Download Exam 2 with Answers - Analytic Geometry and Calculus III | MATH 213 and more Exams Analytical Geometry and Calculus in PDF only on Docsity! jovrn'onJ MATH 213 FaIl,2008, Prof. Sachs Exam 2 N"-" Tt/Ar.; ATvMAY G number o Show all work clearly and complet ely. You may lose poi,nts forincorrect, incomplete, or unclear work, euen i'f your fi,nal answer is corcect' o Do all work in the space provided, or, if extra room is needed, continue your work on the back of thc exam pages, being sure to indicate clearly where the work is. Put your final answer in the box. Extra paper will not be accepted. o This exam has B problems, each of which is worth 10 or 15 points. o You may not use notes, books, or each other' 1. (10 pts.) Let f(r,a, z) : (r + a)-2 * sin(r'z). the linear ap- 0t* tru-- TT at (Ir L) Find the gradient vector i f (",y'z) at all points and proximation to / near the point tr:7,a :2,2: -L' -z[x \)-t I b= --L(*t1) -t, ln*t)-t I + (R"[-)?s h+ l,Lr-t ) t f t r ,lt-t )' -- j-r+Q= L,.r*a- hFftLry nswer: Y -'1"U'r)-Y/r\n) V) = -L(nna)-'i' -'L (ffr\Lt (/vL€ 2. (15 pts.) If u(r,t) : h"-x2/(4t),find z1 - u,,' /O r . / v-*(not" - {*k ,," }= -t6(ut+fry',0rr)' "-"fu' ,*'+(*ntf ',,.fit.) ,?-^7K H=(.{'Ri "' (:#) {"fu ,, # = ( ur,9-,,; I-fr ? :\ a-*,(+ l v-\.J^t hAY.. t 4 U,-*ilL'S;Uk- UXX t 14 \r'n--*-' - 3. (15 p\s.) (a) Suppose h is a function of the variablcs u,'u, and u', sav glL g(r.u,tu) and that each of thcse three quantitics u' u'tr is a function of r,y. Express #,* i" terms of thc derivativcs of g and those of LL,u,w. DO NOT PUT THIS ANSWER IN THE BOX' ,^.5^t-, 9-^J aV^ @) (b) If h : ln(u2 + u2 + w2) and u : r +2y, u : 2r - y, w : 2r'y, find *4, ff at the point r : a : 1' PUT THIS ANSWER rN Box BELOW. \l -, -l_ '7^rtlt,^r a\o*, h-- L! . ^ rS': _''r avr') t^ - -=- ?- / )r fN L+w,u /;*- 2r) fi&"- $:r -N=z ?x , ?g-- L/bt )..f-- -'|,D \Ear-7vtZr1 lf z= 3 F u?+.ltfulL : 2-l -- | q+l+e= 1y vlY'-_ L*--7 tlxu ^L -- u#n-- f/o \b=tL-L-+t = %l/ t--L ) I I /t" (6 k- \ .HF-HB )^ nltr *l! SrA?{ 5y-' ?..^ ), ' ar %i ?t*ny, 7. (15 pts.) Find all the loc,al of t[" function f (",il :V Q= -;fr'r71 l fb= 'l t maxima, Iocai minima, and saddle points /2)r cos(r) f.or -r 12 < r < 3tr f 2.'/ cr,,,[-r^-Q d , U=o Jrr'63 4 ) X" O o/ X= If I.,,J &.b^) l<tt: f*n-' l*, (oto) = -' J ha.(,rg= L 5o SryDD W ot- (q> l) --I-JJ l*b-- -OOS \ L o P*,. ( ha( F"z TT, o) = I AD'L [nrr) = () M,w (/v) [ "rJ ^'',xt t^r- NS: 8. (15 pts.) (a) How are level sets related to the gradient vector field? 0t rt^/t) D '-@ |, t"A Y+-r Ll'^M1 -) 9t w- \7f- € ,1*w (b) Given the level sets picture below, sketch the gradient field on it paying attention to where the gradient is Iarge and where it is smali and to its direction. [n",t^{- + k sV f ...!; v*k-r^ \ tu,lt \.il lr'o /l \ l-v f,'-b Qrj l=T N-- |
