MATH 201 Exam 2, Fall 2010

Math 201 Exam 2 (Fall 2010) B. Shayya and H Yamani - Please write your section number on your booklet. - Please answer each problem on the indicated page(s) of the booklet. Any part of your answer that is not written on the indicated page(s) will not be graded. - Unjustified answers will receive little or no credit. Problem 1 (answer on page 1 of the booklet). at Given f(xy,z)=e* 7 (i) (4 pts) Find the domain and range of the function f. (ii) (8 pts) Determine if the domain of f is an open region, a closed region, or neither and decide if the domain of f is bounded or unbounded. Also describe the function’s level surface. Problem 2 (answer on page 2 of the booklet) (16 pts) Find the equation of the tangent plane and normal line to the surface Inz- (4 aie + y*)= 0 at the point (1,1,1). Problem 3 (answer on pages 3,4 of the booklet) Determine if the limit of each of the following functions exists as (x, y) > (0,0). Explain (8 pts each) 6 D5 Aaa x : xX vv is x as sine @ f(xy)= ly ii) g(x,y)= ——_ Git) A(x, y)= 7 yrx yo =x yey Problem 4 (answer on page 5 booklet) (16 pts) Find the absolute maximum and minimum values of f (x, y, z) = xyz on the sphere x+y? +27 =1. Problem 5 (answer on page 6 and last page of the booklet) Suppose that f (x, y, z) is a differentiable function and given that f(6,2,0)=-1, Vf (6,2,0)=3i- j + 5k, f(2,0,0)=4, VF(2,0,0)=i- j+h Let x=r? +2s y=rs Z—Tins and w= f(x,y,z) (i)(10 pts) Find Sag (r,s)=(2,). or Os (ii)(8 pts) Estimate f(6.1, 2.1, 0.1) » (iii)(4 pts) Let u be a unit vector in space. Which is greatest D, f (6,2,0) or D, f (2,0,0) 2 (iv)(10 pts) Find a plane tangent to the surface w(r,s) = t* in the rst space.