which f ( x ) is increasing or decreasing, assuming that Figure 12 is the graph of the derivative f ' (x).

14. Plot the derivative f ' (x) of f(x) = 3x⁵ − 5x³ and describe the sign changes of f ' (x). Use this to determine the local extreme values of f(x). Then graph f(x) to confirm your conclusions.

In Exercises 15–18, sketch the graph of a function f(x) whose derivative f ' (x) has the given description.

15. f '(x) > 0 for x > 3 and f '(x) < 0 for x < 3.

In Exercises 19–22, use the First Derivative Test to determine whether the function attains a local minimum or local maximum (or neither) at the given critical point.

19. y = 7 + 4x − x²,    c = 2

In Exercises 25–52, find the critical points and the intervals on which the function is increasing or decreasing, and apply the First Derivative Test to each critical point.

25. y = −x² + 7x − 17

53. Show that f(x) = x² + bx + c is decreasing on (−∞, −b/2) and increasing on (−b/2, ∞).

54. Show that f(x) = x³ − 2x² + 2x is an increasing function.

55. Find conditions on a and b that ensure that f(x) = x³ + ax + b is increasing on (−∞, ∞) .

57. Sam made two statements that Deborah found dubious.
(a) “Although the average velocity for my trip was 70 mph, at no point in time did my speedometer read 70 mph.”
(b) “Although a policeman clocked me going 70 mph, my speedometer never read 65 mph.”
In each case, which theorem did Deborah apply to prove Sam’s statement false: the Intermediate Value Theorem or the Mean Value Theorem? Explain.

58. Determine where f(x) = (1000 − x)² + x² is increasing. Use this to decide which is larger: 1000² or 998² + 2².

59. Show that f(x) = 1 − |x| satisfies the conclusion of the MVT on [a, b] if both a and b are positive or negative, but not if a < 0 and b > 0.

60. Which values of c satisfy the conclusion of the MVT on the interval [a, b] if  f(x) is a linear function?

61. Show that if f is a quadratic polynomial, then the midpoint c = (a + b)/2 satisfies the conclusion of the MVT on [a, b] for any a and b.

62.  Suppose that f(0) = 4 and  f ¢(x) ≤ 2 for x > 0. Apply the MVT to the interval [0, 3] to prove that f(3) ≤ 10. Prove more generally that f(x) ≤ 4 + 2x for all x > 0.

63. Suppose that f(2) = −2 and  f ¢( x ) ≥ 5. Show that f(4) ≥ 8.

64. Find the minimum value of f(x) = x^x for x > 0.

65. Show that the cubic function f(x) = x³ + ax² + bx + c is increasing on (−∞, ∞) if b > a²/3.

66. Prove that if f(0) = g(0) and f ' (x) ≤ g ' (x) for x ≥ 0, then f(x) ≤ g(x) for all x ≥ 0.

67. Use Exercise 66 to prove that sin x ≤ x for x ≥ 0.

Section 4.4

19. Sketch the graph of f(x) = x⁴ and state whether f has any points of inflection. Verify your conclusion by showing that f ²(x) does not change sign.

20. Through her website, Leticia has been selling bean bag chairs with monthly sales as recorded below. In a report to investors, she states, “Sales reached a point of inflection when I started using pay-per-click advertising.” In which month did that occur? Explain.

22. Figure 16 shows the graph of the derivative  f ¢(x) on [0, 1.2]. Locate the points of inflection of f(x) and the points where the local minima and maxima occur. Determine the intervals on which  f(x) has the following properties:
(a) Increasing       (b) Decreasing     (c) Concave up       (d) Concave down

In Exercises 23–36, find the critical points of f(x) and use the Second Derivative Test (if possible) to determine whether each corresponds to a local minimum or maximum.

23. f(x) = x³ − 12x² + 45x

In Exercises 37–50, find the intervals on which f is concave up or down, the points of inflection, and the critical points, and determine whether each critical point corresponds to a local minimum or maximum (or neither).

37. f(x) = x³ − 2x² + x

51. An infectious flu spreads slowly at the beginning of an epidemic. The infection process accelerates until a majority of the susceptible individuals are infected, at which point the process slows down.
(a) If R(t) is the number of individuals infected at time t, describe the concavity of the graph of R near the beginning and end of the epidemic.
(b) Write a one-sentence news bulletin describing the status of the epidemic on the day that R(t) has a point of inflection.

52. Water is pumped into a sphere at a constant rate (Figure 17). Let h(t) be the water level at time t. Sketch the graph of h(t) (approximately, but with the correct concavity). Where does the point of inflection occur?

53. Water is pumped into a sphere at a variable rate in such a way that the water level rises at a constant rate c (Figure 17). Let V(t) be the volume of water at time t. Sketch the graph of V(t) (approximately, but with the correct concavity). Where does the point of inflection occur?

Section 4.5

1. Find the dimensions of the rectangle of maximum area that can be formed from a 50-in. piece of wire.
(a) What is the constraint equation relating the lengths x and y of the sides?
(b) Find a formula for the area in terms of x alone.
(c) Does this problem require optimization over an open interval or a closed interval?
(d) Solve the optimization problem.

2. A 100-in. piece of wire is divided into two pieces and each piece is bent into a square. How should this be done in order to minimize the sum of the areas of the two squares?
(a) Express the sum of the areas of the squares in terms of the lengths x and y of the two pieces.
(b) What is the constraint equation relating x and y?
(c) Does this problem require optimization over an open or closed interval?
(d) Solve the optimization problem.

3. Find the positive number x such that the sum of x and its reciprocal is as small as possible. Does this problem require optimization over an open interval or a closed interval?

4. The legs of a right triangle have lengths a and b satisfying a + b = 10. Which values of a and b maximize the area of the triangle?

5. Find positive numbers x, y such that xy = 16 and x + y is as small as possible.

6. A 20-in. piece of wire is bent into an L-shape. Where should the bend be made to minimize the distance between the two ends?

7. Let S be the set of all rectangles with area 100.
(a) What  are  the  dimensions  of  the  rectangle  in  S  with  the  least perimeter?
(b) Is there a rectangle in S with the greatest perimeter? Explain.

8. A box has a square base of side x and height y.
(a) Find the dimensions x, y for which the volume is 12 and the surface area is as small as possible.
(b) Find the dimensions for which the surface area is 20 and the volume is as large as possible.

9. Suppose that 600 ft of fencing are used to enclose a corral in the shape of a rectangle with a semicircle whose diameter is a side of the rectangle as in Figure 10. Find the dimensions of the corral with maximum area.

10. Find the rectangle of maximum area that can be inscribed in a right triangle with legs of length 3 and 4 if the sides of the rectangle are parallel to the legs of the triangle, as in Figure 11.

11. A landscape architect wishes to enclose a rectangular garden on one side by a brick wall costing $30/ft and on the other three sides by a metal fence costing $10/ft. If the area of the garden is 1000 ft², find the dimensions of the garden that minimize the cost.

12. Find the point on the line y = x closest to the point (1, 0).

13. Find the point P on the parabola y = x² closest to the point (3, 0) (Figure 12).

15. A box is constructed out of two different types of metal. The metal for the top and bottom, which are both square, costs $1/ft² and the metal for the sides costs $2/ft². Find the dimensions that minimize cost if the box has a volume of 20 ft³.

16. Find the dimensions of the rectangle of maximum area that can be inscribed in a circle of radius r (Figure 14).

17. Problem of Tartaglia (1500–1557)    Among all positive numbers a , b whose sum is 8, find those for which the product of the two numbers and their difference is largest.

18. Find the angle θ that maximizes the area of the isosceles triangle whose legs have length l (Figure 15).

19. The volume of a right circular cone is and its surface area is . Find the dimensions of the cone with surface area 1 and maximal volume (Figure 16).

20. Rice production requires both labor and capital investment in equipment and land. Suppose that if x dollars per acre are invested in labor and y dollars per acre are invested in equipment and land, then the yield P of  rice  per acre is  given by the formula P =100√x + 150√y. If a farmer invests $40/ acre, how should he divide the $40 between labor and capital investment in order to maximize the amount of rice produced?

22. Find the dimensions x and y of the rectangle inscribed in a circle of radius r that maximizes the quantity xy².

23. Find the angle θ that maximizes the area of the trapezoid with a base of length 4 and sides of length 2, as in Figure 17.

24. Consider a rectangular industrial warehouse consisting of three separate spaces of equal size as in Figure 18. Assume that the wall materials cost $200 per linear ft and the company allocates $2,400,000 for the project.
(a) Which dimensions maximize the total area of the warehouse?
(b) What is the area of each compartment in this case?

