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All Textbook Solutions for Calculus (MindTap Course List)
51EThe marginal cost function C(x) was defined to be the derivative of the cost function. See Sections 2.7 and 3.7. The marginal cost of producing x gallons of orange juice is C(x)=0.820.00003x+0.000000003x2 measured in dollars per gallon. The fixed start-up cost is C(0)=18,000. Use the Net Change Theorem to find the cost of producing the first 4000 gallons of juice.2E3E4E5EThe supply function ps(x) for a commodity gives the relation between the selling price and the number of units that manufacturers will produce at that price. For a higher price, manufacturers will produce more units, so ps is an increasing function of x. Let X be the amount of the commodity currently produced and let P=ps(X) be the current price. Some producers would be willing to make and sell the commodity for a lower selling price and are therefore receiving more than their minimal price. The excess is called the producer surplus. An argument similar to that for consumer surplus shows that the surplus is given by the integral 0x[pps(x)]dx Calculate the producer surplus for the supply function ps(x)=3+0.01x2 at the sales level X = 10. Illustrate by drawing the supply curve and identifying the producer surplus as an area.7E8E9EA camera company estimates that the demand function for its new digital camera is p(x)=312e0.14x and the supply function is estimated to be ps(x)=26e0.2x, where x is measured in thousands. Compute the maximum total surplus.11E12E13E14E15E16E17EA hot, wet summer is causing a mosquito population explosion in a lake resort area. The number of mosquitoes is increasing at an estimated rate of 2200+10e0.8t per week where t is measured in weeks. By how much does the mosquito population increase between the fifth and ninth weeks of summer?19E20EThe dye dilution method is used to measure cardiac output with 6 mg of dye. The dye concentrations, in mg/L, are modeled by c(t)=20te0.6t,0t10, where t is measured in seconds. Find the cardiac output.22EThe graph of the concentration function c(t) is shown after a 7-mg injection of dye into a heart. Use Simpsons Rule to estimate the cardiac output.1E2E3E4ELet f(x)=c/(1+x2). a For what value of c is f a probability density function? b For that value of c, find P(1X1).6E7Ea Explain why the function whose graph is shown is a probability density function. b Use the graph to find the following probabilities: i P(X3) ii P(3X8) c Calculate the mean.Show that the median waiting time for a phone call to the company described in Example 4 is about 3.5 minutes.10EAn online retailer has determined that the average time for credit card transactions to be electronically approved is 1.6 seconds. a Use an exponential density function to find the probability that a customer waits less than a second for credit card approval. b Find the probability that a customer waits more than 3 seconds. c What is the minimum approval time for the slowest 5 of transactions?The time between infection and the display of symptoms for streptococcal sore throat is a random variable whose probabililty density function can be approximated by f(x)=115,676t2e0.05tif0t150 and f(t)=0 otherwise t measured in hours. a What is the probability that an infected patient will display symptoms within the first 48 hours? b What is the probability that an infected patient will not display symptoms until after 36 hours?13E14E15E16EThe speeds of vehicles on a highway with speed limit 100 km/h are normally distributed with mean 112 km/h and standard deviation 8 km/h. a What is the probability that a randomly chosen vehicle is traveling at a legal speed? b If police are instructed to ticket motorists driving 125 km/h or more, what percentage of motorists are targeted?18EFor any normal distribution, find the probability that the random variable lies within two standard deviations of the mean.20EThe hydrogen atom is composed of one proton in the nucleus and one electron, which moves about the nucleus. In the quantum theory of atomic structure, it is assumed that the electron does not move in a well-defined orbit. Instead, it occupies a state known as an orbital, which may be thought of as a cloud of negative charge surrounding the nucleus. At the state of lowest energy, called the ground state, or Is-orbital, the shape of this cloud is assumed to be a sphere centered at the nucleus. This sphere is described in terms of the probability density function p(r)=4a03r2e2r/a0r0 Where a0 is the Bohr radius (a05.591011m). The integral P(r)=0r4a03s2e2s/a0ds gives the probability that the electron will be found within the sphere of radius r meters centered at the nucleus. a Verify that p(r) is a probability density function. b Find limrp(r). For what value of r does p(r) have its maximum value? c Graph the density function. d Find the probability that the electron will be within the sphere of radius 4a0 centered at the nucleus. e Calculate the mean distance of the electron from the nucleus in the ground state of the hydrogen atom.1CC2CC3CCa What is the physical significance of the centre of mass of a thin plate? b If the plate lies between y=f(x) and y=0, where axb, write expressions for the coordinates