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All Textbook Solutions for Calculus: Early Transcendentals (3rd Edition)
Assuming solutions are unique (at most one solution curve passes through each point), explain why a solution curve cannot cross the line y = 2 in Example 1. Example 1 Direction Field For a Linear Differential Equation Figure 9.11 shows the direction field for the equation y(t) = y 2, for t 0 and y 0. For what initial conditions at t = 0 are the solutions constant? Increasing? Decreasing?2QC3QCNotice that the errors in Table 9.1 increase in time for both time steps. Give a possible explanation for this increase in the errors. Table 9.1Explain how to sketch the direction field of the equation y(t) = f(t, y), where f is given.2E3E4EIdentifying direction fields Which of the differential equations ad corresponds to the following direction field? Explain your reasoning. a.y(t) = 0.5(y + 1)(t 1) b.y(t) = 0.5(y + 1)(t 1) c.y(t) = 0.5(y 1)(t + 1) d.y(t) = 0.5(y 1)(t + 1)Direction fields A differential equation and its direction field are shown in the following figures. Sketch a graph of the solution curve that passes through the given initial conditions. 5.y(t)=t2y2+1,y(0)=2 and y(2) = 0.8EDirection fields with technology Plot a direction field for the following differential equation with a graphing utility. Then find the solutions that are constant and determine which initial conditions y(0) = A lead to solutions that are increasing in time. y(t) = 0.5(y + 1)2 (t 1)2, |t| 3 and |y| 310EDirection fields with technology Plot a direction field for the following differential equation with a graphing utility. Then find the solutions that are constant and determine which initial conditions y(0) = A lead to solutions that are increasing in time. 11.y(t)=t(y1),0t2,0y2Sketching direction fields Use the window [2, 2] [2, 2] to sketch a direction field for the following equations. Then sketch the solution curve that corresponds to the given initial condition. A detailed direction field is not needed. 12.y(t) = y 3, y(0) = 1Sketching direction fields Use the window [2, 2] [2, 2] to sketch a direction field for the following equations. Then sketch the solution curve that corresponds to the given initial condition. A detailed direction field is not needed. 13.y(t) = 4 y, y(0) = 1Sketching direction fields Use the window [2, 2] [2, 2] to sketch a direction field for the following equations. Then sketch the solution curve that corresponds to the given initial condition. A detailed direction field is not needed. 14.y(t) = y(2 y), y(0) = 1Sketching direction fields Use the window [2, 2] [2, 2] to sketch a direction field for the following equations. Then sketch the solution curve that corresponds to the given initial condition. A detailed direction field is not needed. 15.y(x) = sin x, y(2) = 2Sketching direction fields Use the window [2, 2] [2, 2] to sketch a direction field for the following equations. Then sketch the solution curve that corresponds to the given initial condition. A detailed direction field is not needed. 16.y(x)=siny,y(2)=12Increasing and decreasing solutions Consider the following differential equations. A detailed direction field is not needed. a.Find the solutions that are constant, for all t 0 (the equilibrium solutions). b.In what regions are solutions increasing? Decreasing? c.Which initial conditions y(0) = A lead to solutions that are increasing in time? Decreasing? d.Sketch the direction field and verify that it is consistent with parts (a)(c). 17.y(t) = (y 1)(1 + y)Increasing and decreasing solutions Consider the following differential equations. A detailed direction field is not needed. a.Find the solutions that are constant, for all t 0 (the equilibrium solutions). b.In what regions are solutions increasing? Decreasing? c.Which initial conditions y(0) = A lead to solutions that are increasing in time? Decreasing? d.Sketch the direction field and verify that it is consistent with parts (a)(c). 18.y(t) = (y 2)(y + 1)Increasing and decreasing solutions Consider the following differential equations. A detailed direction field is not needed. a.Find the solutions that are constant, for all t 0 (the equilibrium solutions). b.In what regions are solutions increasing? Decreasing? c.Which initial conditions y(0) = A lead to solutions that are increasing in time? Decreasing? d.Sketch the direction field and verify that it is consistent with parts (a)(c). 19.y(t) = cos y, for |y|Increasing and decreasing solutions Consider the following differential equations. A detailed direction field is not needed. a.Find the solutions that are constant, for all t 0 (the equilibrium solutions). b.In what regions are solutions increasing? Decreasing? c.Which initial conditions y(0) = A lead to solutions that are increasing in time? Decreasing? d.Sketch the direction field and verify that it is consistent with parts (a)(c). 20.y(t) = y(y + 3)(4 y)Logistic equations Consider the following logistic equations, for t 0. In each case, sketch the direction field, draw the solution curve for each initial condition, and find the equilibrium solutions. A detailed direction field is not needed. Assume t 0 and P 0. 21.P(t)=0.05P(1P500);P(0)=100,P(0)=400,P(0)=700Logistic equations Consider the following logistic equations, for t 0. In each case, sketch the direction field, draw the solution curve for each initial condition, and find the equilibrium solutions. A detailed direction field is not needed. Assume t 0 and P 0. 22.P(t)=0.1P(1P1200);P(0)=600,P(0)=800,P(0)=1600Logistic equations Consider the following logistic equations, for t 0. In each case, sketch the direction field, draw the solution curve for each initial condition, and find the equilibrium solutions. A detailed direction field is not needed. Assume t 0 and P 0. 23.P(t)=0.02P(4P800);P(0)=1600,P(0)=2400,P(0)=4000Logistic equations Consider the following logistic equations, for t 0. In each case, sketch the direction field, draw the solution curve for each initial condition, and find the equilibrium solutions. A detailed direction field is not needed. Assume t 0 and P 0. 24.P(t)=0.05P0.001P2;P(0)=10,P(0)=40,P(0)=80P(0)=4000Two steps of Eulers method For the following initial value problems, compute the first two approximations u1 and u2 given by Eulers method using the given time step. 25.y(t) = 2y, y(0) = 2; t = 0.5Two steps of Eulers method For the following initial value problems, compute the first two approximations u1 and u2 given by Eulers method using the given time step. 26.y(t) = y, y(0) = 1; t = 0.2Two steps of Eulers method For the following initial value problems, compute the first two approximations u1 and u2 given by Eulers method using the given time step. 27.y(t) = 2 y, y(0) = 1; t = 0.1Two steps of Eulers method For the following initial value problems, compute the first two approximations u1 and u2 given by Eulers method using the given time step. 28.y(t) = t + y, y(0) = 4; t = 0.5Errors in Eulers method Consider the following initial value problems. a.Find the approximations to y(0, 2) and y(0.4) using Eulers method with time steps of t = 0.2, 0.1, 0.05, and 0.025. b.Using the exact solution given, compute the errors in the Euler approximations at t = 0.2 and t = 0.4. c.Which time step results in the more accurate approximation? Explain your observations. d.In general, how does halving the time step affect the error at t = 0.2 and t = 0.4? 29.y(t) = y, y(0) = 1; y(t) = etErrors in Eulers method Consider the following initial value problems. a.Find the approximations to y(0, 2) and y(0.4) using Eulers method with time steps of t = 0.2, 0.1, 0.05, and 0.025. b.Using the exact solution given, compute the errors in the Euler approximations at t = 0.2 and t = 0.4. c.Which time step results in the more accurate approximation? Explain your observations. d.In general, how does halving the time step affect the error at t = 0.2 and t = 0.4? 29.y(t) = y, y(0) = 1; y(t) = et31E32E33E34E35E36E37EEquilibrium solutions A differential equation of the form y(t) = f(y) is said to be autonomous (the function f depends only on y). The constant function y = y0 is an equilibrium solution of the equation provided f(y0) = 0 (because then y(t) = 0 and the solution remains constant for all t). Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations. a. Find the equilibrium solutions. b. Sketch the direction field, for t 0. c. Sketch the solution curve that corresponds to the initial condition y (0) = 1. 