MATH 201 Exam 2, Spring 2016

Spring 2015-2016 Prof. M. Egeile} ‘ Quiz 2 Date: March 17 ror. iVl. Egeileh Vath * f tj Math 201 - Sections 5 to 8 Duration: 1 hour Problem l (answer on page | of Hy, booklet) rind the domain and range of the function lei 2 2 mY ; . nge ot the function f(x, y, z) Int] 7 yy z*). Determine if the dom 117) Ol / ] Ml ODEN Teron 1 / ey] ‘ \ j | ] oe ee 1 Ghd iL ; Rion, a Closed region, or neither. Also, determine if the domain is bounded 01 wnbouwnded L Problem 2 1777." er on page Z of th booklet) Find the tangent blane and the al Ii 7 ind the tangent plane and the normal line to the surface - at the poimt (1, 2, 1) I : : ent i PUSUEL ‘ 5 hy : cr’ + 4’ 1z Problem 3 (answer on pages 3 and 4 of the booklet) For each of the following limits, say if it exists or no. iustifvine vour anewer (8 até oncki justifying your answer. (8 pts each) * tn aa eee ae | we; oy ty J/l+y-—2z PS lim ‘ - — b) lim = a C) lim mA Boe 00S Se J 27)—3(0.0) i | j +(0,.0) ./ 72 - 2 (x,y) 3 (0.0 Lc . ve Vy 2 roy (x,y)—(0,0) V 4 ey (2,y)—-(0,0) eo 6 Problem 4 (answer on page 5 of the booklet) Let f(x,y) be a differentiable function. At the point (2,—1), f decreases most rapidly im the direction of ¢ = 31 + 7. In this direction, the derivative of f at (2,—1) is —2\V/10. Find the derivative of f at (2,—1) in the direction of w =7—j. (15 pts) Problem 5 (answer on page 6 of the booklet) Let f be a differentiable function of two variables, (zo, yo) an interior point of the domain of f, and (x,y) a point of the domain of f. Let uv = ai + bj be the unit vector from (29, yo) to (x,y), and At the directed distance between the two points, so that (« — to)t + (y — yo)j = (At)i. 1. Justify briefly the following approximation : (5 pts) Af = (Vf (0, yo) -%) At for small At 2. Let R be the error resulting from the above approximation. es Show that jim, nee 0. (5 pts) Problem 6 (answer on last page of the booklet, and on its back) Suppose f(x,y) is a differentiable function of two variables such that : > Vf(-7,7) = 31-3. f (9, —9) passes also by the point (1,2), and that the tangent line to that level curve at (1,2) passes by the point (7,7). , Finally, let api~gt, yost-r?, and w= fay) 1. Find a and ~ at the point (r,s) = (2,1). (8 pts) v d * » 9. Find the directions of zero change of f at the point (1, 2). (6 pts) 3. Find a plane tangent to the surface w(r, 8) ’ —r+s+t in the rst-space. (9 pts)