First-Order Linear Differential Equations. Find the general solution of y y e x. e e x. This implies that the general solution is

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1 43 CHAPTER 6 Differential Equations Section 6.4 First-Order Linear Differential Equations Solve a first-order linear differential equation. Solve a Bernoulli differential equation. Use linear differential equations to solve applied problems. First-Order Linear Differential Equations In this section, ou will see how to solve a ver important class of first-order differential equations first-order linear differential equations. Definition of First-Order Linear Differential Equation A first-order linear differential equation is an equation of the form P Q d where P and Q are continuous functions of. This first-order linear differential equation is said to be in standard form. NOTE It is instructive to see wh the integrating factor helps solve a linear differential equation of the form When both sides of the equation are multiplied b the integrating factor u e P d, the left-hand side becomes the derivative of a product. P Q. e P d Pe P d Qe P d e P d Qe P d Integrating both sides of this second equation and dividing b u produces the general solution. To solve a linear differential equation, write it in standard form to identif the functions P and Q. Then integrate and form the epression P u ep d Integrating factor which is called an integrating factor. The general solution of the equation is EXAMPLE u Qu d. Find the general solution of e. General solution Solving a Linear Differential Equation Solution For this equation, P and Q e. So, the integrating factor is u e P d Integrating factor d e e. This implies that the general solution is u Qu d e e e d e e C. e Ce. Tr It Eploration A General solution

2 SECTION 6.4 First-Order Linear Differential Equations 433 ANNA JOHNSON PELL WHEELER ( ) Anna Johnson Pell Wheeler was awarded a master s degree from the Universit of Iowa for her thesis The Etension of Galois Theor to Linear Differential Equations in 904. Influenced b David Hilbert, she worked on integral equations while stuing infinite linear spaces. THEOREM 6.3 Solution of a First-Order Linear Differential Equation An integrating factor for the first-order linear differential equation P Q is u e P The solution of the differential equation is e P d d. Qe P d d C. STUDY TIP Rather than memorizing the formula in Theorem 6.3, just remember that multiplication b the integrating factor e P d converts the left side of the differential equation into the derivative of the product e P d. EXAMPLE Solving a First-Order Linear Differential Equation Find the general solution of. Solution The standard form of the given equation is P Q. So, P, and ou have P d d ln e P d eln eln Standard form. Integrating factor. Figure 6.9 C = 0 C = C = 4 C = 3 C = C = C = So, multipling each side of the standard form b ields 3 d d d ln C ln C. General solution Several solution curves for C,, 0,,, 3, and 4 are shown in Figure 6.9. Editable Graph Tr It Eploration A

3 434 CHAPTER 6 Differential Equations EXAMPLE 3 Solving a First-Order Linear Differential Equation Find the general solution of tan t, < t <. Solution The equation is alrea in the standard form So, Pt tan t, and Pt tan t ln cos t Pt Qt. which implies that the integrating factor is π C = C = C = 0 C = π t e Pt e ln cos t A quick check shows that cos t b cos t produces tan t cos t. d cos t cos t cos t cos t Integrating factor is also an integrating factor. So, multipling. Figure 6.0 C = cos t sin t C tan t C sec t. General solution Several solution curves are shown in Figure 6.0. Editable Graph Tr It Eploration A Open Eploration Bernoulli Equation A well-known nonlinear equation that reduces to a linear one with an appropriate substitution is the Bernoulli equation, named after James Bernoulli ( ). P Q n Bernoulli equation This equation is linear if n 0, and has separable variables if n. So, in the following development, assume that n 0 and n. Begin b multipling b n and n to obtain n P n Q n n np n nq d d n np n nq which is a linear equation in the variable n. Letting z n produces the linear equation dz npz nq. d Finall, b Theorem 6.3, the general solution of the Bernoulli equation is n np d e nqe np d d C.

