1. (07.01 MC) Determine the value of k that will make y = 2cos(2x) a solution to ky - y " = y. (10 points) OK= -3 OK = -6 O k = 5 O K = 02. (07.02 MC) A slope field is shown for which of the following differential equations? (10 points) -2 -5 -- 31 --- --- I dy = e dx O dy = Inx dx O dy -X dx O dy dx3. (0703 MC) Let y = f (x) be the solution to the differential equation f'(x) = 2:0! with the initial condition f(1) = 1. What is the approximation for f(2) if Euler's method is used with two steps of equal length? (10 points) 4. (07.07 MC) The number of guests at a theme park can be modeled by function P(t) where t is measured in hours. P is a solution to the logistic differential 2 equation E _ P P where P(0) = 2.500. What is the greatest rate of change of the number of guests at the park, per hour? (10 points) d! 4 60000' O 938 guests per hour 0 7,500 guests per hour 0 2,500 guests per hour 0 511 guests per hour All changes saved 5. (07.05 MC) What is the general solution to the differential equation y = xy + 2x? (10 points) dx y=e2 O x2 y=e2 -2+C O y = e-2x+C O dy y + 2 dx 1 - x 6. (07.05 MC) Let y = f(x) be the particular solution to the differential equation = -4xy3, with the initial condition f(-1) = -1. Which of the following gives dx an expression for f(x) and the domain for which the solution is valid? (10 points) for ( -oo , - 3 ) and (3 . .) Of( x ) = - Vax2 -3 f ( x ) = - 1 V4x2 - 3 for ( -oo , - V3 ) and (V3,00) F( x ) = 1 for ( - oo , -3 , and (3 . . ) V 4 X 2 - 3 O f ( x ) = 1/ 1 for (-0o, -V/3) and (V3,00) V4x2 -37. (0702 MC) Based on the graph of the general solution to the differential equation % = 2x 2y_ which of the following statements is true? (10 points) 0 The slopes along the xaxis are horizontal. O The slopes are all positive in Quadrant 4. O The slopes are all positive in Quadrant 'l. O The slopes along the yaxis are horizontal. a. (0701 MC) A coolant is injected into engine uid, causing the temperature of the engine uid to decrease with respect to time at a rate that is proportional to the difference of the uids instantaneous temperature, T. and the uid's original temperature, To. Select the differential equation that represents the relationship. (10 points) 0 \"Francr) O T=t(T-To) O T=r(To-T) O =c{TT) dt 9. (07.01 MC) Select the general solution to x- dy = (10 points) dx 2 + 3y 1. 2Inly| + 3y = In|x] + C y 1. Lin 2 + 3y = In/ x/+ C OI O II Both ONeither 10. (07.05 MC) Which of the following is the solution to the differential equation = -, with the initial condition y(1) = 2? (10 points) dx x2 + 1 Oy= x2 + 2 Oy = In/x 2 + 11 + In(2) O y= In/x2 + 21 Oy= x2 + 1