Midterm-06-w - University of Waterloo Waterloo Ontario Mathematics 237 Mid-Term Test — Winter 2006 Time 1 hours Date NO AIDS PERMITTED Initials

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Unformatted text preview: § . University of Waterloo Waterloo, Ontario Mathematics 237 Mid-Term Test — Winter 2006 Time: 1 % hours Date: February 14, 2006 NO AIDS PERMITTED Initials: ID. Number; Family Name: ’ Signature: [:1 X. Liu Section 001, 8:30 am. E] D. Wolczuk Section 002, 11:30 am. E1 B. Bodmann Section 003, 11:30 am. Instructors: Your grade will be influenced by how clearly you express your ideas, and how well you organize your solutions. Instructions: 1. Complete the information section above, indicating your instructor’s priate box. 2. Answer all questions. Two marks of Quesmons Mammum Mark each question are for exposition. - 3. This examination has 8 pages. 1 Efifil I 15 ENE Math 237 Mid—Term Test Page 2 of 8 2m [18] 1. A function f is defined by (J/Vflé?‘ flay) = (w — 2y)2 42;, ‘ 4:; a) Sketch the level curves for C = 0, 1, 2. . r 7207) Z 1717;? 71 L/ ff L ‘ -- flmlfl/‘z' CAM/fit fizz J 113/ fa 745244 4/ / ,2 / 6’ .1 0 -: ( pf" -' - f 32’: 7: film /( b) Sketch the cross—sections with the :v = 0 and y = 0 planes. Zf <74”ng L» 7 MM ‘ l 9! 17 V X L @— N”"“\\\_’7 4:) 4W1 :V/ac vwfl gala ~ng ’ 3w 9» 0) Determine the equation of the tangent plane to the surface 2 = f (x, y) at the point (1,1,1). "5 7 l s ' .l " if i “W” i’flwwi‘ IW/ m n f Y; a l l 9 j / 35 C I 4 (-z)/>(43 h (4 3&9le ,2 Q : ‘ ,1(14»~D {- / 6? Math 237 Mid-Term Test Page 3 of 8 [12] 2. The volume of a silo is given by 2 3 2 V(T, h) = gwr + 7T7“ h where 7‘ is the radius and h is the height of its cylindrical wall. a) Suppose the silo has a radius 5 m and a height 25 In. Estimate by linear approxi— mation the change AV in volume if the radius is decreased by 0.1 m and the heigh is increased by 0.1 m. ' r , is" t, if, P} H? i ’ ’ if "QM if f AV {(ZYTXQM): “(1%); (“@934”) fl V“: / of”, 4 (M‘jwml b) Discuss the validity of your estimation, i.e. explain Why the approximation error is expected to be small. Wm r; Mama 62% 46; will [7'22 We age-e ” V a) 551 [pa fit};in 5;” 4/ if,“ j; / [II/£7 u [NW M/‘NXM at? 2M jig/Kw) .. "(its a flaw ya)? 3”“ i/l ._./ ‘4 i 9-. E a ‘1" '15:? Math 237 Mid-Term Test ' Page 4 of 8 [14] 3. A function is defined by ms, 2;) = em siny + My a) Find the directional derivative of f (x, y) at the point (0, Z) in the direction of the 2 vector (3,4). (/ 7, G 7%]. Qéflfii : W133»? X”? + WV 3 6” @3szwa v’{(g:-E) :(H—{Js 0+0“) .: (3,0) 3 :7 % “(5,4311 My}; (35» t r 9 Z (l;0)‘(%ay—E:) : 9’: if g b) Find the direction and the rate of the most rapid change of f at (0, \é Mar-£74m J Mar? WW? I‘mmm {r 7% Amm of 7% flmflr’wz‘ 7? fly Mm " é “ Maj/Judge '( /, ”" #7200, 747/ = m M : , 767 W % inc/2%; wk 8 k W“? V6042 44/ 0 Y0 77’ illkff fig //L9\/1£4/l n // W Li: W), Math 237 Mid-Term Test ' Page 5 of 8 [14] 4. Suppose f(a:,y) is differentiable, and T(r,«9) = f(7" cos 0, rsin 6). 8T (2 7r I ’L r/g a) Calculate — —) if Vf(1,\/§) = (—1,1). 60 ’3 55;; : D.