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All Textbook Solutions for Calculus: Early Transcendentals (3rd Edition)
71E72E73E74E75E76EHarmonic sum In Chapter 10, we will encounter the harmonic sum . Use a left Riemann sum to approximate (with unit spacing between the grid points) to show that . Use this fact to conclude that does not exist.
Probability as an integral Two points P and Q are chosen randomly, one on each of two adjacent sides of a unit square (see figure). What is the probability that the area of the triangle formed by the sides of the square and the line segment PQ is less than one-fourth the area of the square? Begin by showing that x and y must satisfy xy12 in order for the area condition to be met. Then argue that the required probability is 12+1/21dx2x and evaluate the integral.Population A increases at a constant rate of 4%/yr. Population b increases at a constant rate of 500 people/yr. Which population exhibits exponential growth? What kind of growth is exhibited by the other population?Verify that the time needed for y(t) = y0ekt. to double from y0 to 2y0 is the same as the time needed to double from 2y0 to 4y0.Assume y() 100e0.005, 3y (exactly) what percentage does y increase when increases by 1 unit?If a quantity decreases by a factor of 8 every 30 years, what is its half-life?In terms of relative growth rate, what is the defining property of exponential growth?2E3E4E5E6ESuppose a quantity described by the function y(t) = y0ekt, where t is measured in years, has a doubling time of 20 years. Find the rate constant k.Suppose a quantity is described by the function y(t) = 30,000e0.05t, where t is measured in years. Find the half-life of the quantity.Give two examples of processes that are modeled by exponential growth.Give two examples of processes that are modeled by exponential decay.Because of the absence of predators, the number of rabbits on a small island increases at a rate of 11% per month. If y(t) equals the number of rabbits on the island t months from now, find the rate constant k for the growth function y(t) = y0ekt.After the introduction of foxes on an island, the number of rabbits on the island decreases by 4.5% per month. If y(t) equals the number of rabbits on the island t months after foxes were introduced, find the rate constant k for the exponential decay function y(t) = y0ekt.Absolute and relative growth rates Two functions f and g are given. Show that the growth rate of the linear function is constant and the relative growth rate of the exponential function is constant. 9. f(t) = 100 + 10.5t, g(t) = 100et/10Absolute and relative growth rates Two functions f and g are given. Show that the growth rate of the linear function is constant and the relative growth rate of the exponential function is constant. 10. f(t) = 2200 + 400t, g(t) = 4002t/20Designing exponential growth functions Complete the following steps for the given situation. a. Find the rate constant k and use it to devise an exponential growth function that fits the given data. b. Answer the accompanying question. Population The population of a town with a 2016 population of 90,000 grows at a rate of 2.4%/yr. In what year will the population reach 120,000?Designing exponential growth functions Complete the following steps for the given situation. a. Find the rate constant k and use it to devise an exponential growth function that fits the given data. b. Answer the accompanying question. Population The population of Clark County, Nevada, was about 2.115 million in 2015. Assuming an annual growth rate of 1.5%/yr, what will the county population be in 2025?Designing exponential growth functions Complete the following steps for the given situation. a. Find the rate constant k and use it to devise an exponential growth function that fits the given data. b. Answer the accompanying question. Population The current population of a town is 50,000 and is growing exponentially. If the population is projected to be 55,000 in 10 years, then what will be the population 20 years from now?Designing exponential growth functions Complete the following steps for the given situation. a. Find the rate constant k and use it to devise an exponential growth function that fits the given data. b. Answer the accompanying question. Savings account An initial deposit of 1500 is placed in a savings account with an APY of 3.1%. How long will it take until the balance of the account is 2500? Assume the interest rate remains constant and no additional deposits or withdrawals are made.Designing exponential growth functions Complete the following steps for the given situation. a. Find the rate constant k and use it to devise an exponential growth function that fits the given data. b. Answer the accompanying question. Rising costs Between 2010 and 2016, the average rate of inflation was about 1.6%/yr. If a cart of groceries cost 100 in 2010, what will it cost in 2025, assuming the rate of inflation remains constant at 1.6%?Designing exponential growth functions Complete the following steps for the given situation. a. Find the rate constant k and use it to devise an exponential growth function that fits the given data. b. Answer the accompanying question. Cell growth The number of cells in a tumor doubles every 6 weeks starting with 8 cells. After how many weeks does the tumor have 1500 cells?Determining APY Suppose 1000 is deposited in a savings account that icreases exponentially. Determine the APY if the account increases to 1200 in 5 years. Assume the interest rate remains constant and no additional deposits or withdrawals aremade.Tortoise growth In a study conducted at University of New Mexico, It was found that the mass (weight) of juvenile desert tortoises exhibited exponential growth after a diet switch. One of these tortoises had a mass of about 64 g at the time of the diet switch, and 33 days later the mass was 73 g. How many days after the diet switch did the tortoise have a mass of 100 g? (Source: Physiological and Biochemical Zoology, 85, 1, 2012)Projection sensitivity According to the 2014 national population projections published by the U.S. Census Bureau, the U.S. population is projected to be 334.4 million in 2020 with an estimated growth rate of 0.79%/yr. a. Based on these figures, find the doubling time and the projected population in 2050. Assume the growth rate remainsconstant. b. Suppose the actual growth rate is instead 0.7%. What are the resulting doubling time and projected 2050 population?Energy consumption On the first day of the year (t = 0), a city uses electricity at a rate of 2000 MW. That rate is projected to increase at a rate of 1.3% per year. a. Based on these figures, find an exponential growth function for the power (rate of electricity use) for the city. b. Find the total energy (in MW-yr) used by the city over four full years beginning at t = 0. c. Find a function that gives the total energy used (in MW-yr) between t = 0 and any future time t 0.Population of Texas Texas was the third fastest growing state in the United States in 2016. Texas grew from 25.1 million in 2010 to 26.47 million in 2016. Use an exponential growth model to predict the population of Texas in 2025.Oil consumption Starting in 2018 (t = 0), the rate at which oil is consumed by a small country increases at a rate of 1.5%/yr, starting with an initial rate of 1.2 million barrels/yr.
