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Sample Midterm 2 key

Sample Midterm 2 key - SAMPLE Math 16C Sec 1(Malkin Name...

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Unformatted text preview: SAMPLE Math 16C Sec 1 (Malkin) Name: Sample Mid-term exam 2 Student ID: \j Mon Nov 10th 2008 Signature: I ‘ DO NOT TURN OVER THIS PAGE UNTIL INSTRUCTED TO DO so: Write your name, student ID, and signature NOW! NO NOTES, CALCULATORS, OR BOOKS ARE ALLOWED. NO ASSISTANCE FROM CLASSMATES IS ALLOWED. Read directions to each problem carefully. Show all work for full credit. In most cases, a correct answer with no supporting work will NOT receive full credit. Be organized and neat, and use notation appropriately. You will be graded on the proper use of derivative and integral notation. Put units on answers where appropriate. Please write legiblyH I # Student’s Score Total points 50 Page 1 of 6 SAMPLE 1. (a) (10 points) Consider the function f(:c,y) = log2(l — x — y). Find the domain and range of f. Sketch the domain. (b) 6 p ints) Sketch the level curves of the surface 2 = f (30,34) 2 «1:2 —- y2 for o 2:0,12 (c) (4 points) Find the slope of the surface 2 == f(m,y) = V932 — 2y2 at the point (2, 1, M5 in the z-direction and in the y—direction. l“ m? t 52645 \ﬂlx‘lalvaW 2. j, “ﬂ” ( Illﬂ. ’33-... S (“X1 55me 36%;»? is 9% (/4305 “ﬁll” 19;; 2€2r€'£l'£"'ﬁl"l 57‘; "W '5} SAMPLE 2. (a) (5 points) Find all the critical points of the function ﬂat, 3/) = l—l—1n(:z:2+y2 * 1). (Hint: the function has inﬁnitely many critical points). 2% go 1‘} ‘ 0 0&va :0 LAN“ 1:6 an L: l K \ ‘ _ 5 §) “"3"” 4km (ml-s gamed peek c317 2- » x " A \ In») : Egg-7;“ my). cl.\\ {1&5} Sum )Cm‘a :0 0:6; (fallow. 4 ,m \J (b) (10 points) Find a of the critical points of the function f(:1:, y) = 33:2y—4zry—l-y2. (Hint: there are three critical points.) 3.4.... S’Dt : SL3 r Kg : 0 (\$13 (30mm 0 €53.73 or rat: 3 1CD : 378* - 1H", 2%,:- C’ ' ' L, [mm L El: 0 " gizuLhL firm) :06» “(Bi ‘30 (73> 1:0 amt—3 CW 2‘- 1}? 3‘33): “137 W‘Qij’Ot‘ é’ﬁﬂjzowi 5’», Hit: 3 ail-trad falfrg we (awn?) (c) (10 points) The function f(17, 3 — y3 + 3333/ has two critical points: (0,0) and (1,—1). Classify each of the critical points of f. yr : 326+ 33 \ lot 191 “313 : 3.) :‘fgnl‘l 2X, , :13 j Page 3 of 6 SAMPLE 3. (21) Consider the following integral: . /01 [Own + y)dxdy. » i. (4 points) Sketch the region R over which we are integrating. (b) (6 points) Find the function g(x,y) so that 9x = W -1a 9y = 2W + 2y and 9(0, 0) = 0. : MM :: “@45er , 163‘" 1A:- Cw . 3 x} E) Pgﬂpgecf is Somdﬂht’hmkm camel. 3: Elﬁn if: (5‘31"; Q: (I 0 , ( (are 8(1) \3 some gwwiim X; r Pa e 4 of 6 Vi a l V if» pQ€,1—~>( MA 1' -5,” ‘\ (Jim? ( L3 0.. mi PUM> . guys} Hm (3‘20 30 3,): L35 w x4» I SmLQ SAMPLE 4. (a) (10 points) Write down the double integrals expressing the area of the region R given in the diagram for both orders of integration. DO NOT EVALUATE THE INTEGRALS! '3 Hwa 3:“? (3m, 0% 9:3 clgclx. r :5 tr“ X image, \— .a-I 9 o (b) (6 points) Write down a double integral expressing the average value of the function f (x, y) = my over the region R bounded by the curves cc 2 0, y = 1, and y = 290. DO NOT EVALUATE THE INTEGRAL! PM“ O‘l Quiltingﬁsh . " goi'avtlolﬂtkrahaool l (5 v W 1% titt~h~=t\;iuxl tam. M mm D (c) (4 points) Write down a double integral expressing the volume of the solid der the surface 2 = f (2:, y) = 4—312 and above the rectangular region R with vertices (1,2), (4, 2), (1, 3), and (4, 3). DO NOT EVALUATE THE INTEGRAL! Page 5 of 6 SAMPLE 5. (15 points) Use the method of Lagrange multipliers to maximize the function f(\$7yvz)=\$2+y2+22 subject to the constraints 2x + z = 4 and y + z = 8. FLXW) (*2, .3, m 2: 30+ 3103 3’ [21*1“'\B~PL\¢%1-§ Jive 1' Zoe #an 23c) 6:3 3 7; WW. I 6) F3 :2. 23 w” :0 (2:) ﬂu.»:\13 Va. '2: 7.2 “a -fmo C39» \1 \$153 gm (3} ’w @x (3) E.) «x. 2‘} “1-.” ® F’) 2: ~~~ L‘ZDL—t Q; '" 7:0 35:12-14—H‘ Y tho «Lu; . J. [3.19% to?“ ng‘i‘Nl) O‘ @ {:ﬁ 3 is ,2!)ng him“ ® owl Mi“ 3w; - / g, a no 4.; 4+ « .« Aug/“4:0 (2011”) 3L M2 WM) 3;;O.X30§::£1’Wditbs '3‘ Wet acme/x (of “H END on EXAM ‘ 1‘ F . Fag 6 of6 ...
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Sample Midterm 2 key - SAMPLE Math 16C Sec 1(Malkin Name...

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