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20Emidterm1 - 1 Consider t he surface z = f(x y = x 2y2...

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1. Consider the surface z = f (x, y) = x 2 y2 e -TY a) (5pts) Find an equation for the tangent plane to this surface at the point (I, -I, f(l, -1)). b) (5pts) Find the two unit vectors that are perpendicular to this tangent plane. G\ \ jClj-l\::: e L.-Xj ~ 3 -XJ jx (X/~I-- ZrX~ e .- X j e ix.(I)-f) -- 3~ . :ilf .3 'Z- - x :f J d ('it J) -= .z;; X ~ J - X d e- !C I}":'I) ~ .-3~ --QA/r1 r fl ~ ' , .:z. == (2. + 3 e ( x:..- /) - 3 e (~ -+ I) 3e.'I- .- 3 ~ ~ = 5e. \"'l 0 (' YYI a\ " ck R +0 +k .1- ~ -P-:::t 2..3e ) - 3 e )' -I' !.. 3e- - 3 e. -\) + ./ ) 2
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311 (- 2,S\ ~ 2. Consider the path c(t) = (2 sin t, 4 cos t + 5), 0 ::; t ::; 27f . a) (5pts) Draw a picture of this path, clearly indicating starting point, end point and direction of movement. b) (5pts) Find the points (and corresponding values for t) at which the speed is maximal and minimal. c) (5pt s) Find the points (and corresponding values for t) at which the tangent line to this path is parallel to the line y =~x. e.- {\~ J " 7-( l?-0 - LeO) = ( 0 ) q) = C (2/T) C.(!)::: (p ) 2 fi -rS\ C. (1E\ -:= ( :J- ) 5 \ C- (/i) :::: (0) I c.. (3IJ :::- (- 2 I ~\ ~ ') spe ed '. :;:; fi c.,i (-\-) h = \\<-..9. Co s -C ) - 4 Si ~ t '7 II .== 'i 1 Co~ ~ t-!£ & 'fN:.- = fY eo.? i + Lj s; Jt 1- It) 5J ;lt == 'flI +-/ P5 ; ri1..f; '5, ~'-l = 0 So o ( O ,ql 5~ 'nt -= 0 (0) 1\ ( v, q) V) ~i m oJL vJ ~1tV Y\t..t ::::
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