How much oil is consumed over the course of the year 2018 (between t = 0 and t = 1)?
Find the function that gives the amount of oil consumed between t = 0 and any future time t.
How many years after 2018 will the amount of oil consumed since 2018 reach 10 million barrels?
Designing exponential decay functions Devise an exponential decay function that fits the following data; then answer the accompanying questions. Be sure to identify the reference point (t = 0) and units of time. Crime rate The homicide rate decreases at a rate of 3%/yr in a city that had 800 homicides/yr in 2018. At this rate, when will the homicide rate reach 600 homicides/yr?Designing exponential decay functions Devise an exponential decay function that fits the following data; then answer the accompanying questions. Be sure to identify the reference point (t = 0) and units of time. 22. Drug metabolism A drug is eliminated from the body at a rate of 15%/hr. After how many hours does the amount of drug reach 10% of the initial dose?Designing exponential decay functions Devise an exponential decay function that fits the following data; then answer the accompanying questions. Be sure to identify the reference point (t = 0) and units of time. 23. Valium metabolism The drug Valium is eliminated from the bloodstream with a half-life of 36 hr. Suppose that a patient receives an initial dose of 20 mg of Valium at midnight. How much Valium is in the patients blood at noon the next day? When will the Valium concentration reach 10% of its initial level?Designing exponential decay functions Devise an exponential decay function that fits the following data; then answer the accompanying questions. Be sure to identify the reference point (t = 0) and units of time. 24. Chinas population Chinas one-child policy was implemented with a goal of reducing Chinas population to 700 million by 2050 (from 1.2 billion in 2000). Suppose Chinas population declines at a rate of 0.5%/yr. Will this rate of decline be sufficient to meet the goal?Population of West Virginia The population of West Virginia decreased from about 1.853 million in 2010 to 1.831 million in 2016. Use an exponential model to predict the population in 2025. Explain why an exponential (decay) model might not be anappropriate long-term model of the population of West Virginia.32EAtmospheric pressure The pressure of Earths atmosphere at sea level is approximately 1000 millibars and decreases exponentially with elevation. At an elevation of 30,000 ft (approximately the altitude of Mt. Everest), the pressure is one-third the sea-level pressure. At what elevation is the pressure half the sea-level pressure? At what elevation is it 1% of the sea-level pressure?Carbon dating The half-life of C-14 is about 5730 yr. a. Archaeologists find a piece of cloth painted with organic dyes. Analysis of the dye in the cloth shows that only 77% of the C-14 originally in the dye remains. When was the cloth painted? b. A well-preserved piece of wood found at an archaeological site has 6.2% of the C-14 that it had when it was alive. Estimate when the wood was cut.Uranium dating Uranium-238 (U-238) has a half-life of 4.5 billion years. Geologists find a rock containing a mixture of U-238 and lead, and determine that 85% of the original U-238 remains: the other 15% has decayed into lead. How old is the rock?Radioiodine treatment Roughly 12,000 Americans are diagnosed with thyroid cancer every year, which accounts for 1% of all cancer cases. It occurs in women three times as frequently as in men. Fortunately, thyroid cancer can be treated successfully in many cases with radioactive iodine, or I-131. This unstable form of iodine has a half-life of 8 days and is given in small doses measured in millicuries. a. Suppose a patient is given an initial dose of 100 millicuries. Find the function that gives the amount of I-131 in the body after t 0 days. b. How long does it take the amount of I-131 to reach 10% of the initial dose? c. Finding the initial dose to give a particular patient is a critical calculation. How does the time to reach 10% of the initial dose change if the initial dose is increased by 5%?Caffeine After an individual drinks a beverage containing caffeine, the amount of caffeine in the bloodstream can be modeled by an exponential decay function, with a half-life that depends on several factors, including age and body weight. For the sake of simplicity, assume the caffeine in the following drinks immediately enters the bloodstream upon consumption.
