Calculus II, Spring 2009 (http://www.math.nthu.edu.tw/˜ wangwc/)

Homework Assignment for Week 08

Assigned April 16, 2009.

1. Section 13.3: Problems 44.

2. Section 13.4: Problems 13, 19, 21, 29.

3. Section 13.5: Problems 29, 31, 39, 43.

4. Section 13.6: Problem 15 (Read Example 5 for the meaning of direction of A), 21, 23, 25, 27, 37, 39, 47, 49, 51, 53, 54, 58.

5. Find the Taylor expansion of f(x, y, z) around (x_{0}, y_{0}, z_{0}) up to quadratic terms of x, y and z. Give an expression of the remainder term, \frac{1}{3!}(⋯).

6. True or False?

(a) If f(x, y) is differentiable at (x, y) = (0, 0), then f(x, 0) is differentiable at x = 0 and f(0, y) is differentiable at y = 0.

(b) If f(x, y) is differentiable at (x, y) = (0, 0), then for any θ ∈ [0, 2π), f(t cos θ, t sin θ) is differentiable at t = 0.

(c) If f(x, 0) is differentiable at x = 0 and f(0, y) is differentiable at y = 0, then f(x, y) is differentiable at (x, y) = (0, 0).

(d) If for any θ ∈ [0, 2π), f(t cos θ, t sin θ) is differentiable at t = 0, then f(x, y) is differentiable at (x, y) = (0, 0).

(e) If for any θ ∈ [0, 2π), f(t cos θ, t sin θ) is differentiable at t = 0, and let

g(θ) = \frac{d}{dt}f(t cos θ, t sin θ)|_{t=0}.

If in addition, g(θ) is a continous fucntion of θ then f(x, y) is differentiable at (x, y) = (0, 0). Hint: If f(x, y) has a tangent line at every direction θ, do these tangent lines necessarily form a tangent plane?

(f) Consider the function

f(x, y) = \{ \begin{matrix}\frac{x^{3}+y^{3}}{x^{2}+y^{2}} , & (x, y) ≠ (0, 0) \\ 0, & (x, y) = (0, 0) \end{matrix}

Which of the assumptions in (c-e) above does f satisfy? Is f differentiable at (0, 0)?