Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Single Variable Calculus: Early Transcendentals
19EThe standard deviation for a random variable with probability density function f and mean is defined by =[(x)2f(x)dx]1/2 Find the standard deviation for an exponential density function with mean .21E1RCCWhat can you say about the solutions of the equation y = x2 + y2 just by looking at the differential equation?3RCC4RCC5RCC6RCC7RCC8RCC9RCCDetermine 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. 1. All solutions of the differential equation y = 1 y4 are decreasing functions.2RQDetermine 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. 3. The equation y = x + y is separable.4RQ5RQDetermine 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. 6. The equation y + xy = ey is linear.7RQ1RE2RE3RE4RESolve the differential equation. 5. y = xesin x y cos x6RESolve the differential equation. 7. 2yey2y=2x+3x8RE9RE10RE11RE12RE13RE14RE15RE16RE17REA tank contains 100 L of pure water. Brine that contains 0.1 kg of salt per liter enters the tank at a rate of 10 L/min. The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt is in the tank after 6 minutes?One model for the spread of an epidemic is that the rate of spread is jointly proportional to the number of infected people and 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 take for 80% of the population to become infected?The Brentano-Stevens Law in psychology models the way that a subject reacts to a stimulus. It slates that if R represents the reaction to an amount S of stimulus, then the relative rates of increase are proportional: 1RdRdt=kSdSdt where k is a positive constant. Find R as a function of S.The transport of a substance across a capillary wall in lung physiology has been modeled by the differential equation dhdt=RV(hk+h) where h is the hormone concentration in the bloodstream, t is time, R is the maximum transport rate, V is the volume of the capillary, and k is a positive constant that measures the affinity between the hormones and the enzymes that assist the process. Solve this differential equation to find a relationship between h and t.Populations of birds and insects are modeled by the equations dxdt=0.4x0.002xydydt=0.2y+0.000008xy (a) Which of the variables, x or y, represents the bird population and which represents the insect population? Explain. (b) Find the equilibrium solutions and explain their significance. (c) Find an expression for dy/dx. (d) The direction field for the differential equation in part (c) is shown. Use it to sketch the phase trajectory corresponding to initial populations of 100 birds and 40,000 insects. Then use the phase trajectory to describe how both populations change. (e) Use part (d) to make rough sketches of the bird and insect populations as functions of time. How are these graphs related to each other?23REBarbara weighs 60 kg and is on a diet of 1600 calories per day, of which 850 are used automatically by basal metabolism. She spends about 15 cal/kg/day times her weight doing exercise. If 1 kg of fat contains 10,000 cal and we assume that the storage of calories in the form of fat is 100% efficient, formulate a differential equation and solve it to find her weight as a function of time. Does her weight ultimately approach an equilibrium weight?1P2P3PFind all functions f that satisfy the equation (f(x)dx)(1f(x)dx)=15P6P7PSnow began to fall during the morning of February 2 and continued steadily into the afternoon. At noon a snowplow began removing snow from a road at a constant rate. The plow traveled 6 km from noon to 1 pm but only 3 km from 1 pm to 2 pm. When did the snow begin to fall? [Hints: To get started, let t be the time measured in hours after noon; let x(t) be the distance traveled by the plow at time t; then the speed of the plow is dx/dt. Let b be the number of hours before noon that it began to snow. Find an expression for the height of the snow at time t. Then use the given information that the rate of removal R (in m3/h) is constant.]9P10P11P12P13P14P15PShow that y=23ex+e2x is a solution of the differential equation y + 2y = 2ex.2E(a) For what values of r does the function y = erx satisfy the differential equation 2y + y y = 0? (b) if r1 and r2 are the values of r that you found in pint (a), show that every member of the family of functions y=aer1x+ber2x is also a solution.(a) For what values of k does the function y = cos kt satisfy the differential equation 4y = 25y? (b) For those values of k, verify that every member of the family of functions y = A sin kt + B cos kt is also a solution.Which of the following functions are solutions of the differential equation y + y = sin x? (a) y = sin x (b) y = cos x (c) y=12xsinx (d) y=12xcosx(a) Show that every member of the family of functions y = (ln x + C)/x 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) = 1.7E8E9E10EExplain why the functions with the given graphs cant be solutions of the differential equation dydt=et(y1)2 (a) (b)12E13E14EPsychologists interested in learning theory study learning curves, A learning curve is the graph of a function P(t), the performance of someone learning a skill as a function of the training time t. The derivative dP/dt represents the rate at which performance improves. (a) When do you think P increases most rapidly? What happens to dP/dt as t increases? Explain. (b) If M is the maximum level of performance of which the learner is capable, explain why the differential equation dPdt=k(MP)kapositiveconstant is a reasonable model for learning. (c) Make a rough sketch of a possible solution of this differential equation.Von 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.17EA direction field for the differential equation y = x cos y 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.2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19EA 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) = 1.24E25E26E27E28ESolve the differential equation. 1. dydx=3x2y22E3E4ESolve the differential equation. 5. (ey 1)y = 2 + cos x6E7E8E9E10E11E12E13E14E15E16E17E18EFind an equation of the curve that passes through the point (0, 2) and whose slope at (x, y) is x/y.20ESolve the differential equation y = x + y by making the change of variable u = x + y.Solve the differential equation xy = y + xey/x by making the change of variable v = y/x.23E24E25E26E27E28E29E30E31E32E33EAn 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.] 