4 SECTION 6.4 First-Order Linear Differential Equations 435 EXAMPLE 4 Solving a Bernoulli Equation Find the general solution of e 3. Solution For this Bernoulli equation, let n 3, and use the substitution z 4 Let z n 3. Differentiate. Multipling the original equation b 3 43 produces Write original equation. z 4 3. e e Multipl each side b 4 3. z 4z 4e. Linear equation: This equation is linear in z. Using P 4 produces P d 4 d which implies that factor produces e z Pz Q is an integrating factor. Multipling the linear equation b this. z 4z 4e ze 4ze 4e d 4e d ze ze 4e d ze Finall, substituting z 4, the general solution is 4 e Ce. e C z e Ce. Linear equation Multipl b integrating factor. Write left side as derivative. Integrate each side. Divide each side b e. General solution Tr It Eploration A So far ou have studied several tpes of first-order differential equations. Of these, the separable variables case is usuall the simplest, and solution b an integrating factor is ordinaril used onl as a last resort. Summar of First-Order Differential Equations Method Form of Equation. Separable variables: Md N 0. Homogeneous: M, d N, 0, where M and N are nth-degree homogeneous 3. Linear: P Q 4. Bernoulli equation: P Q n

5 436 CHAPTER 6 Differential Equations Applications One tpe of problem that can be described in terms of a differential equation involves chemical mitures, as illustrated in the net eample. EXAMPLE 5 A Miture Problem 4 gal/min Figure gal/min A tank contains 50 gallons of a solution composed of 90% water and 0% alcohol. A second solution containing 50% water and 50% alcohol is added to the tank at the rate of 4 gallons per minute. As the second solution is being added, the tank is being drained at a rate of 5 gallons per minute, as shown in Figure 6.. Assuming the solution in the tank is stirred constantl, how much alcohol is in the tank after 0 minutes? Solution Let be the number of gallons of alcohol in the tank at an time t. You know that 5 when t 0. Because the number of gallons of solution in the tank at an time is 50 t, and the tank loses 5 gallons of solution per minute, it must lose 5 50 t gallons of alcohol per minute. Furthermore, because the tank is gaining gallons of alcohol per minute, the rate of change of alcohol in the tank is given b 5 50 t To solve this linear equation, let Pt 550 t and obtain Pt 5 50 t 5 ln 50 t. Because t < 50, ou can drop the absolute value signs and conclude that e Pt e 5 ln50t So, the general solution is 50 t 5 50 t5 50 t C 4 Because 5 when t 0, ou have 5 50 which means that the particular solution is Finall, when t 0, the amount of alcohol in the tank is 50 t C t C50 t t t t C gal which represents a solution containing 33.6% alcohol. Tr It Eploration A

6 SECTION 6.4 First-Order Linear Differential Equations 437 In most falling-bo problems discussed so far in the tet, air resistance has been neglected. The net eample includes this factor. In the eample, the air resistance on the falling object is assumed to be proportional to its velocit v. If g is the gravitational constant, the downward force F on a falling object of mass m is given b the difference mg kv. But b Newton s Second Law of Motion, ou know that F ma mdv which ields the following differential equation. m dv mg kv dv k m v g EXAMPLE 6 A Falling Object with Air Resistance An object of mass m is dropped from a hovering helicopter. Find its velocit as a function of time t, assuming that the air resistance is proportional to the velocit of the object. Solution The velocit v satisfies the equation dv kv m g where g is the gravitational constant and k is the constant of proportionalit. Letting b km, ou can separate variables to obtain dv g bv dv g bv b ln g bv t C. ln g bv bt bc g bv Ce bt. Because the object was dropped, v 0 when t 0; so g C, and it follows that bv g ge bt v g gebt b mg k ektm. Tr It Eploration A NOTE Notice in Eample 6 that the velocit approaches a limit of mgk as a result of the air resistance. For falling-bo problems in which air resistance is neglected, the velocit increases without bound. E S R I A simple electric circuit consists of electric current I (in amperes), a resistance R (in ohms), an inductance L (in henrs), and a constant electromotive force E (in volts), as shown in Figure 6.. According to Kirchhoff s Second Law, if the switch S is closed when t 0, the applied electromotive force (voltage) is equal to the sum of the voltage drops in the rest of the circuit. This in turn means that the current I satisfies the differential equation Figure 6. L L di RI E.

