MATH 212 - Vector Calculus
Exam 1 - Friday, September 19th
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Instructions. This exam is closed-book, closed-notes. Use of calculators is not permitted.
Please show all your work for full credit. You have 55 minutes to finish the exam.
Upon finishing please sign the pledge below: On my honor I have neither given nor received
any aid on this exam.
(1) Given three vectors u = (12, −3, 2), v = (3, 2, 1), w = (−2, 4, 1) calculate
(a) u · v
(b) length of w
(c) orthogonal projection of u onto v
(d) v × w
(2) Find the line through the origin which perpendicularly intersects the line
l(t) = (−6, 0, 2) + t(1, −3, 4).
(3) Sketch the level surfaces of the function f (x, y, z) = 4x2 + y 2 + 9z 2 .
(4) Find the equation of the plane tangent to the surface
z = x2 + y 3
at (3, 1, 10).
(5) Calculate, using the chain rule, the derivative of the composition f ◦ g where
f (x, y) = (2x + ey , x sin y),
What is D(f ◦ g)(1, 0)?
1
g(u, v) = (eu , uv).