Calculus II, Spring 2019 (http://www.math.nthu.edu.tw/˜ wangwc/)

Homework Assignment for Week 09

1. Section 14.4: Problems 29, 31, 43, 51.

2. Suppose that F (x, y, z) = 0 can implicitly define x = f(y, z), or y = g(z, x), or z = h(x, y) near some point (x_{0}, y_{0}, z_{0}) with F (x_{0}, y_{0}, z_{0}) = 0. (for example, F (x, y, z) = x + 2y + 3z − 4 can). Show that, at any such point (x_{0}, y_{0}, z_{0}),

\frac{∂f}{∂y} \frac{∂g}{∂z} \frac{∂h}{∂x} = \frac{∂f}{∂z} \frac{∂g}{∂x} \frac{∂h}{∂y} = −1

3. Section 14.5: Problems 9, 15, 19, 25, 27, 29, 35, 36, 40 (See page 807).

Note: in problem 15, “direction of u” refers to a unit vector.

4. 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) Are f_{x} and f_{y} continuous at (0, 0)?

(d) Evaluate Df_{(cos θ,sin θ)}(0, 0), i.e. the directional derivative of f at (x_{0}, y_{0}) = (0, 0) in the direction (cos θ, sin θ), if it exists.

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