7 438 CHAPTER 6 Differential Equations EXAMPLE 7 An Electric Circuit Problem Find the current I as a function of time t (in seconds), given that I satisfies the differential equation LdI RI sin t, where R and L are nonzero constants. TECHNOLOGY The integral in Eample 7 was found using smbolic algebra software. If ou have access to Derive, Maple, Mathcad, Mathematica, or the TI-89, tr using it to integrate e L RLt sin t. In Chapter 8 ou will learn how to integrate functions of this tpe using integration b parts.. Solution In standard form, the given linear equation is di R L I sin t. L Let Pt RL, so that e Pt e RLt, and, b Theorem 6.3, Ie RLt L e RLt sin t So the general solution is I e RLt 4L R erlt R sin t L cos t C I 4L R erlt R sin t L cos t C. 4L R R sin t L cos t CeRLt. Tr It Eploration A

8 438 CHAPTER 6 Differential Equations Eercises for Section 6.4 The smbol Click on Click on indicates an eercise in which ou are instructed to use graphing technolog or a smbolic computer algebra sstem. to view the complete solution of the eercise. to print an enlarged cop of the graph. In Eercises 4, determine whether the differential equation is linear. Eplain our reasoning.. 3 e. ln In Eercises 5 4, solve the first-order linear differential equation Slope Fields In Eercises 5 and 6, (a) sketch an approimate solution of the differential equation satisfing the initial condition b hand on the slope field, (b) find the particular solution that satisfies the initial condition, and (c) use a graphing utilit to graph the particular solution. Compare the graph with the handdrawn graph of part (a). To print an enlarged cop of the graph, select the MathGraph button cos d 0 Differential Equation d e sin 0,, 0 3 d cos d 0 0. sin d e 3 3 e 3 cos Initial Condition 4 4 Figure for 5 Figure for 6 In Eercises 7 4, find the particular solution of the differential equation that satisfies the boundar condition. Differential Equation Boundar Condition e 0 d 0 5 e cos 0 tan sec cos sec sec

9 SECTION 6.4 First-Order Linear Differential Equations 439 In Eercises 5 30, solve the Bernoulli differential equation Slope Fields In Eercises 3 34, (a) use a graphing utilit to graph the slope field for the differential equation, (b) find the particular solutions of the differential equation passing through the given points, and (c) use a graphing utilit to graph the particular solutions on the slope field e 3 Differential Equation d d 43 3 cot d d 35. Population Growth When predicting population growth, demographers must consider birth and death rates as well as the net change caused b the difference between the rates of immigration and emigration. Let P be the population at time t and let N be the net increase per unit time resulting from the difference between immigration and emigration. So, the rate of growth of the population is given b dp kp N, N is constant. Solve this differential equation to find P as a function of time if at time t 0 the size of the population is P Investment Growth A large corporation starts at time t 0 to invest part of its receipts continuousl at a rate of P dollars per ear in a fund for future corporate epansion. Assume that the fund earns r percent interest per ear compounded continuousl. So, the rate of growth of the amount A in the fund is given b da ra P where A 0 when t 0. Solve this differential equation for A as a function of t. Investment Growth Eercise 36. Points e, 4,, 8 0, 7, 0,,, 3, 0, 3, 0, In Eercises 37 and 38, use the result of 37. Find A for the following. (a) P $00,000, r 6%, and t 5 ears (b) P $50,000, r 5%, and t 0 ears 38. Find t if the corporation needs $800,000 and it can invest $75,000 per ear in a fund earning 8% interest compounded continuousl. 39. Intravenous Feeding Glucose is added intravenousl to the bloodstream at the rate of q units per minute, and the bo removes glucose from the bloodstream at a rate proportional to the amount present. Assume that Qt is the amount of glucose in the bloodstream at time t. (a) Determine the differential equation describing the rate of change of glucose in the bloodstream with respect to time. (b) Solve the differential equation from part (a), letting Q Q 0 when t 0. (c) Find the limit of Qt as t. 40. Learning Curve The management at a certain factor has found that the maimum number of units a worker can produce in a da is 40. The rate of increase in the number of units N produced with respect to time t in das b a new emploee is proportional to 40 N. (a) Determine the differential equation describing the rate of change of performance with respect to time. (b) Solve the differential equation from part (a). (c) Find the particular solution for a new emploee who produced 0 units on the first da at the factor and 9 units on the twentieth da. Miture In Eercises 4 46, consider a tank that at time t 0 contains v 0 gallons of a solution of which, b weight, q 0 pounds is soluble concentrate. Another solution containing q pounds of the concentrate per gallon is running into the tank at the rate of r gallons per minute. The solution in the tank is kept well stirred and is withdrawn at the rate of gallons per minute. 4. If Q is the amount of concentrate in the solution at an time t, show that dq r Q v 0 r r t q r. 4. If Q is the amount of concentrate in the solution at an time t, write the differential equation for the rate of change of Q with respect to t if r r r. 43. A 00-gallon tank is full of a solution containing 5 pounds of concentrate. Starting at time t 0, distilled water is admitted to the tank at a rate of 0 gallons per minute, and the well-stirred solution is withdrawn at the same rate. (a) Find the amount of concentrate Q in the solution as a function of t. (b) Find the time at which the amount of concentrate in the tank reaches 5 pounds. (c) Find the quantit of the concentrate in the solution as t. 44. Repeat Eercise 43, assuming that the solution entering the tank contains 0.04 pound of concentrate per gallon. 45. A 00-gallon tank is half full of distilled water. At time t 0, a solution containing 0.5 pound of concentrate per gallon enters the tank at the rate of 5 gallons per minute, and the well-stirred miture is withdrawn at the rate of 3 gallons per minute. (a) At what time will the tank be full? (b) At the time the tank is full, how man pounds of concentrate will it contain? r

