MAT B41 — Week 3

Posted in 2018 -- MAT B41, Lecture Notes, Uncategorized by pgadey on 2018/05/22

Homework 2 is available. Due next week!

Suggested Exercises:

• Section 1.5 n-Dimensional Euclidean Space: 1, 2, 7, 9, 10, 11, 15, 16, 17
• Section 2.2 Limits and Continuity: 1, 2, 4b, 8, 11a, 16, 18, 20, 32, 33

Past Term Tests:

2016 Term Test: Calculate the following limits, showing all your steps, or show that the limit does not exist:

1. $\lim_{(x,y) \rightarrow (0,0)} \frac{x^4y^4}{(x^2+y^4)^3}$
2. $\lim_{(x,y) \rightarrow (0,0)} \frac{x^3 - x^2y}{\sqrt{x} + \sqrt{y}}$

Define $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ by $f(0,0) = 0$ and $f(x,y) = \frac{2x^2y}{x^2 + y^2}$ when $(x,y) \neq (0,0)$.
Find all points $(x,y)$ such that $f$ is continuous at $(x,y)$.

2015 Term Test:

Define $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ by $f(0,0) = 2$ and $f(x,y) = \frac{2x^2 - 2xy + 4y^2}{x^2 + 2y^2}$ when $(x,y) \neq (0,0)$.
Find all points $(x,y)$ such that $f$ is continuous at $(x,y)$.

Notes:

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