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20EPracticeMTSolns!!!

# 20EPracticeMTSolns!!! - \j C\l =-2S"l~~ 2 c.~D;b 2...

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1. (8 points) Let f:]R2 -+]R2 be given by f(x,y) = (e 2z +.,ez- U ), and let c:]R -+]R2 be 2. (6 points) Verify that the path a path with c(O) = (0,0) and c'(O) = (1,1). c(t) = (e- 3t cos(t), e- 3t sin(t)) (a) Compute Df(c(O)). is a flow line for the vector field F(x,y) = (-y - 3x, -3y + x). (b) Let -y( t) = f (c( t)). What is the tangent vector to the image of -y at t. = apply the chain rule to -y = f 0 c.) ()'(-O):: ",~oC) \ lD') ::: \)~ lc.J~») oc.,'(() l~~ O~ Q'ro-Je.. \ · ~ < '3 \\)

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3. (8 points) The path c : [0,21l"] -+ ]R2 given by c(t) = (cos (2t) , sin (2t) ,2t) is called a helix. (a) Compute the velocity and acceleration vectors for
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Unformatted text preview: \j -::: C \l-\.-) =-(-2S\"l~~) 2. c.~D. .;b-' \ 2') ~ -:: ( -icos2t l -~ S.\()~t , 0) (b) Compute the arc length along c for 0 :'0 t :'0 21l". s:1t [., si,,"-2t. \-L\ ~.C~?"2t -\-!=\ d-b S ~ tI' L-\ ( ~i (\':l-2:t ". (~,2 2:\:. . ') ~ L\ d Co-S;~ ~J< JBt \~~ 5. (6 points) Let f(x, y, z) = x 2 + y2 + z2 -2xyz. Compute the following quantities: (a) V f (b) V· (V J) j\ns2 . \L ~l ~ 1.-~)(. .\'\ ~-2)(-~~ ~-2Y2 2-c-2 >c'1 -Al2~-l"'t \ -t~ 12,"\-2~) L ',~2 _-2) ~-\:l.' l-l-il~ ~ 1 ."'" l -'2. --2') \l. l~~-2.~)-d;J~t2.><-Lvt~ J I _ ~ T \ ~/~~ C2.'1-z. .~~) -al~~(2.x-11~) , -o....
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20EPracticeMTSolns!!! - \j C\l =-2S"l~~ 2 c.~D;b 2...

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