Calculus II, Fall 2014 (http://www.math.nthu.edu.tw/˜ wangwc/)

Homework Assignment for Week 10

1. Let f(x, y) = x^{2}y/(x^{2} + y^{2}) for (x, y) ≠ (0, 0) and f(0, 0) = 0. (a) Is f continuous at (0, 0)?

(b) Do f_{x} and f_{y} exist at (0, 0)?

(c) Evaluate the directional derivative of f in the direction (cos θ, sin θ), i.e. df/ds_{(0,0),(cos θ,sin θ)}, if it exists.

(d) Is f differentiable at (0, 0)?

2. Section 14.7: Problems 1, 31, 35, 39, 43, 44, 49, 57.

Hint for problem 44: try the gradient analysis we introduced in class.

3. Section 14.8: Problems 1, 23, 27, 33, 35.

4. Section 14.9: Problems 7, 9, 11.

5. Derive Taylor’s formula for functions of three variables.