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The october 4th, 2002 math 212 multivariable calculus midterm i exam. The exam is closed book and closed notes, with no calculator usage allowed. It consists of six problems covering topics such as volume of parallelopipeds, curve equations, function composition and differentiation, and surface analysis. Students are required to show all their work for full credit.
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Instructions: This is a closed book, closed notes exam. Use of calculators is not permitted. Showall your work for a full credit.
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Problem Max Points Your Score Problem Max Points Your Score 1 10 4 15 2 20 5 20 3 15 6 20 Total 100
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(1) Compute the volume of the parallelopiped with sides 2i+j−k, 5i− 3 k, and i− 2 j+k. (2) Find the equation of the curve obtained by intersecting θ = 3π/4 and x − 2 y + z = 2. (3) Let f (u, v, w) = (eu−w, cos(v + u) + sin(u + v + w)) and g(x, y) = (ex, cos(y − x), e−y). Calculate f ◦ g and D(f ◦ g)(0, 0). (4) Let f (x, y) = xy. (a) In which direction from (1, −1) does f increase the fastest? (b) Compute the directional derivative of f along v = (0, 1) at (1, −1). (5) Consider the surface given by z = f (x, y) = x^7 y + xy^7 + xy. (a) Show that f is differentiable everywhere. (b) Find the critical points of f and classify them. (c) Find an equation of the tangent plane at each critical point. (d) Compare the surface with z = xy and explain why they have the same type of critical points. (6) Let S be the closed and bounded surface obtained by intersecting x^2 + y^2 ≤ 1 and x x 2 + (^) + 2 y +y z (^2) += 1. Find the absolute maximum and the absolute minimum of zy − y + 1 on S. f (x, y, z) =
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