I. Foundations/Comprehension (No technology allowed, show all work!!) 1. Solve the following differential equations, using the method of separation of variables (4 points each): a) y' = 1-xty-xy [Implicit solution is fine ] b) (1 + cos(x))y' = (1 + e->)sin (x) ; y(0) =0 2. Compute the exact sum of the following series (4 points): 1 2 1 2 1 2 + OIN + ... 27 81 243 729 2187 65613. Sketch carefully the solution curves with initial points (0, 1), (0, 2), (0, 3), (0, 4), and (0, 5), without solving the equation (5 points): dp dt - = (P -3)2 4. Using Maclaurin series, compute the following (3 points): sin (x) - x lim r+0 5. Find the interval of convergence (don't forget to check the endpoints!) of the following power series (5 points): (2x -9)" M Vn n=1II. Applications/Analysis (Technology allowed, show all work!) 1. A cup of hot chocolate has temperature 80 C in a room kept at 20 C. After half an hour, the hot chocolate cools to 60*C. Knowing this: a) What is the temperature of the hot chocolate after another half hour (4 points)? b) After about how long will the hot chocolate have cooled down to 40*C (3points)? 2. Find the Maclaurin series (in summation form) for the exponential function f (x) = 2*(5 points).3. Determine if the following series converges or diverges, using either the Ratio Test or Root Test (4 points): (-5)2n nzon 4. Use Taylor's Inequality to estimate the accuracy of the 5th degree Taylor polynomial approximation to the function f (x) = sin (x), centered at x = =, when 0