25. Suppose, in the previous exercise, that the warehouse consists of n separate spaces of equal size. Find a formula in terms of n for the maximum possible area of the warehouse.

26. The amount of light reaching a point at a distance r from a light source A of intensity I_A is I_A /r². Suppose that a second light source B of intensity I_B = 4I_A  is located 10 ft from A. Find the point on the segment joining  A and B where the total amount of light is at a minimum.

27. Find the area of the largest rectangle that can be inscribed in the region bounded by the graph of y = (4 − x)/(2+ x) and the coordinate axes (Figure 19).

28. According to postal regulations, a carton is classified as “oversized” if the sum of its height and girth (the perimeter of its base) exceeds 108 in. Find the dimensions of a carton with square base that is not oversized and has maximum volume.

29. Find the maximum area of a triangle formed in the first quadrant by the x-axis,  y-axis, and a tangent line to the graph of  y = ( x + 1 )⁻².

30. What is the area of the largest rectangle that can be circumscribed around a rectangle of sides L and H? Hint: Express the area of the circumscribed rectangle in terms of the angle θ (Figure 20).

31. Optimal Price    Let r be the monthly rent per unit in an apartment building with 100 units. A survey reveals that all units can be rented when r = $900 and that one unit becomes vacant with each $10 increase in rent. Suppose that the average monthly maintenance per occupied unit is $100 / month.
(a) Show that the number of units rented is n = 190 − r/10 for 900 ≤ r ≤ 1900.
(b) Find a formula for the net cash intake (revenue minus maintenance) and determine the rent r that maximizes intake.

32. An 8-billion-bushel corn crop brings a price of $2.40/bushel. A commodity broker uses the following rule of thumb: If the crop is reduced by x percent, then the price increases by 10x cents. Which crop size results in maximum revenue and what is the price per bushel?

34. Let P = (a, b) be a point in the first quadrant.
(a) Find the slope of the line through P such that the triangle bounded by this line and the axes in the first quadrant has minimal area.
(b) Show that P is the midpoint of the hypotenuse of this triangle.

35. A truck gets 10 mpg (miles per gallon) traveling along an interstate highway at 50 mph, and this is reduced by 0.15 mpg for each mile per hour increase above 50 mph.
(a)  If the truck driver is paid $30 / hour and diesel fuel costs  P = $3/gal, which speed v between 50 and 70 mph will minimize the cost of a trip along the highway? Notice that the actual cost depends on the length of the trip but the optimal speed does not.
(b) Plot cost as a function of v (choose the length arbitrarily) and verify your answer to part (a).
(c) Do you expect the optimal speed v to increase or decrease if fuel costs go down to P = $2/gal? Plot the graphs of cost as a function of v for P = 2 and P = 3 on the same axis and verify your conclusion.

36. Figure 21 shows a rectangular plot of size 100 × 200 feet. Pipe is to be laid from A to a point P on side BC and from there to C. The cost of laying pipe through the lot is $30/ft (since it must be underground) and the cost along the side of the plot is $15/ft.
(a)  Let f(x) be the total cost, where x is the distance from P to B. Determine f(x), but note that f  is  continuous at x = 0 (when x = 0, the cost of the entire pipe is $15/ft).
(b) What is the most economical way to lay the pipe? What if the cost along the sides is $24/ft?

37. Find the dimensions of a cylinder of volume 1 m³ of minimal cost if the top and bottom are made of material that costs twice as much as the material for the side.

38. In Example 6 in this section, find the x-coordinate of the point P where the light beam strikes the mirror if h₁ = 10, h₂ = 5, and L = 20.

In Exercises 39–41, a box (with no top) is to be constructed from a piece of cardboard of sides A and B by cutting out squares of length h from the corners and folding up the sides (Figure 22).

39. Find the value of h that maximizes the volume of the box if A = 15 and B = 24. What are the dimensions of the resulting box?

40. Which value of h maximizes the volume if A = B?

41. Suppose  that  a  box  of  height  h = 3  in.  is constructed using 144 in.² of cardboard (i.e., AB = 144). Which values A and B maximize the volume?

42. The monthly output P of a light bulb factory is given by the formula P = 350LK, where L is the amount invested in labor and K the amount invested in equipment (in thousands of dollars). If the company needs to produce 10,000 units per month, how should the investment be divided among labor and equipment to minimize the cost of production? The cost of production is L + K.

43. Use calculus to show that among all right triangles with hypotenuse of length 1, the isosceles triangle has maximum area. Can you see more directly why this must be true by reasoning from Figure 23?

44. Janice can swim 3 mph and run 8 mph. She is standing at one bank of a river that is 300 ft wide and wants to reach a point located 200 ft downstream on the other side as quickly as possible. She will swim diagonally across the river and then jog along the river bank. Find the best route for Janice to take.

46. (a) Find the radius and height of a cylindrical can of total surface area A whose volume is as large as possible.
(b) Can you design a cylinder with total surface area A and minimal total volume?

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

48. A billboard of height b is mounted on the side of a building with its bottom edge at a distance h from the street. At what distance x should an observer stand from the wall to maximize the angle of observation θ (Figure 24)?
(a) Find x using calculus.
(b) Solve the problem again using geometry (without any calculation!). There is a unique circle passing through points B and C whichis tangent to the street. Let R be the point of tangency. Show that θ is maximized at the point R. Hint: The two angles labeled ψ are, in fact, equal because they subtend equal arcs on the circle. Let A be the intersection of the circle with PC and show that ψ = θ + PBA > θ.
(c) Prove that the two answers in (a) and (b) agree.

49. Use the result of Exercise 48 to show that θ is maximized at the value of x for which the angles ÐQPB and ÐQCP are equal.

52. A poster of area 6 ft² has blank margins of width 6 in. on the top and bottom and 4 in. on the sides. Find the dimensions that maximize the printed area.

54. Find the minimum length l of a beam that can clear a fence of height h and touch a wall located b ft behind the fence (Figure 27).

55. Let (a, b) be a fixed point in the first quadrant and let S(d) be the sum of the distances from (d, 0) to the points (0, 0) , (a, b) , and (a, −b).
(a) Find the value of d for which S ( d ) is minimal. The answer depends on whether b < √3a or b ≥√3a.
(b) Let a = 1. Plot S(d) for b = 0.5, √3, 3 and describe the position of the minimum. 

56. The minimum force required to drive a wedge of angle α into a block (Figure 28) is proportional to

where f is a positive constant. Find the angle α for which the least force is required, assuming f = 0.4.

 57. In the setting of Exercise 56, show that for any  f  the minimal force required is proportional to .

58. The problem is to put a “roof” of side s on an attic room of height h and width b. Find the smallest length s for which this is possible. See Figure 29.

59. Find the maximum length of a pole that can be carried horizontally around a corner joining corridors of widths 8 ft and 4 ft (Figure 30).

60. Redo Exercise 59 for corridors of arbitrary widths a and b.

61. Find the isosceles triangle of smallest area that circumscribes a circle of radius 1 (from Thomas Simpson’s The Doctrine and Application of Fluxions, a calculus text that appeared in 1750). See Figure 31.

Section 4.8

In Exercises 1–4, use Newton’s Method with the given function and initial value x₀ to calculate x₁, x₂, x₃.

1. f(x) = x² − 2,    x₀ = 1

5. Use Figure 6 to choose an initial guess x₀ to the unique real root of x³ + 2x + 5 = 0. Then compute the first three iterates of Newton’s Method.

6. Use Newton’s Method to find a solution to sin x = cos 2x in the interval [0, π/2] to three decimal places. Then guess the exact solution and compare with your approximation.

7. Use Newton’s Method to find the two solutions of e^x = 5x to three decimal places (Figure 7).

8. Use Newton’s Method to approximate the positive solution to the equation ln(x + 4) = x to three decimal places.

In Exercises 9–12, use Newton’s Method to approximate the root to three decimal places and compare with the value obtained from a calculator.

9. √10

13. Use Newton’s Method to approximate the largest positive root of f(x) = x⁴ − 6x² + x + 5 to within an error of at most 10⁻⁴. Refer to Figure 5.

14. Sketch the graph of f(x) = x³ − 4x + 1 and use Newton’s Method to approximate the largest positive root to within an error of at most 10⁻³.

15. Use a graphing calculator to choose an initial guess for the unique positive root of x⁴ + x² − 2x − 1 = 0. Calculate the first three iterates of Newton’s Method.