38.y(t) = 2y + 439E40EEquilibrium solutions A differential equation of the form y(t) = f(y) is said to be autonomous (the function f depends only on y). The constant function y = y0 is an equilibrium solution of the equation provided f(y0) = 0 (because then y(t) = 0 and the solution remains constant for all t). Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations. a. Find the equilibrium solutions. b. Sketch the direction field, for t 0. c. Sketch the solution curve that corresponds to the initial condition y (0) = 1. 41.y (t) = y (y 3)Equilibrium solutions A differential equation of the form y(t) = f(y) is said to be autonomous (the function f depends only on y). The constant function y = y0 is an equilibrium solution of the equation provided f(y0) = 0 (because then y(t) = 0 and the solution remains constant for all t). Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations. a. Find the equilibrium solutions. b. Sketch the direction field, for t 0. c. Sketch the solution curve that corresponds to the initial condition y(0) = 1. 43.y (t) = y (y 3) (y + 2)Direction field analysis Consider the first-order initial value problem y(t)=ay+b,y(0)=A, for t 0,where a, b, and A are real numbers. a.Explain why y = b/a is an solution and corresponds to a horizontal line in the direction field. b.Draw a representative direction fields in the case that a 0. Show that if A b/a, then the solution increases for t 0 and if A b/a, then the solution decreases for t 0 c.Draw a representative direction fields in the case that a 0. Show that if A b/a, then the solution decreases for t 0 and if A b/a, then the solution increases for t 0Eulers method on more general grids Suppose the solution of the initial value problem y(t)=f(t,y),y(a)=A is to be approximated on the interval [a, b]. a.If N + 1 grid points are used (including the endpoints), what is the time step t? b.Write the first step of Eulers method to compute u1. c.Write the general step of Eulers method that applies, for k = 0,1,, N 1.Analyzing models The following models were discussed in Section 9.1 and reappear in later sections of this chapter. In each case, carry out the indicated analysis using direction fields. Drug infusion The delivery of a drug (such as an antibiotic) through an intravenous line may be modeled by the differential equation m(t) + km(t) = I. where m(t) is the mass of the drug in the blood at time t 0, K is a constant that describes the rate at which the drug is absorbed, and I is the infusion rate. Let I = 10 mg/hr and k = 0.05 hr1. a. Draw the direction field, for 0 t 100, 0 y 600. b. For what initial values m(0) = A are solutions increasing? Decreasing? c. What is the equilibrium solution?47EAnalyzing models The following models were discussed in Section 9.1 and reappear in later sections of this chapter. In each case, carry out the indicated analysis using direction fields. Chemical rate equations Consider the chemical rate equations y(t) = ky(t) and y(t) = ky2(t), where y(t) is the concentration of the compound for t 0, and k 0 is a constant that determines the speed of the reaction. Assume the initial concentration of the compound is y(0) = y0 0. a. Let k = 0.3 and make a sketch of the direction fields for both equations. What is the equilibrium solution in both cases? b. According to the direction fields, which reaction approaches its equilibrium solution faster?Convergence of Eulers method Suppose Eulers method is applied to the initial value problem y(t)=ay,y(0)=1, which has the exact solution y(t) = eat. For this exercise, let h denote the time step (rather than t). The grid points are then given by tk = kh. We let uk be the Euler approximation to the exact solution y(tk), for k = 0, 1, 2,. a.Show that Eulers method applied to this problem can be written u0=1,uk+1=(1+ah)uk, for k = 0, 1, 2, . b.Show by substitution that uk = (1 + ah)k is a solution of the equations in part (a), for k = 0, 1, 2, . c.Recall that limh0(1+ah)a/h=ea. Use this fact to show that as the time step goes to zero (h 0. with tk = kh fixed), the approximations given by Euler's method approach the exact solution of the initial value problem; that is, limh0uk=limh0(1+ah)k=y(tk)=eatk.Stability of Eulers method Consider the initial value problem y'(t) = ay, y(0) = 1, where a 0; it has the exact solution y(t) = eat, which is a decreasing function. a.Show that Eulers method applied to this problem with time step h can be written u0 = 1, uk+l = (1 ah)uk, for k = 0, 1, 2,. b.Show by substitution that uk = (1 ah)k is a solution of the equations in part (a), for k = 0,1,2, . c.Explain why as k increases the Euler approximations uk = (1 ah)k decrease in magnitude only if |1 ah| 1. d.Show that the inequality in part (c) implies that the time step must satisfy the condition 0h2a. If the time step does not satisfy this condition, then Eulers method is unstable and produces approximations that actually increase in time.Which of the following equations are separable? (A) y(t) = y + t, (B) y(t)=tyt+1, and (C) y(x) = ex + yWrite y(t) = (t2 + 1)/y3 in separated form.Find the value of the constant C in Example 2 with the initial condition y() = 0, and then explain why the domain of the corresponding solution is (/2, 3/2). Example 2 Another Separable Equation Find the solutions of the equation y(x) = ey sin x subject to the three different initial conditionsFind the value of the constant C in Example 3 with the initial conditiony(6)=0. Example 3 An Implicit Solution Find and graph the solution of the initial value problem (cosy)y(t)=sin2tcost,y(0)=6.What is a separable first-order differential equation?Is the equation t2y(t)=t+4y2 separable?Is the equation y(t)=2yt separable?Explain how to solve a separable differential equation of the form g(y) y'(t) = h (t).Solving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. 5.t3y(t)=1Solving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. 6.e4ty(t)=5Solving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. 7.dydt=3t2ySolving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. 8.dydt=y(x2+1)Solving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. 9.y(t)=ey/2sintSolving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. 10.x2dwdx=w(3x+1),x0Solving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. 11.x2y(x)=y2,x0Solving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. 12.(t2+1)3yy(t)=t(y2+4)Solving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. 13.y(t)csct=y32Solving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. 14.y(t)et/2=y2+4Solving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. 15.u(x)=e2xuSolving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. 16.xu(x)=u24,x0Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem. 19.2yy(t)=3t2,y(0)=9Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem. 20.y(t)=ety,y(0)=1Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem. 21.dydtty+2,y(1)=2Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem. 22.y(t)=y(4t3+1),y(0)=4Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem. 21. y(t) = yet, y(0) = 1Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem. 22. y(x) = y cos x, y(0) = 3Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem. 25.dydx=exy,y(0)=ln3Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem. 26.y(t)=cos2y,y(1)=4Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem. 25. y(t)=ln3ttey,y(1)=0Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem. 26. ty(t) = y(y + 1), y(3) = 1Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem. 27. y(t)=sec2t2y,y(/4)=1Solutions of separable equations Solve the following initial value problems. When possible, give the solution as an explicit function of t. 39.y(t)=y+35t+6,y(2)=0Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem. 29. y(t)=ty,y(1)=2Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem. 30. y(t) = y3 sin t, y(0) = 1Solving initial value problems Determine whether the following equations are separable. If so. solve the initial value problem 31. ty(t) = 1, y(0) = 2Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem. 18.secty(t)=1,y(0)=1Solutions in implicit form Solve the following initial value problems and leave the solution in implicit form. Use graphing software to plot the solution. If the implicit solution describes more than one function, be sure to indicate which function corresponds to the solution of the initial value problem. 