MATH 308-301,302 Summer 2008 Practice Test I 1. A large tank initially contains 10 L of fresh water. A brine containing 20 g/L of salt flows into the tank at a rate of 3 L/min. The solution inside the tank is kept well stirred and flows out of the tank at the rate 2 L/min. Determine the concentration of salt in the tank as a function of time. 2. A 400-lb object is released from rest 500 ft above the ground and allowed to fall under the influence of gravity. Assume that the force in pounds due to air resistance is −10v, where v is the velocity of the object in ft/sec. Determine the equation of motion of the object. 3. An object with temperature 1500 is placed in a freezer whose temperature is 300 . Assume that the temperature of the freezer remains essentially constant. (a) If the object is cooled to 1200 after 8 min, what will its temperature be after 18 min? (b) When will its temperature be 600 ? 4. Draw the isoclines with their direction markers and sketch the solution to the initial value problem yy ′ = −x, y(0) = 4. 5. Determine (without solving the problem) an interval in which the solution to the initial value problem y ′ + y tan x = sec x, y(0) = 3 is certain to exist. 6. Solve the following initial value problem √ ydx + (1 + x)dy = 0 y(0) = 1. 7. Find the general solution to the equation (x2 − 1)y ′ + 2xy + 3 = 0 8. Show that the relation y = x + ey is an implicit solution to the equation (x + y − 1)y ′ = 1.