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Unformatted text preview: FIRST PRACTICE MIDTERM The following ﬁve questions are worth 20 points each, although many have
multiple parts. Show ALL your work in your answers to each question. You
have 75 minutes for the examination. Calculators are NOT permitted.
1. Let P, Q, and R be points in R3 with coordinates P = (0, 2, 1), Q =
(2, 0, 1), R = (0, 0, −1).
1a. Give a parametrization (in terms of auxiliary variables t and u) for the
plane containing P , Q, and R.
1b. Give an equation of the form ax + by + cz = d for the plane containing
P , Q, and R.
2. Draw the level sets f (x, y ) = c for the given values of c, in the given
2a. f (x, y ) = |x| + |y |, −2 ≤ x ≤ 2, −2 ≤ y ≤ 2, c = −1, 0, 1, 2, 3, 4.
2b. f (x, y ) = xy , −2 ≤ x ≤ 2, −2 ≤ y ≤ 2, c = −2, −1, 0, 1, 2.
3. Deﬁne vectors u, v in R3 by u = (−2, 1, 0), v = (0, 2, −1). Let θ be the
angle between the two vectors.
3a. Compute the lengths of u and v .
3b. Compute cos θ.
3c. Give a unit vector orthogonal to both u and v .
4. Let f (x, y ) = x3 + x2 y + xy 2 + y 3 .
4a. Find ∂f
∂x and ∂f
∂y . 4b. Find the directional derivative of f at P = (1, 1) in the direction of
v = (1, −1).
4c. Give the linear approximation to f near the point (1, −1).
5. Let f : R2 → R3 be deﬁned by f (x, y ) = (x2 , xy, y 2 ), and let g : R3 → R2
be deﬁned by g (t, u, v ) = (t + 2u + v, t − 2u + v ).
5a. Compute the derivative matrices Df and Dg .
5b. Compute g ◦ f : R2 → R2 .
5c. Compute D(g ◦ f ) directly, using your answer to 5b.
5d. Compute D(g ◦ f ) by the chain rule and your answer to 5a. 1 ...
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This note was uploaded on 03/29/2012 for the course M 325k taught by Professor Schurle during the Spring '08 term at University of Texas at Austin.
- Spring '08