{H~Slvle)l+02frcogé *WJ ~910,2)vszsmeM mam? : “7"?4243? " 99 ’ _ m: e 7 2"; + I: L l-rre' / WM?) : K m, w): (47 z) a» <%i-‘>2+;% (2—3)? = (292+ (2—32 95: : ®.Fm59+ Mme "g ‘— ~anfllfl9 “f szmmg ‘ "T 1 ' » - 12M ‘ 1 < .0... ) : (Moss) + .{ 21(605g‘m94 (viking) 2r (fill : (W mm? ~ WWJ r‘ slag (650 «e (M “47931 L z 2:; mme am; m ‘ V (gm 1 j + *Uoéffiw‘eWWfi (\‘UCQ C (03‘ )1”! 39 4 @maellflava 4 @0141“? % @zfll will? : (OlfflCoCél 3: 4 @2102 (Comkfiwfl : r / I; ‘6 filmy! “ gwnm Q? “fl/Mg, Math 237 Mid-Term Test Page 6 of 8 '1 [27] 5. Let f be defined by 2 m y 1 7 O? 0 f(x,y):{ $2+y2 (1' ) ' 0» (way) = (0, 0) a %,W<;m /.',41»2fl E A/V/é/6h7lfé/k if /V/ a) State the definition of differentiability of a function f : R2 —> R at a given point a= (a, b). it) ‘fl/z‘ ml WWI 6W} 5) m 1% aw \(¢,3)—%(4,b:fl’—j‘[€q@~ [M7 7% WW (it/15>) b) Show that the given function f is continuous at (0,0). WE M/l/K/ flflf .’ 4) ,Zkfl/fl); ré'X/77‘ (Mai—(m - é) /M,0/‘ : fll’WZ/JZMJ (“8290;0) m A 79% .72? 3 57:529) ,3 a 75 +31 Lfig _——— ( flwe MI W” f‘azfivn' Mmi m N13“ >4 ) M529?) 75‘”? fly 4/ [xlwlwi : {5‘ 717%?» 761,992 , 2%. {éflmzf 56%; i O 'i 947; > x/ J firm/Q ,WéJ/ngi «véfwfiifé/ fir/A’éé z/M/f fl if %»7‘ ///7/M) /0/ é MM?) 749V 9% ’ (2% 7%! f gCi/szx c “NV” Math 237 Mid-Term Test Page 7 of 8 r a“ . . 8f 5. c) For the given function f, calculate —(O, 0) and 1 c ) 6:1: {MW ” §xm N (WWW 0 be 06:3):(010) QfO/w fee»; “ 0 8f 6—y(0, 0). MH l/flr/éy [719/0 7é/W IL! 749/ ’4’ )fflflj/ “47/ fl// W / fl/l/ /” % am/ wil/ 97% 48 fl 4 v‘ rfi l4 f V ’ {fl/fl) _ />’ éé‘éw’é’ gay/Gm; /‘~ WWW [7/ 44 W' . . V. 1%] d) Is f dlfferentiable at (0,0)? “kg; MI M/M/ fi/ f 3 {/5} MM V ,' " / 7[ (1/9)» UM? / ‘ g-.. ir % m m a y , WHO?) [fl/76y), (070) “15/ Zfim / 2 747.7 flwabg4 71/8, 5%” [WI/W1 Way/j - QM) : flMH 00M) r o/QJQ) : 25 Wm): ///M)/ t “53' ‘ c FWD/L (22+31),;)){W\f{ I . a ‘ #1 [(747) I) '29? i 171M m4 Z)», {(7%) par/“W 7’29? W M, ,1 ,7 W, I w ./ w “f (/th We, . lflwfirz Mm‘f / 057% 615] 6. Answer true (T) or false for the following statements. You don’t have to justify your (0’ answer, but you’ll get a penalty of a negative mark of 2 for each wrong answer and zero marks for no answer. 8f Bf [ a) If $(a,b) and—83m, b) exist, then f is continuous at (a, b). [ T ] b) If f : R2 —> R and Vf(a, b) = (0,0) for all (a, b) E R2, then f($,y) is a constant. [ c) If both gig:— and 2—: are continuous 61’ (a, b), then f is differentiable at (a, b). [ d) ( b)f(:z:,y) = L if and only if ’f(:c,y) tends to L when (my) approaches (a,b) my —r a, along any straight line. 1%,” (mm 2 . t2 NW“ e) The plane curve F(t) = it t — 3, — t + 5) intersects the/level curve of f (I, y) = 5x2 +_ 333 orthogonally when t z: 0. 51 Z 1’ \ ,,,, J/ ...
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This note was uploaded on 07/28/2009 for the course MATH MATH137 taught by Professor Oancea during the Spring '08 term at Waterloo.

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Midterm-06-w - University of Waterloo Waterloo Ontario Mathematics 237 Mid-Term Test — Winter 2006 Time 1 hours Date NO AIDS PERMITTED Initials

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