1. Let I(u) = F(u'(x), u(x), x) dx. Find the first variation and Euler-Lagrange equations for each choice of F below. Then solve the equations to find a general solution for extremals of I(u). [Note: your solutions should have two constants of integration.] (a) F(p, u, x) = 1+p² U (b) F(p, u, x) = (c) F(p, u, x) = ² - /p² √p² + u²

3) Jim rides a second Merry-Go-Round shown below. The distance, d, that Jim is from the wall can be described by the equation d = 38 + 22sin(8t), where t is the number of seconds since Jack was at the Start point. Answer the questions below (just enter the numbers), show ALL work, AND upload your work paper (or show your work) in the next question. The Merry-Go-Round makes one rotation every 45 seconds. Start 22 m 38 m Wall meters a) What is Jim's distance from the wall when t = 41 seconds? Round your answer to the nearest hundredth of a meter. b) The owner of the Merry-Go-Round decides to make the ride more exciting by tripling the rotational speed. Rewrite the equation for d given above to reflect this change. sin(

B. Find all solution(s) to the equation shown below. Show all work. y+3=-Vy+ 15 %3D U Font Family -AA A F- E • E uestion in this part. The question is worth 6 pojints. Clearly indicate the necessary steps and ex no work shown will receive only 1 credit. (You may attach a picture of your work or put your we