34.y(x)=2+1xdtty(t), x 035EFind a function f such that f(3) = 2 and (t2+1)f(t)+[f(t)]2+1=0 t 1 [Hint: Use the addition formula for tan(x + y) on Reference Page 2.]Solve the initial-value problem in Exercise 9.2.27 to find an expression for the charge at time t. Find the limiting value of the charge.38EIn Exercise 9.1.15 we formulated a model for learning in the form of the differential equation dPdt=k(MP) where P(t) measures the performance of someone learning a skill after a training time t, M is the maximum level of performance, and k is a positive constant. Solve this differential equation to find an expression for P(t). What is the limit of this expression?40E41EA sphere with radius 1 m has temperature 15C. It lies inside a concentric sphere with radius 2 m and temperature 25C. The temperature T(r) at a distance r from the common center of the spheres satisfies the differential equation d2Tdr2+2rdTdr=0 If we let S = dT/dr, then S satisfies a first-order differential equation. Solve it to find an expression for the temperature T(r) between the spheres.A glucose solution is administered intravenously into the bloodstream at a constant rate r. As the glucose is added, it is converted into other substances and removed from the bloodstream at a rate that is proportional to the concentration at that time. Thus a model for the concentration C = C(t) of the glucose solution in the bloodstream is dCdt=rkC where k is a positive constant. (a) Suppose that the concentration at time t = 0 is C0. Determine the concentration at any time t by solving the differential equation. (b) Assuming that C0 r/k, find limt C(t) and interpret your answer.A 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?45E46E47E48E49E50E51E52E53E54E1EA 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 10 weeks? 2. dPdt=0.02P0.0004P2,P(0)=403E4EThe 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 lime t (measured in years), the carrying capacity is estimated to be M = 8 107 kg, and k = 0.71 per year. (a) If y(0) = 2 107 kg, find the biomass a year later. (b) How long will it take for the biomass to reach 4 107 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?7E8E9E10E11EBiologists 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?13E14E15E16E17ELet c be a positive number. A differential equation of the form dydt=ky1+c where k is a positive constant, is called a doomsday equation because the exponent in the expression ky1+c 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 y(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?19E20EThere 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 (1 m/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 m P M and decreasing if 0 P m. (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 P0 m, then the species will become extinct. [Hint: Show that the numerator in your expression for P(t) is 0 for some value of t.]Another model for a growth function for a limited population is given by the Gompertz function, which is a solution of the differential equation dPdt=cln(MP)P where c is a constant and M is the carrying capacity. (a) Solve this differential equation. (b) Compute limt P(t). (c) Graph the Gompertz growth function for M = 1000, P0 = 100, and c = 0.05, and compare it with the logistic function in Example 2. What are the similarities? What are the differences? (d) We know from Exercise 13 that the logistic function grows fastest when P = M/2. Use the Gompertz differential equation to show that the Gompertz function grows fastest when P = M/e.23E24E25E1E2E3E4E5E6E7E8E9E10E11E12ESolve the differential equation. 13. t2dydt+3ty=1+t2, t 0Solve the differential equation. 14. tlntdrdt+r=tet15E16E17E18E19E20E21E22E23E24EUse the method of Exercise 23 to solve the differential equation. 25. y+2xy=y3x226EIn the circuit shown in Figure 4, a battery supplies a constant voltage of 40 V, the inductance is 2 H, the resistance is 10 , and I(0) = 0. (a) Find I(t). (b) Find the current after 0.1 seconds.In the circuit shown in Figure 4, a generator supplies a voltage of E(t) = 40 sin 60t volts, the inductance is 1 H, the resistance is 20 , and I(0) = 1 A. (a) Find I(t). (b) Find the current after 0.1 seconds. (c) Use a graphing device to draw the graph of the current function.29E30ELet P(t) be the performance level of someone learning a skill as a function of the training time t. The graph of P is called a learning curve. In Exercise 9.1.15 we proposed the differential equation dPdt=k[MP(t)] as a reasonable model for learning, where k is a positive constant. Solve it as a linear differential equation and use your solution to graph the learning curve.32EIn 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+2t Solve this equation and find the concentration after 20 minutes.A tank with a capacity of 400 L is full of a mixture of water and chlorine with a concentration of 0.05 g of chlorine per liter. In order to reduce the concentration of chlorine, fresh water is pumped into the tank at a rate of 4 L/s. The mixture is kept stirred and is pumped out at a rate of 10 L/s. Find the amount of chlorine in the tank as a function of time.35E36E37E38E1EEach 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+0.00001xydydt=0.08x+0.00004xy (b) dxdt=0.15x0.0002x20.0006xydydt=0.2y0.00008y20.0002xy3ELynx eat snowshoe hares and snowshoe hares eat woody plants like willows. Suppose that, in the absence of hares, the willow population will grow exponentially and the lynx population will decay exponentially. In the absence of lynx and willow, the hare population will decay exponentially. If L(t), H(t), and W(t) represent the populations of these three species at time t, write a system of differential equations as a model for their dynamics. If the constants in your equation are all positive, explain why you have used plus or minus signs.5E6E7E8E9E10EIn 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)0.001RWdWdt=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.12E1RCC2RCC3RCC4RCC5RCC6RCC7RCC8RCC9RCC1RQ2RQ3RQ4RQ5RQ6RQ7RQ8RQ9RQ10RQ1RE2RE3RE4RE5RE6RE7RE8RE9RE10RE11RE12RE13RESketch the polar curve. 14. r = 2 cos(/2)15RE16RE17RE18RE19RE20RE21RE22RE23RE24RE25RE26RE27RE28RE29RE30REFind the area enclosed by the curve r2 = 9 cos 5.32RE33RE34RE35RE36RE37RE38RE39REFind the length of the curve. 40. r = sin3(/3), 041RE42RE43RE44RE45RE46RE47RE48RE49RE