of the centre of mass.5CC6CC7CC8CC9CC10CC1E2E3Ea Find the length of the curve y=x416+12x21x2 b Find the area of the surface obtained by rotating the curve in part a about the y-axis.5E6E7E8E9EFind the area of the surface obtained by rotating the curve in Exercise 9 about the y-axis.A gate in an irrigation canal is constructed in the form of a trapezoid 3 ft wide at the bottom, 5 ft wide at the top, and 2 ft high. It is placed vertically in the canal so that the water just covers the gate. Find the hydrostatic force on one side of the gate.A trough is filled with water and its vertical ends have the shape of the parabolic region in the figure. Find the hydrostatic force on one end of the trough.13E1314 Find the centroid of the region shown.15E16E17E18E19E20E21ELengths of human pregnancies are normally distributed with mean 268 days and standard deviation 15 days. What percentage of pregnancies last between 250 days and 280 days?The length of time spent waiting in line at a certain bank is modeled by an exponential density function with mean 8 minutes. a What is the probability that a customer is served in the first 3 minutes? b What is the probability that a customer has to wait more than 10 minutes? c What is the median waiting time?Find the area of the region S={(x,y)|x0,y1,x2+y24y}.Find the centroid of the region enclosed by the loop of the curve y2=x3x4.If a sphere of radius r is sliced by a plane whose distance from the centre of the sphere is d, then the sphere is divided into two pieces called segments of one base see the first figure. The corresponding surfaces are called spherical zones of one base. aDetermine the surface areas of the two spherical zones indicated in the figure. bDetermine the approximate area of the Arctic Ocean by assuming that it is approximately circular in shape, with center at the North Pole and circumference at 75 north latitude. Use r = 3960 mi for the radius of the earth. cA sphere of radius r is inscribed in the right circular cylinder of radius r. Two planes perpendicular to the central axis of the cylinder and a distance h apart cut off a spherical zone of two bases on the sphere see the second figure. Show that the surface area of the spherical zone equals the surface area of the region that the two planes cut off on the cylinder. d The Torrid Zone is the region on the surface of the earth that is between the Tropic of Cancer 23.45 north latitude and the Tropic of Capricorn 23.45 south latitude.What is the area of the Torrid Zone?a Show that an observer at height H above the north pole of a sphere of radius r can see a part of the sphere that has area 2r2Hr+H b Two spheres with radii r and R are placed so that the distance between their centers is d, where dr+R. Where should a light be placed on the line joining the centers of the spheres in order to illuminate the largest total surface?5PThe figure shows a semicircle with radius 1, horizontal diameter PQ, and tangent lines at P and Q. At what height above the diameter should the horizontal line be placed so as to minimize the shaded area?7PConsider a flat metal plate to be placed vertically underwater with its top 2 m below the surface of the water. Determine a shape for the plate so that if the plate is divided into any number of horizontal strips of equal height, the hydrostatic force on each strip is the same.9PA triangle with area 30cm2 is cut from a corner of a square with side 10 cm, as shown in the figure. If the centroid of the remaining region is 4 cm from the right side of the square, how far is it from the bottom of the square?11PIf the needle in Problem 11 has length hL, its possible for the needle to intersect more than one line. a If L=4, find the probability that a needle of length 7 will intersect at least one line. Hint: Proceed as in Problem 11. Define y as before; then the total set of possibilities for the needle can be identified with the same rectangular region 0yL, 0. What portion of the rectangle corresponds to the needle intersecting a line? b If L=4, find the probability that a needle of length 7 will intersect two lines. c If 2Lh3L, find a general formula for the probability that the needle intersects three lines.13PShow that y=23ex+e2x is a solution of the differential equation y+2y=2ex.Verify that y=tcostt is a solution of the initial-value problem tdydt=y+t2sinty()=0a For what values of r does the function y=erx satisfy the differential equation 2y+yy=0? b If r1 and r2 are the values of r that you found in part a, show that every member of the family of functions y=aer1x+ber2x is also a solution.4EWhich of the following functions are solutions of the differential equation y+y=sinx? a y=sinx b y=cosx c y=12xsinx d y=12xcosxa Show that every member of the family of functions y=(lnx+C)/x is a solution of the differential equation x2y+xy=1. b Illustrate part a by graphing several members of the family of solutions on a common screen. c Find a solution of the differential equation that satisfies the initial condition y(1)=2. d Find a solution of the differential equation that satisfies the initial condition y(2)=1.a What can you say about a solution of the equation y=y2 just by looking at the differential equation? b Verify that all members of