Can you please help me with the flowing problems. Thanks so much! :)

Section 3.4: 21, 27, 36, 39, 44

160 CHAPTER 3 DIFFERENTIATION Exercises 12 and 13 refer to the data in Example 1. Approximate the 20. At the start of the 27th century, the population of Zosania was an derivative with the symmetric difference quotient (SDQ) approximation: proximately 40 million. Early-century prosperity saw the population nearly T'(1) ~ (+ 20) - T(1 - 20) double in the first three decades, but the growth slowed in the 30s and 40s 40 and then leveled off completely during the war years in the 50s. A postwar 12. (a) At what ? does the SDQ approximation give the fastest rate of boom saw another rapid population increase, but that turned around in a increase of temperature? What is the rate of change? major decline resulting from the great famine of the 70s. c of the 70s. A slow end-of- century rebound resulted in an increase of the population to approximately b) At what ? does the SDQ approximation give the fastest rate of decrease 90 million at century's end. Let P represent the population in millions and of temperature? What is the rate of change? I represent time in years since the start of the century. Make sketches of 13. At what / does the SDQ approximation give the smallest (i.e., closest the graphs of P and P' as functions of : for Zosania's population during the century to 0) rate of change of temperature? What is the rate of change? 21. The velocity (in centimeters per second) of blood molecules flowing Exercises 14-16 refer to the four graphs of s as a function of t in through a capillary of radius 0.008 cm is v = 6.4 x 10-8 - 0.001r2. where Figure 7. r is the distance from the molecule to the center of the capillary. Find the 14. Sketch s' for each of the four graphs of s. rate of change of velocity with respect to r when r = 0.004 cm. 22. Figure 13 displays the voltage V across a capacitor as a function of 15. Match each situation with the graph that best represents it. time while the capacitor is being charged. Estimate the rate of change of (a) Rocky slowed down his car as it approached the moose in the road. voltage at 1 = 20 s. Indicate the values in your calculation and include The distance from the car to the moose is s and the time since he spotted proper units. Does voltage change more quickly or more slowly as time the moose is t. goes on? Explain in terms of tangent lines. (b) The rocket's speed increased after liftoff until the fuel was used up. The distance from the rocket to the launchpad is s and the time since liftoff V (V) is f. c) The increase in college costs slowed for the fourth year in a row. The cost of college is s and the time since the start of the 4-year period is t. 16. Match each situation with the graph that best represents it. (a) Dusty's batting average increased over the first 10 games of the sea- N son but from game to game the amount of increase went down. Dusty's batting average is s and the time since the beginning of the season is f. b) In performing a crash test, the car continued to speed up until it hit the 10 - 1 (s ) 20 30 40 wall. The distance between the car and the wall is s and the time since the car started moving is f. FIGURE 13 (c) The hurricane strengthened at an increasing rate over the first day of is development. The strength of the hurricane is s, and the time since it started developing is 1. 23. Use Figure 14 to estimate dT /dh at h = 30 and 70, where T is atmo- spheric temperature (in degrees Celsius) and h is altitude (in kilometers). 17. Sketch a graph of velocity as a function of time for the shuttle train in Where is dT /dh equal to zero? Example 6 T( C) 18. Figure 12 shows the height y of a mass oscillating at the end of a spring, through one cycle of the oscillation. Sketch the graph of velocity 250 as a function of time. 200 150 Stratosphere Troposphere Mesosphere Thermosphere 100 - A 50 0 -50 100 - h (km) 10 50 100 150 FIGURE 12 FIGURE 14 Atmospheric temperature versus altitude. 19. Fred X has to make a book delivery from his warehouse, 15 mi north of the city, to the Amazing Book Store 10 mi south of the city. Traffic is 24. The earth exerts a gravitational force of F(r) = (2.99 x 1016)/r) usually congested within 5 mi of the city. He leaves at noon, traveling due newtons on an object with a mass of 75 kg located r meters from the center south through the city, and arrives at the store at 12:50. After 15 min at the of the earth. Find the rate of change of force with respect to distance ra store. he makes the return trip north to his warehouse, arriving at 2:00. Let the surface of the earth. s represent the distance from the warehouse in miles and I represent time 25. For the escape velocity relationship, vesc = (2.82 x 107)r -1/Z mis. in minutes since noon. Make sketches of the graphs of s and s' as functions calculate the rate of change of the escape velocity with respect to distance of t for Fred's trip. r from the center of the earth.SECTION 3.4 Rates of Change 161 26. The power delivered by a battery to an apparatus of resistance R (in 40. The atmospheric CO2 level A() at Mauna Loa, Hawaii, at time ? ohms) is P = 2.25R/(R + 0.5)2 watts (W). Find the rate of change of (in parts per million by volume) is recorded by the Scripps Institution power with respect to resistance for R = 3 $ and R = 5 02. of Oceanography. Reading across, the annual values for the 4-year inter- 27. A particle moving along a line has position s(!) = /4 - 1812 m at vals are time / seconds. At which times does the particle pass through the origin? At which times is the particle instantaneously motionless (i.e., it has zero 1960 4 1968 1972 1976 1980 1984 velocity)? 