16. The first positive solution of sin x = 0 is x = π. Use Newton’s Method to calculate π to four decimal places.

Section 4.9

69. Show that f(x) = tan²x and g(x) = sec²x have the same derivative. What can you conclude about the relation between f and g? Verify this conclusion directly.

70. Show, by computing derivatives, that  for some constant C. Find C by setting x = 0.

71. A particle located at the origin at t = 0 begins moving along the x-axis with velocity  ft/s. Let s(t) be its position at time t. State the differential equation with initial condition satisfied by s(t) and find s(t).

72. Repeat Exercise 71, but replace the initial condition s(0) = 0 with s(2) = 3.

73. A particle moves along the x-axis with velocity v(t) = 25t −t² ft/s. Let s(t) be the position at time t.
(a) Find s(t), assuming that the particle is located at x = 5 at time t = 0.
(b) Find s(t), assuming that the particle is located at x = 5 at time t = 2.

74. A particle located at the origin at t = 0 moves in a straight line with acceleration  ft / s. Let v(t) be the velocity and s(t) the position at time t.
(a) State and solve the differential equation for v( t ) assuming that the particle is at rest at t = 0.
(b) Find s(t).

75. A car traveling 84 ft/s begins to decelerate at a constant rate of 14 ft/s². After how many seconds does the car come to a stop and how far will the car have traveled before stopping?

76. Beginning at rest, an object moves in a straight line with constant acceleration a, covering 100 ft in 5 s. Find a.

77. A 900-kg rocket is released from a spacecraft. As the rocket burns fuel, its mass decreases and its velocity increases. Let v( m ) be the velocity (in meters per second) as a function of mass m. Find the velocity when m = 500 if dv/dm = −50m^−1/2. Assume that v(900) = 0.

78. As water flows through a tube of radius R = 10 cm, the velocity of an individual water particle depends on its distance r from the center of the tube according to the formula dv/dr = −0.06r. Determine v(r), assuming that particles at the walls of the tube have zero velocity.

79. Find constants c₁ and c₂ such that F(x) = c₁x sin x + c₂ cos x is an antiderivative of f(x) = x cos x.

80. Find the general antiderivative of (2x + 9)¹⁰.

Section 5.1

1. An athlete runs with velocity 4 mph for half an hour, 6 mph for the next hour, and 5 mph for another half-hour. Compute the total distance traveled and indicate on a graph how this quantity can be interpreted as an area.

2. Figure 14 shows the velocity of an object over a 3-min interval. Determine the distance traveled over the intervals [0, 3] and [1, 2.5] (remember to convert from miles per hour to miles per minute).

3. A rainstorm hit Portland, Maine, in October 1996, resulting in record rainfall. The rainfall rate R(t) on October 21 is recorded, in inches per hour, in the following table, where t is the number of hours since midnight. Compute the total rainfall during this 24-hour period and indicate on a graph how this quantity can be interpreted as an area.

4. The velocity of an object is v(t) = 32t ft/s. Use Eq. (2) and geometry to find the distance traveled over the time intervals [0, 2] and [2, 5].

5. Compute R₆, L₆, and M₃ to estimate the distance traveled over [0, 3] if the velocity at half-second intervals is as follows:

6. Use the following table of values to estimate the area under the graph of f (x) over [0,1] by computing the average of R₅ and L₅.

7. Consider f (x) = 2x + 3 on [0, 3].
(a) Compute R₆ and L₆ over [0, 3].
(b) Find the error in these approximations by computing the area exactly using geometry.

Section 5.5

1. An airplane makes the 350-mile trip from Los Angeles to San Francisco in 1 hour. Assuming that the plane’s velocity at time t is v(t) mph, what is the value of the integral ?

2. A hot metal object is submerged in cold water. The rate at which the object cools (in degrees per minute) is a function f (t) of time. Which quantity is represented by the integral ?

3. Which of the following quantities would be naturally represented as derivatives and which as integrals?
(a) Velocity of a train
(b) Rainfall during a 6-month period
(c) Mileage per gallon of an automobile
(d) Increase in the population of Los Angeles from 1970 to 1990

4. Two airplanes take off at t = 0 from the same place and in the same direction. Their velocities are v₁(t) and v₂(t), respectively. What is the physical interpretation of the area between the graphs of v₁(t) and v₂(t) over an interval [0, T ]? 

1. Water flows into an empty reservoir at a rate of 3,000 + 5t gal/hour. What is the quantity of water in the reservoir after 5 hours?

2. Find the displacement of a particle moving in a straight line with velocity v(t) = 4t − 3 ft/s over the time interval [2, 5].

3. A population of insects increases at a rate of 200 + 10t + 0.25t² insects per day. Find the insect population after 3 days, assuming that there are 35 insects at t = 0. 

4. A survey shows that a mayoral candidate is gaining votes at a rate of 2,000t + 1,000 votes per day, where t is the number of days since she announced her candidacy. How many supporters will the candidate have after 60 days, assuming that she had no supporters at t = 0?

5. A factory produces bicycles at a rate of 95 + 0.1t² − t bicycles per week (t in weeks). How many bicycles were produced from day 8 to 21?

6. Find the displacement over the time interval [1, 6] of a helicopter whose (vertical) velocity at time t is v(t) = 0.02t² + t ft/s.

7. A cat falls from a tree (with zero initial velocity) at time t = 0. How far does the cat fall between t = 0.5 and t = 1 s? Use Galileo’s formula v(t) = −32t ft/s.

8. A projectile is released with initial (vertical) velocity 100 m/s. Use the formula v(t) = 100 − 9.8t for velocity to determine the distance traveled during the first 15 s.

In Exercises 9–12, assume that a particle moves in a straight line with given velocity. Find the total displacement and total distance traveled over the time interval, and draw a motion diagram like Figure 3 (with distance and time labels).

9. 12 − 4t ft/s, [0, 5]

13. The rate (in liters per minute) at which water drains from a tank is recorded at half-minute intervals. Use the average of the left- and right-endpoint approximations to estimate the total amount of water drained during the first 3 min.

14. The velocity of a car is recorded at half-second intervals (in feet per second). Use the average of the left- and right-endpoint approximations to estimate the total distance traveled during the first 4 s. 

15. Let a(t) be the acceleration of an object in linear motion at time t. Explain why  is the net change in velocity over [t₁, t₂]. Find the net change in velocity over [1, 6] if a(t) = 24t − 3t² ft/s².

16. Show that if acceleration a is constant, then the change in velocity is proportional to the length of the time interval.

17. The traffic flow rate past a certain point on a highway is q(t) = 3,000 + 2,000t − 300t², where t is in hours and t = 0 is 8 AM. How many cars pass by during the time interval from 8 to 10 AM?

18. Suppose that the marginal cost of producing x video recorders is 0.001x² −0.6x + 350 dollars. What is the cost of producing 300 units if the setup cost is $20,000 (see Example 4)? If production is set at 300 units, what is the cost of producing 20 additional units?

19. Carbon Tax To encourage manufacturers to reduce pollution, a carbon tax on each ton of CO₂ released into the atmosphere has been proposed. To model the effects of such a tax, policymakers study the marginal cost of abatement B(x), defined as the cost of increasing CO₂ reduction from x to x + 1 tons (in units of ten thousand tons—Figure 4). Which quantity is represented by ?

20. Power is the rate of energy consumption per unit time. A megawatt of power is 10⁶ W or 3.6 × 10⁹ J/hour. Figure 5 shows the power supplied by the California power grid over a typical 1-day period. Which quantity is represented by the area under the graph?

21. Figure 6 shows the migration rate M(t) of Ireland during the period 1988–1998. This is the rate at which people (in thousands per year) move in or out of the country.

(a) What does  represent?
(b) Did migration over the 11-year period 1988–1998 result in a net influx or outflow of people from Ireland? Base your answer on a rough estimate of the positive and negative areas involved.
(c) During which year could the Irish prime minister announce, “We are still losing population but we’ve hit an inflection point—the trend is now improving.”

22. Figure 7 shows the graph of Q(t), the rate of retail truck sales in the United States (in thousands sold per year).
(a) What does the area under the graph over the interval [1995, 1997] represent?
(b) Express the total number of trucks sold in the period 1994–1997 as an integral (but do not compute it).
(c) Use the following data to compute the average of the right- and left-endpoint approximations as an estimate for the total number of trucks sold during the 2-year period 1995–1996.

23. Heat Capacity The heat capacity C(T ) of a substance is the amount of energy (in joules) required to raise the temperature of 1 g by 1◦C at temperature T .
(a) Explain why the energy required to raise the temperature from T₁ to T₂ is the area under the graph of C(T ) over [T₁, T₂].

