MATH 201 Exam 2, Fall 2012

Math 201 Exam 2 (Fall 2012) 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). (16 pts) Find the oe of the tangent plane and normal line to the surface 5x? +4e” =12~In (c*x) at the point (1,0,e). Problem 2 (answer on page 2 of the booklet). Given f(s.y.z)=Inle— 2? -y¥" =!) Ai) (7 pts) Find the domain and range of the function /. ii) (4 pts) Determine if the domain of f is an open region, a closed region, or neither and decide if the domain of / is bounded or unbounded. Problem 3 (answer on page 3 of the booklet). Determine if the limit of each of the following functions exists as(x,y)—> (0,0). Explain (8 pts sees @) sy) =F Ai) s(e9)= eZ Gi) Wx y) = 22H) x! +5sin’y xy Problem 4 (answer on page 4 of the booklet). Suppose that f(x, y,z)is a differentiable function and given that es 4, “ eo ae Let x=r'+s “peta Por and w= f(x,y,z) r i) (8 pts) Find = and x at (r.s)= (2.4). Gii-(8 pts) Estimate f(8.1,2.9,3.9) (8 pts) Let w= w(r,s). Find a plane tangent to the surface w(r,s)=3r+5s—int-6 in the rsf-space . Problem 5 (answer on page 5 of the booklet). (13 pts) Use Lagrange method to find the absolute maximum and minimum values of f(x y)ae’ on x? 4y?=1. Problem 6 (answer on page 6 of the booklet). Consider the function f(x,y)=x+ytx?°+y? G¥(4 pts) Suppose f(x,y)—2z° = sin z? defines z implicitly as a function of (x,y).Find G2/@x G58 pts) Prove that the function (x, y)is differentiable everywhere in the plane.