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All Textbook Solutions for Calculus: An Applied Approach (MindTap Course List)
39E40ESlope of a Tangent Line In Exercises 41 and 42, find the slope of the tangent line to the given sine function at the origin. What can you conclude about the slope of the sine function sin ax at the origin? (a)y=sinx(b)y=sin2x42E43E44E45E46E47EFinding an Equation of a Tangent Line In Exercises 4350, find an equation of the tangent line to the graph of the function at the given point. FunctionPointy=sinxcosx(32,0)49E50E51E52E53E54E55E56E57E58EFinding Relative Extrema In Exercises 53-62, find the relative extrema of the trigonometric function in the interval (0,2). Use a graphing utility to confirm your results. See Examples 6 and 7. y=2sinx+sin2x60E61EFinding Relative Extrema In Exercises 53-62, find the relative extrema of the trigonometric function in the interval (0,2). Use a graphing utility to confirm your results. See Examples 6 and 7. y=exsinx63E64E65E66E67E68ETransportation Workers The number W (in thousands) of scenic and sightseeing transportation workers employed in the United States during 2014 can be modeled by W=31.184+8.447sin(0.568t-2.595) where t is the time (in months), with t = 1 corresponding to January. Approximate the month t in which the number of scenic and sightseeing transportation workers employed was a maximum. What was the maximum number of scenic and sightseeing transportation workers employed? (Source: U.S. Bureau of Labor Statistics)Employment The number W (in thousands) of golf course and country club workers employed in the United States during 2014 can be modeled by W=362.909+112.127sin(0.549t-2.380) where t is the time (in months), with t = 1 corresponding to January. Approximate the month t in which the number of golf course and country club workers employed was a maximum. What was the maximum number of golf course and country club workers employed? (Source: U.S. Bureau of Labor Statistics)Biology Plants do not grow at constant rates during a normal 24-hour period because their growth is affected by sunlight. Suppose that the growth of a certain plant species in a controlled environment is modeled by h=0.2t+0.03sin2t where h is the height of the plant (in inches) and t is the time (in days), with t = 0 corresponding to midnight of day 1. During what time of day is the rate of growth of this plant (a) the greatest? (b) the least?Tides Throughout the day, the depth of water D (in meters) at the end of a dock varies with the tides. The depth for one particular day can be modeled by D=3.5+1.5cost6,0t24 where t is the time (in hours), with t = 0 corresponding to midnight. (a) Determine dD/dt. (b) Evaluate dD/dt for t = 4 and t = 20, and interpret the meaning of these values in the context of the problem. (c) Find the time(s) when the water depth is the greatest and the time(s) when the water depth is the least. (d) What is the greatest depth? What is the least depth? (e) When is the rate of change of the water depth the greatest? the least?73EHOW DO YOU SEE IT? The graph shows the height h (in feet) above ground of a seat on a Ferris wheel at time t (in seconds). (a) What is the period of the model? What does the period tell you about the ride? (b) Find the open intervals on which the height is increasing and decreasing. (c) Estimate the relative extrema in the interval [0. 