8 ( Due to April 27th) 1 (20 points). State the Fundamental Theorem of Calculus (including both parts). 2 (20 points). State the Leibniz's formula on differentiating under the integral sign and prove the formula. 3 (15 points). Let K(t) be the capital stock of a firm at time t, p(t) be the purchase price per unit of capital, and R(t) be the rental price per unit of capital. In capital theory, one principle for determining the acquisition value V (t) of the firm capital implies that Z V (t) = p(t)K(t) = R( )K( )er( t) d (0.1) t where r is the constant real interest rate. Find R(t) by differentiating (0.1). 4 (25 points). Use the Leibniz's formula to answer following questions. (a) Define the function F for all x 0 by Z x F (x) = f (t)er(tx) dt 0 0 where f is a given function and r is a given number. Find expression for F (x) and show that 0 F (x) rF (x) = f (x). (b) Define 2y Z F (y) = ert dt. 0 0 Find an expression for F (y). (c) Define the function F for all x > 0 by Z F (x) = 0 0 00 Find expression for F (x) and F (x). 1 x 2 ext dt. 5 (20 points). Compute the following double integrals. Z 1Z 1 xe dydx, (a) I = Z0 Z0 (c) y M= Z (b) 1Z 4 J= 0 yex dxdy, ( xy + 2x + y) dydx 0 where A = {(x, y) : 0 y 1, 0 x y 2 } A Bonus Questions 1 (5 points). Sketch the set S in the xyplane bounded by the y-axis, the curve y = y and the straight line x/4 + 1. Compute Z Z J= xdxdy S using two different order of integration, i.e., (i) first take integration w.r.t x and then w.r.t y, (ii) first take integration w.r.t y and then w.r.t x. 2 (5 points). Find Z 1 \u0012Z x \u0013 (x 1)dy dx. I= 0 x4 Sketch the domain of integration, reverse the order of integration and check that you get the same result. 2