33. y(t)=2t2y21,y(0)=0Solutions in implicit form Solve the following initial vainc problems and leave the solution in implicit form. Use graphing software to plot the solution. If the implicit solution describes more than one curve, be sure to indicate which curve corresponds to the solution of the initial value problem. 28.y(t)=1+x2y,y(1)=1Solutions in implicit form Solve the following initial vainc problems and leave the solution in implicit form. Use graphing software to plot the solution. If the implicit solution describes more than one curve, be sure to indicate which curve corresponds to the solution of the initial value problem. 29.u(x)=cscucosx2,u()=2Solutions in implicit form Solve the following initial vainc problems and leave the solution in implicit form. Use graphing software to plot the solution. If the implicit solution describes more than one curve, be sure to indicate which curve corresponds to the solution of the initial value problem. 30.yy(x)=2x(2+y2)2,y(1)=1Solutions in implicit form Solve the following initial value problems and leave the solution in implicit form. Use graphing software to plot the solution. If the implicit solution describes more than one function, be sure to indicate which function corresponds to the solution of the initial value problem. 37. y(x)y+4=x+1,y(3)=5Solutions in implicit form Solve the following initial vainc problems and leave the solution in implicit form. Use graphing software to plot the solution. If the implicit solution describes more than one curve, be sure to indicate which curve corresponds to the solution of the initial value problem. 32.z(x)=z2+4x2+16,z(4)=2Logistic equation for a population A community of hares on an island has a population of 50 when observations begin (at t = 0). The population is modeled by the initial value problem dPdt=0.08P(1P200),P(0)=50. a. Find and graph the solution of the initial value problem, for t 0. b. What is the steady-state population?Logistic equation for an epidemic When an infected person is introduced into a closed and otherwise healthy community, the number of people who contract the disease (in the absence of any intervention) may be modeled by the logistic equation dPdt=kp(1PA),P(0)=P0, where k is a positive infection rate. A is the number of people in the community, and P0 is the number of infected people at t = 0. The model also assumes no recovery. a. Find the solution of the initial value problem, for t 0, in terms of k, A, and P0. b. Graph the solution in the case that k = 0.025, A = 300, and P0 = 1. c. For a fixed value of k and A, describe the long-term behavior of the solutions, for any P0 with 0 P0 A.Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The equation u(x)=(x2u7)1 is separable. b. The general solution of the separable equation y(t)=ty7+10y4 can be expressed explicitly with y in terms of t. c. The general solution of the equation yy(x)=xey can be found using integration by parts.Implicit solutions for separable equations For the following separable equations, carry out the indicated analysis a. Find the general solution of the equation. b. Find the value of the arbitrary constant associated with each Initial condition. (Each Initial condition requires a different constant.) c. Use the graph of the general solution that is provided to sketch the solution curve for each initial condition. 42. y(t)=t2y2+1;y(1)=1,y(0)=0,y(1)=1Implicit solutions for separable equations For the following separable equations, carry out the indicated analysis. a. Find the general solution of the equation. b. Find the value of the arbitrary constant associated with each initial condition. (Each initial condition requires a different constant.) c. Use the graph of the general solution that is provided to sketch the solution curve for each initial condition. 41. ey/2y(x)=4xsinx2x;y(0)=0,y(0)=ln(14),y(2)=0Orthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection. A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. Use the following steps to find the orthogonal trajectories of the family of ellipses 2x2+y2=a2. a. Apply implicit differentiation to 2x2 + y2 = a2 to show that dydx=2xy. b. The family of trajectories orthogonal to 2x2 + y2 = a2 satisfies the differential equation dydx=y2x Why? c. Solve the differential equation in part (b) to verify that y2=eC|x| and then explain why it follows that y2 = kx. Therefore, the family of parabolas y = kx forms the orthogonal trajectories of the family of ellipses 2x2 + y2 = a2.Orthogonal trajectories Use the method in Exercise 44 to find the orthogonal trajectories for the family of circles x2 + y2 = a2. Orthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular a1 each point of intersection. A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. Use the following steps to find the orthogonal trajectories of the family of ellipses 2x2 + y2 = a2. a. Apply implicit differentiation to 2x2 + y2 = a2 to show that dydx=2xy. b. The family of trajectories orthogonal to 2x2 + y2 = a2 satisfies the differential equation dydx=y2x. Why? c. Solve the differential equation in part (b) to verify that y2 = eC |x| and then explain why it follows that y2 = kx where k is an arbitrary constant Therefore, the family of parabolas y2 = kx forms the orthogonal trajectories of the family of ellipses 2x2 + y2 = a2.Applications 44.Logistic equation for spread of rumors Sociologists model the spread of rumors using logistic equations. The key assumption is that at any given time, a fraction y of the population, where 0 y 1. knows the rumor, while the remaining fraction 1 y does not. Furthermore, the rumor spreads by interactions between those who know the rumor and those who do not. The number of such interactions is proportional to y(1 y). Therefore, the equation that describes the spread of the rumor is y(t) = ky(1 y), where k is a positive real number. The number of people who initially know the rumor is y(0) = y0, where 0 y0 = 0.1 a. Solve this initial value problem and give the solution in terms of k and y0. b. Assume k = 0.3 weeks1 and graph the solution for y0 = 0.1 and y0 = 0.7. c. Describe and interpret the long-term behavior of the rumor function, for any 0 y0 1.Free fall An object in free fall may be modeled by assuming that the only forces at work are the gravitational force and air resistance. By Newtons Second Law of Motion (mass acceleration = the sum of the external forces), the velocity of the object satisfies the differential equation mmassv(t)acceleration=mg+f(v)externalforces, where f is a function that models the air resistance (assuming the positive direction is downward). One common assumption (often used for motion in air) is that f(v) = kv2 , where k 0 is a drag coefficient. a.Show that the equation can be written in the form v'(t) = g av2, where a = k/m. b.For what (positive) value of v is v(t) = 0? (This equilibrium solution is called the terminal velocity.) c.Find the solution of this separable equation assuming v(0) = 0 and 0 v2 g/a. d.Graph the solution found in part (c) with g = 9.8 m/s2, m = 1, and k = 0.1, and verify that the terminal velocity agrees with the value found in part (b).Free fall Using the background given in Exercise 47, assume the resistance is given by f(v) = Rv, for t 0, where R 0 is a drag coefficient (an assumption often made for a heavy medium such as water or oil). a. Show that the equation can be written in the form v(t) = g bv, where b=Rm. b. For what value of v is v(t) = 0? (This equilibrium solution is called the terminal velocity.) c. Find the solution of this separable equation assuming v(0) = 0 and 0vgb. d. Graph the solution found in part (c) with g = 9.8 m/s2, m = 1, and R = 0.1, and verify that the terminal velocity agrees with the value found in part (b). 