10 440 CHAPTER 6 Differential Equations 46. Repeat Eercise 45, assuming that the solution entering the tank contains pound of concentrate per gallon. Falling Object In Eercises 47 and 48, consider an eight-pound object dropped from a height of 5000 feet, where the air resistance is proportional to the velocit. 47. Write the velocit as a function of time if its velocit after 5 seconds is approimatel 0 feet per second. What is the limiting value of the velocit function? 48. Use the result of Eercise 47 to write the position of the object as a function of time. Approimate the velocit of the object when it reaches ground level. Electric Circuits In Eercises 49 and 50, use the differential equation for electric circuits given b L di RI E. In this equation, I is the current, R is the resistance, L is the inductance, and E is the electromotive force (voltage). 49. Solve the differential equation given a constant voltage E Use the result of Eercise 49 to find the equation for the current if I0 0, E 0 0 volts, R 600 ohms, and L 4 henrs. When does the current reach 90% of its limiting value? Writing About Concepts 5. Give the standard form of a first-order linear differential equation. What is its integrating factor? 5. Give the standard form of the Bernoulli equation. Describe how one reduces it to a linear equation. In Eercises 53 56, match the differential equation with its solution. Differential Equation (a) Ce 54. (b) Ce 55. (c) C 56. (d) Ce 0 0 In Eercises 57 68, solve the first-order differential equation b an appropriate method. 57. e 58. d e d 59. cos cos d d 0 d 0 e d 0 d 0 4 d d d 0 d e 0 True or False? In Eercises 69 and 70, determine whether the statement is true or false. If it is false, eplain wh or give an eample that shows it is false. 69. is a first-order linear differential equation. 70. is a first-order linear differential equation. e Solution

11 REVIEW EXERCISES 44 Review Eercises for Chapter 6 The smbol Click on Click on indicates an eercise in which ou are instructed to use graphing technolog or a smbolic computer algebra sstem. to view the complete solution of the eercise. to print an enlarged cop of the graph.. Determine whether the function 3 is a solution of the differential equation Determine whether the function sin is a solution of the differential equation In Eercises 3 8, use integration to find a general solution of the differential equation. 3. d cos d Slope Fields In Eercises 9 and 0, a differential equation and its slope field are given. Determine the slopes (if possible) in the slope field at the points given in the table d Slope Fields In Eercises 6, (a) sketch the slope field for the differential equation, and (b) use the slope field to sketch the solution that passes through the given point d 3e3 d /d Differential Equation Point, 0, 0, 3, 0, 0, sin d d 3 d sin In Eercises 7, solve the differential equation d d d d. 0. In Eercises 3 6, find the eponential function Ce kt that passes through the two points , 5, 5, Air Pressure Under ideal conditions, air pressure decreases continuousl with the height above sea level at a rate proportional to the pressure at that height. The barometer reads 30 inches at sea level and 5 inches at 8,000 feet. Find the barometric pressure at 35,000 feet. 8. Radioactive Deca Radioactive radium has a half-life of approimatel 599 ears. The initial quantit is 5 grams. How much remains after 600 ears? 9. Sales The sales S (in thousands of units) of a new product after it has been on the market for t ears is given b S Ce kt. ( 3 0, 4 (a) Find S as a function of t if 5000 units have been sold after ear and the saturation point for the market is 30,000 units that is, lim S 30. t (b) How man units will have been sold after 5 ears? (c) Use a graphing utilit to graph this sales function. 30. Sales The sales S (in thousands of units) of a new product after it has been on the market for t ears is given b S 5 e kt. (5, 5) t 3 4 5, 9, 6, (a) Find S as a function of t if 45,000 units have been sold after ear. (b) How man units will saturate this market? (c) How man units will have been sold after 5 ears? (d) Use a graphing utilit to graph this sales function. 3. Population Growth A population grows continuousl at the rate of.5%. How long will it take the population to double? ( 3, ( ( 0 (4, 5) t