(b) How much energy is required to raise the temperature from 50 to 100◦C if C(T ) = 6 + 0.2√T ?

In Exercises 24 and 25, consider the following. Paleobiologists have studied the extinction of marine animal families during the phanerozoic period, which began 544 million years ago. A recent study suggests that the extinction rate r(t) may be modeled by the function r (t) = 3,130/(t + 262) for 0 ≤ t ≤ 544. Here, t is time elapsed (in millions of years) since the beginning of the phanerozoic period. Thus, t = 544 refers to the present time, t = 540 is 4 million years ago, etc.

24. Use R_N or L_N with N = 10 (or their average) to estimate the total number of families that became extinct in the periods 100 ≤ t ≤ 150 and 350 ≤ t ≤ 400.

25. Estimate the total number of extinct families from t = 0 to the present, using M_N with N = 544.

26. Cardiac output is the rate R of volume of blood pumped by the heart per unit time (in liters per minute). Doctors measure R by injecting A mg of dye into a vein leading into the heart at t = 0 and recording the concentration c(t) of dye (in milligrams per liter) pumped out at short regular time intervals (Figure 8).

(a) The quantity of dye pumped out in a small time interval [t, t + Δt] is approximately Rc(t) Δt. Explain why.

(b) Show that , where T is large enough that all of the dye is pumped through the heart but not so large that the dye returns by recirculation.
(c) Use the following data to estimate R, assuming that A = 5 mg:

27. A particle located at the origin at t = 0 moves along the x-axis with velocity v(t) = (t + 1)⁻². Show that the particle will never pass the point x = 1.

28. A particle located at the origin at t = 0 moves along the x-axis with velocity v(t) = (t + 1)^−1/2. Will the particle be at the point x = 1 at any time t? If so, find t.

Section 5.8

1. Two quantities increase exponentially with growth constants k = 1.2 and k = 3.4, respectively. Which quantity doubles more rapidly?

2. If you are given both the doubling time and the growth constant of a quantity that increases exponentially, can you determine the initial amount?

3. A cell population grows exponentially beginning with one cell. Does it take less time for the population to increase from one to two cells than from 10 million to 20 million cells?

4. Referring to his popular book A Brief History of Time, the renowned physicist Stephen Hawking said, “Someone told me that each equation I included in the book would halve its sales.” If this is so, write a differential equation satisfied by the sales function S(n), where n is the number of equations in the book.

5. Carbon dating is based on the assumption that the ratio R of C₁₄ to C₁₂ in the atmosphere has been constant over the past 50,000 years. If R were actually smaller in the past than it is today, would the age estimates produced by carbon dating be too ancient or too recent?

6. Which is preferable: an interest rate of 12% compounded quarterly, or an interest rate of 11% compounded continuously?

7. Find the yearly multiplier if r = 9% and interest is compounded (a) continuously and (b) quarterly.

8. The PV of N dollars received at time T is (choose the correct answer):
(a) The value at time T of N dollars invested today
(b) The amount you would have to invest today in order to receive N dollars at time T

9. A year from now, $1 will be received. Will its PV increase or decrease if the interest rate goes up?

10. Xavier expects to receive a check for $1,000 1 year from today. Explain, using the concept of PV, whether he will be happy or sad to learn that the interest rate has just increased from 6% to 7%.

1. A certain bacteria population P obeys the exponential growth law P(t) = 2,000e^1.3t (t in hours).
(a) How many bacteria are present initially?
(b) At what time will there be 10,000 bacteria? 

2. A quantity P obeys the exponential growth law P(t) = e^5t (t in years).
(a) At what time t is P = 10?
(b) At what time t is P = 20?
(c) What is the doubling time for P?

3. A certain RNA molecule replicates every 3 minutes. Find the differential equation for the number N(t) of molecules present at time t (in minutes). Starting with one molecule, how many will be present after 10 min?

4. A quantity P obeys the exponential growth law P(t) = Ce^kt (t in years). Find the formula for P(t), assuming that the doubling time is 7 years and P(0) = 100.

5. The decay constant of Cobalt-60 is 0.13 years⁻¹. What is its halflife?

6. Find the decay constant of Radium-226, given that its half-life is 1,622 years.

7. Find all solutions to the differential equation y ' = −5y. Which solution satisfies the initial condition y(0) = 3.4?

8. Find the solution to y ' = √2y satisfying y(0) = 20.

9. Find the solution to y '  = 3y satisfying y(2) = 4.

10. Find the function y = f (t) that satisfies the differential equation y '  = −0.7y and initial condition y(0) = 10.

11. The population of a city is P(t) = 2 · e^0.06t (in millions), where t is measured in years.
(a) Calculate the doubling time of the population.
(b) How long does it take for the population to triple in size?
(c) How long does it take for the population to quadruple in size?

12. The population of Washington state increased from 4.86 million in 1990 to 5.89 million in 2000. Assuming exponential growth,
(a) What will the population be in 2010?
(b) What is the doubling time?

13. Assuming that population growth is approximately exponential, which of the two sets of data is most likely to represent the population (in millions) of a city over a 5-year period?

14. Light Intensity The intensity of light passing through an absorbing medium decreases exponentially with the distance traveled. Suppose the decay constant for a certain plastic block is k = 2 when the distance is measured in feet. How thick must the block be to reduce the intensity by a factor of one-third?

15. The Beer–Lambert Law is used in spectroscopy to determine the molar absorptivity α or the concentration c of a compound dissolved in a solution at low concentrations (Figure 12). The law states that the intensity I of light as it passes through the solution satisfies ln(I /I₀) = αcx, where I₀ is the initial intensity and x is the distance traveled by the light. Show that I satisfies a differential equation dI/dx = −kx for some constant k.

16. An insect population triples in size after 5 months. Assuming exponential growth, when will it quadruple in size?

17. A 10-kg quantity of a radioactive isotope decays to 3 kg after 17 years. Find the decay constant of the isotope.

18. Measurements showed that a sample of sheepskin parchment discovered by archaeologists had a C¹⁴ to C¹² ratio equal to 40% of that found in the atmosphere. Approximately how old is the parchment?

20. A paleontologist has discovered the remains of animals that appear to have died at the onset of the Holocene ice age. She applies carbon dating to test her theory that the Holocene age started between 10,000 and 12,000 years ago. What range of C¹⁴ to C¹² ratio would she expect to find in the animal remains?

21. Atmospheric Pressure The atmospheric pressure P(h) (in pounds per square inch) at a height h (in miles) above sea level on earth satisfies a differential equation P ' = −kP for some positive constant k.

(a) Measurements with a barometer show that P(0) = 14.7 and P(10) = 2.13. What is the decay constant k?
(b) Determine the atmospheric pressure 15 miles above sea level.

22. Inversion of Sugar When cane sugar is dissolved in water, it converts to invert sugar over a period of several hours. The percentage f (t) of unconverted cane sugar at time t decreases exponentially. Suppose that f ' = −0.2f. What percentage of cane sugar remains after 5 hours? After 10 hours?

23. A quantity P increases exponentially with doubling time 6 hours. After how many hours has P increased by 50%?

24. Two bacteria colonies are cultivated in a laboratory. The first colony has a doubling time of 2 hours and the second a doubling time of 3 hours. Initially, the first colony contains 1,000 bacteria and the second colony 3,000 bacteria. At what time t will sizes of the colonies be equal?

25. Moore’s Law In 1965, Gordon Moore predicted that the number N of transistors on a microchip would increase exponentially.
(a) Does the table of data below confirm Moore’s prediction for the period from 1971 to 2000? If so, estimate the growth constant k.
(b) Plot the data in the table.
(c) Let N(t) be the number of transistors t years after 1971. Find an approximate formula N(t) ≈ Ce^kt, where t is the number of years after 1971.
(d) Estimate the doubling time in Moore’s Law for the period from 1971 to 2000.
(e) If Moore’s Law continues to hold until the end of the decade, how many transistors will a chip contain in 2010?
(f) Can Moore have expected his prediction to hold indefinitely?

26. Assume that in a certain country, the rate at which jobs are created is proportional to the number of people who already have jobs. If there are 15 million jobs at t = 0 and 15.1 million jobs 3 months later, how many jobs will there be after two years?

28. To model mortality in a population of 200 laboratory rats, a scientist assumes that the number P(t) of rats alive at time t (in months) satisfies the Gompertz equation with M = 204 and k = 0.15 months⁻¹ (Figure 13). Find P(t) [note that P(0) = 200] and determine the population after 20 months.

