MATH 201 Exam 2, Fall 2012

Math 201 Quiz II (Time: 65 minutes) (Fall 2012) N. Nahlus, R. Nassif, F. Mroue, Z. Rawas Name: ……………………………………….. (VERY CLEARLY) I.D ........................................... Circle your Section number ( -3 points if incorrect) Sec 8 (12:30 T) — Sec 9 (2:00 T) — Sec 10 (11:00 T) — Sec 11 (5:00 T) --- Sec 12 (9:00 M) — Sec 13 (3:30 R) — Sec 14 (10:00 M) — Sec 15 (5:00 R) --- 1. (24%) The function at a point P decreases most rapidly in the direction of v = i+j+3k. In this direction, the value of the derivative is Find (i) (ii) Find where w = 2i+2j+k (iii) Is there a unit vector u such that | | | | | | | | | | | | | Use page 2 (if necessary) Problem 1 24% Problem 2 26% Problem 3 25% Problem 4 25% Total over 100 2a) (7%) Given that and the components of are never zero, Find given that (Justify your answer). (Hint: Use the known formulas about 2b) (8%) Suppose 2 . Find the Tangent plane to the level surface of F at P. (Hint: You need f(P)=2). | | | | | | | | | | | | | 2c) (11%) Use Lagrange multipliers to find the absolute maximum and minimum of the function. | | | | | | | | | Continue on Back (if necessary) 3a) (10%) Investigate any 2 critical points (any 2 critical points )(of your choice) of the function for local maxima, local minima, saddle points. 3b) (10%) Check that Then show that 3c) (5%) Find, if it exists, 4ab) (5%, 6%) Let a) By any method, show that (0,0) =5. Then, by symmetry, we have (0,0) =5 (Do not prove it) b) From definitions, show that f is differentiable at (0, 0). ----------------------------------------------------------------------------------------------------------------------------- 4c) (6%) Given a differentiable function f(x,y) such that f(1,2)=3i+4j. Approximate/Estimate the change in f when we move from the point by using any method. 4d) (8%) Let (where f (u,v) is a sufficiently differentiable function). Find