Disc2 - Math 17C Kouba Discussion Sheet 2 1.) Evaluate the...

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Unformatted text preview: Math 17C Kouba Discussion Sheet 2 1.) Evaluate the following limits or determine that the limit does not exist. .2 2 __ 4 __ __ . a_) 11m b.) hm W (w.y)~+(0.o> :v + y + 2 (aw—41.1) .r — 1 _ 4 ~ ..2 2 Q) hm W d_) m w cum—M22) x/m + y * 2 (x.y)—>(0.0) m2 + :y2 3 , 2 e.) lim 3: f.) lim try g.) lim my (Luz/HM) \$3 + 113 (mm—Mom 3:2 + y2 (aw-aw) m2 + 3/4 2.) Compute 21. and 2y for each of the following functions. a.) z 2 \$y2 + lan‘ + 6y + 5 b.) 2 2 \$629 arctana‘ c.) z 2 w — y2 ..3 . 4 , d.) z 2 + sin(.1:y) e.) z 2 :yQ f.) z 2 {exzy + tan(3y+4:c)}5 f.) z 2 y“’””3 Determine functions 2 whose partial derivatives are given, or state that this is impos- sible. ' 2w 2 2a: and 2y 2 33/2 +1 b.) zw 2 231/2 —— y and zy 2 3323/ —— :1: c.) 21.. 2 easy—1 and 2y 2 (am—:1: (1.) 2x 2 y2 cos(a:y) and 2y 2 my cos(xy)+sin(.ry) 4.) Plane A, parallel to the mz—plane, and plane B, parallel to the yz-plane, pass through the surface determined by the equation 2 2 acyz —— .733 + 7 . Both planes include the point (1, 0,6) , which lies on the surface. a.) Determine the slope of the line tangent to the surface at the point (1,0,6) if the line lies in i.) plane A. ii.) plane B. b.) Determine an equation of the plane tangent to the surface at the point (1, 0, 6) . 5.) Determine an equation of the plane tangent to the surface at the given point. a.) z 2 1:2 +312 , point (1, ~1) b.) 3 2 my , point (3,4) 6.) Determine the linearization, L(:c), for f(a:) 2 :L’2(;I: — 1) at :r 2 2. Use to estimate the value of f at :L' 2 1.9. 7.) Determine the linearization7 L(:L',y), for f(:L',y) 2 3:2 + 23/2 at (\$.31) 2 (1, ~1). Use L(.1;. y) to estimate the value of f at (3:. y) 2 (1.1, —0.9). 1 8.) Determine the linearization, L(:c,y), for f(m,y) = at (.1331) = (3, 2). Use L(:I:, to estimate the value of f at (33,31) 2 (2.9,1.8). ' 9.) Write a precise 6/ 6 proof for each limit. a. lim \$2 + 2 2 = 0 ) (ma'y)~>(0,0)( y ) b.) lim vii—~33? —y2 = (mam-M00) c. lim .1:- +3 =5 )mwhhp( y ) 10.) Write a precise 6/6 proof for each statement. a.) f(a:,y) = 39!:2 + 4312 is continuous at (0,0). b.) f(:c, y) = 51:2 + y2 is continuous at (1,1). t . z (x2 + mm , (as, 2/) ¢ W» 11.) Show that the functlon f(a:,y) { 125 , (Ly) : (0, 0) a.) is NOT continuous at (0,0) . b.) is continuous at (3,4) . 4 +++++++++++++++++++++++++++++++++ ” Whether you think you can, or that you can’t, you are usually right.” — Henry Ford (1863-1947) ...
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This note was uploaded on 02/17/2012 for the course MATH 17C taught by Professor Lewis during the Summer '09 term at UC Davis.

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Disc2 - Math 17C Kouba Discussion Sheet 2 1.) Evaluate the...

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