37. An individual consumes an energy drink that contains caffeine. If 80% of the caffeine from the energy drink is still in the bloodstream 2 hours later, find the half-life of caffeine for this individual.
Caffeine After an individual drinks a beverage containing caffeine, the amount of caffeine in the bloodstream can be modeled by an exponential decay function, with a half-life that depends on several factors, including age and body weight. For the sake of simplicity, assume the caffeine in the following drinks immediately enters the bloodstream upon consumption.
An individual consumes two cups of coffee, each containing 90 mg of caffeine, two hours apart. Assume the half-life of caffeine for this individual is 5.7 hours.
Determine the amount of caffeine in the bloodstream 1 hour after drinking the first cup of coffee.
Determine the amount of caffeine in the bloodstream 1 hour after drinking the second cup of coffee.
39E40ETumor growth Suppose the cells of a tumor are idealized as spheres each with a radius of 5 m (micrometers). The number of cells has a doubling time of 35 days. Approximately how long will it take a single cell to grow into a multi-celled spherical tumor with a volume of 0.5 cm3 (1 cm = 10,000 m)? Assume that the tumor spheres are tightly packed.Tripling time A quantity increases according to the exponential function y(t) = y0ekt. What is the tripling time for the quantity? What is the time required for the quantity to increase p-fold?Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. A quantity that increases at 6%/yr obeys the growth function y(t) = y0e0.06t. b. If a quantity increases by 10% /yr, it increases by 30% over 3 years. c. A quantity decreases by one-third every month. Therefore, it decreases exponentially. d. If the rate constant of an exponential growth function is increased, its doubling time is decreased. e. If a quantity increases exponentially, the time required to increase by a factor of 10 remains constant for all time.A running model A model for the startup of a runner in a short race results in the velocity function v(t) = a(1 et/c), where a and c are positive constants and v has units of m/s. (Source: A Theory of Competitive Running, Joe Keller, Physics Today 26, Sep 1973) a. Graph the velocity function for a = 12 and c = 2. What is the runners maximum velocity? b. Using the velocity in part (a) and assuming s(0) = 0, find the position function s(t), for t 0. c. Graph the position function and estimate the time required to run 100 m.45E46EA slowing race Starting at the same time and place, Abe and Bob race, running at velocities u(t) = 4/(t + 1) mi/hr and v(t) = 4et/2 mi/hr, respectively, for t 0. a. Who is ahead after t = 5 hr? After t = 10 hr? b. Find and graph the position functions of both runners. Which runner can run only a finite distance in an unlimited amount of time?48ECompounded inflation The U.S. government reports the rate of inflation (as measured by the Consumer Price Index) both monthly and annually. Suppose that for a particular month, the monthly rate of inflation is reported as 0.8%. Assuming that this rate remains constant, what is the corresponding annual rate of inflation? Is the annual rate 12 times the monthly rate? Explain.Acceleration, velocity, position Suppose the acceleration of an object moving along a line is given by a(t) = kv(t), where k is a positive constant and v is the objects velocity. Assume that the initial velocity and position are given by v(0) = 10 and s(0) = 0, respectively. a. Use a(t) = v(t) to find the velocity of the object as a function of time. b. Use v(t) = s(t) to find the position of the object as a function of time. c. Use the fact that dv/dt = (dv/ds)(ds/dt) (by the Chain Rule) to find the velocity as a function of position.Air resistance (adapted from Putnam Exam, 1939) An object moves in a straight line, acted on by air resistance, which is proportional to its velocity; this means its acceleration is a(t) = kv(t). The velocity of the object decreases from 1000 ft/s to 900 ft/s over a distance of 1200 ft. Approximate the time required for this deceleration to occur. (Exercise 38 may be useful.)General relative growth rates Define the relative growth rate of the function f over the time interval T to be the relative change in f over an interval of length T: RT=f(t+T)f(t)f(t). Show that for the exponential function y(t) = y0ekt, the relative growth rate RT is constant for any T; that is, choose any T and show that RT is constant for all t.Equivalent growth functions The same exponential growth function can be written in the forms y(t) = y0ekt, y(t) = y0(l + r)2, and y(t)=y02t/T2. Write k as a function of r, r as a function of T2, and T2 as a function of k.Geometric means A quantity grows