the family y=1/(x+C) are solutions of the equation in part a. c Can you think of a solution of the differential equation y=y2 that is not a member of the family in part b? d Find a solution of the initial-value problem y=y2y(0)=0.5a What can you say about the graph of a solution of the equation y=xy3 when x is close to 0? What if x is large? b Verify that all members of the family y=(cx2)1/2 are solutions of the differential equation y=xy3. c Graph several members of the family of solutions on a common screen. Do the graphs confirm what you predicted in part a? d Find a solution of the initial-value problem y=xy3y(0)=29EThe Fitzhugh-Nagumo model for the electrical impulse in a neuron states that, in the absence of relaxation effects, the electrical potential in a neuron v(t) obeys the differential equation dvdt=v[v2(1+a)v+a] where a is a positive constant such that 0a1. a For what values of v is v unchanging that is, dv/dt=0? b For what values of v is v increasing? c For what values of v is v decreasing?Explain why the functions with the given graphs cant be solutions of the differential equation dydt=et(y1)2The function with the given graph is a solution of one of the following differential equations. Decide which is the correct equation and justify your answer. A. y=1+xy B. y=2xy C. y=12xyMatch the differential equations with the solution graphs labeled IIV. Give reasons for your choices. a y=1+x2+y2 b y=xex2y2 a y=11+ex2+y2 a y=sin(xy)cos(xy)Suppose you have just poured a cup of freshly brewed coffee with temperature 95C in a room where the temperature is 20C. a When do you think the coffee cools most quickly? What happens to the rate of cooling as time goes by? Explain. b Newtons Law of Cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings, provided that this difference is not too large. Write a differential equation that expresses Newtons Law of Cooling for this particular situation. What is the initial condition? In view of your answer to part a, do you think this differential equation is an appropriate model for cooling? c Make a rough sketch of the graph of the solution of the initial-value problem in part b.15EVon Bertalanffys equation states that the rate of growth in length of an individual fish is proportional to the difference between the current length L and the asymptotic length L in centimeters. a Write a differential equation that expresses this idea. b Make a rough sketch of the graph of a solution of a typical initial-value problem for this differential equation.Differential equations have been used extensively in the study of drug dissolution for patients given oral medications. One such equation is the Weibull equation for the concentration c(t) of the drug: dcdt=ktb(csc) where k and cs are positive constants and 0b1. Verify that c(t)=cs(1et1b) is a solution of the Weibull equation for t0, where =k/(1b). What does the differential equation say about how drug dissolution occurs?A direction field for the differential equation y=xcosy is shown. a Sketch the graphs of the solutions that satisfy the given initial conditions. i y(0)=0 ii y(0)=0.5 iii y(0)=1 iv y(0)=1.6 b Find all the equilibrium solutions.A direction field for the differential equation y=tan(12y) is shown. a Sketch the graphs of the solutions that satisfy the given initial conditions. i y(0)=1 ii y(0)=0.2 iii y(0)=2 iv y(1)=3 b Find all the equilibrium solutions.3E4E36 Match the differential equation with its direction field labeled IIV. Give reasons for your answer. y=x+y136 Match the differential equation with its direction field labeled IIV. Give reasons for your answer. y=sinxsiny7E8E9E910 Sketch a direction field for the differential equation. Then use it to sketch three solution curves. y=xy+111E12E13E14E15E16EUse a computer algebra system to draw a direction field for the differential equation y=y34y. Get a printout and sketch on it solutions that satisfy the initial condition y(0)=c for various values of c. For what values of c does limty(t) exist? What are the possible values for this limit?Make a rough sketch of a direction field for the autonomous differential equation y=f(y), where the graph of f is as shown. How does the limiting behavior of solutions depend on the value of y(0)?a Use Eulers method with each of the following step sizes to estimate the value of y(0.4), where y is the solution of the initial-value problem y=y,y(0)=1. i h=0.4 ii h=0.2 iii h=0.1 b We know that the exact solution of the initial-value problem in part a is y=ex. Draw, as accurately as you can, the graph of y=ex,0x0.4, together with the Euler approximations using the step sizes in part a. Your sketches should resemble Figures 11, 12, and 13. Use your sketches to decide whether your estimates in part a are underestimates or overestimates. c The error in Eulers method is the difference between the exact value and the approximate value. Find the errors made in part a in using Eulers method to estimate the true value of y(0.4), namely, e0.4. What happens to the error each time the step size is halved?A direction field for a differential equation is shown. Draw, with a ruler, the graphs of the Euler approximations to the solution curve that passes through the origin. Use step sizes h=1 and