29. A certain quantity increases quadratically: P(t) = P₀t².
(a) Starting at time t₀ = 1, how long will it take for P to double in size? How long will it take starting at t₀ = 2 or 3?
(b) In general, starting at time t₀, how long will it take for P to double in size?

30. Verify that the half-life of a quantity that decays exponentially with decay constant k is equal to ln 2/k. 

31. Compute the balance after 10 years if $2,000 is deposited in an account paying 9% interest and interest is compounded (a) quarterly, (b) monthly, and (c) continuously. 

32. Suppose $500 is deposited into an account paying interest at a rate of 7%, continuously compounded. Find a formula for the value of the account at time t. What is the value of the account after 3 years? 

33. A bank pays interest at a rate of 5%. What is the yearly multiplier if interest is compounded
(a) yearly? (b) three times a year? (c) continuously?

34. How long will it take for $4,000 to double in value if it is deposited in an account bearing 7% interest, continuously compounded?

35. Show that if interest is compounded continuously at a rate r, then an account doubles after (ln 2)/r years.

36. How much must be invested today in order to receive $20,000 after 5 years if interest is compounded continuously at the rate r = 9%?

37. An investment increases in value at a continuously compounded rate of 9%. How large must the initial investment be in order to build up a value of $50,000 over a seven-year period?

38. Compute the PV of $5,000 received in 3 years if the interest rate is (a) 6% and (b) 11%. What is the PV in these two cases if the sum is instead received in 5 years?

39. Is it better to receive $1,000 today or $1,300 in 4 years? Consider r = 0.08 and r = 0.03.

40. Find the interest rate r if the PV of $8,000 to be received in 1 year is $7,300.

41. If a company invests $2 million to upgrade its factory, it will earn additional profits of $500,000/year for 5 years. Is the investment worthwhile, assuming an interest rate of 6% (assume that the savings are received as a lump sum at the end of each year)?

42. A new computer system costing $25,000 will reduce labor costs by $7,000/year for 5 years.
(a) Is it a good investment if r = 8%?
(b) How much money will the company actually save?

43. After winning $25 million in the state lottery, Jessica learns that she will receive five yearly payments of $5 million beginning immediately.
(a) What is the PV of Jessica’s prize if r = 6%?
(b) How much more would the prize be worth if the entire amount were paid today?

44. An investment group purchased an office building in 1998 for $17 million and sold it 5 years later for $26 million. Calculate the annual (continuously compounded) rate of return on this investment.

45. Use Eq. (3) to compute the PV of an income stream paying out R(t) = $5,000/year continuously for 10 years and r = 0.05. 

46. Compute the PV of an income stream if income is paid out continuously at a rate R(t) = $5,000e^0.1t /year for 5 years and r = 0.05. 

47. Find the PV of an investment that produces income continuously at a rate of $800/year for 5 years, assuming an interest rate of r = 0.08.

48. The rate of yearly income generated by a commercial property is $50,000/year at t = 0 and increases at a continuously compounded rate of 5%. Find the PV of the income generated in the first four years if r = 8%.

49. Show that the PV of an investment that pays out R dollars/year continuously for T years is R(1 − e^−rT )/r, where r is the interest rate.

50. Explain this statement: If T is very large, then the PV of the income stream described in Exercise 49 is approximately R/r .

51. Suppose that r = 0.06. Use the result of Exercise 50 to estimate the payout rate R needed to produce an income stream whose PV is $20,000, assuming that the stream continues for a large number of years.

53. Use Eq. (6) to compute the PV of an investment that pays out income continuously at a rate R(t) = (5,000 + 1,000t)e^0.02t dollars/year for 10 years and r = 0.08.

54. Banker’s Rule of 70 Bankers have a rule of thumb that if you receive R percent interest, continuously compounded, then your money doubles after approximately 70/R years. For example, at R = 5%, your money doubles after 70/5 or 14 years. Use the concept of doubling time to justify the Banker’s Rule.

55. Isotopes for Dating Which of the following isotopes would be most suitable for dating extremely old rocks: Carbon-14 (half-life 5,570 years), Lead-210 (half-life 22.26 years), and Potassium-49 (half-life 1.3 billion years)? Explain why.

56. Let P = P(t) be a quantity that obeys an exponential growth law with growth constant k. Show that P increases m-fold after an interval of (lnm)/k years.

57. Average Time of Decay Physicists use the radioactive decay law R = R₀e^−kt to compute the average or mean time M until an atom decays. Let F(t) = R/R₀ = e^−kt be the fraction of atoms that have survived to time t without decaying.
(a) Find the inverse function t (F).
(b) The error in the following approximation tends to zero as N →∞:

39. On a typical day, a city consumes water at the rate of r (t) = 100 + 72t − 3t² (in thousands of gallons per hour), where t is the number of hours past midnight. What is the daily water consumption? How much water is consumed between 6 PM and midnight?

40. The learning curve for producing bicycles in a certain factory is L(x) = 12x^−1/5 (in hours per bicycle), which means that it takes a bike mechanic L(n) hours to assemble the nth bicycle. If 24 bicycles are produced, how long does it take to produce the second batch of 12?

41. Cost engineers at NASA have the task of projecting the cost P of major space projects. It has been found that the cost C of developing a projection increases with P at the rate dC/dP ≈ 21P^−0.65, where C is in thousands of dollars and P in millions of dollars. What is the cost of developing a projection for a project whose cost turns out to be P = $35 million?

42. The cost of jet fuel increased dramatically in 2005. Figure 6 displays Department of Transportation estimates for the rate of percentage price increase R(t) (in units of percentage per year) during the first 6 months of the year. Express the total percentage price increase I during the first 6 months as an integral and calculate I . When determining the limits of integration, keep in mind that t is in years since R(t) is a yearly rate.

Section 6.1

1. Find the area of the region between y = 3x2 +12 and y = 4x +4 over [−3, 3] (Figure 8).

2. Compute the area of the region in Figure 9(A), which lies between y = 2 − x² and y = −2 over [−2, 2].

3. Let f (x) = x and g(x) = 2 − x2 [Figure 9(B)].
(a) Find the points of intersection of the graphs.
(b) Find the area enclosed by the graphs of f and g.

4. Let f (x) = 8x − 10 and g(x) = x² − 4x + 10.
(a) Find the points of intersection of the graphs.
(b) Compute the area of the region below the graph of f and above the graph of g.

19. Find the area of the region enclosed by the curves y = x³ − 6x and y = 8 − 3x².

20. Find the area of the region enclosed by the semicubical parabola y² = x³ and the line x = 1.

23. Find the area of the region lying to the right of x = y² + 4y − 22 and the left of x = 3y + 8.

24. Find the area of the region lying to the right of x = y² − 5 and the left of x = 3 − y².

25. Calculate the area enclosed by x = 9− y² and x = 5 in two ways: as an integral along the y-axis and as an integral along the x-axis.

26. Figure 15 shows the graphs of x = y³ − 26y + 10 and x = 40 − 6y² − y³. Match the equations with the curve and compute the area of the shaded region.

50. Find the area enclosed by the curves y = c − x² and y = x² − c as a function of c. Find the value of c for which this area is equal to 1.

57. Find the line y = mx that divides the area under the curve y = x(1 − x) over [0, 1] into two regions of equal area.

58. Let c be the number such that the area under y = sin x over [0,π] is divided in half by the line y = cx (Figure 18). Find an equation for c and solve this equation numerically using a computer algebra system.

Section 6.2

1. What is the average value of f (x) on [1, 4] if the area between the graph of f (x) and the x-axis is equal to 9?

2. Find the volume of a solid extending from y = 2 to y = 5 if the cross section at y has area A(y) = 5 for all y.

1. Let V be the volume of a pyramid of height 20 whose base is a square of side 8.
(a) Use similar triangles as in Example 1 to find the area of the horizontal cross section at a height y.
(b) Calculate V by integrating the cross-sectional area.

2. Let V be the volume of a right circular cone of height 10 whose base is a circle of radius 4 (Figure 16).
(a) Use similar triangles to find the area of a horizontal cross section at a height y.
(b) Calculate V by integrating the cross-sectional area.

3. Use the method of Exercise 2 to find the formula for the volume of a right circular cone of height h whose base is a circle of radius r (Figure 16). 