exponentially according to y(t) = y0ekt. What is the relationship between m, n, and p such that y(p)=y(m)y(n)?Constant doubling time Prove that the doubling time for an exponentially increasing quantity is constant for all time.Use the definition of the hyperbolic sine to show that sinh x is an odd function.Explain why the graph of tanh x has the horizontal asymptotes y = 1 and y = 1.Find both the derivative and indefinite integral of f(x) = 4 cosh 2x.4QC5QC6QCExplain why longer waves travel faster than shorter waves in deep water.State the definition of the hyperbolic cosine and hyperbolic sine functions.Sketch the graphs of y = cosh x, y sinh x, and y = tanh x (include asymptotes), and state whether each function is even, odd, or neither.What is the fundamental identity for hyperbolic functions?4EExpress sinh1 x in terms of logarithms.6E7EOn what interval is the formula d/dx (tanh1 x) = 1/(x2 1) valid?9E10EVerifying identities Verify each identity using the definitions of the hyperbolic functions. 11. tanhxe2x1e2x+1Verifying identities Verify each identity using the definitions of the hyperbolic functions. 12. tanh(x) = tanh xVerifying identities Verify each identity using the definitions of the hyperbolic functions. 13. cosh 2x = cosh2x + sinh2x (Hint: Begin with the right side of the equation.)Verifying identities Verify each identity using the definitions of the hyperbolic functions. 14. 2 sinh(ln (sec x)) = sin x tan xVerifying identities Verify each identity using the definitions of the hyperbolic functions. 15. cosh x + sinh x = exVerifying identities Use the given identity to verify the related identity. 16. Use the fundamental identity cosh2 x sinh2 x = 1 to verify the identity coth2 x 1 = csch2 x.Verifying identities Use the given identity to verify the related identity. 17. Use the identity cosh 2x = cosh2 x + sinh2 x to verify the identities cosh2x=cosh2x+12 and sinh2x=cosh2x12.18EDerivative formulas Derive the following derivative formulas given that d/dx(cosh x) = sinh x and d/dx(sinh x) = cosh x. 19. d/dx(coth x) = csch2 xDerivative formulas Derive the following derivative formulas given that d/dx(cosh x) = sinh x and d/dx(sinh x) = cosh x. 20. d/dx(sech x) = sech x tanh xDerivative formulas Derive the following derivative formulas given that d/dx(cosh x) = sinh x and d/dx(sinh x) = cosh x. 21. d/dx(csch x) = csch x coth xDerivatives Compute dy/dx for the following functions. 22. y = sinh 4xDerivatives Compute dy/dx for the following functions. 23. y = cosh2 xDerivatives Compute dy/dx for the following functions. 24. y = sinh3 4xDerivatives Compute dy/dx for the following functions. 25. y = tanh2 xDerivatives Compute dy/dx for the following functions. 26. y=coth3xDerivatives Compute dy/dx for the following functions. 27. y = ln sech 2xDerivatives Compute dy/dx for the following functions. 28. y = x tanh xDerivatives Compute dy/dx for the following functions. 29. y = x2 cosh2 3x30EDerivatives Find the derivatives of the following functions. 47. f(x) = cosh1 4xDerivatives Find the derivatives of the following functions. 48. f(t)=2tanh1tDerivatives Find the derivatives of the following functions. 49. f(v) = sinh1 v2Derivatives Find the derivatives of the following functions. 50. f(x) = csch1 (2/x)35E36EIndefinite integrals Determine each indefinite integral. 31. cosh2xdxIntegrals Evaluate each integral. sech2wtanhwdwIndefinite integrals Determine each indefinite integral. 33. sinhx1+coshxdxIndefinite integrals Determine each indefinite integral. 34. coth2xcsch2xdxIndefinite integrals Determine each indefinite integral. 35. tanh2xdx (Hint: Use an identity.)Indefinite integrals Determine each indefinite integral. 36. sinh2xdx (Hint: Use an identity.)Definite integrals Evaluate each definite integral. 37. 01cosh33xsinh3xdxIntegrals Evaluate each integral. 0ln2sech2xxdxDefinite integrals Evaluate each definite integral. 39. ln0ln2tanhxdxDefinite integrals Evaluate each definite integral. 40. ln2ln3cschydyIndefinite integrals Determine the following indefinite integrals. 53. dx8x2,x22Integrals Evaluate each integral. 48.dxx216,x4Indefinite integrals Determine the following indefinite integrals. 55. ex36e2xdx,xln650EIndefinite integrals Determine the following indefinite integrals. 57. dxx4x852EAdditional integrals Evaluate the following integrals. 89.coshzsinhzdzAdditional integrals Evaluate the following integrals. 90. cos9sin2d55EAdditional integrals Evaluate the following integrals. 92.25225dxx2+25x (Hint: x2+25x=xx+25.)Two ways Evaluate the following integrals two ways. a. Simplify the integrand first and then integrate. b. Change variables (let u = ln x), integrate, and then simplify your answer. Verify that both methods give the same answer. 