h=0.5. Will the Euler estimates be underestimates or overestimates? Explain.21E22EUse Eulers method with step size 0.1 to estimate y(0.5), where y(x) is the solution of the initial-value problem y=y+xy,y(0)=124Ea Program a calculator or computer to use Eulers method to compute y(1), where y(x) is the solution of the initial-value problem dydx+3x2y=6x2y(0)=3 i h=1 ii h=0.1 iii h=0.01 iv h=0.001 b Verify that y=2+ex3 is the exact solution of the differential equation. c Find the errors in using Eulers method to compute y(1) with the step sizes in part a. What happens to the error when the step size is divided by 10?a Program your computer algebra system, using Eulers method with step size 0.01, to calculate y(2), where y is the solution of the initial-value problem y=x3y3y(0)=1 b Check your work by using the CAS to draw the solution curve.The figure shows a circuit containing an electromotive force, a capacitor with a capacitance of C farads F, and a resistor with a resistance of R ohms . The voltage drop across the capacitor is Q/C, where Q is the charge in coulombs, C, so in this case Kirchhoffs Law gives RI+QC=E(t) But I=dQ/dt, so we have RdQdt+1CQ=E(t) Suppose the resistance is 5 , the capacitance is 0.05 F, and a battery gives a constant voltage of 60 V. a Draw a direction field for this differential equation. b What is the limiting value of the charge? c Is there an equilibrium solution? d If the initial charge is Q(0)=0C, use the direction field to sketch the solution curve. e If the initial charge is Q(0)=0C, use Eulers method with step size 0.1 to estimate the charge after half a second.In Exercise 9.1.14 we considered a 95C cup of coffee in a 20C room. Suppose it is known that the coffee cools at a rate of 1C per minute when its temperature is 70C. a What does the differential equation become in this case? b Sketch a direction field and use it to sketch the solution curve for the initial-value problem. What is the limiting value of the temperature? c Use Eulers method with step size h=2 minutes to estimate the temperature of the coffee after 10 minutes.110 Solve the differential equation. dydx=3x2y22E3E110 Solve the differential equation. y+xey=05E6E7E8E9E10E1118 Find the solution of the differential equation that satisfies the given initial condition. dydx=xey,y(0)=012E13E14E1118 Find the solution of the differential equation that satisfies the given initial condition. xlnx=y(1+3+y2)y,y(1)=116E17E18EFind an equation of the curve that passes through the point (0,2) and whose slope at (x,y) is x/y.Find the function f such that f(x)=xf(x)xandf(0)=2.21E22E23E24E25E26Ea Use a computer algebra system to draw a direction field for the differential equation. Get a printout and use it to sketch some solution curves without solving the differential equation. b Solve the differential equation. c Use the CAS to draw several members of the family of solutions obtained in part b. Compare with the curves from part a. y=y22728 a Use a computer algebra system to draw a direction field for the differential equation. Get a printout and use it to sketch some solution curves without solving the differential equation. b Solve the differential equation. c Use the CAS to draw several members of the family of solutions obtained in part b. Compare with the curves from part a. y=xy2932 Find the orthogonal trajectories of the family of curves. Use a graphing device to draw several members of each family on a common screen. x2+2y2=k22932 Find the orthogonal trajectories of the family of curves. Use a graphing device to draw several members of each family on a common screen. y2=kx32932 Find the orthogonal trajectories of the family of curves. Use a graphing device to draw several members of each family on a common screen. y=kx2932 Find the orthogonal trajectories of the family of curves. Use a graphing device to draw several members of each family on a common screen. y=1x+k3335 An integral equation is an equation that contains an unknown function y(x) and an integral that involves y(x). Solve the given integral equation. Hint: Use an initial condition obtained from the integral equation. y(x)=2+x2[tty(t)]dt3335 An integral equation is an equation that contains an unknown function y(x) and an integral that involves y(x). Solve the given integral equation. Hint: Use an initial condition obtained from the integral equation. y(x)=2+x1dtty(t),x035EFind a function f such that f(3)=2 and (t2+1)f(t)+[f(t)]2+1=0t1 Hint: Use the addition formula for tan(x+y) on Reference Page 2.37EIn Exercise 9.2.28 we discussed a differential equation that models the temperature of a 95C cup of coffee in a 20C room. Solve the differential equation to find an expression for the temperature of the coffee at time t.39EIn an elementary chemical reaction, single molecules of two reactants A and B form a molecule of the product C:A+BC. The law of mass action states that the rate of reaction is proportional to the product of the concentrations of A and B: d[C]dt=k[A][B] See Example 2.7.4. Thus, if the initial concentrations are [A]=amoles/L and [B]=bmoles/L and we write x=[C], then we have dxdt=k(ax)(bx) a Assuming that ab, find x as a function of t. Use the fact that the initial concentration of C is 0. b Find x(t) assuming