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exam2sol - Math 16C(Fall 2005 Kouba Exam 2 Please PRINT...

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Unformatted text preview: Math 16C (Fall 2005) Kouba Exam 2 Please PRINT your name here : __________________________________________________________ Your Exam ID Number ___________ 1. PLEASE DO NOT TURN THIS PAGE UNTIL TOLD TO DO SO. 2. IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO, IN ANY WAY, ASSIST ANOTHER PERSON IN THE COMPLETION OF THIS EXAM. PLEASE KEEP YOUR OWN WORK COVERED UP AS MUCH AS POSSIBLE DURING THE EXAM SO THAT OTHERS WILL NOT BE TEMPTED OR DISTRACTED. THANK YOU FOR YOUR COOPERATION. 3. COPYING ANSWERS FROM A CLASSMATE’S EXAM IS A VIOLATION OF THE UNIVERSITY HONOR CODE 4. No notes, books, or handouts may be used as resources for this exam. YOU MAY USE A CALCULATOR ON THIS EXAM. 5. 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. What you write down and how you write it are the most important means of your getting a good score on this exam. Neatness and organization are also important. 6. You will be graded on proper use of derivative and integral notation. 7. Put units on answers where units are appropriate. 8. Make sure that you have 7 pages, including the cover page. 9. You have until 8:50 am. sharp to finish the exam. 1.) (1‘2 pts‘) Compute 31., 2y, and Zn: for z : 1173 + sing + (23’5” . XV Ex: 3xL+ YE 1 a: 2.) (12 pts.) Evaluate the double integral / / (2a: — 33/2) dy dry . 0 .0 I YCX '3 SO CRXV“* y3j{y:a $1, :3:—( 3 H :35. (9x .2 3.) (12 pts.) Evaluate the double integral / / 0—1;? (1:1: dy . (HINT : Switch the order 0 y 1 /2 of integration.) 4.) (12 pts.) SET BUT DO NOT EVALUATE the double integrai(s) for the average value of the function f(x, 3/) = my over the region R given in the diagram. 1/; y»: Am £2380 1435‘? X W/Q 5.) (14 pts.) Find and Classify each critical point as that which determines a relative maximum value, a relative minimum value, or a saddle point for the function f($»y)=$3-313+3:vy : lxz 3x2+ 3v : o —-r 7: “xky CM] *7: ~3y*+ 3x:o ~—-» x: V‘ 9\ 72:22:11 2 D= 8% 57 ~ (ng :Cé/Cél-‘KW’EWM M gXK : (o > O M Cf/~ W 4 WV] WM (1 .2: _( 6.) (14 pts.) Use the Method of Lagrange Multipliers to minimize f(.1:, y, z) : 11:2 + y2 + Z2 subject to the constraint m' — 2y + 62 = 82 Fng/e/A) : ><z+ y"+ 3‘» A LXV‘WWE‘J‘K) 3x4+XA+7§A~AX+AAyV®A~Zi+XRA "’7 FA: ~><+ay—®%+8oz: 0 76km ~x+a62>0~ Mex/Jr fol : O a ~><~H><~ 36X z—Yo? *9 *Q!X:”é7&~> K: / tbs/“9192) : 02"[email protected]}°102fi : 7.) Consider the function f(x, y) = 3 + x/y ’ 11:2 . _ a.) (5 pts.) Determine the domain of f and sketch the domain on the given axes. $4620 Mag-ma y MW.Lx/yjmmMW 7 7 t K t b.) (5 pts.) Determine the range of f. - .— \‘ ~ '-' r, fl ’ W '— R ) ACq M, / 8.) (14 pts‘) You are to construct an open (no top) rectangular box of volume 4 ft.3 What should be the length 3:, width y, and height 2 of the box if the box is to have a minimum surface area ? (You need only determine the critical point and surface area. You need NOT verify that it determines a minimum value.) 3 ; x7+ szl+ :2wa— : KY 4, (39% 51V]- % C306] :, Ky + Qz><+ 0W}- i The following EXTRA CREDIT PROBLEM is worth 10 points. This problem is OP~ TIONAL. 1.) Find a function z : f(;z:, y) for which I 2 7 , 2 3, 2 x : 3:1:2z/3exsy + 3/8002 my + 3 and f1 : 21:3er‘1” + 63‘ y + .‘ESCC2 my) — l , i J or explain why it is not possible. My" +X=V°<3><2fe /’j'flfi—QQQ<Y)+3 ———=> x3)? @{why'e +75Mx0<yj+3><+~302a 3 1 3x1 , “H = J" <27><3€X 9/4, ex /4, Malay/X + 3((\// 3 .2 3 .2 : QXXJ‘RCX), +€Ky+Xmley)+3/(>7 . 3 .2 34 W}: Rxaygfix Hewichxw ~—1 —>> ...
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