MATH 214: Section A1 – Practice Final Exam
π
.
6
(b) (5 points) Compute the interval of convergence of the Taylor series for sin(2x) centered
π
at a = .
6
1. (a) (15 points) Compute the Taylor series of sin(2x) centered at the point a =
2. (20 points) Find an equation of the normal plane to the curve given by parametric equations
x = t, y = cos(3t), z = sin(3t) for t ∈ R
at the point (π, −1, 0).
3. Compute the indicated derivative.
4xyz + ye2xz
(a) (5 points) Compute fx for f (x, y, z) =
xy
(b) (5 points) Compute the directional derivative of f (x, y, z) = xexy
in the direction of h1, −1, 1i.
∂z
∂z
and
.
(c) (10 points) If cos(xyz) = 1 + x2 y 2 + z 2 compute
∂x
∂y
2z
at the point (−1, 1, 0)
4. (15 points) Consider w = f (x, y, z) where f is a differentiable function of three variables and
∂w
∂w ∂w
,
and
x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ. Compute
∂ρ ∂φ
∂θ
5. (10 points) Show that every normal line to the sphere x2 + y 2 + z 2 = r2 passes through the
center of the sphere.
6. Consider the function f (x, y) = xy 2 .
(a) (5 points) Prove that f attains an absolute maximum and minimum value over the set
K = {(x, y) : x ≥ 0, y ≥ 0, x2 + y 2 ≤ 3}.
(b) (15 points) Find the absolute maximum and minimum value of f (x, y) = xy 2 on the set
K.
7. (20 points) Find the minimum distance from the origin to the curve of intersection between
the surfaces x − y = 1 and y 2 − z 2 = 1
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