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

Homework Assignment for Week 07

Assigned April 09, 2009.

1. Section 13.2: Problems 27, 29, 31, 34, 35, 45, 47.

2. Section 13.3: Problems 27, 35, 39.

3. Read page 825 on definition of Linearization and think about its relation with differentiability.

4. Section 13.4: Problems 5, 7, 8, 23, 35, 36.

5. Section 13.5: Problems 7, 11, 21, 23.

6. If f(x, y) is continuous at (0, 0) along each straight line and each parabola that passes

through (0, 0), i.e., if for each a ∈ R,

lim_{t→0} f(t, at) = lim_{t→0} f(at, t) = lim_{t→0} f(t, at^{2}) = lim_{t→0} f(at^{2}, t) = f(0, 0).

Does it mean that lim_{(x,y)→(0,0)} f(x, y) = f(0, 0)? Explain.

7. Define

g(r) = max_{\sqrt{(x−x_{0})^{2}+(y−y_{0})^{2}}=r} |f(x, y) − f(x_{0}, y_{0})|

If “f(x, y) is continuous at (x_{0}, y_{0})” equivalent to “g(r) continuous at r = 0^{+}”? Explain.

8. Let f(t) = |t|^{p}. For what values of p > 0 is f(t) continuous at t = 0? For what values of p > 0 is f(t) differentiable at t = 0?

9. Let f(x, y) = \sqrt{x^{2} + 2y^{2}}. Is f(x, y) continuous at (x, y) = (0, 0)? Is it differentiable at (0, 0)?

10. Show that if f(x, y) = o(1)|x − x_{0}| + o(1)|y − y_{0}| as (x, t) → (x_{0}, y_{0}) then f(x, y) = o(1)\sqrt{(x − x_{0})^{2} + (y − y_{0})^{2}} and vice versa (the converse). Note that all three o(1) refer to 2D limits as (x, y) → (x_{0}, y_{0}).