Mathematics 524 Homework #2

1. Most of the class will be concerned with function of two or more variable rather than just the wimpy little case of functions of one variable like we just finished. Our immediate goal is to extend to functions of several variables the first and second derivative test, and the second order Taylor’s theorem. The easiest way to state these results is with vectors, dot products, and matrices. The book does a very nice job of reviewing this material. So by next Tuesday you should have read sections 1.2 (which I hope is mostly review) and 1.3 (which may be new to some of you). As before we will be working through this in class and so you should be prepared to present parts of it at the board.

2. Review the definition of partial derivative. As practice let

f(x_{1}, x_{2}, x_{2}) = x_{1}x_{2}^{3}e^{x_{1}x_{3}^{2}+x_{2}}

and compute all there first partial derivatives and all six second partial derivatives (due Thursday have questions on Tuesday.)

3. Write out a reasonably detailed argument of what for a function H(x, y) of two variables that any local maximizer or minimizer must be a critical point. (Due Thursday and ask questions on Tuesday.)

4. Read over the definition of positive definite, negative definite, and semi-definite. Then do problems 2, 3, 4, and 5 on pages 31–32. (Due Thursday and ask questions on Tuesday. Also as there is a fair amount of calculation to be done here you should get started over the weekend.)