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All Textbook Solutions for Calculus (MindTap Course List)
51REEnvironment A conservation department releases 1200 brook trout into a lake. It is estimated that the carrying capacity of the lake for the species is 20,400. After the first year, there are 2000 brook trout in the lake. (a) Write a logistic equation that models the number of brook trout in the lake. (b) Find the number of brook trout in the lake after 8 years. (c) When will the number of brook trout reach 10,000? (d) Write a logistic differential equation that models the growth rate of the brook trout population. Then repeat part (b) using Eulers method with a step size of h=1. Compare the approximation with the exact answer. (e) At what time is the brook trout population growing most rapidly? Explain.53RE54RE55RE56RE57RE58RE59RE60RE61RE62RE1PSSales Let S represent sales of a new product (in thousands of units), let L. represent the maximum level of sales (in thousands of units), and let t represent time (in months). The rate of change of S with respect to t is proportional to the product of S and LS. (a) Write the differential equation for the sales model when L=100,S=10whent=0,andS=20whent=1. Verify that S=L1+Cekt (b) At what time is the growth in sales increasing most rapidly? (c) Use a graphing utility to graph the sales function. (d) Sketch the solution from part (a) on the slope field below. To print an enlarged copy of the graph, go to MathGraphs.com. (e) Assume the estimated maximum level of sales is correct. Use the slope Field to describe the shape of the solution curves for sales when, at some period of time, sales exceed L.3PS4PSTorricelli's Law Torricellis Law states that water will flow from an opening at the bottom of a tank with the same speed that it would attain falling from the surface of the water to the opening. One of the forms of Torricellis Law is A(h)dhdt=k2gh where h is the height of the water in the tank, k is the area of the opening at the bottom of the tank, A(h) is the horizontal cross-sectional area at height h, and g is the acceleration due to gravity (g32 feet per second per second). A hemispherical water tank has a radius of 6 feet. When the tank is full, a circular valve with a radius of 1 inch is opened at the bottom, as shown in the figure. How long will it take for the tank to drain completely?Torricelli's Law The cylindrical water tank shown in the figure has a height of 18 feet. When the tank is full, a circular valve is opened at the bottom of the tank. After 30 minutes, the depth of the water is 12 feet. (a) Using Torricellis Law, how long will it take for the tank (b) What is the depth of the water in the tank after 1 hour?Torricelli's Law A tank similar to the one in Exercise 6 has a height of 20 feet and a radius of 8 feet, and the valve is circular with a radius of 2 inches. The tank is full when the valve is opened. How long will it take for the tank to drain completely?8PS9PS10PS11PS12PSIn Exercises 11 and 12, it was assumed that there was a single initial injection of the tracer drug into the compartment. Now consider the case in which the tracer is continuously injected (beginning at t=0) at the rate of Q moles per minute. Considering Q to be negligible compared with R, use the differential equation dCdt=QV(RV)C where C=0 when t=0. (a) Solve this differential equation to find the concentration C as a function of time t. (b) Find the limit of C as t.Verifying a Solution Describe how to determine whether a function y=f(x) is a solution of a differential equation.2E3E4EVerifying a Solution In Exercises 510, verify that the function is a solution of the differential equation. Function Differential Equation y=Ce5xy=5yVerifying a Solution In Exercises 510, verify that the function is a solution of the differential equation. Function Differential Equation y=e2x3y+5y=e2x7E8EVerifying a Solution In Exercises 510, verify that the function is a solution of the differential equation. Function Differential Equation y=(cosx)lnsecx+tanxy+y=tanx10E11E12E13E14E15E16E17EDetermining a Solution In Exercises 1522, determine whether the function is a solution of the differential equation y(4)-16y=0. y = 2 sin x19EDetermining a Solution In Exercises 1522, determine whether the function is a solution of the differential equation y(4)-16y=0. y=5lnx21E22E23E24EDetermining a Solution: In Exercises 23-30, determine whether the function is a solution of the differential equation xy'-2y=x3ex;x0. y=x2ex26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46E47E48E49E50E51E52E53E54E55E56E57E58E59E60E61ESlope Field In Exercises 6164, (a) sketch the slope field for the differential equation, (b) use the slope field to sketch the solution that passes through the given point, and (c) discuss the graph of the solution as and Use a graphing utility to verify your results. To print a blank graph, go to MathGraphs.com. y=13x212x, (1,1)Slope Field In Exercises 6164, (a) sketch the slope field for the differential equation, (b) use the slope field to sketch the solution that passes through the given point, and (c) discuss the graph of the solution as and Use a graphing utility to verify your results. To print a blank graph, go to MathGraphs.com. y=y4x, (2,2)64E65ESlope Field Use the slope field for the differential equation y=1/y, where y0, to sketch the graph of the solution that satisfies each given initial condition. Then make a conjecture about the behavior of a particular solution of y=1/y as x. To print an enlarged copy of the graph, go to MathGraphs.com. a). (0,1) b). (1,1)Slope Field In Exercises 6772, use a computer algebra system to (a) graph the slope field for the differential equation and (b) graph the solution satisfying the specified initial condition. dydx=0.25y, y(0)=468E69E70E71E72EEuler's Method In Exercises 73-78, use Eulers Method to make a table of values for the approximate solution of the differential equation with the specified initial value. Use n steps of size h. y=x+y,y(0)=2,n=10,h=0.174E75EEuler's Method In Exercises 73-78, use Eulers Method to make a table of values for the approximate solution of the differential equation with the specified initial value. Use n steps of size h. y=0.5x(3y),y(0)=1,n=5,h=0.477E78E79EEuler's Method In Exercises 79-81, complete the table using the exact solution of the differential equation and two approximations obtained using Eulers Method to approximate the particular solution of the differential equation. Use h=0.2 and h=0.1 , and compute each approximation to four decimal places. x 0 0.2 0.4 0.6 0.8 1 y(x) (exact) y(x)(h=0.2) y(x)(h=0.1) Differential Equation Initial Exact Condition Solution dydx=2xy (0.2) y=2x2+4Euler's Method In Exercises 79-81, complete the table using the exact solution of the differential equation and two approximations obtained using Eulers Method to approximate the particular solution of the differential equation. Use h=0.2 and h=0.1 , and compute each approximation to four decimal places. x 0 0.2 0.4 0.6 0.8 1 y(x) (exact) y(x)(h=0.2) y(x)(h=0.1) Differential Equation Initial Exact Condition Solution dydx=y+cosx (0,0) y=12(sinxcosx+ex)82E83E84E85EEXPLORING CONCEPTS Finding Values II is known that y=Cekx is a solution of the differential equation y=0.07y. Is it possible to determine C or k from the information given? Explain.87E88E89E90EElectric Circuit The diagram shows a simple electric circuit consisting of a power source, a resistor, and an inductor. A model of the current I, in amperes (A) at time t is given by the first-order differential equation LdIdt+RI=E(t) where E(t) is the voltage (V) produced by the power source, R is the resistance, in ohms and is the inductance, in henrys (H). Suppose the electric circuit consists of a 24-V power source, a 12- resistor, and a 4-H inductor. (a) Sketch a slope field for the differential equation. (b) What is the limiting value of the current? Explain.92E93E94EPUTNAM EXAM CHALLENGE Let f be a twice-differentiable real-valued function satisfying f(x)+f(x)=xg(x)f(x). where g(x)0 for all real x. Prove that |f(x)| is bounded.96ECONCEPT CHECK Describing Values Describe what the values of C and k represent in the exponential growth and decay model y=Cekt.CONCEPT CHECK Growth and Decay For y=Cekt, explain why exponential growth occurs when k0 and exponential decay occurs when k0.Solving a Differential Equation In Exercises 3-12, find the general solution of the differential equation. dydx=x+3Solving a Differential Equation In Exercises 3-12, find the general solution of the differential equation. dydx=58xSolving a Differential Equation In Exercises 3-12, find the general solution of the differential equation. dydx=y+3Solving a Differential Equation In Exercises 3-12, find the general solution