47. Free fall An object in free fall may be modeled by assuming the only forces at work are the gravitational force and air resistance. By Newtons Second Law of Motion {mass acceleration = the sum of external forces), the velocity of the object satisfies the differential equation where f is a function that models the air resistance (assuming the positive direction is downward). One common assumption (often used for motion in air) is that f(v) = kv2, for t 0, where k 0 is a drag coefficient. a. Show that the equation can be written in the form v(t) = g av2, where a=km. b. For what (positive) value of v is v(t) = 0? (This equilibrium solution is called the terminal velocity.) c. Find the solution of this separable equation assuming v(0) = 0 and 0v2ga. d. Graph the solution found in part (c) with g = 9.8 m/s2, m= 1, and k = 0.1, and verify that the terminal velocity agrees with the value found in part (b).Torricellis law An open cylindrical tank initially filled with water drains through a hole in the bottom of the tank according to Torricellis law (see figure), if h(t) is the depth of water in the tank for t 0 s, then Torricellis law implies h(t)=kh, where K is a constant that includes g = 9.8 m/s2, the radius of the tank, and the radius of the drain. Assume the initial depth of the wafer is h(0) = H m. a. Find the solution of the Initial value problem. b. Find the solution in the case that k = 0.1 and H = 0.5 m. c. In part (b), how long does It take for the tank to drain? d. Graph the solution in part (b) and check that it Is consistent with part (c).Chemical rate equations Let y(t) be the concentration of a substance in a chemical reaction (typical units are moles/liter). The change in the concentration, under appropriate conditions, is modeled by the equation dydt=kyn where k 0 is a rate constant and the positive integer n is the order of the reaction. a. Show that for a first-order reaction (n = 1), the concentration obeys an exponential decay law. b. Solve the initial value problem for a second-order reaction (n = 2) assuming y(0) = y0. c. Graph the concentration for a first-order and second-order reaction with k = 0.1 and y0 = 1Tumor growth The Gompertz growth equation is often used to model the growth of tumors. Let M(t) be the mass of a tumor at time t 0. The relevant initial value problem is dMdt=rMln(MK),M(0)=M0, where r and K are positive constants and 0 M0 K. a. Graph the growth rate function R(M)=rMln(MK) (which equals M(t)) assuming r = 1 and K = 4. For what values of M is the growth rate positive? For what value of M is the growth rate a maximum? b. Solve the initial value problem and graph the solution for r = 1, K =4, and M0 = 1. Describe the growth pattern of the tumor. Is the growth unbounded? If not, v/hat Is the limiting size of the tumor?Blowup in finite time Consider the initial value problem y(t)=yn+1,y(0)=y0 where n is a positive integer. a. Solve the initial value problem with n = 1 and y0 = 1. b. Solve the initial value problem with n = 2 and y0=12. c. Solve the problem for positive integers n and y0=n1/n. How do solutions behave as t 1?Analysis of a separable equation Consider the differential equation yy(t)=12et+t and carry out the following analysis. a.Find the general solution of the equation and express it explicitly as a function of t in two cases: y 0 and y 0. b.Find the solutions that satisfy the initial conditions y(1) = 1 and y( 1) = 2. c.Graph the solutions in part (b) and describe their behavior as t increases. d.Find the solutions that satisfy the initial conditions y(1) = 1 and y( 1) = 2. e.Graph the solutions in part (d) and describe their behavior as t increases.Verify by substitution that y(t) = Cekt b/k is a solution of y(t) = ky + b, for real numbers b and k 0.2QC3QC4QCIn general, what is the equilibrium temperature for any Newton cooling problem? Is it a stable or unstable equilibrium?The general solution of a first-order linear differential equation is y(t)=Ce10t13. What solution satisfies the initial condition y(0) = 4?2EWhat is the general solution of the equation y'(t) = 4y + 6?4EFirst-order linear equations Find the general solution of the following equations. 5.y(t)=3y4First-order linear equations Find the general solution of the following equations. 6.y(x)=y+2First-order linear equations Find the general solution of the following equations. 7.y(t)=2y4First-order linear equations Find the general solution of the following equations. 8.y(x)=2y+6First-order linear equations Find the general solution of the following equations. 9.u(t)+12u=15First-order linear equations Find the general solution of the following equations. 10.v(y)v2=14Initial value problems Solve the following initial value problems. 11.y(t)=3y6,y(0)=9Initial value problems Solve the following initial value problems. 12.y(x)=y+6,y(0)=2Initial value problems Solve the following initial value problems. 13.y(t)2y=8,y(0)=0Initial value problems Solve the following initial value problems. 14.u(x)=2u+6,u(0)=6Initial value problems Solve the following initial value problems. 15.y(t)3y=12,y(1)=4Initial value problems Solve the following initial value problems. 16.z(t)+z2=6,z(1)=0Stability of equilibrium points Find the equilibrium solution of the following equations, make a sketch of the direction field, for t 0, and determine whether the equilibrium solution is stable. The direction field needs to indicate only whether solutions are increasing or decreasing on either side of the equilibrium solution. 17.y(t)=12y18Stability of equilibrium points Find the equilibrium solution of the following equations, make a sketch of the direction field, for t 0, and determine whether the equilibrium solution is stable. The direction field needs to indicate only whether solutions are increasing or decreasing on either side of the equilibrium solution. 18.y(t)=6y+12Stability of equilibrium points Find the equilibrium solution of the following equations, make a sketch of the direction field, for t 0, and determine whether the equilibrium solution is stable. The direction field needs to indicate only whether solutions are increasing or decreasing on either side of the equilibrium solution. 19.y(t)=y31Stability of equilibrium points Find the equilibrium solution of the following equations, make a sketch of the direction field, for t 0, and determine whether the equilibrium solution is stable. The direction field needs to indicate only whether solutions are increasing or decreasing on either side of the equilibrium solution. 20.y(t)=y41=0Stability of equilibrium points Find the equilibrium solution of the following equations, make a sketch of the direction field, for t 0, and determine whether the equilibrium solution is stable. The direction field needs to indicate only whether solutions are increasing or decreasing on either side of the equilibrium solution. 21.u(t)+7u+21=0Stability of equilibrium points Find the equilibrium solution of the following equations, make a sketch of the direction field, for t 0, and determine whether the equilibrium solution is stable. The direction field needs to indicate only whether solutions are increasing or decreasing on either side of the equilibrium solution. 22.u(t)=4u=3Loan problems The following initial value problems model the payoff of a loan. In each case, solve the initial value problem, for t 0, graph the solution, and determine the first month in which the loan balance is zero. 23.B(t)=0.005B500,B(0)=50,000Loan problems The following initial value problems model the payoff of a loan. In each case, solve the initial value problem, for t 0, graph the solution, and determine the first month in which the loan balance is zero. 24.B(t)=0.01B750,B(0)=45,000Loan problems The following initial value problems model the payoff of a loan. In each case, solve the initial value problem, for t 0, graph the solution, and determine the first month in which the loan balance is zero. 25.B(t)=0.0075B1500,B(0)=100,000Loan problems The following initial value problems model the payoff of a loan. In each case, solve the initial value problem, for t 0, graph the solution, and determine the first month in which the loan balance is zero. 26.B(t)=0.004B800,B(0)=40,000Newtons Law of Cooling Solve the differential equation for Newtons Law of Cooling to find the temperature in the following cases. Then answer any additional questions. 27. A cup of coffee has a temperature of 90C when it is poured and allowed to cool in a room with a temperature of 25C. One minute after the coffee is poured, its temperature is 85C. How long must you wait until the coffee is cool enough to drink, say 30C?Newton's Law of Cooling Solve the differential equation for Newtons Law of Cooling to find the temperature in the following cases. Then answer any additional questions. 28.An iron rod is removed from a blacksmiths forge at a temperature of 900C. Assume that k = 0.02 and the rod cools in a room with a temperature of 30C. When does the temperature of the rod reach 100C?Newtons Law of Cooling Solve the differential equation for Newtons Law of Cooling to find the temperature in the following cases. Then answer any additional questions. 