12 44 CHAPTER 6 Differential Equations 3. Fuel Econom An automobile gets 8 miles per gallon of gasoline for speeds up to 50 miles per hour. Over 50 miles per hour, the number of miles per gallon drops at the rate of percent for each 0 miles per hour. (a) s is the speed and is the number of miles per gallon. Find as a function of s b solving the differential equation (b) Use the function in part (a) to complete the table. In Eercises 33 38, solve the differential equation d e d 3 e d d 39. Verif that the general solution C C 3 satisfies the differential equation Then find the particular solution that satisfies the initial condition 0 and when. 40. Vertical Motion A falling object encounters air resistance that is proportional to its velocit. The acceleration due to gravit is 9.8 meters per second per second. The net change in velocit is dv kv 9.8. (a) Find the velocit of the object as a function of time if the initial velocit is v 0. (b) Use the result of part (a) to find the limit of the velocit as t approaches infinit. (c) Integrate the velocit function found in part (a) to find the position function s. 4 ds 0.0, 0 Slope Fields In Eercises 4 and 4, sketch a few solutions of the differential equation on the slope field and then find the general solution analticall. To print an enlarged cop of the graph, select the MathGraph button d 4 d 4 s > 50. Speed Miles per Gallon e sin 0 4 In Eercises 43 and 44, the logistic equation models the growth of a population. Use the equation to (a) find the value of k, (b) find the carring capacit, (c) find the initial population, (d) determine when the population will reach 50% of its carring capacit, and (e) write a logistic differential equation that has the solution Pt Environment A conservation department releases 00 brook trout into a lake. It is estimated that the carring capacit of the lake for the species is 0,400. After the first ear, there are 000 brook trout in the lake. (a) Write a logistic equation that models the number of brook trout in the lake. (b) Find the number of brook trout in the lake after 8 ears. (c) When will the number of brook trout reach 0,000? 46. Environment Write a logistic differential equation that models the growth rate of the brook trout population in Eercise 45. Then repeat part (b) using Euler s Method with a step size of h. Compare the approimation with the eact answers. In Eercises 47 56, solve the first-order linear differential equation sin d tan e d In Eercises 57 60, solve the Bernoulli differential equation. 57. Hint: Pt Pt 8 4 e 4 5 e 5 a b 4 e d e e 0.55t e 0.5t 3 e 4e 5 d In Eercises 6 64, write an eample of the given differential equation. Then solve our equation. 6. Homogeneous differential equation 6. Logistic differential equation 63. First-order linear differential equation 64. Bernoulli differential equation