4. Calculate the volume of the ramp in Figure 17 in three ways by integrating the area of the cross sections:
(a) Perpendicular to the x-axis (rectangles)
(b) Perpendicular to the y-axis (triangles)
(c) Perpendicular to the z-axis (rectangles)

5. Find the volume of liquid needed to fill a sphere of radius R to height h (Figure 18).

6. Find the volume of the wedge in Figure 19(A) by integrating the area of vertical cross sections.

7. Derive a formula for the volume of the wedge in Figure 19(B) in terms of the constants a, b, and c.

8. Let B be the solid whose base is the unit circle x² + y² = 1 and whose vertical cross sections perpendicular to the x-axis are equilateral triangles. Show that the vertical cross sections have area  and compute the volume of B.

In Exercises 9–14, find the volume of the solid with given base and cross sections.

9. The base is the unit circle x² + y² = 1 and the cross sections perpendicular to the x-axis are triangles whose height and base are equal.

10. The base is the triangle enclosed by x + y = 1, the x-axis, and the y-axis. The cross sections perpendicular to the y-axis are semicircles.

11. The base is the semicircle , where −3 ≤ x ≤ 3. The cross sections perpendicular to the x-axis are squares.

12. The base is a square, one of whose sides is the interval [0,l] along the x-axis. The cross sections perpendicular to the x-axis are rectangles of height f (x) = x².

13. The base is the region enclosed by y = x² and y = 3. The cross sections perpendicular to the y-axis are squares.

14. The base is the region enclosed by y = x² and y = 3. The cross sections perpendicular to the y-axis are rectangles of height y³.

15. Find the volume of the solid whose base is the region |x| +|y| ≤ 1 and whose vertical cross sections perpendicular to the y-axis are semicircles (with diameter along the base).

16. Show that the volume of a pyramid of height h whose base is an equilateral triangle of side s is equal to.

17. Find the volume V of a regular tetrahedron whose face is an equilateral triangle of side s (Figure 20).

18. The area of an ellipse is π ab, where a and b are the lengths of the semimajor and semiminor axes (Figure 21). Compute the volume of a cone of height 12 whose base is an ellipse with semimajor axis a = 6 and semiminor axis b = 4.

20. A plane inclined at an angle of 45◦ passes through a diameter of the base of a cylinder of radius r. Find the volume of the region within the cylinder and below the plane (Figure 23).

21. Figure 24 shows the solid S obtained by intersecting two cylinders of radius r whose axes are perpendicular.
(a) The horizontal cross section of each cylinder at distance y from the central axis is a rectangular strip. Find the strip’s width.
(b) Find the area of the horizontal cross section of S at distance y.
(c) Find the volume of S as a function of r.

22. Let S be the solid obtained by intersecting two cylinders of radius r whose axes intersect at an angle θ . Find the volume of S as a function of r and θ.

23. Calculate the volume of a cylinder inclined at an angle θ = 30◦ whose height is 10 and whose base is a circle of radius 4 (Figure 25).

24. Find the total mass of a 1-m rod whose linear density function is ρ(x) = 10(x + 1)⁻² kg/m for 0 ≤ x ≤ 1.

25. Find the total mass of a 2-m rod whose linear density function is ρ(x) = 1 + 0.5sin(πx) kg/m for 0 ≤ x ≤ 2.

26. A mineral deposit along a strip of length 6 cm has density s(x) = 0.01x(6 − x) g/cm for 0 ≤ x ≤ 6. Calculate the total mass of the deposit.

27. Calculate the population within a 10-mile radius of the city center if the radial population density is ρ(r) = 4(1 + r²)^1/3 (in thousands per square mile).

28. Odzala National Park in the Congo has a high density of gorillas. Suppose that the radial population density is ρ(r) = 52(1 + r²)⁻² gorillas per square kilometer, where r is the distance from a large grassy clearing with a source of food and water. Calculate the number of gorillas within a 5-km radius of the clearing.

9. Table 1 lists the population density (in people per squared kilometer) as a function of distance r (in kilometers) from the center of a rural town. Estimate the total population within a 2-km radius of the center by taking the average of the left- and right-endpoint approximations.

10. Find the total mass of a circular plate of radius 20 cm whose mass density is the radial function ρ(r) = 0.03 + 0.01cos (πr²) g/cm².

31. The density of deer in a forest  is  the  radial  function ρ(r) = 150(r² + 2)⁻² deer per km², where r  is the distance  (in kilometers) to a small meadow. Calculate the number of deer in the region 2 ≤ r ≤ 5 km.

32. Show that a circular plate of radius 2 cm with radial mass density ρ(r) = 4/r g/cm has finite total mass, even though the density becomes infinite at the origin.

33. Find the flow rate through a tube of radius 4 cm, assuming that the velocity of fluid particles at a distance r  cm from the center is v(r) = 16 − r² cm/s.

34. Let v(r) be the velocity of blood in an arterial capillary of radius R = 4 × 10⁻⁵ m. Use Poiseuille’s Law (Example 6) with k = 10⁶ (m-s)⁻¹to determine the velocity at the center of the capillary and the flow rate (use correct units).

35. A solid rod of radius 1cm is placed in a pipe of radius 3cm so that their axes are aligned. Water flows through the pipe and around the rod. Find the flow rate if the velocity of the water is given by the radial function v(r) = 0.5(r − 1)(3 − r) cm/s.

36. To estimate the volume V of Lake Nogebow, the Minnesota Bureau of Fisheries created the depth contour map in Figure 26 and determined the area of the cross section of the lake at the depths recorded in the table below. EstimateV by taking the average of the right- and left-endpoint approximations to the integral of cross-sectional area.

53. The temperature T(t) at time t (in hours) in an art museum varies according to T(t) = 70 + 5cos(πt/12). Find the average over the time periods [0, 24] and [2, 6].

54. A ball is thrown in the air vertically from ground level with initial velocity 64 ft/s. Find the average height of the ball over the time interval extending from the time of the ball’s release to its return to ground level. Recall that the height at time t is h(t) = 64t − 16t².

5. What is the average area of the circles whose radii vary from 0 to 1?

6. An object with zero initial velocity accelerates at a constant rate of 10 m/s². Find its average velocity during the first 15 s.

57. The acceleration of a particle is a(t) = t − t³ m/s² for 0 ≤ t ≤ 1. Compute the average acceleration and average velocity over the time interval [0, 1], assuming that the particle’s initial velocity is zero.

58. Let M be the average value of  f(x) = x⁴ on [0, 3] . Find a value of c in [0, 3] such that  f(c) = M.

Section 6.4

29. Use both the Shell and Disk Methods to calculate the volume of the solid obtained by rotating the region under the graph of f(x) = 8 − x³ for 0 ≤ x ≤ 2 about:
(a) the x-axis                (b) the y-axis

46. Use the Shell Method to calculate the volume V of the “bead” formed by removing a cylinder of radius r from the center of a sphere of radius R (compare with Exercise 51 in Section 6.3).

50. The surface area of a sphere of radius r is 4πr². Use this to derive the formula for the volume V of a sphere of radius R in a new way.
(a) Show that the volume of a thin spherical shell of inner radius r and thickness Dx is approximately 4πr²Dx.
(b) Approximate V by decomposing the sphere of radius R into N thin spherical shells of thickness Dx = R/N.
(c) Show that the approximation is a Riemann sum which converges to an integral. Evaluate the integral.

Section 6.5

1. How much work is done raising a 4-kg mass to a height of 16 m above ground?

2. How much work is done raising a 4-lb mass to a height of 16 ft above ground?

In Exercises 3–6, compute the work (in joules) required to stretch or compress a spring as indicated, assuming that the spring constant is k = 150 kg/s².

3. Stretching from equilibrium to 12 cm past equilibrium

4. Compressing from equilibrium to 4 cm past equilibrium

5. Stretching from 5 to 15 cm past equilibrium

6. Compressing the spring 4 more cm when it is already compressed 5 cm

7. If 5 J of work are needed to stretch a spring 10 cm beyond equilibrium, how much work is required to stretch it 15 cm beyond equilibrium?

8. If 5 J of work are needed to stretch a spring 10 cm beyond equilibrium,  how  much  work  is  required  to  compress  it 5 cm beyond equilibrium?

9. If 10 ft-lb of work are needed to stretch a spring 1 ft beyond equilibrium, how far will the spring stretch if a 10-lb weight is attached to its end?

10. Show that the work required to stretch a spring from position a to position b is , where k is the spring constant. How do you interpret the negative work obtained when |b| < |a|?

In Exercises 11–14, calculate the work against gravity required to build the structure out of brick using the method of Examples 2 and 3. Assume that brick has density 80 lb/ft³.