60].Analyzing a Function In Exercises 75-80, use a graphing utility to (a) graph f and f in the same viewing window over the specified interval, (b) find the critical numbers of f, (c) find the open interval(s) on which f' is positive and the open interval(s) on which f' is negative, and (d) find the relative extrema in the interval. Function Interval f(t)=t2sint(0,2)76E77E78E79E80E81E82E83E84EFind 5sinxdx.2CP3CP4CP5CP6CP7CP8CP9CP1SWU2SWU3SWU4SWU5SWU6SWU7SWU8SWU9SWU10SWU11SWU12SWU13SWU14SWUIn Exercises 9-16, simplify the expression using the trigonometric identities listed in Section 8.2. cotxsecx16SWU17SWU18SWU19SWU20SWU1E2E3E4E5E6E7E8E9E10E11E12EIntegrating a Trigonometric Function In Exercises 132, find the indefinite integral. See Examples 1, 2, 3, 4, 5, and 8. sec2xtanxdx14E15E16E17E18E19E20E21E22EIntegrating a Trigonometric Function In Exercises 132, find the indefinite integral. See Examples 1, 2, 3, 4, 5, and 8. exsinexdx24E25E26E27E28E29E30E31E32E33EIntegration by Parts In Exercises 33-38, use integration by parts to find the indefinite integral. xsin5xdx35E36E37E38E39E40E41E42E43E44E45E46E47E48E49E50E51E52E53EFinding the Area of a Region In Exercises 53-56, sketch the region bounded by the graphs of the functions and find the area of the region. y=sinx,y=cos2x,x=2,x=655E56E57E58E59E60ECost The temperature T (in degrees Fahrenheit) in a house is given by T=72+12sin(t8)12 where t is the time (in hours), with t=0 corresponding to midnight. The hourly cost of cooling a house is $0.30 per degree. Find the cost C of cooling this house between 8 A.M. and 8 P.M. when the thermostat is set at 72F (see figure) by evaluating the integral C=0.3820[ 72+12sin(t8)1272 ]dt. (b) Find the saving realized by resetting the thermostat to 78F (see figure) by evaluating the integral C=0.31018[ 72+12sin(t8)1278 ]dt.Water Supply The flow rate R (in thousands of gallons per hour) of water at a pumping station during a day can be modeled by R=53+7sin(t6+3.6)+9cos(t12+8.9), 0t24 where t is the time (in hours), with t = 0 corresponding to midnight. Find the average hourly flow rate from midnight to noon (0t12). Find the average hourly flow rate from noon to midnight (12t24). Find the total volume of water pumped in one day.63E64E67E68E1RE2RE3RE4RE5RE6RE7RE8RE9RE10RE11RE12RE13RE14RE15RE16RERevenue A publishing company introduces a new weekly magazine that sells for $3.95. The marketing group of the company estimates that the sales x (in thousands) will be approximated by the following probability function. x 10 15 20 30 40 P(x) 0.10 0.20 0.50 0.15 0.05 Find E(x) and . Find the expected revenue.18RE19RE20RE21RE22RE23RE24RE25RE26RE27RE28RE29RE30RE31RE32RE33RE34RE35RE36REWaiting Time The waiting time t (in minutes) for patients arriving at a health clinic is described by the probability density function f(t)=112et/12,[0,) Find the probability that a patient will wait (a) no more than 5 minutes and (b) between 9 and 12 minutes.38RE39RE40RE41RE42RE43RE44RE45RE46RE47RE48RE49RE50RE51RE52RE53RE54RE55RE56RE57RE58RE59RE60RE61RE62RE1TYS2TYS3TYS4TYS5TYS6TYS7TYS8TYS9TYS10TYS11TYS12TYS13TYS14TYS15TYS16TYSCheckpoint 1 Worked-out solution available at LarsonAppliedCalculus.com An experiment consists of rolling a twelve-sided die, similar to the one shown at the left. a. What is the sample space? b. Describe the event corresponding to rolling a number greater than 7.2CP3CP4CP5CP6CP1SWU2SWU3SWU4SWU5SWU6SWU7SWU8SWU9SWU10SWU1E2E3E4E5E6E7ERandom Selection A card is chosen at random from a standard 52-card deck of playing cards. What is the probability that the card is (a) a face card? (b) not a face card? (c) a black face card?9E10EIdentifying Probability Distributions In Exercises 11-14, determine whether the table represents a probability distribution. If it is a probability distribution, sketch its graph. If it is not a probability distribution, state any properties that are not satisfied. x 0 1 2 3 P(x) 0.10 0.45 0.30 0.1512E13E14E15EUsing Probability Distributions In Exercises 1518, sketch a graph of the probability distribution and find each probability. x 0 1 2 3 4 5 P(x) 1030 830 630 330 230 130 (a) P(1x4) (b) P(x4)17E18E19EChildren The table shows the probability distribution of the numbers of children per family in the United States in 2014. (Source: U.S. Census Bureau) Children, x 0 1 2 3 or more P(x) 0.570 0.183 0.161 0.086 (a) Sketch the probability distribution. (b) Find the probability that a family has at least 2 children. (c) Find the probability that a family has at most 2 children. (d) Find the probability that a family has at least 1 child.21EDie Roll Consider the experiment of rolling a six-sided die twice. (a) Complete the set to form the sample space of 36 elements. Note that each element is an ordered pair in which the entries are the numbers of points on the first and second rolls, respectively. 