of the differential equation. dydx=6ySolving a Differential Equation In Exercises 3-12, find the general solution of the differential equation. y=5xySolving a Differential Equation In Exercises 3-12, find the general solution of the differential equation. y=x4ySolving a Differential Equation In Exercises 3-12, find the general solution of the differential equation. y=xySolving a Differential Equation In Exercises 3-12, find the general solution of the differential equation. y=x(1+y)Solving a Differential Equation In Exercises 3-12, find the general solution of the differential equation. (1+x2)y2xy=0Solving a Differential Equation In Exercises 3-12, find the general solution of the differential equation. xy+y=100xWriting and Solving a Differential Equation In Exercises 13 and 14, write and find the general solution of the differential equation that models the verbal statement The rate of change of Q with respect to t is inversely proportional to the square of t.Writing and Solving a Differential Equation In Exercises 13 and 14, write and find the general solution of the differential equation that models the verbal statement The rate of change of P with respect to t is proportional to 25t.Slope Field In Exercises 15 and 16, a differential equation, a point, and a slope field are given. (a) Sketch two approximate solutions of the differential equation on the slope field, one of which passes through the given point (To print an enlarged copy of the graph, go to MathGraphs.com.) (b) Use integration and the given point to find the particular solution of the differential equation and use a graphing utility to graph the solution. Compare the result with the sketch in part (a) that passes through the given point. dydx=x(6y),(0,0)Slope Field In Exercises 15 and 16, a differential equation, a point, and a slope field are given. (a) Sketch two approximate solutions of the differential equation on the slope field, one of which passes through the given point (To print an enlarged copy of the graph, go to MathGraphs.com.) (b) Use integration and the given point to find the particular solution of the differential equation and use a graphing utility to graph the solution. Compare the result with the sketch in part (a) that passes through the given point. dydx=xy,(0,12)Finding a Particular Solution In Exercises 17-20, find the function y=f(t) passing through the point (0, 10) with the given differential equation. Use a graphing utility to graph the solution. dydt=12tFinding a Particular Solution In Exercises 17-20, find the function y=f(t) passing through the point (0, 10) with the given differential equation. Use a graphing utility to graph the solution. dydt=9tFinding a Particular Solution In Exercises 17-20, find the function y=f(t) passing through the point (0, 10) with the given differential equation. Use a graphing utility to graph the solution. dydt=12yFinding a Particular Solution In Exercises 17-20, find the function y=f(t) passing through the point (0, 10) with the given differential equation. Use a graphing utility to graph the solution. dydt=34yWriting and Solving a Differential Equation In Exercises 21 and 22, write and find the general solution of the differential equation that models the verbal statement. Evaluate the solution at the specified value of the independent variable. The rate of change of N is proportional to N. When t=0,N=250, and when t=1,N=400. What is the value of N when t=4?Writing and Solving a Differential Equation In Exercises 21 and 22, write and find the general solution of the differential equation that models the verbal statement. Evaluate the solution at the specified value of the independent variable. The rate of change of P is proportional to P. When t=0,P=5000, and when t=1,P=4750. What is the value of P when t=5?Finding an Exponential Function In Exercises 23-26, find the exponential function y=Cekt that passes through the two given points.24E25E26E27E28E29E30ERadioactive Decay In Exercises 29-36, complete the table for the radioactive isotope. Isotope Half-life (in years) Initial Quantity Amount After 1000 Years Amount After 10,000 Years 226Ra 1599 ____ ____ 0.1g32E33E34E35E36ERadioactive Decay Radioactive radium has a half-life of approximately 1599 years. What percent of a given amount remains after 100 years?Carbon Dating Carbon-14 dating assumes that the carbon dioxide on Earth today has