11. A tower of height 20 ft and square base of side 10 ft

12. A cylindrical tower of height 20 ft and radius 10 ft

13. A 20-ft-high tower in the shape of a right circular cone with base of radius 4 ft

14. A structure in the shape of a hemisphere of radius 4 ft

15. Built around 2600 BCE, the Great Pyramid of Giza in Egypt is 485 ft high (due to erosion, its current height is slightly less) and has a square base of side 755.5 ft (Figure 6). Find the work needed to build the pyramid if the density of the stone is estimated at 125 lb/ft³.

In Exercises 16–20, calculate the work (in joules) required to pump all of the water out of the tank. Assume that the tank is full, distances are measured in meters, and the density of water is 1000 kg/m³.

16. The box in Figure 7; water exits from a small hole at the top.

17. The hemisphere in Figure 8; water exits from the spout as shown.

18. The conical tank in Figure 9; water exits through the spout as shown.

19. The horizontal cylinder in Figure 10; water exits from a small hole at the top.

20. The trough in Figure 11; water exits by pouring over the sides.

21. Find the work W required to empty the tank in Figure 7 if it is half full of water.

22. Assume the tank in Figure 7 is full of water and let W be the work required to pump out half of the water. Do you expect W to equal the work computed in Exercise 21? Explain and then compute W.

23. Find the work required to empty the tank in Figure 9 if it is half full of water.

24. Assume the tank in Figure 9 is full of water and find the work required to pump out half of the water.

26. How much work is done lifting a 25-ft chain over the side of a building (Figure 12)? Assume that the chain has a density of 4 lb/ft.

27. How much work is done lifting a 3-m chain over the side of a building if the chain has mass density 4 kg/m?

28. An 8-ft chain weighs 16 lb. Find the work required to lift the chain over the side of a building.

29. A 20-ft chain with mass density 3 lb/ft is initially coiled on the ground. How much work is performed in lifting the chain so that it is fully extended (and one end touches the ground)?

30. How much work is done lifting a 20-ft chain with mass density 3 lb/ft (initially coiled on the ground) so that its top end is 30 ft above the ground?

31. A  1,000-lb wrecking  ball  hangs  from a  30-ft  cable  of  density 10 lb/ft attached to a crane. Calculate the work done if the crane lifts the ball from ground level to 30 ft in the air by drawing in the cable.

In Exercises 32–34, use Newton’s Universal Law of Gravity, according to which the gravitational force between two objects of mass m and M  separated by a distance r  has magnitude GMm/r², where G = 6.67 × 10⁻¹¹ m³kg⁻¹s⁻¹. Although the Universal Law refers to point masses, Newton proved that it also holds for uniform spherical objects, where r is the distance between their centers.

32. Two spheres of mass  M  and m  are separated by a distance r₁. Show that the work required to increase the separation to a distance r₂ is equal to .

33. Use the result of Exercise 32 to calculate the work required to place a 2,000-kg satellite in an orbit 1,200 km above the surface of the earth. Assume that the earth is a sphere of mass M_e = 5.98 × 10²⁴ kg and radius r_e = 6.37 × 10⁶ m. Treat the satellite as a point mass.

34. Use the result of Exercise 32 to compute the work required to move a 1,500-kg satellite from an orbit 1,000 to 1,500 km above the surface of the earth.

35. Assume that the pressure P and volume V of the gas in a 30-in.  cylinder of  radius 3  in. with a  movable  piston  are  related  by PV^1.4 = k, where k  is a constant (Figure 13). When the cylinder is full, the gas pressure is 200 lb/in.².
(a) Calculate k.
(b) Calculate the force on the piston as a function of the length x of the column of gas (the force is PA, where A is the piston’s area).
(c) Calculate the work required to compress the gas column from 30 to 20 in.

36. A 20-ft chain with linear mass density ρ(x) = 0.02x(20 − x) lb/ft lies on the ground.
(a) How much work is done lifting the chain so that it is fully extended (and one end touches the ground)?
(b)  How much work is done lifting the chain so that its top end has a height of 30 ft?

38. A model train of mass 0.5 kg is placed at one end of a straight 3-m electric track. Assume that a force F(x) = 3x − x² N acts on the train at distance x along the track. Use the Work-Kinetic Energy Theorem (Exercise 37) to determine the velocity of the train when it reaches the end of the track.

39. With what initial velocity v₀ must we fire a rocket so it attains a maximum height r above the earth? Hint: Use the results of Exercises 32 and 37. As the rocket reaches its maximum height, its KE decreases from  to zero.

40. With what initial velocity must we fire a rocket so it attains a maximum height of r = 20 km above the surface of the earth?

Section 7.1

54. An  airplane’s velocity is recorded at 5-min intervals during a 1-hour period with the following results, in mph: 550, 575, 600, 580, 610, 640, 625, 595, 590, 620, 640, 640, 630
Use Simpson’s Rule to estimate the distance traveled during the hour.

55. Use Simpson’s Rule to determine the average temperature in a museum over a 3-hour period, if the temperatures (in degrees Celsius), recorded at 15-min intervals, are 21, 21.3, 21.5, 21.8, 21.6, 21.2, 20.8, 20.6, 20.9, 21.2, 21.1, 21.3, 21.2.

Section 7.4

54. Find the average height of a point on the semicircle for − 1 ≤ x ≤ 1.

55. Find the volume of the solid obtained by revolving the graph of over [0, 1] about the y-axis.

56. Find the volume of the solid obtained by revolving the region between the graph of y² − x² = 1 and the line y = 2 about the line y = 2.

57. Find the volume of revolution for the region in Exercise 56, but revolve around y = 3.

58. A charged wire creates an electric field at a point P located at a distance D from the wire (Figure 7). The component E_^ of the field perpendicular to the wire (in volts) is

Section 7.7

75. An investment pays a dividend of $250/year continuously forever. If the interest rate is 7%, what is the present value of the entire income stream generated by the investment?

76. An investment is expected to earn profits at a rate of 10000e^0.01t dollars/year forever. Find the present value of the income stream if the interest rate is 4%.

77. Compute the present value of an investment that generates income at a rate of 5000te^0.01t dollars/year forever, assuming an interest rate of 6%.

78. Find the volume of the solid obtained by rotating the region below the graph of y = e^−x about the x-axis for 0 ≤ x < ∞.

Section 8.1

1. Express the arc length of the curve y = x⁴ between x = 2 and x = 6 as an integral (but do not evaluate).

In Exercises 5–10, calculate the arc length over the given interval.

5. y = 3x + 1, [0, 3]

Section 8.2

1. A box of height 6 ft and square base of side 3 ft is submerged in a pool of water. The top of the box is 2 ft below the surface of the water.
(a) Calculate the fluid force on the top and bottom of the box.
(b) Write a Riemann sum that approximates the fluid force on a side of the box by dividing the side into N  horizontal strips of thickness Dy = 6/N.
(c) To which integral does the Riemann sum converge?
(d) Compute the fluid force on a side of the box.

2. A plate in the shape of an isosceles triangle with base 1 ft and height 2 ft is submerged vertically in a tank of water so that its vertex touches the surface of the water (Figure 7).
(a) Show that the width of the triangle at depth y is f(y) = y/2.
(b) Consider a thin strip of thickness Dy at depth y. Explain why the fluid force on a side of this strip is approximately equal to where w = 62.5 lb/ft³.
(c) Write an approximation for the total fluid force F on a side of the plate as a Riemann sum and indicate the integral to which it converges.
(d) Calculate F.

3. Repeat Exercise 2, but assume that the top of the triangle is located 3 ft below the surface of the water. 

4. The thin plate R in Figure 8, bounded by the parabola y = x² and y = 1, is submerged vertically in water. Let F be the fluid force on one side of R.

(a)  Show that the width of  R at height  y is  f(y) = 2√y and the fluid force on a side of a horizontal strip of thickness Dy at height y is approximately .
(b) Write a Riemann sum that approximates F and use it to explain why .
(c) Calculate F.

5. Let F be the fluid force (in Newtons) on a side of a semicircular plate of radius r meters, submerged in water so that its diameter is level with the water’s surface (Figure 9).

(a) Show that the width of the plate at depth y is .
(b) Calculate F using Eq. (2). 

6. Calculate the force on one side of a circular plate with radius 2 ft, submerged vertically in a tank of water so that the top of the circle is tangent to the water surface. 

7. A semicircular plate of radius r, oriented as in Figure 9, is submerged in water so that its diameter is located at a depth of m feet. Calculate the force on one side of the plate in terms of m and r. 

8. Figure 10 shows the wall of a dam on a water reservoir. Use the Trapezoidal Rule and the width and depth measurements in the figure to estimate the total force on the wall. 

