THE UNIVERSITY OF BRITISH COLUMBIA OKANAGAN MATH 142 - Calculus II 4. Suppose that a population is changed either by immigration or emigration, and the growth model is given by dy = ky + f (t), where y is the population at time t and f (t) is some function of t that describes the net effect of the emigration/immigration. Assume k = 0.02 and y(0) = 10000. Solve this differential equation for y, given the function f(t) = e-t. 5. Let f(x, y) = V:2-3xyty y In(x) Find the following. (a) f(2, -1) (b) f(1, 2021) 6. Let f (I, y) = / ut yt2 + 1 dt. Find the following. (a) f(-3, 1) (b) f(a, 0) for an arbitrary real number a. 7. Sketch the level curves of the function f(x, y) = (x - 2)2 + (y - 3)? for the values k = 0, 1, 4, 7. 8. Sketch the level curves of the function f(x, y) = y + 2 for the values k = -1, 0, 1. 9. Production of a certain item is given by the Cobb-Douglas production z = P(x, y ) = zy, where r is the amount of labor in work-hours and y is the amount of capital. (a) What is the production when 81 hours and 16 units of capital are provided? (b) Determine the relationship between the amounts of labor and the capital re- quired for the production of 20 units. (c) Sketch the level curve at a production of 20 units in the ry-plane. are available, find the amount of capital needed in the produc