29.A glass of milk is moved from a refrigerator with a temperature of 5C to a room with a temperature of 20C. One minute later the milk has warmed to a temperature of 7C. After how many minutes does the milk have a temperature that is 90% of the ambient temperature?30EIntravenous drug dosing The amount of drug in the blood of a patient (in milligrams) due to an intravenous line is governed by the initial value problem y(t) = 0.02y + 3, y(0) = 0, where t is measured in hours. a.Find and graph the solution of the initial value problem. b.What is the steady-state level of the drug? c.When does the drug level reach 90% of the steady-state value?Optimal harvesting rate Let y(t) be the population of a species that is being harvested, for t 0. Consider the harvesting model y(t) = 0.008y h, y(0) = y0, where h is the annual harvesting rate, y0 is the initial population of the species, and t is measured in years. a.If y0 = 2000, what harvesting rate should he used to maintain a constant population of y = 2000, for t 0? b.If the harvesting rate is h = 200/year, what initial population ensures a constant population?Endowment model An endowment is an investment account in which the balance ideally remains constant and withdrawals are made on the interest earned by the account. Such an account may be modeled by the initial value problem B(t) = rB m, for t 0, with B(0) = B0. The constant r 0 reflects the annual interest rate, m 0 is the annual rate of withdrawal, B0 is the initial balance in the account, and t is measured in years. a.Solve the initial value problem with r = 0.05, m = 1000/year, and B0 = 15,000. Does the balance in the account increase or decrease? b.If r = 0.05 and B0 = 50,000, what is the annual withdrawal rate m that ensures a constant balance in the account? What is the constant balance?Explain why or why not Determine whether the following statements are true and give an explanations or counterexample a.The general solution of y(t)=2y18 isy(t)=2e2t+9. b.If k 0 and b 0, then y(t)=0 is never a solution of y(t)=kyb c.The equation y(t)=ty(t)+3 is separable and can be solved using the methods of this section. d.According to Newtons Law of Cooling, the temperature of a hot object will reach the ambient temperature after a finite amount of time36EA bad loan Consider a loan repayment plan described by the initial value problem B(t)=0.03B600,B(0)=40,000, where the amount borrowed is B(0) = 40,000, the monthly payments are 600, and B(t) is the unpaid balance in the loan. a.Find the solution of the initial value problem and explain why B is an increasing function. b.What is the most that you can borrow under the terms of this loan without going further into debt each month? c.Now consider the more general loan repayment plan described by the initial value problem B(t)=rBm,B(0)=B0, where r 0 reflects the interest rate, m 0 is the monthly payment, and B0 0 is the amount borrowed. In terms of m and r, what is the maximum amount B0 that can be borrowed without going further into debt each month?38ESpecial equations A special class of first-order linear equations have the forma(t)y(t)+a(t)y(t)=f(t), where a and f are given functions of t. Notice that the left side of this equation can be written as the derivative of a product, so the equation has the form a(t)y(t)+a(t)y(t)=ddt(a(t)y(t))=f(t). Therefore, the equation can be solved by integrating both sides with respect to t. Use this idea to solve the following initial value problems. 33.ty(t)+y=1+t,y(1)=440ESpecial equations A special class of first-order linear equations have the form a(t)y(t) + a(t)y(t) = f(t), where a and f are given functions of t. Notice that the left side of this equation can be written as the derivative of a product, so the equation has the form a(t)y(t)+a(t)y(t)=ddt(a(t)y(t))=f(t). Therefore, the equation can be solved by integrating both sides with respect to t. Use this idea to solve the following initial value problems. 35.ety(t)ety=e2t,y(0)=442E43E44EGeneral first-order linear equations Consider the general first-order linear equation y(t) + a(t)y(t) = f(t). This equation can be solved, in principle, by defining the integrating factor p(t) = exp(a(t) dt). Here is how the integrating factor works. Multiply both sides of the equation by p (which is always positive) and show that the left side becomes an exact derivative. Therefore, the equation becomes p(t)(y(t)+a(t)y(t))=ddt(p(t)y(t))=p(t)f(t). Now integrate both sides of the equation with respect to t to obtain the solution. Use this method to solve the following initial value problems. Begin by computing the required integrating factor. 45.y(t)+1ty(t)=0,y(1)=646E47EGeneral first-order linear equations Consider the general first-order linear equation y(t) + a(t)y(t) = f(t). This equation can be solved, in principle, by defining the integrating factor p(t) = exp(a(t) dt). Here is how the integrating factor works. Multiply both sides of the equation by p (which is always positive) and show that the left side becomes an exact derivative. Therefore, the equation becomes p(t)(y(t)+a(t)y(t))=ddt(p(t)y(t))=p(t)f(t). Now integrate both sides of the equation with respect to t to obtain the solution. Use this method to solve the following initial value problems. Begin by computing the required integrating factor. 48.y(t) + 2ty(t) = 3t, y(0) = 1Explain why the maximum growth rate for the logistic equation occurs at P = K/2.2QC3QCExplain how the growth rate function determines the solution of a population model.2EExplain how the growth rate function can be decreasing while the population function is increasing.4EIs the differential equation that describes a stirred tank reaction (as developed in this section) linear or nonlinear? What is its order?What are the assumptions underlying the predator-prey model discussed in this section?Describe the solution curves in a predator-prey model in the FH-plane.8ESolving logistic equations Write a logistic equation with the following parameter values. Then solve the initial value problem and graph the solution. Let r be the natural growth rate, K the carrying capacity, and P0 the initial population. 15.r = 0.2, K = 300, P0 = 50Solving logistic equations Write a logistic equation with the following parameter values. Then solve the initial value problem and graph the solution. Let r be the natural growth rate, K the carrying capacity, and P0 the initial population. 16.r = 0.4, K = 5500, P0 = 100Designing logistic functions Use the method of Example 1 to find a logistic function that describes the following populations. Graph the population function. 17.The population increases from 200 to 600 in the first year and eventually levels off at 2000.Designing logistic functions Use the method of Example 1 to find a logistic function that describes the following populations. Graph the population function. 18.The population increases from 50 to 60 in the first month and eventually levels off at 150.19E20ESolving the Gompertz equation Solve the Gompertz equation in Exercise 19 with the given values of r, K, and M0. Then graph the solution to be sure that M(0) and limtM(t)are correct. 21.r = 0.05, K = 1200, M0 = 9022EStirred tank reactions For each of the following stirred tank reactions, carry out the following analysis. a. Write an initial value problem for the mass of the substance. b. Solve the initial value problem. 23. A 500-L tank is initially filled with pure water. A copper sulfate solution with a concentration of 20 g/L flows into the tank at a rate of 4 L/min. The thoroughly mixed solution is drained from the tank at a rate of 4 L/min.Stirred tank reactions For each of the following stirred tank reactions, carry out the following analysis. a. Write an initial value problem for the mass of the substance. b. Solve the initial value problem. 24. A 1500-L tank is initially filled with a solution that contains 3000 g of salt. A salt solution with a concentration of 20 g/L flows into the tank at a rate of 3 L/min. The thoroughly mixed solution is drained from the tank at a rate of 3 L/min.Stirred tank reactions For each of the following stirred tank reactions, carry out the following analysis. a. Write an initial value problem for the mass of the substance. b. Solve the initial value problem. 25. A 2000-L tank is initially filled with a sugar solution with a concentration of 40 g/L. A sugar solution with a concentration of 10 g/L flows into the tank at a rate of 10 L/min. The thoroughly mixed solution is drained from the tank at a rate of 10 L/min.Stirred tank reactions For each of the following stirred tank reactions, carry out the following analysis. a. Write an initial value problem for the mass of the substance. b. Solve the initial value problem. 