Review Problems for Exam 2

Transcription

1 Calculus II Math - Fall 4 Name: Review Problems for Eam In question -6, write a differential equation modeling the given situations, you do not need to solve it.. The rate of change of a population P is proportional to the cube root of P. Solution: dp dt = k P. The rate of change of a population P is inversely proportional to the square of P. Solution: dp dt = k/p. The time rate of change of the velocity v(t) of a boat is proportional to the square of v. Solution: dv dt = kv 4. In a city having a fied population M, the time rate of change of the number N of those persons who have heard a certain rumor is proportional to the number of those who have not yet heard the rumor. Solution: dn dt = k(m N) 5. In a city having a fied population M, the time rate of change of the number N of those persons with a contagious disease is proportional to the product of the number who have the disease and the number who have not. Solution: dn dt = kn(m N) 6. Solve dy d = y, y() =

2 Solution: = / Solve dy d = y y, y() = Solution: = /( + ) 8. Solve dy d = + y, y() = Solution: = /( ) / 9. Solve dy d = y sin, y(π) = Solution: y () = e cos(). Solve dy d + 5y = 7, y() = 5 Solution: y () = / e 5 5 e. Solve d dt = t, () = Solution: (t) = /9 t / 7 t / t + 96 e t 7 e. Solve dy d y cos =, y() = Solution: = sin() + ( sin()). In a certain culture of bacteria, the number of bacteria increased si times in hours. Assume the bacteria follow an eponential growth model, how long did it take to double the population. Page

3 Solution: You are about to take out a 5-year mortgage with annual interest rate of 4.5%, compounded continuously. If the house you are planning to purchase costs $75,, how much will your monthly payments be? Assume payments are made continuously. Solution: $ An upcoming rock star signs a new album deal. He is given the following options: (I) For the net 5 years, his record company will deposit money continuously into an account at a rate of $45, per year. (II) It will pay $,, now and nothing more during the net 5 years. If his account yields % interest, compounded continuously, which option should he choose? Eplain your answer. Solution: $,47,5.64 vs. $,, Better to receive payments in installments. 6. The rate of change of the temperature of a dish placed in a 4F oven is proportional to the difference in temperature of the dish and the oven. If the dish is initially frozen (F) and after min is at 7F, then when will it be at 75F? Solution: 9.5min 7. When a cake is removed from the oven, the temperature of the cake is F. The cake is left to cool at room temperature at 75F. After min the temperature of the cake is F. When will it be at 9F? Solution:.75min 8. In the year the population of a country is 75 million and about 5 births and 7 deaths per thousand of population were occurring annually. In addition a new migration of about 9 thousand people per year was occurring. What is the population of this country today (in the year 4)? Page

4 Solution: 9.7 million 9. How much money should you deposit monthly at a bank account giving 5% interest rate annually compounded continuously in order to pay cash for a $, car after 5 years? Solution: $44.. A tank of capacity L contains 5L of a solution containing 4kg of salt dissolved. Water containing kg/l is pumped in the tank at a rate of L/min and the mied solution is then pumped out at 5L/min. How much salt is in the tank when it is full? Solution: 77 kg. A 4 gallon tank initially containing 5 gallons of water with lb of salt dissolved is being filled with brine at the rate of 4gal/min containing lb/gal of salt. As the solution is being mied, gal/min is pumped out. When full, how much salt will be in the tank? Solution: lb. Suppose the a population (in millions) of bacteria grows according to the following logistic equation dp =.5P.P dt After a long period of time, what do you epect the population to be? Solution: 5 million. A colony of ants initially containing (in thousands) ants grows according to the the following logistic model dp dt =.5P.5P What do you epect to happen to the population size during net year? Page 4

5 Solution: decrease 4. Classify the following equations according to order, linearity, and homogeneity. (a) d y dy + d d = y (b) d y d + d y dy d d = y (c) y y + sin = (d) y y + y sin = (e) y e y = Solution: (a) nd Order Linear Homogeneous (b) rd Order Nonlinear Homogeneous (c) st Order Linear Nonhomogeneous (d) st Order Linear Homogeneous (e) nd Order NonLinear Nonhomogeneous 5. Show that the ODE y + y = is linear by showing that its associated operator T (y) = y + y is linear. Solution: Proof 6. Show that the ODE y + y y + = is not linear by looking at its associated operator T (y) = y + y y +. Solution: Proof In each of the problems bellow find the general solution and particular solution (when applicable) of the constant coefficient homogeneous second order linear equation 7. y y 6y = Page 5

6 Solution: = c e + c e 8. y y + y = Solution: = c e sin + c e cos 9. y + y + 6y = Solution: = c e 5 sin + c e 5 cos. y + 6y + 64y = Solution: = c e 8 + c e 8. y + y + 6y =, y() =, y () = Solution: = 5 5e sin( 5) + e cos( 5). y + y y =, y() =, y () = 6 Solution: = 4 e 5 4 e. y + y + y =, y() =, y () = 5 Solution: = e + 8e 4. y 5y + y =, y() =, y () = 4 Solution: = 8 cos( e 5/ ) sin( e 5 ) + 8 sin( e 5/ ) cos( e ) 5 Page 6

7 5. Match the following directional fields with the following ordinary Differential Equation (write the corresponding letter near each directional field). (a) dy d = + y (e) dy d = y Solution: f,g,b,d,c,e,h,a (b)dy d = y (f)dy d = y y (c)dy d = y (g) dy d = y y (d)dy d = y (h) dy d = y Page 7