41. sinh(lnx)xdxTwo ways Evaluate the following integrals two ways. a. Simplify the integrand first and then integrate. b. Change variables (let u = ln x), integrate, and then simplify your answer. Verify that both methods give the same answer. 42. 13sech(lnx)xdxVisual approximation a. Use a graphing utility to sketch the graph of y = coth x and then explain why 510cothxdx5. b. Evaluate 510cothxdx analytically and use a calculator to arrive at a decimal approximation to the answer. How large is the error in the approximation in part (a)?60E61EPoints of intersection and area a. Sketch the graphs of the functions f and g and find the x-coordinate of the points at which they intersect. b. Compute the area of the region described. 46. f(x) = sinh x, g(x) = tanh x; the region bounded by the graphs of f, g, and x = ln 3Definite integrals Evaluate the following definite integrals. Use Theorem 7.7 to express your answer in terms of logarithms. swDefinite integrals Evaluate the following definite integrals. Use Theorem 7.7 to express your answer in terms of logarithms. 535dxx29Definite integrals Evaluate the following definite integrals. Use Theorem 7.7 to express your answer in terms of logarithms. 22dtt2966E67E68ECatenary arch The portion of the curve y=1716coshx that lies above the x-axis forms a catenary arch. Find the average height of the arch above the x-axis.Length of a catenary Show that the arc length of the catenary y = cosh x over the interval [0, a] is L = sinh a.Power lines A power line is attached at the same height to two utility poles that are separated by a distance of 100 ft; the power line follows the curve f(x) = a cosh(x/a). Use the following steps to find the value of a that produces a sag of 10 ft midway between the poles. Use a coordinate system that places the poles at x = 50. a. Show that a satisfies the equation cosh(50/a) 1 = 10/a. b. Let t = 10/a, confirm that the equation in part (a) reduces to cosh 5t 1 = t, and solve for t using a graphing utility. Report your answer accurate to two decimal places. c. Use your answer in part (b) to find a and then compute the length of the power line.Sag angle Imagine a climber clipping onto the rope described in Example 7 and pulling himself to the ropes midpoint. Because the rope is supporting the weight of the climber, it no longer takes the shape of the catenary y = 200 cosh(x/200). Instead, the rope (nearly) forms two sides of an isosceles triangle. Compute the sag angle illustrated in the figure, assuming that the rope does not stretch when weighted. Recall from Example 7 that the length of the rope is 101 ft. EXAMPLE 7 Length of a catenary A climber anchors a rope at two points of equal height, separated by a distance of 100 ft, in order to perform a Tyrolean traverse. The rope follows the catenary f(x) = 200 cosh(x/200) over the interval [50, 50] (Figure 6.98). Find the length of the rope between the two anchor points.Wavelength The velocity of a surface wave on the ocean is given by v=g2tanh(2d) (Example 8). Use a graphing utility or root finder to approximate the wavelength of an ocean wave traveling at v = 7 m/s in water that is d = 10 m deep. EXAMPLE 8 Velocity of an ocean wave The velocity v (in meters/second) of an idealized surface wave traveling on the ocean is modeled by the equation v=g2tanh(2d), where g = 9.8 m/s2 is the acceleration due to gravity, is the wavelength measured in meters from crest to crest, and d is the depth of the undisturbed water, also measured in meters (Figure 6.99). a. A sea kayaker observes several waves that pass beneath her kayak, and she estimates that = 12 m and v = 4 m/s. How deep is the water in which she is kayaking? b. The deep-water equation for wave velocity is v=g2, which is an approximation to the velocity formula given above. Waves are said to be in deep water if the depth-to-wavelength ratio d/ is greater than 12. Explain why v=g2 is a good approximation when d/12.74E75E76EExplain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. ddx(sinh(ln3))=cosh(ln3)3. b. ddx(sinhx)=coshx and ddx(coshx)=sinhx. c. ln(1+2)=ln(1+2). d. 01dx4x2=12(coth112coth10).Evaluating hyperbolic functions Use a calculator to evaluate each expression or state that the value does not exist. Report answers accurate to four decimal places. a. coth 4 b. tanh1 2 c. csch1 5 d. cschx1/22 e. ln|tanh(x2)||110 f. tan1(sinhx)|33 g. 14coth1x4|2036Evaluating hyperbolic functions Evaluate each expression without using a calculator or state that the value does not exist. Simplify answers to the extent possible. a. cosh 0 b. tanh 0 c. csch 0 d. sech(sinh 0) e. coth(ln 5) f. sinh(2ln 3) g. cosh2 1 h. sech1 (ln 3) i. cosh1 (17/8) j. sinh1(e212e)80ECritical points Find the critical points of the function f(x) = sinh2 x cosh x.Critical points a. Show that the critical points of f(x)=coshxx satisfy x = coth x. b. Use a root finder to approximate the critical points of f.Points of inflection Find the x-coordinate of the point(s) of inflection of f(x) = tanh2 x.84EArea of region Find the area of the region bounded by y = sech x, x = 1, and the unit circle.86ELHpital loophole Explain why lHpitals Rule fails when applied to the limit limxsinhxcoshx and then find the limit another way.Limits Use lHpitals Rule to evaluate the following limits. 