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Vector Calculus Independent Study

Unit 3 Sample Test

1. Let f(x, y, z) = x^{2} + y^{2} − z^{2}.

(a) [20 points] Graph at least three level sets of f.

(b) [20 points] Find the tangent plane to the surface x^{2} +y^{2} −z^{2} = 12 at the point (3, 2, 1).

(c) [20 points] Compute \frac{∂f}{∂v⃗} where v⃗ = \frac{1}{\sqrt{3}}(1, 1, 1).

(d) [20 points] When I die, my soul will spiral to nirvana along the helix σ(t) = (cos t, sin t, t). If my karma at a point is given by f, use the chain rule to determine my \frac{dkarma}{dt}.

2. [20 points] If f is a function of x, y, and z, and f’s zeroth, first, second, and third partial derivatives are all continuous, then how many of f’s third derivatives can be different?