math1 - FIRST PRACTICE MIDTERM The following five...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 domains. 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 ...
View Full Document

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.

Ask a homework question - tutors are online