| course |
course_year |
question_number |
tags |
title |
year |
Analysis II |
IB |
2 |
|
Paper 2, Section I, G |
2016 |
(a) What does it mean to say that the function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ is differentiable at the point $x=\left(x_{1}, x_{2}, \ldots, x_{n}\right) \in \mathbb{R}^{n}$ ? Show from your definition that if $f$ is differentiable at $x$, then $f$ is continuous at $x$.
(b) Suppose that there are functions $g_{j}: \mathbb{R} \rightarrow \mathbb{R}^{m}(1 \leqslant j \leqslant n)$ such that for every $x=\left(x_{1}, \ldots, x_{n}\right) \in \mathbb{R}^{n}$,
$$f(x)=\sum_{j=1}^{n} g_{j}\left(x_{j}\right) .$$
Show that $f$ is differentiable at $x$ if and only if each $g_{j}$ is differentiable at $x_{j}$.
(c) Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be given by
$$f(x, y)=|x|^{3 / 2}+|y|^{1 / 2}$$
Determine at which points $(x, y) \in \mathbb{R}^{2}$ the function $f$ is differentiable.