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The fall 2012 exam for a multivariable calculus course. It includes 13 problems covering topics such as unit vectors, perpendicular planes, cauchy-schwarz inequality, limits, differentiability, and gradient. Students are required to find solutions, justify answers, and complete problems without using calculators.
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Name: Honor Code Statement:
Directions: Complete all problems. Justify all answers/solutions. Calculators are not permitted. Best of luck.
(1) Find a unit vector that is perpendicular to both 2i + j − 3 k and i + k.
(2) If a · b = a · c and a 6 = 0 , does it follow that b = c? Explain.
Date: October 11, 2012. 1
(3) Give an equation for the plane containing the point (9, 5 , −1) and perpen- dicular to i − 2 k.
Now give the set of parametric equations for this plane.
(8) Determine several (say 3 with c ≥ 0 and 3 with c < 0) level curves of the given function f (and make sure to indicate the height c of each curve). Use this information to describe the graph of f , be as specific as you can. (A sketch would be appropriate, but not required.)
f (x, y) = xy
(9) Show that the following limit does not exist.
lim (x,y)→(0,0)
(x + y)^2 x^2 + y^2
(11) Find the gradient Of (a), where f (x, y) = exy^ + ln(x − y) and a = (2, 1).
(12) Find the matrix of partial derivatives of the function f (s, t) = (st, t sin(s), set).
(13) A rectangular stick of butter (that is a right parallelepiped with square base) is placed in a microwave to melt. When the butter’s length is 6 inches and its square cross-section measures 1.5 inches on a side, its length is decreasing at a rate of 0.25 inches per minute and its cross-sectional edge is decreasing at a rate of 0.125 inches per minute. How fast is the butter melting (i.e. at what rate is the solid volume of butter turning to liquid) at that instant?
