Math 524  University of South Carolina
D. Meade  Fall 1995

Calculus and Linear Algebra Review Problems

1. Let f(x, y, z) = xy^{3} − 5x^{2}yz^{4}. Compute ∇f.

2. In what direction is f(x, y) = 9x^{4} + 4y^{2} increasing most rapidly at (1, 2)?

3. Define f(t) = ∥F(t)∥ where F is a differentiable function. Compute f′(t).

4. Find and classify all critical points of f(x, y) = xy^{2} − 6x^{2} − 3y^{2} and of g(x, y) = e^{−(x^{2}+y^{2}−4y)}.

5. A rectangular box whose edges are parallel to the coordinate axes is inscribed in the ellipsoid 36x^{2} + 4y^{2} + 9z^{2} = 36. What is the greatest possible volume for the box?

6. Are the vectors

v_{1} = ( \begin{matrix}1 \\ 0 \\ 0 \end{matrix} ) ,  v_{2} = ( \begin{matrix}1 \\ 1 \\ 0 \end{matrix} ) ,  v_{3} = ( \begin{matrix}2 \\ 3 \\ 4 \end{matrix} ) .

linearly dependent or independent? Are these vectors a basis for R^{3}?

7. Find the inverse of the matrix

A = ( \begin{matrix}2 & −1 & 0 \\ −1 & 2 & −1 \\ 0 & −1 & 2 \end{matrix} ) .

8. Let A and B be invertible n × n matrices. What is the inverse of their product, AB?

9. Define A^{†} = (A^{T} A)^{−1}A^{T}. Compute A^{†}, A^{†}A, and AA^{†} when

A = ( \begin{matrix}−1 & −1 \\ 1 & 0 \\ 0 & 1 \end{matrix} ) .