that a=b. How does this expression for x(t) simplify if it is known that [C]=12a after 20 seconds?In contrast to the situation of Exercise 40, experiments show that the reaction H2+Br22HBr satisfies the rate law d[HBr]dt=k[H2][Br2]1/2 and so for this reaction the differential equation becomes dxdt=k(ax)(bx)1/2 where x=[HBr] and a and b are the initial concentrations of hydrogen and bromine. a Find x as a function of t in the case where a=b. Use the fact that x(0)=0. b If ab, find t as a function of x. Hint: In performing the integration, make the substitution u=bx.42E43EA certain small country has 10 billion in paper currency in circulation, and each day 50 million comes into the countrys banks. The government decides to introduce new currency by having the banks replace old bills with new ones whenever old currency comes into the banks. Let x=x(t) denote the amount of new currency in circulation at time t, with x(0)=0. a Formulate a mathematical model in the form of an initial-value problem that represents the flow of the new currency into circulation. b Solve the initial-value problem found in part a. c How long will it take for the new bills to account for 90 of the currency in circulation?45E46EA vat with 500 gallons of beer contains 4 alcohol by volume. Beer with 6 alcohol is pumped into the vat at a rate of 5 gal/min and the mixture is pumped out at the same rate. What is the percentage of alcohol after an hour?A tank contains 1000 L of pure water. Brine that contains 0.05 kg of salt per liter of water enters the tank at a rate of 5 L/min. Brine that contains 0.04 kg of salt per liter of water enters the tank at a rate of 10 L/min. The solution is kept thoroughly mixed and drains from the tank at a rate of 15 L/min. How much salt is in the tank a after t minutes and b after one hour?49EAn object of mass m is moving horizontally through a medium which resists the motion with a force that is a function of the velocity; that is, md2sdt2=mdvdt=f(v) where v=v(t) and s=s(t) represent the velocity and position of the object at time t, respectively. For example, think of a boat moving through the water. a Suppose that the resisting force is proportional to the velocity, that is, f(v)=kv, k a positive constant. This model is appropriate for small values of v. Let v(0)=v0 and s(0)=s0 be the initial values of v and s. Determine v and s at any time t. What is the total distance that the object travels from time t=0? b For larger values of v a better model is obtained by supposing that the resisting force is proportional to the square of the velocity, that is, f(v)=kv2,k0. This model was first proposed by Newton. Let v0 and s0 be the initial values of v and s. Determine v and s at any time t. What is the total distance that the object travels in this case?51E52ELet A(t) be the area of a tissue culture at time t and let M be the final area of the tissue when growth is complete. Most cell divisions occur on the periphery of the tissue and the number of cells on the periphery is proportional to A(t). So a reasonable model for the growth of tissue is obtained by assuming that the rate of growth of the area is jointly proportional to A(t) and MA(t). a Formulate a differential equation and use it to show that the tissue grows fastest when A(t)=13M. b Solve the differential equation to find an expression for A(t). Use a computer algebra system to perform the integration.54E12 A population grows according to the given logistic equation, where t is measure in weeks. a What is the carrying capacity? What is the value of k? b Write the solution of the equation. c What is the population after 10 weeks? dPdt=0.04P(1P1200),P(0)=601-2 A population grows according to the given logistic equation, where t is measured in weeks. a What is the carrying capacity? What is the value of k? b Write the solution of the equation. c What is the population after 10weeks? dpdt=0.02p0.0004p2,p(0)=40Suppose that a population develops according to the logistic equation dpdt=0.05P0.0005P2 where t is measured in weeks. a What is the carrying capacity? What is the value of k? b A direction field for this equation is shown. Where are the slopes close to 0? Where are they largest? Which solutions are increasing? Which solutions are decreasing? c Use the direction field to sketch solutions for initial populations of 20, 40, 60, 80, 120, and 140. What do these solutions have in common? How do they differ? Which solutions have inflection points? At what population levels do they occur? d What are the equilibrium solutions? How are the other solutions related to these solutions?Suppose that a population grows according to a logistic model with carrying capacity 6000 and k=0.0015 per year. a Write the logistic differential equation for these data. b Draw a direction field either by hand or with a computer algebra system. What does it tell you about the solution curves? c Use the direction field to sketch the solution curves for initial populations of 1000, 2000, 4000, and 8000. What can you say about the concavity of these curves? What is the significance of the inflection points? d Program a calculator or computer to use Eulers method with step size h=1 to estimate the population after 50 years if the initial