84. limx1cothx1tanhxLimits Use lHpitals Rule to evaluate the following limits. 85. limx0tanh1xtan(x/2)90E91E92EKiln design Find the volume interior to the inverted catenary kiln (an oven used to fire pottery) shown in the figure.94EFalling body When an object falling from rest encounters air resistance proportional to the square of its velocity, the distance it falls (in meters) after t seconds is given by d(t)=mkln(cos(kgmt)), where m is the mass of the object in kilograms, g = 9.8 m/s2 is the acceleration due to gravity, and k is a physical constant. a. A BASE jumper (m = 75 kg) leaps from a tall cliff and performs a ten-second delay (she free-falls for 10 s and then opens her chute). How far does she fall in 10 s? Assume k = 0.2. b. How long does it take her to fall the first 100 m? The second 100 m? What is her average velocity over each of these intervals?96E97E98EDifferential equations Hyperbolic functions are useful in solving differential equations (Chapter 9). Show that the functions y = A sinh kx and y = B cosh kx, where a, b, and k are constants, satisfy the equation y(x) k2y(x) = 0.100E101E102E103E104E105ETheorem 7.8 a. The definition of the inverse hyperbolic cosine is y=cosh1xx=coshy, for x 1, 0 y . Use implicit differentiation to show that ddx(cosh1x)=1x21. b. Differentiate sinh1x=ln(x+x2+1) to show that ddx(sinh1x)=1x2+1.107E108EArc length Use the result of Exercise 108 to find the arc length of f(x) = ln |tanh (x/2) | on [ln 2, ln 8]. 108. Integral formula Carry out the following steps to derive the formula cschxdx=lntanh(x/2)+C (Theorem 6.9). a. Change variables with the substitution u = x/2 to show that cschxdx=2dusinh2u. b. Use the identity for sinh 2u to show that 2sinh2u=sech2utanhu. c. Change variables again to determine sech2utanhudu and then express your answer in terms of x.110E111EDefinitions of hyperbolic sine and cosine Complete the following steps to prove that when the x- and y-coordinates of a point on the hyperbola x2 y2 = 1 are defined as cosh t and sinh t, respectively, where t is twice the area of the shaded region in the figure, x and y can be expressed as x=cosht=et+et2 and y=sinht=etet2. a. Explain why twice the area of the shaded region is given by t=2(12xy1zz21dz)=xx2121zz21dz. b. In chapter 8, the formula for the integral in part (a) is derived: z21dz=z2z2112lnz+z21+C. Evaluate this integral on the interval (1, x]. explain why the absolute value can be dropped, and combine the result with part (a) to show that t=ln(x+x21). c. Solve the final equation from part (b) for x to show that x=et+et2. d. Use the fact that y=x21 in combination with part (c). to show that y=etet2.Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The variable y = t + 1 doubles in value whenever t increases by 1 unit. b. The function y = Ae0.1t increases by 10% when t increases by 1 unit. c. ln xy = (ln x)(ln y). d. sinh(lnx)=x212x.Integrals Evaluate the following integrals. 56. ex4ex+6dxIntegrals Evaluate the following integrals. 57. e2e8dxxlnxIntegrals Evaluate the following integrals. 58. 1410xxdxIntegrals Evaluate the following integrals. 59. x+4x2+8x+25dxIntegrals Evaluate the following integrals. 60. ln2ln3cothsdsIntegrals Evaluate the following integrals. 61. dxx29,x3Integrals Evaluate the following integrals. 62. exe2x+4dxIntegrals Evaluate the following integrals. 63. 01x29x6dxDerivatives Find the derivatives of the following functions. 10. f(x) = ln(3 sin2 4x)Derivatives Find the derivatives of the following functions. 11. g(x)x3x2+1Derivatives Find the derivatives of the following functions. 12. f(x)=sinhx1+sinhxDerivatives Find the derivatives of the following functions. 13. f(t) = cosh t sinh t14RE15REDerivatives Find the derivatives of the following functions. 