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

Homework Assignment for Week 08

1. Section 14.2: Problems 43, 48, 51, 57, 67.

2. Section 14.3: Problems 19, 21, 27, 51, 60, 65, 70, 81, 91.

3. Show that if g(x, y) = ε_{1} · (x − x_{0}) + ε_{2} · (y − y_{0}) as (x, y) → (x_{0}, y_{0}) then g(x, y) = ε · \sqrt{(x − x_{0})^{2} + (y − y_{0})^{2}} and vice versa (the converse). Here lim_{(x,y)→(x_{0},y_{0})}(ε_{1}, ε_{2}, ε) = (0, 0, 0).

Remark: A brief version of the above statement reads: Show that o(1)·Δx+o(1)·Δy = o(1)·\sqrt{Δx^{2} + Δy^{2}} where the o(1)’s refer to 2D limits as (Δx, Δy) → (0, 0). Hint: \sqrt{Δx^{2} + Δy^{2}} = \frac{Δx}{\sqrt{Δx^{2}+Δy^{2}}} Δx + \frac{Δy}{\sqrt{Δx^{2}+Δy^{2}}} Δy

4. Section 14.4: Problems 1, 7, 10, 21, 24.

Remark: Quiz on April 18 covers homework 07 and 08.