population is 1000. f If the initial population is 1000, write a formula for the population after t years. Use it to find the population after 50 years and compare with your estimate in part d. g Graph the solution in part e and compare with the solution curve you sketched in partThe Pacific halibut fishery has been modeled by the differential equation dydt=ky(1yM) where y(t) is the biomass the total mass of the members of the population in kilograms at time t measured in years, the carrying capacity is estimated to be M=8107 kg, k=0.71 per year. a If y(0)=2107 kg, find the biomass a year later. b How long will it take for the biomass to reach 4107 kg?Suppose a population P(t) satisfies dpdt=0.4P0.001P2P(0)=50 where t is measured in years. a What is the carrying capacity? b What is P(0)? c When will the population reach 50 of the carrying capacity?Suppose a population grows according to a logistic model with initial population 1000 and carrying capacity 10, 000. If the population grows to 2500 after one year, what will the population be after another three years?The table gives the number of yeast cells in a new laboratory culture. Time hours Yeast cells 0 18 2 39 4 80 6 171 8 336 10 509 12 597 14 640 16 664 18 672 a Plot the data and use the plot to estimate the carrying capacity for the yeast population. b Use the data to estimate the initial relative growth rate. c Find both an exponential model and a logistic model for these data. d Compare the predicted values with the observed values, both in a table and with graphs. Comment on how well your models fit the data. e Use your logistic model to estimate the number of yeast cells after 7 hours.9E10EOne model for the spread of a rumor is that the rate of spread is proportional to the product of the fraction y of the population who have heard the rumor and the fraction who have not heard the rumor. a Write a differential equation that is satisfied by y. b Solve the differential equation. c A small town has 1000 inhabitants. At 8 am, 80 people have heard a rumor. By noon half the town has heard it. At what time will 90 of the population have heard the rumor?Biologists stocked a lake with 400 fish and estimated the carrying capacity the maximal population for the fish of that species in that lake to be 10, 000. The number of fish tripled in the first year. a Assuming that the size of the fish population satisfies the logistic equation, find an expression for the size of the population after t years. b How long will it take for the population to increase to 5000?13E14E15E16EConsider a population P=P(t) with constant relative birth and death rates and , respectively, and a constant emigration rate m, where ,, and m are positive constants. Assume that Then the rate of change of the population at time t is modeled by the differential equation dPdt=kPmwherek=- a Find the solution of this equation that satisfies the initial condition P(0)=P0. b What condition on m will lead to an exponential expansion of the population? c What condition on m will result in a constant population? A population decline? d In 1847, the population of Ireland was about 8 million and the difference between the relative birth and death rates was 1.6 of the population. Because of the potato famine in the 1840s and 1850s, about 210, 000 inhabitants per year emigrated from Ireland. Was the population expanding or declining at that time?Let c be a positive number. A differential equation of the form dydt=ky1+e where k is a positive constant, is called a doomsday equation because the exponent in the expression ky1+e is larger than the exponent 1 for natural growth. a Determine the solution that satisfies the initial condition y(0)=y0.. b Show that there is a finite time t=T doomsday such that limtT(t)=. c An especially prolific breed of rabbits has the growth term ky1.01. If 2 such rabbits breed initially and the warren has 16 rabbits after three months, then when is doomsday?Lets modify the logistic differential equation of Example 1 as follows: dPdt=0.08P(1P1000)15 a Suppose P(t) represents a fish population at time t, where t is measured in weeks. Explain the meaning of the final term in the equation (15). b Draw a direction field for this differential equation. c What are the equilibrium solutions? d Use the direction field to sketch several solution curves. Describe what happens to the fish population for various initial populations. e Solve this differential equation explicitly, either by using partial fractions or with a computer algebra system. Use the initial populations 200 and 300. Graph the solutions and compare with your sketches in part d.Consider the differential equation dPdt=0.08P(1P1000)c as a model for a fish population, where t is measured in weeks and c is a constant. a Use a CAS to draw direction fields for various values of c. b From your direction fields in part a, determine the values of c for which there is at least one equilibrium solution. For what values of c does the fish population always die out? c Use the differential equation to prove what you discovered graphically in part b. d What would you recommend for a limit to the weekly catch of this fish population?There is considerable evidence to support the theory that