16. h(z) = ln sinh z z coth z17RE18RE19REPopulation growth The population of a large city grows exponentially with a current population of 1.3 million and a predicted population of 1.45 million 10 years from now. a. Use an exponential model to estimate the population in 20 years. Assume the annual growth rate is constant. b. Find the doubling time of the population.Caffeine An adult consumes an espresso containing 75 mg of caffeine. If the caffeine has a half-life of 5.5 hours, when will the amount of caffeine in her bloodstream equal 30 mg?Two cups of coffee A college student consumed two cups of coffee, each containing 80 mg of caffeine, 90 minutes apart. Suppose that just before he consumed his second cup of coffee, 65 mg of caffeine was still in his system from the first cup of coffee. How much caffeine remains in his system 7 hours after he drank his first cup of coffee?Moores Law In 1965, Gordon Moore observed that the number of transistors that could be placed on an integrated circuit was approximately doubling each year, and he predicted that this trend would continue for another decade. In 1975. Moore revised the doubling time to every two years, and this prediction became known as Moores Law. a. In 1979, Intel introduced the Intel 8088 processor; each of its integrated circuits contained 29,000 transistors. Use Moores revised doubling time to find a function y(t) that approximates the number of transistors on an integrated circuit t years after 1979. b. In 2000, the Pentium 4 integrated circuit was introduced, it contained 42 million transistors. Compare this value to the value obtained using the function found in part (a).Radioactive decay The mass of radioactive material in a sample has decreased by 30% since the decay began. Assuming a half-life of 1500 years, how long ago did the decay begin?Population growth Growing from an initial population of 150,000 at a constant annual growth rate of 4%/yr, how long will it take a city to reach a population of 1 million?26RE27RECurve sketching Use the graphing techniques of Section 4.4 to graph the following functions on their domains. Identify local extreme points inflection points, concavity, and end behavior Use a graphing utility only to check your work. 28. f(x) = ln x ln2x29RE30RELinear approximation Find the linear approximation to f(x) = cosh x at a = ln 3 and then use it to approximate the value of cosh 1.Limit Evaluate limx(tanhx)x.Derivatives of hyperbolic functions Compute the following derivatives. a. d6/dx6(cosh x) b. d/dx(x sech x)Arc length Find the arc length of the curve y = ln x between x = 1 and x = b 1 given that x2+a2xdx=x2+a2aln(a+x2+a2x)+C. Use any means to approximate the value of b for which the curve has length 2.What change of variable would you use for the integral sec2xtan3xdx?Explain how to simplify the integrand of x3+xx3/2dx before integrating.Explain how to simplify the integrand of x+1x1dx before integrating.Express x2 + 6x + 16 in terms of a perfect square.What change of variables would you use for the integral (47x)6dx?Evaluate (secx+1)2dx. (Hint: Expand (sec x + 1)2 first.)What trigonometric identity is useful in evaluating sin2xdx?Let f(x)=4x3+x+24x+2x2+1. Use long division to show that f(x)=4x+1+1x2+1 and use this result to evaluate f(x)dx.Describe a first step in integrating 10x24x9dx.Evaluate 2x+1x2+1dx using the following steps. a. Fill in the blanks: By splitting the integrand into two fractions, we have 2x+1x2+1dx=dx+dx. b. Evaluate the two integrals on the right side of the equation in part (a).Substitution Review Evaluate the following integrals. 7. dx(35x)4Substitution Review Evaluate the following integrals. 8. (9x2)3dxSubstitution Review Evaluate the following integrals. 9. 03/8sin(2x4)dxSubstitution Review Evaluate the following integrals. 10. e34xdxSubstitution Review Evaluate the following integrals. 11. ln2xxdxSubstitution Review Evaluate the following integrals. 12. 50dx4xSubstitution Review Evaluate the following integrals. 13. exex+1dxIntegration review Evaluate the following integrals. 14. 01x3x2+1dxSubtle substitutions Evaluate the following integrals. 17. 1e2ln2(x2)xdxIntegration review Evaluate the following integrals. 16. 01t21+t6dtIntegration review Evaluate the following integrals. 17. 12s(s9)9dsIntegration review Evaluate the following integrals. 18. 37(t6)t3dtIntegration review Evaluate the following integrals. 19. (lnw1)7lnwwdwIntegration review Evaluate the following integrals. 20. ex(1+ex)9(1ex)dxIntegration review Evaluate the following integrals. 21. x+2x2+4dxIntegration review Evaluate the following integrals. 22. sinx+1cosxdxIntegration review Evaluate the following integrals. 23. excsc(3ex+4)dxSplitting fractions Evaluate the following integrals. 24. 49x5/2x1/2x3/2dxIntegration review Evaluate the following integrals. 25. 0/4sec+cscseccscdSplitting fractions Evaluate the following integrals. 26. 4+e2xe3xdxSplitting fractions Evaluate the following integrals. 27. 23x1x2dxSplitting fractions Evaluate the following integrals. 28. 3x+14x2dxIntegration review Evaluate the following integrals. 29. /4/21+cot2xdxIntegration review Evaluate the following integrals. 30. e/4e/3cot(lnx)xdxCompleting the square Evaluate the following integrals. 33. dxx22x+10Completing the square Evaluate the following integrals. 