Exam 2 Solution Spring 2015 on Calculus III - MAC 2313 Exam...

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MAC 2313Mar 5, 2015Exam IIProf. S. Hudson1) [10 pts] Evaluate the integral:R20he3t,sin(πt/2)idt.2) [10 pts] Find the equation of the osculating plane for the curve cos(t)i+ sin(t)j+katt=π/4. Explain your method briefly and circle your answer.3) [10 pts] A particle moves along a curve with speed||v||=t2+e-3t. Find the scalartangential component of acceleration whent= 0.4) [10 pts] Evaluate the limit (if it exists) as (x, y)(0,0) by converting to polar coordi-nates, limpx2+y2ln(x2+y2). Remember to show all your work.5) [15 pts] Letf(x, y, z) = 2xy2z3,P(1,-1,2) andQ(0.99,-1.02,2.02). Use a total differ-ential to approximate the change in the value offfromPtoQ. Simplify completely.6) [10 pts] ComputeDufatP, givenf(x, y) =e2xy,P(4,0),u=35i+45j.7) [15 pts] Answer T or F; you do not have to justify your answers (but this sometimeshelps, if there is some minor misunderstanding):Iffx(a, b) andfy(a, b) exist thenfis continuous at (a, b).Gradients are normal vectors to level curves.Iffis differentiable atPand||u||= 1 then|Duf(P)| ≤ ||∇f(P)||.IfDis a closed set inR2then every point inDis a boundary point.Iffxy(a, b) andfyx(a, b) exist then they are equal.8) [10 pts] State the definition ofdifferentiablefor a functionf(x, y) at a point (x0, y0). Ifyou usedf

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GRANTCHAROV
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