Topics on the final (with practice problems)

differentials

section 6.4, problems 1, 5

chain rule, implicit differentiation

section 6.5, problems 7, 8, 10; section 6.6, problems 3, 10, 11

Lagrange multipliers

section 6.8, problems 1, 2, 5, 7

Theorems from Sections 7.1, 7.2, 7.4

sample problem: under what conditions on the partial derivatives of f(x,y) can we conclude that f is differentiable?

second derivative test

section 7.6, problem 1

inverse and implicit function theorems

section 8.2, problems 1, 2, 4, 6, 17; section 8.3, problems 1, 3, 4

linear algebra

sample problems:

1. Prove that the two vectors (1,1) and (2,3) form a basis for R².
2. The function T defined by T(x,y) = (x-y, x+y) is a linear transformation. Find its matrix (a) with respect to the standard basis, and (b) with respect to the basis given in the previous problem.
3. Is the linear transformation in the previous problem invertible?
4. Which of the following are vector spaces: C = {(x,y) : x² + y² = 1}, P = {(x,y) : 3x - 7y = 0}?