13 P.S. Problem Solving 443 P.S. Problem Solving The smbol Click on Click on indicates an eercise in which ou are instructed to use graphing technolog or a smbolic computer algebra sstem. to view the complete solution of the eercise. to print an enlarged cop of the graph.. The differential equation k where k and are positive constants, is called the doomsda equation. (a) Solve the doomsda equation.0 given that 0. Find the time T at which lim t. t T (b) Solve the doomsda equation k given that 0 0. Eplain wh this equation is called the doomsda equation.. A thermometer is taken from a room at 7F to the outdoors, where the temperature is 0F. The reading drops to 48F after minute. Determine the reading on the thermometer after 5 minutes. 3. Let S represent sales of a new product (in thousands of units), let L represent the maimum level of sales (in thousands of units), and let t represent time (in months). The rate of change of S with respect to t varies jointl as the product of S and L S. (a) Write the differential equation for the sales model if L 00, S 0 when t 0, and S 0 when t. Verif that S (b) At what time is the growth in sales increasing most rapidl? (c) Use a graphing utilit to graph the sales function. (d) Sketch the solution from part (a) on the slope field shown in the figure below. To print an enlarged cop of the graph, select the MathGraph button L Ce kt. S 3 4 t 4. Another model that can be used to represent population growth is the Gompertz equation, which is the solution of the differential equation k ln L where k is a constant and L is the carring capacit. (a) Solve the differential equation. (b) Use a graphing utilit to graph the slope field for the differential equation when k 0.05 and L 000. (c) Describe the behavior of the graph as t. (d) Graph the equation ou found in part (a) for L 5000, 0 500, and k 0.0. Determine the concavit of the graph and how it compares with the general solution of the logistical differential equation. 5. Show that the logistic equation L be kt can be written as L tanh k ln b t k. What can ou conclude about the graph of the logistic equation? 6. Torricelli s Law states that water will flow from an opening at the bottom of a tank with the same speed that it would attain falling from the surface of the water to the opening. One of the forms of Torricelli s Law is Ah dh kgh where h is the height of the water in the tank, k is the area of the opening at the bottom of the tank, Ah is the horizontal crosssectional area at height h, and g is the acceleration due to gravit g 3 feet per second per second. A hemispherical water tank has a radius of 6 feet. When the tank is full, a circular valve with a radius of inch is opened at the bottom, as shown in the figure. How long will it take for the tank to drain completel? 6 ft 6 h h (e) If the estimated maimum level of sales is correct, use the slope field to describe the shape of the solution curves for sales if, at some period of time, sales eceed L.

14 444 CHAPTER 6 Differential Equations 7. The clindrical water tank shown in the figure has a height of 8 feet. When the tank is full, a circular valve is opened at the bottom of the tank. After 30 minutes, the depth of the water is feet. r In Eercises 3, a medical researcher wants to determine the concentration C (in moles per liter) of a tracer drug injected into a moving fluid. Solve this problem b considering a singlecompartment dilution model (see figure). Assume that the fluid is continuousl mied and that the volume of the fluid in the compartment is constant. 8 ft h Tracer injected Flow R (pure) Volume V (a) How long will it take for the tank to drain completel? (b) What is the depth of the water in the tank after hour? 8. Suppose the tank in Eercise 7 has a height of 0 feet, a radius of 8 feet, and the valve is circular with a radius of inches. The tank is full when the valve is opened. How long will it take for the tank to drain completel? 9. In hill areas, radio reception ma be poor. Consider a situation where an FM transmitter is located at the point, behind a hill modeled b the graph of and a radio receiver is on the opposite side of the hill. (Assume that the -ais represents ground level at the base of the hill.) (a) What is the closest position, 0 the radio can be to the hill so that reception is unobstructed? (b) Write the closest position, 0 of the radio with represented as a function of h if the transmitter is located at, h. (c) Use a graphing utilit to graph the function for in part (b). Determine the vertical asmptote of the function and interpret the result. 0. Biomass is a measure of an amount of living matter in an ecosstem. Suppose the biomass st in a given ecosstem increases at a rate of about 3.5 tons per ear, and decreases b about.9% per ear. This situation can be modeled b the differential equation ds s. (a) Solve the differential equation. (b) Use a graphing utilit to graph the slope field for the differential equation. What do ou notice? (c) Eplain what happens as t. Figure for 3. If the tracer is injected instantaneousl at time t 0, then the concentration of the fluid in the compartment begins diluting according to the differential equation dc R V C, C C 0 when t 0. (a) Solve this differential equation to find the concentration C as a function of time t. (b) Find the limit of C as t.. Use the solution of the differential equation in Eercise to find the concentration C as a function of time t, and use a graphing utilit to graph the function. (a) V liters, R 0.5 liter per minute, and C mole per liter (b) V liters, R.5 liters per minute, and C mole per liter 3. In Eercises and, it was assumed that there was a single initial injection of the tracer drug into the compartment. Now consider the case in which the tracer is continuousl injected beginning at t 0 at the rate of Q moles per minute. Considering Q to be negligible compared with R, use the differential equation dc Q V R V C, Flow R (concentration C) C 0 when t 0. (a) Solve this differential equation to find the concentration C as a function of time t. (b) Find the limit of C as t.