26. A one-million-liter pond is contaminated by a chemical pollutant with a concentration of 20 g/L. The source of the pollutant is removed, and pure water is allowed to flow into the pond at a rate of 1200 L/hr. Assuming the pond is thoroughly mixed and drained at a rate of 1200 L/hr, how long does it take to reduce the concentration of the solution in the pond to 10% of the initial value?31EGrowth rate functions a.Show that the logistic growth rate function f(P)=rP(1PK) has a maximum value of rK4at the point P=K2. b.Show that the Gompertz growth rate function f(M)=rMln(MK) has a maximum value of rKe at the point M=Ke.Solution of the logistic equation Use separation of variables to show that the solution of the initial value problem P(t)=rP(1PK),P(0)=P0 is P(t)=K(KP01)ert+1.Properties of the Gompertz solution Verify that the function M(t)=K(M0K)exp(rt) satisfies the properties M(0)=M0andlimtM(t)=K.Properties of stirred tank solutions a.Show that for general positive values of R, V, Ci, and m0, the solution of the initial value problem m(t)=RVm(t)+CiR,m(0)=m0 is m(t) = (m0 CiV)eRt/V + CiV. b.Verify that m(0) = m0. c.Evaluate limtm(t) and give a physical interpretation of the result. d.Suppose m0 and V are fixed. Describe the effect of increasing R on the graph of the solution.36ERC circuit equation Suppose a battery with voltage V is connected in series to a capacitor (a charge storage device) with capacitance C and a resistor with resistance R. As the charge Q in the capacitor increases, the current I across the capacitor decreases according to the following initial value problems. Solve each initial value problem and interpret the solution. a.I(t)+1RCI(t)=0,I(0)=VR b.Q(t)+1RCQ(t)=VR,Q(0)=0U.S. population projections According to the U.S. Census Bureau, the nations population (to the nearest million) was 296 million in 2005 and 321 million in 2015. The Bureau also projects a 2050 population of 398 million. To construct a logistic model, both the growth rate and the carrying capacity must be estimated. There are several ways to estimate these parameters. Here is one approach: a. Assume t = 0 corresponds to 2005 and that the population growth is exponential for the first ten years; that is, between 2005 and 2015, the population is given by P(t) = P(0)ert. Estimate the growth rate r using this assumption. b. Write the solution of the logistic equation with the value of r found in part (a). Use the projected value P(45) = 398 million to find a value of the carrying capacity K. c. According to the logistic model determined in parts (a) and (b), when will the U.S. population reach 95% of its carrying capacity? d. Estimations of this kind must be made and interpreted carefully. Suppose the projected population for 2050 is 410 million rather than 398 million. What is the value of the carrying capacity in this case? e. Repeat part (d) assuming the projected population for 2050 is 380 million rather than 398 million. What is the value of the carrying capacity in this case? f. Comment on the sensitivity of the carrying capacity to the 35-year population projection.Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a.The differential equation y + 2y = t is first-order, linear, and separable. b.The differential equation yy = 2t2 is first-order, linear, and separable. c.The function y = t + 1/t satisfies the initial value problem ty + y = 2t, y(1) = 2. d.The direction field for the differential equation y(t) = t + y(t) is plotted in the ty-plane. e.Eulers method gives the exact solution to the initial value problem y = ty2, y(0) = 3 on the interval [0, a] provided a is not too large.2REGeneral solutions Use the method of your choice to find the general solution of the following differential equations. 3.y(t) + 2y = 6General solutions Use the method of your choice to find the general solution of the following differential equations. 4.p(x) = 4p + 8General solutions Use the method of your choice to find the general solution of the following differential equations. 5.y(t) = 2ty6REGeneral solutions Use the method of your choice to find the general solution of the following differential equations. 7.y(t)=yt2+1General solutions Use the method of your choice to find the general solution of the following differential equations. 8.y(x)=sinx2yGeneral solutions Use the method of your choice to find the general solution of the following differential equations. 9.y(t) = (2t + 1) (y2 + 1)10RESolving initial value problems Use the method of your choice to find the solution of the following initial value problems. 11.y(t) = 2t + cos t, y(0) = 112RESolving initial value problems Use the method of your choice to find the solution of the following initial value problems. 13.Q(t) = Q 8, Q(1) = 014RESolving initial value problems Use the method of your choice to find the solution of the following initial value problems. 15.u(t)=(ut)1/3,u(1)=8Solving initial value problems Use the method of your choice to find the solution of the following initial value problems. 16.y(x) = 4x csc y, y(0) = /2Solving initial value problems Use the method of your choice to find the solution of the following initial value problems. t(t2 + 1) s(t) = s, s(1) = 1Solving initial value problems Use the method of your choice to find the solution of the following initial value problems. 18.(x) = 4x cos2 , (0) = /4Direction fields Consider the direction field for the equation y = y(2 y) shown in the figure and initial conditions of the form y(0) = A. a.Sketch a solution on the direction field with the initial condition y(0) = 1. b.Sketch a solution on the direction field with the initial condition y(0) = 3. c.For what values of A are the corresponding solutions increasing, for t 0? d.For what values of A are the corresponding solutions decreasing, for t 0? e.Identify the equilibrium solutions for the differential equation.Direction fields The direction field for the equation y'(t) = t − y, for |t| ≤ 4 and |y| ≤ 4, is shown in the figure.
Use the di1rection field to sketch the solution curve that passes through the point (0, 1/2).
Use the di1rection field to sketch the solution curve that passes through the point (0, −1/2).
In what region of the ty-plane are solutions increasing? Decreasing?
Complete the following sentence. The solution of the differential equation with the initial condition y(0) = A, where A is a real number, approaches the line _________ as t → ∞.
Eulers method Consider the initial value problem y(t)=12y,y(0)=1. a.Use Eulers method with t = 0.1 to compute approximations to y(0.1) and y(0.2). b.Use Eulers method with t = 0.05 to compute approximations to y(0.1) and y(0.2). c.The exact solution of this initial value problem is y=t+1. Compute the errors in the approximations to y(0.2) found in parts (a) and (b). Which approximation gives the smaller error?Equilibrium solutions Find the equilibrium solutions of the following equations and determine whether each solution is stable or unstable. 22.y(t) = y(2 y)Equilibrium solutions Find the equilibrium solutions of the following equations and determine whether each solution is stable or unstable. 23.y(t) = y(3 + y) (y 5)Equilibrium solutions Find the equilibrium solutions of the following equations and determine whether each solution is stable or unstable. 24.y(t) = sin 2 y, for |y|Equilibrium solutions Find the equilibrium solutions of the following equations and determine whether each solution is stable or unstable. 