9. Calculate the total force (in Newtons) on a side of the plate in Figure 11(A), submerged in water. 

10. Calculate the total force (in Newtons) on a side of the plate in Figure 11(B), submerged in a fluid of mass density ρ = 800 kg/m³. 

11. The plate in Figure 12 is submerged in water with its top level with the surface of the water. The left and right edges of the plate are the curves y = x^1/3 and y = − x^1/3. Find the fluid force on a side of the plate.

12. Let R be the plate in the shape of the region under y = sin x for 0 ≤ x ≤ π/2 in Figure 13(A). Find the fluid force on a side of R if it is rotated counterclockwise by 90◦ and submerged in a fluid of density 140 lb/ft³ with its top edge level with the surface of the fluid as in (B).

13. In the notation of Exercise 12, calculate the fluid force on a side of the plate R if it is oriented as in Figure 13(A). You may need to use Integration by Parts and trigonometric substitution.

14. Let A be the region under the graph of y = ln x for 1 ≤ x ≤ e (Figure 14). Calculate the fluid force on one side of a plate in the shape of region A if the water surface is at y = 1.

15. Calculate the fluid force on one side of the “infinite” plate B in Figure 14.

16. A square plate of side 3 m is submerged in water at an incline of 30◦ with the horizontal. Its top edge is located at the surface of the water. Calculate the fluid force (in Newtons) on one side of the plate.

17. Repeat Exercise 16, but assume that the top edge of the plate lies at a depth of 6 m.

18. Figure 15(A) shows a ramp inclined at 30◦ leading into a swimming pool. Calculate the fluid force on the ramp.

19. Calculate the fluid force on one side of the plate (an isosceles triangle) shown in Figure 15(B).

20. The trough in Figure 16 is filled with corn syrup, whose density is 90 lb/ft³. Calculate the force on the front side of the trough.

21. Calculate the fluid pressure on one of the slanted sides of the trough in Figure 16, filled with corn syrup as in Exercise 20.

22. Figure 17 shows an object whose face is an equilateral triangle with 5-ft sides. The object is 2 ft thick and is submerged in water with its vertex 3 ft below the water surface. Calculate the fluid force on both a triangular face and a slanted rectangular edge of the object.

23. The end of the trough in Figure 18 is an equilateral triangle of side 3. Assume that the trough is filled with water to height y. Calculate the fluid force on each side of the trough as a function of the level y and the length l of the trough.

24. A rectangular plate of side l is submerged vertically in a fluid of density w, with its top edge at depth h. Show that if the depth is increased by an amount Dh, then the force on a side of the plate increases by wADh, where A is the area of the plate.

25. Prove that the force on the side of a rectangular plate of area A submerged vertically in a fluid is equal to p₀A, where p₀ is the fluid pressure at the center point of the rectangle.

26. If the density of a fluid varies with depth, then the pressure at depth y is a function p ( y ) (which need not equal w y as in the case of constant density). Use Riemann sums to argue that the total force F on the flat side of a submerged object submerged vertically is , where f(y) is the width of the side at depth y.

Section 8.3

1. Four particles are located at points (1, 1), (1, 2), (4, 0), (3, 1)
(a) Find the moments M_x  and M_y  and the center of mass of the system, assuming that the particles have equal mass m.
(b) Find the center of mass of the system, assuming the particles have mass 3, 2, 5, and 7, respectively.

2. Find the center of mass for the system of particles of mass 4, 2, 5, 1 located at (1, 2), (−3, 2), (2, −1), (4, 0).

3. Point masses of equal size are placed at the vertices of the triangle with coordinates (a, 0), (b, 0), and (0, c). Show that the center of mass of the system of masses has coordinates .

4. Point masses of mass m₁, m₂, and m₃  are placed at the points (−1, 0), ( 3, 0), and (0, 4).
(a) Suppose that m₁ = 6. Show that there is a unique value of m² such that the center of mass lies on the y-axis.
(b) Suppose that m₁ = 6 and m₂ = 4. Find the value of m³ such that y_CM = 2. 

5. Sketch the lamina S of constant density ρ = 3 g/cm² occupying the region beneath the graph of y = x² for 0 ≤ x ≤ 3.
(a) Use formulas (1) and (2) to compute M_x and M_y.
(b) Find the area and the center of mass of S.

6. Use Eqs. (1) and (3) to find the moments and center of mass of the lamina S of constant density ρ = 2 g/cm² occupying the region between y = x² and y = 9x over [0, 3]. Sketch S, indicating the location of the center of mass.

7. Find the moments and center of mass of the lamina of uniform density ρ occupying the region underneath y = x³ for 0 ≤ x ≤ 2.

8. Calculate M_x (assuming ρ  = 1) for the region underneath the graph of y = 1 − x² for 0 ≤ x ≤ 1 in two ways, first using Eq. (2) and then using Eq. (3).

9. Let T be the triangular lamina in Figure 17.

(a) Show that the horizontal cut at height y has length and use Eq. (2) to compute M_x (with ρ = 1).
(b) Use the Symmetry Principle to show that M_y = 0 and find the center of mass.

In Exercises 10–17, find the centroid of the region lying underneath the graph of the function over the given interval.

10. f(x) = 6 − 2x, [0, 3] 

18. Calculate the moments and center of mass of the lamina occupying the region between the curves y = x and y = x² for 0 ≤ x ≤ 1. 

19. Sketch the region between y = x + 4 and y = 2 − x for 0 ≤ x ≤ 2. Using symmetry, explain why the centroid of the region lies on the line y = 3. Verify this by computing the moments and the centroid. 

In Exercises 34–36, use the additivity of moments to find the COM of the region.

34.  Isosceles triangle of height 2 on top of a rectangle of base 4 and height 3 (Figure 19)

35. An ice cream cone consisting of a semicircle on top of an equilateral triangle of side 6 (Figure 20)

Section 8.4

In Exercises 1–14, calculate the Taylor polynomials T₂(x) and T₃(x) centered at x = a for the given function and value of a.

1. f(x) = sin x,    a = 0

19. Show that the nth Maclaurin polynomial for  f(x) = e^x is

27. Plot  y = e^x together with the Maclaurin polynomials T_n(x) for n = 1, 3, 5 and then for n = 2, 4, 6 on the interval [−3, 3]. What difference do you notice between the even and odd Maclaurin polynomials?

29. Use the Error Bound to find the maximum possible size of |cos 0.3 − T₅(0.3)|, where T₅(x) is the Maclaurin polynomial. Verify your result with a calculator.

32. Let and let T_n(x) be the Taylor polynomial centered at a = 8.
(a) Find T₃(x) and calculate T₃(8.02).
(b) Use the Error Bound to find a bound for |T₃(8.02) − √9.02|. 

37. Find n such that |T_n(1.3) − ln(1.3)| ≤ 10⁻⁴, where T_n  is the Taylor polynomial for f(x) = ln x at a = 1.

42. Verify that the third Maclaurin polynomial for f(x) = e^xsin x is equal to the product of the third Maclaurin polynomials of e^x and sin x (after discarding terms of degree greater than 3 in the product).

43. Find the fourth Maclaurin polynomial for  f(x) = sin x cos x by multiplying the fourth Maclaurin polynomials for  f(x) = sin x and f(x) = cos x.

44. Find the Maclaurin polynomials T_n(x) for f(x) = cos (x²). You may use the fact that T_n(x) is equal to the sum of the terms up to degree n obtained by substituting x²for x in the nth Maclaurin polynomial of cos x.

Section 9.1

45. A cylindrical tank filled with water has height 10 ft and a base of area 30 ft². Water leaks through a hole in the bottom of area
1/3 ft². How long does it take (a) for half of the water to leak out and (b) for the tank to empty?

46. A conical tank filled with water has height 12 ft [Figure 7(A)]. Assume that the top is a circle of radius 4 ft and that water leaks through
a hole in the bottom of area 2 in². Let y ( t ) be the water level at time t.
(a) Show that the cross-sectional area of the tank at height y is A ( y ) = (π/9)y².
(b) Find the differential equation satisfied by y ( t ) and solve for y ( t ). Use the initial condition y(0) = 12.
(c) How long does it take for the tank to empty?

47. The tank in Figure 7(B) is a cylinder of radius 10 ft and length 40 ft. Assume that the tank is half-filled with water and that water leaks through a hole in the bottom of area B = 3 in². Determine the water level y(t) and the time t_e when the tank is empty.

 

Online homework help for college and high school students. Get homework help and answers to your toughest questions in math, algebra, trigonometry, precalculus, calculus, physics. Homework assignments with step-by-step solutions.