34. 02xx2+4x+8dxIntegration review Evaluate the following integrals. 33. x3+2x2+5x+3x2+x+2dxDivision with rational functions Evaluate the following integrals. 30. 24x2+2x1dxIntegration review Evaluate the following integrals. 35. 01t4+t3+t2+t+1t2+1dtDivision with rational functions Evaluate the following integrals. 31. t32t+1dtCompleting the square Evaluate the following integrals. 35. d2762Completing the square Evaluate the following integrals. 36. xx4+2x2+1dxMultiply by 1 Evaluate the following integrals. 37. d1+sinMultiply by 1 Evaluate the following integrals. 38. 1x1xdxMultiply by 1 Evaluate the following integrals. 39. dxsecx1Multiply by 1 Evaluate the following integrals. 40. d1cscIntegration review Evaluate the following integrals. 43. cosh3x1+sinh3xdxIntegration reviewEvaluate the following integrals 44. 036x3x2+1dxIntegration reviewEvaluate the following integrals 45. exex2exdxIntegration reviewEvaluate the following integrals. 46. e2ze2z4ezdzIntegration reviewEvaluate the following integrals. 47. dxx1+1Integration reviewEvaluate the following integrals. 48. dyy1+y3Integration reviewEvaluate the following integrals. 49. 9+t+1dtMiscellaneous integrals Use the approaches discussed in this section to evaluate the following integrals. 42. 49dx1xMiscellaneous integrals Use the approaches discussed in this section to evaluate the following integrals. 43. 10xx2+2x+2dxIntegration reviewEvaluate the following integrals. 52. /6/2dysinyIntegration reviewEvaluate the following integrals. 53. exsec(ex+1)dxMiscellaneous integrals Use the approaches discussed in this section to evaluate the following integrals. 44. 011+xdxMiscellaneous integrals Use the approaches discussed in this section to evaluate the following integrals. 45. sinxsin2xdxMiscellaneous integrals Use the approaches discussed in this section to evaluate the following integrals. 46. 0/21+cos2xdxMiscellaneous integrals Use the approaches discussed in this section to evaluate the following integrals. 47. dxx1/2+x3/2Miscellaneous integrals Use the approaches discussed in this section to evaluate the following integrals. 48. 01dp4pMiscellaneous integrals Use the approaches discussed in this section to evaluate the following integrals. 49. x2x2+6x+13dxMiscellaneous integrals Use the approaches discussed in this section to evaluate the following integrals. 50. 0/431+sin2xdxMiscellaneous integrals Use the approaches discussed in this section to evaluate the following integrals. 51. exe2x+2ex+1dxIntegration reviewEvaluate the following integrals. 62. x5x42x3+4x+3x2+x+1dxMiscellaneous integrals Use the approaches discussed in this section to evaluate the following integrals. 53. 132x2+2x+1dxMiscellaneous integrals Use the approaches discussed in this section to evaluate the following integrals. 54. 022s3+3s2+3s+1dsFurther Explorations 41. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. 3x2+4dx=3x2dx+34dx. b. Long division simplifies the evaluation of the integral x3+23x4+xdx. c. dxsinx+1=lnsinx+1+C. d. dxex=lnex+C.Use a change of variables to prove that cotxdx=ln|sinx|+C.Prove that cscxdx=ln|cscx+cotx|+C.(Hint: See Example 2.)Different methods a. Evaluate cotxcsc2xdx using the substitution u = cot x. b. Evaluate cotxcsc2xdx using the substitution u = csc x. c. Reconcile the results in parts (a) and (b).Different substitutions a. Evaluate tanxsec2xdx using the substitution u = tan x. b. Evaluate tanxsec2xdx using the substitution u = sec x. c. Reconcile the results in parts (a) and (b).Different methodsLet I=x+2x+4dx. a. Evaluate I after first performing long division on the integrand. b. Evaluate I without performing long division on the integrand. c. Reconcile the results in parts (a) and (b).Different methods a. Evaluate x2x+1dx using the substitution u = x + 1. b. Evaluate x2x+1dx after first performing long division on the integrand. c. Reconcile the results in parts (a) and (b).Area of a region between curves Find the area of the entire region bounded by the curves y=x3x2+1 and y=8xx2+1.Area of a region between curves Find the area of the region bounded by the curves y=x2x33x and y=1x33x on the interval [2, 4].Volume of a solidConsider the region R bounded by the graph of f(x)=1x+2 and the x-axis on the interval [0, 3]. Find the volume of the solid formed when R is revolved about the y-axis.Volume of a solidConsider the Region R bounded by the graph of f(x)=x2+1 and the x-axis on the interval [0, 2]. Find the volume of the solid formed when R is revolved about the y-axis.Different substitutions a. Show that dxxx2=sin1(2x1)+C using either u = 2x 1 or u=x12. b. Show that using dxxx2=2sin1x+C using u=x. c. Prove the identity 2sin1xsin1(2x1)=2. (Source: The College Mathematics Journal 32, 5, Nov 2001)