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

Homework Assignment for Week 08

1. Show that if f(x, y) = o(1) · |x − x_{0}| + o(1) · |y − y_{0}| as (x, y) → (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}).

Hint: \sqrt{Δx^{2} + Δy^{2}} = \frac{Δx}{\sqrt{Δx^{2}+Δy^{2}}} Δx + \frac{Δy}{\sqrt{Δx^{2}+Δy^{2}}} Δy

2. Section 14.4: Problems 7, 29, 31, 43, 51.

3. 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, for any such point (x_{0}, y_{0}, z_{0}), we have

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

4. Section 14.5: Problems 9, 15, 19, 25, 29, 35.

(Do as many as you can, we will go through the rest on Tuesday)