for some species there is a minimum population m such that the species will become extinct if the size of the population falls below m. This condition can be incorporated into the logistic equation by introducing the factor (1m/P). Thus the modified logistic model is given by the differential equation dPdt=kP(1PM)(1mP) a Use the differential equation to show that any solution is increasing if mPM and decreasing if 0Pm. b For the case where k=0.08, M=1000, and m=200, draw a direction field and use it to sketch several solution curves. Describe what happens to the population for various initial populations. What are the equilibrium solutions? c Solve the differential equation explicitly, either by using partial fractions or with a computer algebra system. Use the initial population P0. d Use the solution in part c to show that if p0m, then the species will become extinct. Hint: Show that the numerator in your expression for P(t) is 0 for some value of t.22EIn a seasonal-growth model, a periodic function of time is introduced to account for seasonal variations in the rate of growth. Such variations could, for example, be caused by seasonal changes in the availability of food. a Find the solution of the seasonal-growth model dPdt=kPcos(rt)P(0)=P0 where k, r, and are positive constants. b By graphing the solution for several values of k, r, and , explain how the values of k, r, and affect the solution. What can you say about limtP(t)?24E25E14 Determine whether the differential equation is linear y+xy=x22E3E4E514 Solve the differential equation. y+y=1514 Solve the differential equation. yy=ex514 Solve the differential equation. y=xy514 Solve the differential equation. 4x3y+x4y=sin3x9E10E11E12E13E14E15E16E17E18E1520 Solve the initial-value problem. xy=y+x2sinx,y()=020E21E22EA Bernoulli differential equation named after James Bernoulli is of the form dydx+P(x)y=Q(x)yn Observe that, if n=0 or 1, the Bernoulli equation is linear. For other values of n, show that the substitution u=y1n transforms the Bernoulli equation into the linear equation dudx+(1n)P(x)u=(1n)Q(x)2425 Use the method of Exercise 23 to solve the differential equation. xy+y=xy225E26E27E28E29E30E31E32EIn Section 9.3 we looked at mixing problems in which the volume of fluid remained constant and saw that such problems give rise to separable differentiable equations. See Example 6 in that section. If the rates of flow into and out of the system are different, then the volume is not constant and the resulting differential equation is linear but not separable. A tank contains 100 L of water. A solution with a salt concentration of 0.4 kg/L is added at a rate of 5 L/min. The solution is kept mixed and is drained from the tank at a rate of 3 L/min. If y(t) is the amount of salt in kilograms after t minutes, show that y satisfies the differential equation dydt=23y100+2r Solve this equation and find the concentration after 20 minutes.34EAn object with mass m is dropped from rest and we assume that the air resistance is proportional to the speed of the object. If s(t) is the distance dropped after t seconds, then the speed is v=s(t) and the acceleration is a=v(t). If g is the acceleration due to gravity, then the downward force on the object is mgcv, where c is a positive constant, and Newtons Second Law gives mdvdt=mgcv a Solve this as a linear equation to show that v=mgc(1ect/m) b What is the limiting velocity? c Find the distance the object has fallen after t seconds.If we ignore air resistance, we can conclude that heavier objects fall no faster than lighter objects. But if we take air resistance into account, our conclusion changes. Use the expression for the velocity of a falling object in Exercise 35a to find dv/dm and show that heavier objects do fall faster than lighter ones.37ETo account for seasonal variation in the logistic differential equation, we could allow k and M to be functions of t: dPdt=k(t)P(1PM(t)) a Verify that the substitution z=1/P transforms this equation into the linear equation dzdt+k(t)z=k(t)M(t) b Write an expression for the solution of the linear equation in part a and use it to show that if the carrying capacity M is constant, then P(t)=MCMek(t)dt Deduce that if 0k(t)dt=, then limtP(t)=M. This will be true if k(t)=k0+acosbt with k00, which describes a positive intrinsic growth rate with a periodic seasonal variation. c If k is constant but M varies, show that z(t)=ekt0tkeksM(s)ds+Cekt and use lHospitals Rule to deduce that if M(t) has a limit as t, then P(t) has the same limit.For each predator-prey system, determine which of the variables, x or y, represents the prey population and which represents the predator population. Is the growth of the prey restricted just by the predators or by other factors as well? Do the predators feed only on the prey or do they have additional food sources? Explain. a dxdt=0.05x+0.0001xy dydt=0.1y0.005xy b dxdt=0.2x0.0002x20.006xy dydt=0.015y+0.00008xyEach system of differential equations is a model for two species that either compete for the same resources or cooperate for mutual benefit flowering plants and insect pollinators, for instance. Decide whether each system describes competition or cooperation and explain why it is a reasonable model. Ask