the same radioactive content as it did centuries ago. If this is true, the amount of 14C absorbed by a tree that grew several centuries ago should be the same as the amount of 14C absorbed by a tree growing today. A piece of ancient charcoal contains only 15% as much of the radioactive carbon as a piece of modem charcoal. How long ago was the tree burned to make the ancient charcoal? (The half-life of 14C is 5715 years.)39E40E41E42E43E44E45E46E47E48E49E50E51E52E53E54E55EBacteria Growth The number of bacteria in a culture is increasing according to the law of exponential growth. There are 125 bacteria in the culture after 2 hours and 350 bacteria after 4 hours. (a) Find the initial population. (b) Write an exponential growth model for the bacteria population. Let t represent the time in hours. (c) Use the model to determine the number of bacteria after 8 hours. (d) After how many hours will the bacteria count be 25,000?57E58E59E60E61EForestry The value of a tract of timber is V(t)=100,000e0.8t where t is the time in years, with t=0 corresponding to 2010. If money earns interest continuously at 10%, then the present value of the timber at any time t is A(t)=V(t)e0.10t. Find the year in which the timber should be harvested to maximize the present value function.63ENoise Level With the installation of noise suppression materials, the noise level in an auditorium was reduced from 93 to 80 decibels. Use the function in Exercise 63 to find the percent decrease in the intensity level of the noise as a result of the installation of these materials.Newton's Law of Cooling When an object is removed from a furnace and placed in an environment with a constant temperature of 80F, its core temperature is 1500F. One hour after it is removed, the core temperature is 1120F. (a) Write an equation for the core temperature y of the object t hours after it is removed from the furnace. (b) What is the core temperature of the object 6 hours after it is removed from the furnace?Newton's Law of Cooling A container of hot liquid is placed in a freezer that is kept at a constant temperature of 20F. The initial temperature of the liquid is 160F. After 5 minutes, the liquids temperature is 60F. (a) Write an equation for the temperature y of the liquid t minutes after it is placed in the freezer. (b) How much longer will it take for the temperature of the liquid to decrease to 25F?67ETrue or False? In Exercises 67 and 68, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If prices are rising at a rate of 0.5% per month, then they are rising at a rate of 6% per year.Separation of Variaoles Determine whether each differential equation is separable. (a) y=2x5yy (b) yx=x2y+12E4E5EFinding a General Solution Using Separation of Variables In Exercises 5-18, find the general solution of the differential equation. dydx=3x2y27E8E9E10E11E12E13E14E15E16E17EFinding a General Solution Using Separation of Variables In Exercises 5-18, find the general solution of the differential equation. 12yy7ex=019E20E21E22E23E25E26E27EFinding a Particular Solution Using Separation of Variables In Exercises 19-28, find the particular solution of the differential equation that satisfies the initial condition. Differential Equation Initial Condition dT+k(T70)dt=0 T(0)=140Finding a Particular Solution Curve In Exercises 29-32, find an equation of the curve that passes through the point and has the given slope. (0,2),y=x4y30EFinding a Particular Solution Curve In Exercises 29-32, find an equation of the curve that passes through the point and has the given slope. (3,1),y=y5x32EUsing Slope In Exercises 33 and 34, find all functions f having the indicated property. The tangent to the graph of f at the point (x, y) intersects the x-axis at (x+2,0).34E35E36E37E38E39EChemical Reaction In a chemical reaction a certain compound changes into another compound at a rite proportional to the unchanged amount. There is 40 grans of the original compound initially and 35 grains after 1 hour. When will 75 percent of the compound be changedWeight Gain A calf that weighs 60 pounds at birth gains weight at the rate dw/dt=k(1200w) where w is the weight in pounds and t is the time in years. (a) Find the general solution of the differential equation. (b) Use a graphing utility to graph the particular solutions for k=0.8,0.9, and 1. (c) The animal is sold when its weight reaches 800 pounds. Find the time