316.91 319.20 323.05 327.45 332.15 338.69 344.42 28. GU Plot the position of the particle in Exercise 27. What is the far- 1988 1992 1996 2000 2004 2008 2012 thest distance to the left of the origin attained by the particle? 351.48 356.37 362.64 369.48 377.38 385.34 393.87 29. A projectile is launched in the air from the ground with an initial velocity up = 60 m/s. What is the maximum height that the projectile reaches? (Compare your result with Exercise 37 in Section 2.5, where we (a) Estimate A'() in 1962, 1970. 1978. 1986, 1994, 2002, and 2010. considered maximum height when air resistance is included and we inves- (b) In which of the years from (a) did the approximation to A'() take on tigated the result of letting the air resistance go to 0.) its largest and smallest values? 30. Find the velocity of an air conditioner accidentally dropped from a (c) In which of these years does the approximation suggest that the CO2 height of 300 m at the moment it hits the ground. level was the most constant? 31. A ball tossed in the air vertically from ground level returns to Earth 41. The tangent lines to the graph of f(x) = x grow steeper as x in- 4 s later. Find the initial velocity and maximum height of the ball. creases. At what rate do the slopes of the tangent lines increase? 32. Olivia is gazing out a window from the 10th floor of a building when a bucket (dropped by a window washer) passes by. She notes that it hits the 42. According to Kleiber's Law, the metabolic rate P (in kilocalories per ground 1.5 s later. Determine the floor from which the bucket was dropped day) and body mass m (in kilograms) of an animal are related by a three- if each floor is 5 m high and the window is in the middle of the 10th floor. quarter-power law P = 73.3m3/4. Estimate the increase in metabolic rate Neglect air friction. when body mass increases from 60 to 61 kg. 33. Show that for an object falling according to Galileo's formula, the av- 43. The dollar cost of producing x bagels is given by the function erage velocity over any time interval [(1, (2] is equal to the average of the C(x) = 300 + 0.25x - 0.5(x/1000). Determine the cost of producing instantaneous velocities at f, and f2. 2000 bagels and estimate the cost of the 2001st bagel. Compare your esti- 34. An object falls under the influence of gravity near the earth's mate with the actual cost of the 2001st bagel. surface. Which of the following statements is true? Explain. 44. Suppose that for x 2 1000. the dollar cost of producing x video cam- (a) Distance traveled increases by equal amounts in equal time intervals. eras is C(x) = 500x - 0.003x2 + 10-8x'. (b) Velocity increases by equal amounts in equal time intervals. (a) Estimate the marginal cost at production level x = 5000 and compare (c) The derivative of velocity increases with time. it with the actual cost C(5001) - C(5000). 35. By Faraday's Law, if a conducting wire of length & meters moves at (b) Compare the marginal cost at x = 5000 with the average cost per cam- velocity u m/s perpendicular to a magnetic field of strength B (in teslas), a era, defined as C(x)/x. voltage of size V = - Blu is induced in the wire. Assume that B = 2 and ( = 0.5. 45. According to Stevens's Law in psychology, the perceived mag- (a) Calculate d V /du. nitude of a stimulus is proportional (approximately) to a power of the (b) Find the rate of change of V with respect to time : if v(1) = 4 + 9. actual intensity / of the stimulus. Experiments show that the perceived 36. The voltage V, current /, and resistance R in a circuit are related by brightness B of a light satisfies B = 12/3, where I is the light intensity. Ohm's Law: V = / R, where the units are volts, amperes, and ohms. As- whereas the perceived heaviness H of a weight W satisfies H = KW3/2 sume that voltage is constant with V = 12 volts (V). Calculate (specifying (k is a constant that is different in the two cases). Compute d B/d / and dH /dW and state whether they are increasing or decreasing functions. units): Then explain the following statements: (a) The average rate of change of / with respect to R for the interval from R = 8 to R = 8.1 (a) An increase in light intensity is felt more strongly when / is small than when I is large. (b) The rate of change of I with respect to R when R = 8 (c) The rate of change of R with respect to / when / = 1.5 (b) An increase in load W is felt more strongly when W is large than when W is small. 37. Ethan finds that with h hours of tutoring, he is able to answer correctly S(h) percent of the problems on a math exam. Which would you 46. Let M(1) be the mass (in kilograms) of a plant as a function of time expect to be larger: S'(3) or S'(30)? Explain. (in years). Recent studies by Niklas and Enquist have suggested that a re- markably wide range of plants (from algae and grass to palm trees) obey a 38. Suppose @(1) measures the angle between a clock's minute and hour three-quarter-power growth law-that is. hands. What is O'(t) at 3 o'clock? 39. To determine drug dosages, doctors estimate a person's body surface dM area (BSA) (in meters squared) using the formula BSA = hm/60, where di = CM3/4 for some constant C is the height in centimeters and m the mass in kilograms. Calculate the rate of change of BSA with respect to mass for a person of constant height (a) If a tree has a growth rate of 6 kg/yr when M = 100 kg. what is its h = 180. What is this rate at m = 70 and m = 80? Express your result in growth rate when M = 125 kg? the correct units. Does BSA increase more rapidly with respect to mass at (b) If M = 0.5 kg, how much more mass must the plant acquire to double lower or higher body mass? its growth rate