Richard_Exam_2_Practice_Winter_2007

Richard_Exam_2_Practice_Winter_2007 - PLEASE S HOW A LL W...

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L Name and roughlY sketch translations of the origin' Shou'that 2 X, lirn -;- . does not exlst' (x,y)-+(0.0) x' + y- PLEASE SHOW ALL WORK the surfaces in three-dimensional space. Label any vertices and -1)z =x2 +(z+2)2 .11 b. -2xt -2Y'+z'=16 d. +y2 + 12 - 2r+22=0 e. 4x+8Y+22:16 -2 T. Y=X (l' 1 x- J *vo Flnd llfn --;-- ) : ' (x,Y)+(0,0) (x- + )'".)- For the fru-rction f(x' )'): 1'sxY' ' find f"' f'' and f'"' Do not simplify' 5. Suppose t: "E? . i. where x: ln(u2r') and y: sin(u2 11')' State and use an appropriate chain rule to find 9x ' Do not simplifv' 6.Findtherateol.changeofthevolumeofarightcircularc)4inderwhoseradiusisincreasingattherate of 0.5 nvsec and rvhose height is decreasing'at tl-re rate of2 m/sec a1 the instant that the radius is 2 m' and the height is 3 m' 7. Find rhe toral differential of ftx' y) - x:y + 2x' 8. Find and sketch the domain of the function f(x' y) g. Graph the functicxr f(x' .v) : 4 + x ' * yt ' Also' give the range of f' 10. RoughlysketchthelevelsurfaceofthefunctionF(x. y.z): *2 +y2 -22 thatcontainsthe point P( 1, 1, 0). 1 1. Find the equation of the plane that is tangent to the surface z: x'y +
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Answers 1a. Cone with vertex at (0' 1 ' -2) oriented in the direction of the y- axis lb. Hyperboloid of two sheets along the z-axis intercepts (0' 0' 4)' (0' 0' -4) 1c. Lower half of a sphere with center C(0' 0' 0)and radius r = 2 1d. SPhere, C(l, 0, -l)'r: Ji le. Plane: Axis intercepts (4' 0' 0)' (0' 2' 0)' (0' 0' 8) 1f. CYlinder alorrgthe z-axis 3. 0 O. rla-rt ,2xy2e*" +9*Y',2xyoe*" +3y2e*t' ' lplt*l . [# ] (zu "o'tu' * ur) 6. -2x 1. (2xY + 2)dx + x' dY 8. Domain(f): {(x,V) I V -2x>0}:Setof allpointsinthexy-platreabove g.Paraboloidopeninginthepositivez-direction.vertex(0,0,4).Range(f)= *' * y' - 12 :2' Hyperboloid of one sheet along tlie z-axls' ,= Qy +5! the line Y: {z\z> a\ 2x. 10.
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I v MATH-203 EXAM 2 l. Name and roughly sketch translations of the origin. a. 4x+ 2v + 3z:12 PLEASE SHOW ALL WORK the surfaces in three-dirnensional space. Label any vertices and l- -- y2 +27 =4 d. x2 +y2 +22 +2x-4y-62=0 -A -_y C. e. -2x2 +3u2 -422 =12 f. z=t - 1x2 + y2; 2. Find and sketch the dornain of rhe function f(x, y) : ln(1,2 _ x). Graph the function flx, y) : J" t + y 2 . Also, sive the donrain and ranse of f. Roughly sketch the level surface of the function F(x, y, z): - 2x2 + y2 + 322 that contains the point P(0. l. I ). 3. A a. For the function f(x, y) : 2y sin(x 2y; . nna t. Suppose z: ln(ts2 + t) , where s : e"Y' and t chain rule to find @ . Do not simrrlifv. ay 1,. and f^,. Do rrot sinrDlifv. f--- " : tl2y + x' . State and use an appropriate 7. 8. 9. l0
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Richard_Exam_2_Practice_Winter_2007 - PLEASE S HOW A LL W...

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