of sale for each of the models in part (b). (d) What is the maximum weight of the animal for each of the models in part (b)?42E43E44E45E46E47E48EEXPLORING CONCEPTS Separation of Variables Is an equation of the form dydx=f(x)g(y)f(x)h(y),(g)yh(y) separable? Explain.64E66ESailing Ignoring resistance, a sailboat starting from rest accelerates (dv/dt) at a rate proportional to the difference between the velocities of the wind and the boat. (a) The wind is blowing at 20 knots, and after a half-hour, the boat is moving at 10 knots. Write the velocity v as a function of time t (b) Use the result of put (a) to write the distance traveled by the boat as a function of time.Determining if a Function Is Homogeneous In Exercises 69-76, determine whether the function is homogeneous. and if it is, determine its degree. A function f ( x, y ) is homogeneous of degree n if f(tx,ty)=tnf(x,y). f(x,y)=x3+4xy2+y370E71E72E73E74E75E76E77E78E79ESolving a Homogeneous Differential Equation In Exercises 77-82, solve the homogeneous differential equation in terms of x and y. A homogeneous differential equation is an equation of the form M(x,y)dx+N(x,y)dy=0 where M and N are homogeneous functions of the same degree. To solve an equation of this form by the method of separation of variables, use the substitutions y=vxanddy=xdv+vdx. (x2+y2)dx2xydy=081E82E83E84E85E86E3E24E49EMatching In Exercises 49-52, match the logistic equation with its graph. [The graphs are labeled (a), (b), (c), and (d).] y=121+3ex51E52EUsing a Logistic Equation In Exercises 53 and 54, the logistic equation models the growth of a population. Use the equation to (a) find the value of k, (b) find the carrying capacity, (c) find the initial population, (d) determine when the population will reach 50% of its carrying capacity, and (e) write a logistic differential equation that has the solution P(t). P(t)=21001+29e0.75t54E55EUsing a Logistic Differential Equation In Exercises 55 and 56, the logistic differential equation models the growth rate of a population. Use the equation to (a) find the value of k, (b) find the carrying capacity, (c) graph a slope field using a computer algebra system, and (d) determine the value of P at which the population growth rate is the greatest. dPdt=0.1P0.0004P2Solving a Logistic Differential Equation In Exercises 57-60, find the logistic equation that passes through the given point. dydt=y(1y36),(0,4)58E59E60EEndangered Species A conservation organization releases 25 Florida panthers into a game preserve. After 2 years, there are 39 panthers in the preserve. The Florida preserve has a carrying capacity of 200 panthers. (a) Write a logistic equation that models the population of panthers in the preserve. (b) Find the population after 5 years. (c) When will the population reach 100? (d) Write a logistic differential equation that models the growth rate of the panther population. Then repeat part (b) using Eulers Method with a step size of h=1. Compare the approximation with the exact answer. (e) At what time is the panther population growing most rapidly? Explain.Bacteria Growth At time t=0. a bacterial culture weighs 1 gram. Two hours later, the culture weighs 4 grams. The maximum weight of the culture is 20 grams. (a) Write a logistic equation that models the weight of the bacterial culture. (b) Find the cultures weight after 5 hours. (c) When will the cultures weight reach 18 grams? (d) Write a logistic differential equation that models the growth rate of the cultures weight. Then repeat part (b) using Eulers Method with a step size of h=1. Compare the approximation with the exact answer. (e) At what time is the cultures weight increasing most rapidly? Explain.65E68E1E2EDetermining Whether a Differential Equation Is Linear In Exercises 3-6, determine whether the differential equation is linear. Explain your reasoning. x3y+xy=ex+14E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19EFinding a Particular Solution In Exercises 17-24, find the particular solution of the first-order linear differential equation for x0 that satisfies the initial condition. Differential Equation Initial Condition y+ysecx=secx y(0)=421E22EFinding a Particular Solution In Exercises 17-24, find the particular solution of the first-order linear differential equation for x0 that satisfies the initial condition. Differential Equation Initial Condition xdy=(x+y+2)dx y(1)=1024E25E