test4-sol - |||l|||||||l|lll|||l| lllllll llllllllll p 0007...

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Unformatted text preview: |||l|||||||l|lll|||l| lllllll llllllllll p 0007 3" _ 8a42 < 03 Test 4 9-10 November 2005 Math 119, Section 1, Fall 2005 Jason Grout No calculators, notes, or books. Instructions: Read the questions carefully. Put your answers in the provided boxes. In order to receive full credit, you will need to neatly show your work on these pages and simplify your answers appropriately. Do not attach extra pages. If the provided space is insufficient, use the blank sides of adjacent pages. 1. (5 pts) Which of the given graphs corresponds to the function 2 = $4 — 312? Circle the correct graph. (Warning: do not pay attention to the scale of the axes in the graphs below). 2. (5 pts each) Let f (it), y) = —m2 — 41:31; + 63””? Find the following. (a) f(—1,2) (b) Mar, .71) (e) fszv, y) (f) Z—yfi"<x,y> ~2 ’L/x 1'2 6 I: T ® (B 4 3 (3a ?x+25 3x +23 X ,\ , . H + 4 “V I: “r gs _ . W 15 3. (5 pts each) Let f($,y) = (3y — :5)? (a) Find fm(2, 1). Write a sentence or two interpreting how this value affects the graph of f (9:, y)- .gttzaej-x) 14H (2,0, as M MMM W )éidt‘rM/Iém [13,} [mpg/Jab flea X'flwl‘s Wd‘fifugml' |\Y\ 'WFOEJ'VW X' W‘J’M (b) Find fm(2,1). Write a sentence or two interpreting how this value affects the graph of f(a:,y). w XX Cgllxflé MW lh4l¢£ xiél/I‘I‘fih‘l Jrcrl’t‘g WWVW' L : ;.'£Mc§/.w flu lax-MM;pr j>// #wfl «malice Mp2. n; L. 4. (4 pts) Suppose fz(a:,y) = 2x — 6y and fy(x,y) = 12y2 — 6513. Find the critical points of f. Do not test to see if those critical points are relative extrema or saddle points. ’; : X33¢ (0)0) Mm 0 > I 5. (8 pts) Suppose we have the following table of values for the partial derivatives of f(:r, y). For each of these points (a, b), fz(a, b) = 0 and fy(a, b) = 0. Find and classify the relative extrema and saddle points. If there is not enough information to determine whether a point is a relative extrema or saddle point, then put “Not enough information” (61,19) fxz<aab) fyy(avb) fmy(aab) 1 (—1, ) 0 1 —3 (2,1) —1 —4 (3,2) 3 —2 (1,4) 2 2 l wa ./ ,‘,_ :— ' (—1,1)2 D{_"\) ,0| ‘1 740 5AM? T. D c2,» = L-nwu'fw <2a1>= Nat-W74 a; 7 Mal) ': 36$) ’06) <0 (3 2 ' V6. 91 L0 21*! 9370 (1,4); Kala/>3 \, 7 70 g‘ 6. (8 pts) Let {2” W n 3 f($’y)=$2—2y$+y§+22. Find and classify the locations of relative extrema and saddlepoints of f(a:, (DID) 5,4DDL6 1 2y '2? :0 '5) X'Tj £1,13flfibkflrlg MIN. x 7. (8 pts) Using Lagrange multipliers, find the location of possible extrema of f(:v,y,2) = 332 +312 + 22 subject to the constraint 32: + 2y + z = 14. A "L 'L’ l got/mt) ’- 3x myrz ""l _ , . ,,x lull/m MD a) V2" W2 5 : 21 : «if-*5? 11*0 ‘19 3% ’r 1C?) ‘LCEXYH Ufi‘fll / {j :7 l"\ :[L( 90 XZB' / 2 3 {hr/l fly/“VA 8. (3 pts) The rate of change of velocity v of a boat in water is proportional to 100 minus the current velocity. Set up a differential equation modeling the speed of the boat. dy _ 3$2+1 d$_ y 9. (8 pts) Use Euler’s method to approximate the value of y(6) if and y(0) = 1. Use a stepsize of h = 3. 10. (8 pts) Find the general solution to the differential equation gift-q = yfi. d2: Please show every step of your work. if; 1.. : It 5 ’ m6 )\ l \ -: ” g 07 \ Ax ‘ w / 5 a M ' 3 r *L/ , if; . I; l“ L U 2 m zfi & l \‘ 5 6° ‘5 M wk N‘ I 0‘ V” M” fl 037?; w le’k gl “8 ’ 11. (8 pts) Find the particular solution of the initial value problem dy a: _ _ _ EE~E, y—2whenx—1. You do not have to solve for y in your solution. ‘5 \\ M734 J‘— 24W M '1 x C/ j; “1* ...
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