review-chapter 15

Review-chapter 15 - A.V a 1 ’.— A 2> z_3 221/3_V_V,W,WW_~ W.‘2:§£,sz:zm> 2c 60:6 k1:1” c9 4 401,2 gig Q 123>Qggfl\7(a{cogggpw —

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Unformatted text preview: , _ _. . A .V/ _ _ _ _ a. 1 ’ .— ._ A .. 2;} :> z :_3 221/3 _ ,_V.__V..,W,WW__~..____________ W_..‘2(_:§£,sz:zm;>. .... .. 2c : 60:6, k1; :1” c9 4 401,2 gig Q 123>Qggfl \7 (a) {cogggpw — V3 > 0< a «9.17 .__...—_.. “Kai-311%:ij Z ‘ / “4:42.134 sh-..“ J~ w .7 “a,” {LL}:f/X11:45.11."32*"-jxfif.,;.1é.,._ga , HQ: 1/, “flaw”, meant), F. 2 not . a. +* 73:51, h_____,‘ , V Jflléfloml (3/13.) _- _4_ , * 41:2, 3.9.1 ._ 01M; n: _ / .5“? _ _ , _ _ 9y n+2 (50339 #3 _ H ’ h / h—ba " ,5 * ” :‘mfiy w '“::#;‘*2;;g;1@-*§§<* A i: _, . ,,.-,~,-_,_,V3_ , -v- ., I : 3 -3 #6,)”:in :flaéméfalmma , _ , _ -4 Saga)” )6 1L * (W) “L “a £1“ .2: / m" I J a} V/m“ m _. r / r} dJ/Vdf /+M’~ Jar—“#— .— I W Wm“, M _._————— WWW WW” .._._,.____._______._. .___.___.____. m WWW_M_.—. a... _._..__n__.. ..., w_____.M—_WW~ .._ ._._......._....—— .______._..__—_ u.“ w ....._._L....____. ..._ MM " C NV ,5 {I , 0d)». % “4.4,..42). H ¥ , ¥ V710 .Vgigg J- 1 a. m1lo,2§m,.4z.;_ ,, . .Lty. 7. “131:1: _,=, / W K17”: . w ., m. ‘,,--.‘,‘.-,W,_.m,.w .,“_Ww~.ww::;flmm_mm X” ' 'W (ff; w ‘..,.,_,LILQJJLLWQWWWWWW—w .V 12. Problem: Consider the function f (x, y) = x2 + y2 subject to the constraint: g(x, y) : x2 — y2 =1 . a. Sketch the graph of the level curves of g(x, y) = 1, and f (x, y) = k, k = i, 1, 4, 9. y / " “\Aa} / , “M \ K7“! \ b. Compute Vf and Vg. c. Evaluate the gradients at (i1, 0) . Can you find a relationship between these two gradients at those points? Does the relationship holds at any other point? Sketch these gradients in the graph of part a. d. Find the minimum value of the function f constrained to g and the point at which this occurs. 13. Consider the following problem: Find the volume of the largest rectangular solid inscribed in a sphere, that is, maximize V(x, y, z) = (2x)(2 y)(22) given that x2 + y2 + 22 = a2. a. Set up the problem using the method discussed in section 15.7. b. Set up the problem using Lagrange Multipliers. That is write down all the equations needed to solve the problem using Lagrange Multipliers. c. Solve the problem using the method of your preference. 14. Find the tangent plane and the normal line to the surface x2 + y2 — 22 =18, at P0 = (3,5,—4). V1.3; _ . ., , .., ,. .: __ M. _, k .. A W _. _, . ‘ _, . 7, Ag” ,mangn V = “#2 _ , Z++£f2_;:a/+ V, > v . __ W _V E“, Y Z = M76177. v 7 y, ‘ _, _._,,_,,«_;sz aixgm 41;»). 4 _ , _ M _ ,, “.3291 ’— V4712 “Tuiéiffia‘fZJLQ/d: A _ V, 4 1.4/1wrflhflzfityui’itzfla = M vi 3 5 k- ‘11“: _________ ,, w é CL ’3)- Had rsééfiy) ; a “2;: J3“ fig 1: _ 7. y Em; 5/0 I: v - H 2 t m ...
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This note was uploaded on 01/22/2011 for the course MATH 208 taught by Professor Riosadams during the Spring '10 term at Clemson.

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Review-chapter 15 - A.V a 1 ’.— A 2> z_3 221/3_V_V,W,WW_~ W.‘2:§£,sz:zm> 2c 60:6 k1:1” c9 4 401,2 gig Q 123>Qggfl\7(a{cogggpw —

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