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Second Exam, Math 243, Spring 2011, Section 13 - Rakesh

• Points deducted for poorly presented solutions or untidy work.

• Show all important steps. Correct answers with incorrect or incomplete arguments tosupport them will receive no credit.

1. (a) [8] Define carefully what is meant by the length of a curve. Solutions giving anequivalent formula for computing it will receive no credit.

(b) [10] The acceleration of a particle at time t is a(t) = .{sint,cost, !};• given thatv(0] = (1,1,1), compute the velocity of the particle at time t.

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2. [15] Let z = z ( x , y ) be the solution of x3 + y3 + 3xyz = 12 at P(l,2,1). Compute dz/dyat P(l ,2,1).

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3. [20] Find the tangent line, at P(l, 1, 2), to the curve of intersection of z2 — 3x2 + y2 and£ + ? + 2z = 6.

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4. [22] Find the largest and the smallest value of xy + 9x — Qy on the region enclosed byy — x2 and the line y = 4.

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5. [25] Find the point on the ellipsoid — + — + — = 1 which is closest to (1, 4, 3). Showall the important intermediate steps. I can advise you whether you have setup theproblem correctly or not. I can also setup the problem for you for a 5 point deduction.

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