25.y(t) = y3 y2 2yLogistic growth The population of a rabbit community is governed by the initial value problem P(t)=0.2P(1P1200),P(0)=50. a.Find the equilibrium solutions. b.Find the population, for all times t 0. c.What is the carrying capacity of the population? d.What is the population when the growth rate is a maximum?Logistic growth parameters A cell culture has a population of 20 when a nutrient solution is added. After 20 hours, the cell population is 80 and the carrying capacity of the culture is estimated to be 1600 cells. a.Use the population data at t = 0 and t = 20 to find the natural growth rate of the population. b.Give the solution of the logistic equation for the cell population. c.After how many hours does the population reach half of the carrying capacity?Logistic growth in India The population of India was 435 million in 1960 (t = 0) and 487 million in 1965 (t = 5). The projected population for 2050 is 1.57 billion. a.Assume that the population increased exponentially between 1960 and 1965, and use the populations in these years to determine the natural growth rate in a logistic model. b.Use the solution of the logistic equation and the 2050 projected population to determine the carrying capacity. c.Based on the values of r and K found in parts (a) and (b), write the logistic growth function for Indias population (measured in millions of people). d.In approximately what year does the population of India first exceed 2 billion people? e.Discuss some possible shortcomings of this model. Why might the carrying capacity be either greater than or less than the value predicted by the model?Stirred tank reaction A 100-L tank is filled with pure water when an inflow pipe is opened and a sugar solution with a concentration of 20 gm/L flows into the tank at a rate of 0.5 L/min. The solution is thoroughly mixed and flows out of the tank at a rate of 0.5 L/min. a.Find the mass of sugar in the tank at all times after the inflow pipe is opened. b.What is the steady-state mass of sugar in the tank? c.At what time does the mass of sugar reach 95% of its steady-state level?Newtons Law of Cooling A cup of coffee is removed from a microwave oven with a temperature of 80C and allowed to cool in a room with a temperature of 25C. Five minutes later, the temperature of the coffee is 60C. a.Find the rate constant k for the cooling process. b.Find the temperature of the coffee, for t 0. c.When does the temperature of the coffee reach 50C?A first-order equation Consider the equation t2y(t) + 2ty(t) = et. a.Show that the left side of the equation can be written as the derivative of a single term. b.Integrate both sides of the equation to obtain the general solution. c.Find the solution that satisfies the condition y(1) = 0.A second-order equation Consider the equation t2y(t) + 2ty(t) 12y(t) = 0. a.Look for solutions of the form y(t) = tp, where p is to be determined. Substitute this trial solution into the equation and find two values of p that give solutions; call them p1 and p2. b.Assuming the general solution of the equation is y(t) = C1 tp1 + C2 tp2, find the solution that satisfies the conditions y(1) = 0, y(1) = 7.Find a10 for the sequence {1, 4, 7, 10, ...} using the recurrence relation and then again using the explicit formula for the nth term.Find an explicit formula for the sequence {1, 3, 7, 15, ...} (Example 2).Reasoning as in Example 7, what is the value of 0.3 + 0.03 + 0.003 + .? Example 7 Working with Series Consider the infinite series 0.9 + 0.09 + 0.009 + 0.0009 + , where each term of the sum is 110 of the previous term. a. Find the sum of the first one, two, three, and four terms of the series. b. What value would you assign to the infinite series 0.9 + 0.09 + 0.009 + ? Solution a. Let Sn denote the sum of the first n terms of the given series. Then S1 = 0.9, S2 = 0.9 + 0.09 = 0.99, S3 = 0.9 + 0.09 + 0.009 = 0.999, and S4 = 0.9 + 0.09 + 0.009 + 0.0009 = 0.9999. b. The sums S1, S2 , Sn form a sequence {Sn}, which is a sequence of partial sums. As more and more terms are included, the values of Sn approach 1. Therefore, a reasonable conjecture for the value of the series is 1:Do the series k=11 and k=1k converge or diverge?Find the first four terms of the sequence of partial sums for the series k=1(1)kk. Does the series converge or diverge?Define sequence and give an example.Suppose the sequence {an} is defined by the explicit formula an = 1/n, for n = 1, 2, 3, . Write out the first five terms of the sequence.Suppose the sequence {an} is defined by the recurrence relation an + 1 = nan, for n = 1, 2, 3, , where a1 = 1. Write out the first five terms of the sequence.Find two different explicit formulas for the sequence {1, 1, 1, 1, 1, 1, }.Find two different explicit formulas for the sequence {1, 2, 3, 4, 5, ...}.The first ten terms of the sequence {2tan110n}n=1 are rounded to 8 digits right of the decimal point (see table). Make a conjecture about the limit of the sequence.The first ten terms of the sequence {(1+110n)10n}n=1 are rounded to 8 digits right of the decimal point (see table). Make a conjecture about the limit of the sequence.Dees the sequence {1, 2, 1, 2, 1, 2, ...} converge? Explain.The terms of a sequence of partial sums are defined by Sn=k=1nk2, for n = 1, 2, 3,. Evaluate the first four terms of the sequence.Given the series k=1k, evaluate the first four terms of its sequence of partial sums Sn=k=1nk.Use sigma notation to write an infinite series v/hose first four successive partial sums are 10, 20, 30, and 40.Consider the infinite series k=11k. Evaluate the first four terms of the sequence of partial sums.Explicit formulas Write the first four terms of the sequence {an}n=1. 9. an = 1/10nExplicit formulas Write the first four terms of the sequence {an}n=1. 10. an = 3n + 1Explicit formulas Write the first four terms of the sequence {an}n=1. 11. an=(1)n2nExplicit formulas Write the first four terms of the sequence {an}n=1. 12. an = 2 + (1)nExplicit formulas Write the first four terms of the sequence {an}n=1. 13. an=2n+12n+1Explicit formulas Write the first four terms of the sequence {an}n=1. 14. an = n + 1/nExplicit formulas Write the first four terms of the sequence {an}n=1. 15. an = 1 + sin (n/2)Explicit formulas Write the first four terms of the sequence {an}n=1. an = n! (Hint: Recall that n! = n(n 1)(n 2) 2 1.)Recurrence relations Write the first four terms of the sequence {an} defined by the following recurrence relations. 17. an + 1 = 2an; a1 = 2Recurrence relations Write the first four terms of the sequence {an} defined by the following recurrence relations. 18. an + 1 = an/2; a1 = 32Recurrence relations Write the first four terms of the sequence {an} defined by the following recurrence relations. 19. an + 1 = 3an 12; a1 = 10Recurrence relations Write the first four terms of the sequence {an} defined by the following recurrence relations. 20. an+1=an21; a1 = 1Recurrence relations Write the first four terms of the sequence {an} defined by the following recurrence relations. an+111+an;a01Recurrence relations Write the first four terms of the sequence {an} defined by the following recurrence relations. 22. an + 1 = an + an 1; a1 = 1, a0 = 1Working with sequences Several terms of a sequence {an}n=1 are given. a. Find the next two terms of the sequence. b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence). c. Find an explicit formula for the nth term of the sequence. 23. {1,12,14,18,116,}Working with sequences Several terms of a sequence {an}n=1 are given. a. Find the next two terms of the sequence. b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence). c. Find an explicit formula for the nth term of the sequence. 26. {2, 5, 8, 11, }Working with sequences Several terms of a sequence {an}n=1 are given. a. Find the next two terms of the sequence. b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence). c. Find an explicit formula for the nth term of the sequence. 27. {1, 2, 4, 8, 16, }Working with sequences Several terms of a sequence {an}n=1 are given. a. Find the next two terms of the sequence. b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence). c. Find an explicit formula for the nth term of the sequence. 30. {64, 32, 16, 8, 4, }Working with sequences Several terms of a sequence {an}n=1 are given. a. Find the next two terms of the sequence. b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence). c. Find an explicit formula for the nth term of the sequence. 29. {1, 3, 9, 27, 81, }Working with sequences Several terms of a sequence {an}n=1 are given. a. Find the next two terms of the sequence. b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence). c. Find an explicit formula for the nth term of the sequence. 28. {1, 4, 9, 16, 25, }Working with sequences Several terms of a sequence {an}n=1 are given. a. Find the next two terms of the sequence. b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence). c. Find an explicit formula for the nth term of the sequence. 25. {5, 5, 5, 5, }Working with sequences Several terms of a sequence {an}n=1 are given. a. Find the next two terms of the sequence. b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence). c. Find an explicit formula for the nth term of the sequence. 34. {1, 0, 1, 0, 1, 0, 1, ...