MATH 201 Exam 2, Spring 2011

Spring 2010-2011 Quiz 2 ‘Date: April 15 Prof: M. Egeileh Math 201 - Sections 5 to 8 Duration: 1 hour Problem 1 (answer on page 1 of the booklet) Find the domain and range of the function f(a,y,z) = : j= y— 2 domain of f is an open region, a closed region, or neither, Also, decide if the domain is bounded or unbounded. (12 pts) . Determine if the Problem 2 (answer on page 2 of the booklet) Find the tangent plane and the normal line to the surface z = e?#+8u+52 at the point (-1,-1,1). (21 pts) Problem 3 (answer on page 3 of the booklet) For each of the following limits, say if it exists or no, providing full justification to your answer. (7+6+8 pts) ‘ . 2 ay? ey ty li i b) li > li a) (ey) (0,0) = (= + 3) ) (eu)—0,0) x8 + Sy? a (e.v)(0,0) 8r+y Problem 4 (answer on pages 4 and 5 of the booklet) Find the nature (local minimum, local maximum or saddle point) of the critical point(s) of the function f(x,y) defined by He,3) = 1 ~ Fay SBA DCE D) ayn EB n=2 goed for all (x,y) such that |xy| <1. (15 pts) Problem 5 (answer on page 6, on last sheet of the booklet, and on its back) Suppose f(z, y) is a differentiable function of two variables such that VF(3,-3) = 68-27, VF(-7,71) = 37-7, 90,2) =7 +7, —" f(-7,7) =1, f(9,-9) =5, {G;2):=5. a Let c=r?-s?, y=s5?-7? and w = f(z,y). { age gt Z 1. Find ou and = at the point (r,s) = (2,1). (12 pts) 2 SW - eh | 2. Show that ie om gh yh 2 ss—- +r— = = Or Os a . everywhere in the rs-plane. (4 pis) sn 3. Find a plane tangent to the surface w(r,s) =r +s+¢ in the rst-space. (9 pts) 4. Find a line tangent to the curve w(r,s) = 5 in the rs-plane. (6 pis) Bs a” js a t <a