# thq_04 - A04/OLL/Ol Use T.C 16s MATH 2451 thqr04 Exam...

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Unformatted text preview: .. A04/OLL/Ol . Use T.C. 16s MATH 2451 thqr04 Exam Multivariable Calculus Course 2016—02-02 (Tue) McCary Due Date Instructor Net ID ﬁvmwﬂ mnueeeeaee ®©@[email protected] F \ (k ®@@@@® )\ @ﬁ 1 .x. J hawmw Q “we @ mm. C @ ® ® PM» W lieeaessssseesa @©@@@@ @@®® [email protected]® {\le .esst L®®@®@® w®®®® E®©® (K. @ a} A- J. .I\ N mw @®@®[email protected]®m .\ ./....\ Awe .,.,.. Y n @ \ : A“ eat NW,( émﬁgé t©®0®@®@®@@®@ ,. A. t. @@ n.1,. [email protected]@®@@®@ ®3®@®®@ Nix... y\ @ﬁ@@ r- x ;@ (Last Name) 19 2016—01-26 14 .. AO4/0LL/02 . ' Let A C R" be open, let if E R”, and recall the deﬁnition of the translation of a set= 55 + A = {55 + if I g E A} ) Show that :35 + A is open- Lg/Efﬂffj/ .- [3k é IP14 Q IA x i G ’7.” .// A H2 >< nil .7 Show (75,32) X (—5, E) is open in RZLJLJZ 1,:"1‘Jr7L/ ‘ thy ,"r ‘ :‘(j‘s‘I- ‘ .——-.‘ w / Q” }| \‘7’ p o In; OWE; ' J ‘P f .5 {k T :rnZTéE/P'IV *3"! Olga—N - / ‘ \. J / k 44? a} \$Grader0nly¢ O 06) o .. AO4/OLL/OL . .. ( 10 points) Find the natural domain of f , and plot this domain on the provided axis. fa) J, Grader Only J, J x.» .0 AO4/OLL/04 O 3. (10 points) This question is related to the derivation of the sum of a geometric series from Calculus II: 1 0C 27‘": 1—?” for |r§ <1 k=0 You should go review your Calculus II text (or Wikipedia) to make sure you are familiar with this important technique. Let B be the matrix given below. (a) Compute the matrix R such that B = I - R. ‘3' r. _ - \ " (b) Cpmpute every positive integer power of R.’ That is, compute R, R2, R3, 124%”. . .. ,1 m (c) (Pompute the inﬁnite sum S = I + R + R2 + R3 + R4 + . . .. \\_-/‘ (d) After reviewing the geometric sum ﬁom Calculus II, and with the above parts, you now have enough to determine 3—1, 50 what iS 3—1 and K \ I' if? __ é , r/x: ‘I Yn/KF ‘- .l J x «I»: If, ‘v ‘ */ 1-117 /‘T 6,—1- f f} I '- / ‘1":5‘ l e 5 /\ “r” * B: 0 1 5 '0’ / O 0 1 , , ,z-f. ,,= .- — 50~g—£;i c9 03"} in I; - q ’05:— .. — A ’n‘ ‘. r ' f- 0 c? 0 :2 a q v r” O L/ L) j“ Hi ‘ I ' .. N“- . ‘ k / “if ,1 j ”' ./" ' “VT—T 4 i , . * u ‘/ - -'"i f‘ u r rm — J ‘ J'““" I r I ‘3 _ K“ Ki 1‘\/ I. J ’ — 7 i/(I ' _ k l / i J L — ' 7— ‘ '[7 1 i - if “ti / _ UL”: ——' l , 1;; i“ "If y - ‘ {r x/ ._ f ‘ A "I —’ ' 7 ' l /__r I a“) l .-. . I j,- T ’1' ii I"?! 4 2L]; {3 _: v/i til/iii 1 00 6‘ ~ 4 f 1 r: u ; a, i .921" A:// l 1 ' D I 5 r; —' ,ﬁ 1 C; . ‘ ,—— , 'd—u r- -" L1 ,ﬂ") :1 v /‘ _ V L {a i if r. '—_-}_ ‘ _ z / i ‘ f i \- — i . L)! T ‘ I‘m/I , J} + I H, 1 k . -* Q Q 3: [I 0 J I I S //- n .L '9 9 ~ % ‘0 J (:15 ‘ 2 if / a in ~ -4 v J '. ' /‘ 3' i GraderOnly J, O O o (a e .. A04/0LL/05 . ( 10 points) 15, 6 is not required — use limit techniques from scalar calculus. ,3 (a) You may not use derivative — you must calculate it directly. “ . W ‘ r :- x m j’il’ﬂ if ‘ r? in f ( _ f ( ) ("1 ‘l‘ ;i‘/X{J ‘ \ I / 9—49 3:392 2 M. lim _ . h—m _.h where f ( / \ ﬂ ' r' [I .4 ':- _-"_ ' V ' ' -. ' ) 5 " ‘ r ’4 I I x , _ - r , ,' " —*** V - '__ ‘ ‘ __V , (b) Compute the following limit by converting to spherical coordinates. - 351927 I? ' { 7 ,N _ . K ' " hm 0 \$2 + 9.2 + 22 "_ l‘fl/ l. ag/jé’ “"1 F" I M ‘- U U U” 7“ “f . ‘ ‘ I: _ x“ ‘3 F—tya F —> 0 ,l l y fl ,- / L r ‘ ( z 0 5/ W '4 “v ~ 'k ~+ r v. " J" i r, i. 5‘ = m j” J , 13 r‘ I 3’” J V {fl/La { 1‘233M5‘yc u n/ {a ’/C,/ / 9/ l, GraderOnlyJ, . O® @@@@@[email protected]@® . J CI .0 AO4/0LL/06 . 5. (10 points) Ungraded questions which you should be familiar with. 1. With 5,6. :62 lim I 1 sc+y y _} 2 2. With 5,6. lim 5x2 ...
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