yourself what effect an increase in one species has on the growth rate of the other. a dxdt=0.12x0.0006x2+0.00001xy dydt=0.08x+0.00004xy b dxdt=0.15x0.0002x20.0006xy dydt=0.2y0.00008y20.0002xy3E4E56 A phase trajectory is shown for populations of rabbits R and foxes F. a Describe how each population changes as time goes by. b Use your description to make a rough sketch of the graphs of R and F as functions of time.56 A phase trajectory is shown for populations of rabbits R and foxes F. a Describe how each population changes as time goes by. b Use your description to make a rough sketch of the graphs of R and F as functions of time. .78 Graphs of populations of two species are shown. Use them to sketch the corresponding phase trajectory.8E9EPopulations of aphids and ladybugs are modeled by the equations dAdt=2A0.01AL dLdt=0.5L+0.0001AL a Find the equilibrium solutions and explain their significance. b Find an expression for dL/dA. c The direction field for the differential equation in part b is shown. Use it to sketch a phase portrait. What do the phase trajectories have in common? d Suppose that at time t = 0 there are 1000 aphids and 200 ladybugs. Draw the corresponding phase trajectory and use it to describe how both populations change. e Use part d to make rough sketches of the aphid and ladybug populations as functions of t. How are the graphs related to each other?In Example 1 we used Lotka-Volterra equations to model populations of rabbits and wolves. Lets modify those equations as follows: dRdt=0.08R(10.0002R)0.001RW dWdt=0.02W+0.00002RW a According to these equations, what happens to the rabbit population in the absence of wolves? b Find all the equilibrium solutions and explain their significance. c The figure shows the phase trajectory that starts at the point 1000, 40. Describe what eventually happens to the rabbit and wolf populations. d Sketch graphs of the rabbit and wolf populations as functions of time.In Exercise 10 we modeled populations of aphids and ladybugs with a Lotka-Volterra system. Suppose we modify those equations as follows: dAdt=2A(10.0001A)0.01AL dLdt=0.5L+0.0001AL a In the absence of ladybugs, what does the model predict about the aphids? b Find the equilibrium solutions. c Find an expression for dL/dA. d Use a computer algebra system to draw a direction field for the differential equation in part c. Then use the direction field to sketch a phase portrait. What do the phase trajectories have in common? e Suppose that at time t = 0 there are 1000 aphids and 200 ladybugs. Draw the corresponding phase trajectory and use it to describe how both populations change. f Use part e to make rough sketches of the aphid and ladybug populations as functions of t. How are the graphs related to each other?1CC2CC3CC4CC5CC6CC7CC8CCa Write Lotka-Volterra equations to model populations of food-fish F and sharks S. b What do these equations say about each population in the absence of the other?1TFQ2TFQ3TFQDetermine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The equation y=3y2x+6xy1 is separable.5TFQDetermine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The equation y+xy=ey is linear.7TFQ1Ea Sketch a direction field for the differential equation y=x/y. Then use it to sketch the four solutions that satisfy the initial conditions y(0)=1,y(0)=1,y(2)=1, and y(2)=1. b Check your work in part a by solving the differential equation explicitly. What type of curve is each solution curve?a A direction field for the differential equation y=x2y2 is shown. Sketch the solution of the initial-value problem y=x2y2y(0)=1 Use your graph to estimate the value of y(0.3). b Use Eulers method with step size 0.1 to estimate y(0.3), where y(x) is the solution of the initial-value problem in part a. Compare with your estimate from part a. c On what lines are the centers of the horizontal line segments of the direction field in part a located? What happens when a solution curve crosses these fines?4E5E6E58 Solve the differential equation. 2yey2y=2x+3x58 Solve the differential equation. x2yy=2x3e1/x911 Solve the initial-value problem. drdt+2tr=r,r(0)=5911 Solve the initial-value problem. (1+cosx)y=(1+ey)sinx,y(0)=011E12E1314 Find the orthogonal trajectories of the family of curves. y=kex14E15Ea The population of the world was 6.1 billion in 2000 and 6.9 billion in 2010. Find an exponential model for these data and use the model to predict the world population in the year 2020. b According to the model in part a, when will the world population exceed 10 billion? c Use the data in part a to find a logistic model for the population. Assume a carrying capacity of 20 billion. Then use the logistic model to predict the population in 2020. Compare with your prediction from the exponential model. d According to the logistic model, when will the world population exceed 10 billion? Compare with your prediction in part b.17E18EOne model for the spread of an epidemic is that the rate of spread is jointly proportional to the number of infected peopleand the number of uninfected people. In an isolated town of 5000 inhabitants, 160 people have a disease at the beginning of the week and 1200 have it at the end of the week. How long does it for 80 of the population to become infected?