}Limits of sequences Write the terms a1, a2, a3, and a4 of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why. 31. an = 10n 1; n = 1, 2, 3,Limits of sequences Write the terms a1, a2, a3, and a4 of the following sequences. If the sequence appears to converge, make a conjecture about its limit, if the sequence diverges, explain why. 36. an+1=10an;a1=1Limits of sequences Write the terms a1, a2, a3, and a4 of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why. 33. an=110n; n = 1, 2, 3,Limits of sequences Write the terms a1, a2, a3, and a4 of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why. 34. an+1=an10; a0 = 1Limits of sequences Write the terms a1, a2, a3, and a4 of the following sequences. If the sequence appears to converge, make a conjecture about its limit, if the sequence diverges, explain why. 39. an = 3 + cos n; n = 1, 2, 3, ...Limits of sequences Write the terms a1, a2, a3, and a4 of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why. 36. an = 1 10n; n = 1, 2, 3,Limits of sequences Write the terms a1, a2, a3, and a4 of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why. 37. an+1=1+an2; a0 = 2Limits of sequences Write the terms a1, a2, a3, and a4 of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why. 38. an+1=1+an2;a0=23Limits of sequences Write the terms a1, a2, a3, and a4 of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why. 39. an+1=an11+50; a0 = 50Limits of sequences Write the terms a1, a2, a3, and a4 of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why. 40. an + 1 = 10an 1 ; a0 = 0Explicit formulas for sequences Consider the formulas for the following sequences. Using a calculator, make a table with at least ten terms and determine a plausible value for the limit of the sequence or state that the sequence diverges. 45. an=5n5n+1; n = 1, 2, 3,Explicit formulas for sequences Consider the formulas for the following sequences. Using a calculator, make a table with at least ten terms and determine a plausible value for the limit of the sequence or state that the sequence diverges. 46. an = 2n sin (2n); n = 1, 2, 3,Explicit formulas for sequences Consider the formulas for the following sequences {an}n=1 Make a table with at least ten terms and determine a plausible limit of the sequence or state that the sequence diverges. 47. an = n2 + nExplicit formulas for sequences Consider the formulas for the following sequences. Using a calculator, make a table with at least ten terms and determine a plausible value for the limit of the sequence or slate that the sequence diverges. 44. an=100n110n; n = 1, 2, 3,Limits from graphs Consider the following sequences. a. Find the first four terms of the sequence. b. Based on part (a) and the figure, determine a plausible limit of the sequence. 47. an = 2 + 2n; n = 1, 2, 3,Limits from graphs Consider the following sequences. a. Find the first four terms of the sequence. b. Based on part (a) and the figure, determine a plausible limit of the sequence. 48. an=n2n21; n = 1, 2, 3,Recurrence relations Consider the following recurrence relations. Make a table with at least ten terms and determine a plausible limit of the sequence or state that the sequence diverges. 51. an+1=12an+2;a1=3Recurrence relations Consider the following recurrence relations. Using a calculator, make a table with at least ten terms and determine a plausible limit of the sequence or state that the sequence diverges. 50. an=14an13; a0 = 1Recurrence relations Consider the following recurrence relations. Make a table with at least ten terms and determine a plausible limit of the sequence or state that the sequence diverges. 53. an+1 4an + 1; a0 1Recurrence relations Consider the following recurrence relations. Using a calculator, make a table with at least ten terms and determine a plausible limit of the sequence or state that the sequence diverges. 52. an+1=an10+3; a0 = 10Recurrence relations Consider the following recurrence relations. Make a table with at least ten terms and determine a plausible limit of the sequence or state that the sequence diverges. 55. an+1=12an+3;a1=8Recurrence relations Consider the following recurrence relations. Make a table with at least ten terms and determine a plausible limit of the sequence or state that the sequence diverges. 56. an+1=8an+9;a1=10Heights of bouncing balls A ball is thrown upward to a height of h0 meters. After each bounce, the ball rebounds to a fraction r of its previous height. Let hn be the height after the nth bounce. Consider the following values of h0 and r. a. Find the first four terms of the sequence of heights {hn}. b. Find an explicit formula for the nth term of the sequence {hn}. 55. h0 = 20, r = 0.5Heights of bouncing balls A ball is thrown upward to a height of h0 meters. After each bounce, the ball rebounds to a fraction r of its previous height. Let hn be the height after the nth bounce. Consider the following values of h0 and r. a. Find the first four terms of the sequence of heights {hn}. b. Find an explicit formula for the nth term of the sequence {hn}. 56. h0 = 10, r = 0.9Heights of bouncing balls A ball is thrown upward to a height of h0 meters. After each bounce, the ball rebounds to a fraction r of its previous height. Let hn be the height after the nth bounce. Consider the following values of h0 and r. a. Find the first four terms of the sequence of heights {hn}. b. Find an explicit formula for the nth term of the sequence {hn}. 57. h0 = 30, r = 0.25Heights of bouncing balls A ball is thrown upward to a height of h0 meters. After each bounce, the ball rebounds to a fraction r of its previous height. Let hn be the height after the nth bounce. Consider the following values of h0 and r. a. Find the first four terms of the sequence of heights {hn}. b. Find an explicit formula for the nth term of the sequence {hn}. 58. h0 = 20, r = 0.75Sequences of partial sums For the following infinite series, find the first four terms of the sequence of partial sums. Then make a conjecture about the value of the infinite series. 59. 0.3+0.03+0.003+Sequences of partial sums For the following infinite series, find the first four terms of the sequence of partial sums. Then make a conjecture about the value of the infinite series. 60. 0.6+0.06+0.006+Sequences of partial sums For the following infinite series, find the first four terms of the sequence of partial sums. Then make a conjecture about the value of the infinite series. 61. 4+0.9+0.09+0.009+Sequences of partial sums For the following infinite series, find the first four terms of the sequence of partial sums. Then make a conjecture about the value of the infinite series or state that the series diverges. 64. k=110kSequences of partial sums For the following infinite series, find the first four terms of the sequence of partial sums. Then make a conjecture about the value of the infinite series or state that the series diverges. 65. k=1610kSequences of partial sums For the following infinite series, find the first four terms of the sequence of partial sums. Then make a conjecture about the value of the infinite series or state that the series diverges. 66. k=1coskFormulas for sequences of partial sums Consider the following infinite series. a. Find the first four partial sums S1, S2, S3, and S4 of the series. b. Find a formula for the nth partial sum Sn of the infinite series. Use this formula to find the next four partial sums S5, S6, S7, and S8 of the infinite series. c. Make a conjecture for the value of the series. 67. k=12(2k1)(2k+1)Formulas for sequences of partial sums Consider the following infinite series. a. Find the first four partial sums S1, S2, S3, and S4 of the series. b. Find a formula for the nth partial sum Sn of the infinite series. Use this formula to find the next four partial sums S5, S6, S7, and S8 of the infinite series. c. Make a conjecture for the value of the series. 68. k=112kFormulas for sequences of partial sums Consider the following infinite series. a. Find the first four partial sums S1, S2, S3, and S4 of the series. b. Find a formula for the nth partial sum Sn of the infinite series. Use this formula to find the next four partial sums S5, S6, S7, and S8 of the infinite series. c. Make a conjecture for the value of the series. 69. k=190(0.1)kFormulas for sequences of partial sums Consider the following infinite series. a. Find the first four partial sums S1, S2, S3, and S4 of the series. b. Find a formula for the nth partial sum Sn of the infinite series. Use this formula to find the next four partial sums S5, S6, S7, and S8 of the infinite series. c. Make a conjecture for the value of the series. 70. k=123k1Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The sequence of partial sums for the series 1+2+3+ is {1, 3, 6, 10, }. b. If a sequence of positive numbers converges, then the terms of the sequence must decrease in size. c. If the terms of the sequence {an} are positive and increasing, then the sequence of partial sums for the series k=1ak diverges.