5 = {(1, 1), (1, 2),, (2, 1), (2, 2),} (b) Complete the table and sketch the graph of the probability distribution, in which the random variable x is the sum of the two rolls. x 2 3 4 5 6 7 8 9 10 11 12 P(x) (c) Use the table in part (b) to find P(10x12).23E24E25E26E27E28E29EPersonal Income The probability distribution of the random variable x, the annual income of a family (in thousands of dollars) in a certain section of a large city, is shown in the table. Find E(x) and . x 30 40 50 60 80 P(x) 0.10 0.20 0.50 0.15 0.05Insurance An insurance company needs to determine the annual premium required to break even on fire protection policies with a face value of $90,000. The random variable x is the claim size on these policies, and the analysis is restricted to the losses $30,000, $60,000, and $90,000. The probability distribution of x is as shown in the table. What premium should customers be charged for the company to break even? x 0 30,000 60,000 90,000 P(X) 0.995 0.0036 0.0011 0.0003Insurance An insurance company needs to determine the annual premium required to break even for collision protection for cars with a value of $10,000. The random variable x is the claim size on these policies, and the analysis is restricted to the losses $1000, $5000, and $10,000. The probability distribution of x is as shown in the table. What premium should customers be charged for the company to break even? x 0 1000 5000 10,000 P(x) 0.936 0.040 0.020 0.004Baseball A baseball fan examined the record of a favorite baseball players performance during his last season. The numbers of games in which the player had zero, one, two, three, and four hits are recorded in the table shown below. Number of hits 0 1 2 3 4 Frequency 31 58 45 19 5 (a) Complete the table below, where x is the number of hits. x 0 1 2 3 4 P(x) (b) Use the table in part (a) to sketch the graph of the probability distribution. (c) Use the table in part (a) to find P(1x3). (d) Determine the mean and interpret the result in the context of the problem. (e) Determine the variance and standard deviation.Games of Chance If x is a players net gain in a game of chance, then E(x) is usually negative. This value gives the average amount per game the player can expect to lose over the long run. In Exercises 35 and 36, find the players expected net gain for one play of the specified game. Roulette In roulette, the wheel has the 38 numbers 00, 0, 1,2,, 34, 35, and 36, marked on equally spaced slots. If a player bets $1 on a number and wins, then the player keeps the dollar and receives an additional $35. Otherwise, the dollar is lost.Games of Chance If x is a players net gain in a game of chance, then E(x) is usually negative. This value gives the average amount per game the player can expect to lose over the long run. In Exercises 35 and 36, find the players expected net gain for one play of the specified game. Raffle A service organization is selling $2 raffle tickets as part of a fundraising program. The first prize is a boat valued at $2950, and the second prize is a camping tent valued at $400. In addition to the first and second prizes, there are 25 $20 gift certificates to be awarded. The number of tickets sold is 3000.37E38E1CP2CP3CP4CP5CP1SWU2SWU3SWU4SWU5SWU6SWU7SWU1E2E3E4E5E6E7E8E9E10E11E12E13E14E15EMaking a Probability Density Function In Exercises 1318, find the constant k such that the function f is a probability density function over the given interval. f(x)=kx(1x),[ 0,1 ]