Continuity of $r\left(t\right)=\cos^2\left(2t\right)\hat{i}+\sin\left(3t\right)\hat{j}$ along $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$

Question

I had a bonus on a quiz that gave me the vector function \begin{align} r\left(t\right)=\cos^2\left(2t\right)\hat{i}+\sin\left(3t\right)\hat{j},\tag{1} \end{align} and asked me if it was continuous along $\left[\frac{-\pi}{2},\frac{\pi}{2}\right]$. Because differentiability implies continuity (e.g. this proof) my thoughts were that it was continuous over the entire interval except at the endpoints, where I don't believe it is differentiable. Here is a graph of the function from Mathematica:

enter image description here

It appears to oscillate back and forth over this graph no matter what the bounds.

Thank you for your time,

score 0 · Answer 1 · answered Feb 09 '15 at 20:06

0

This graph is differentiable with respect to $t$. Don't mix up graphs of vector valued functions with functions like $y=f(x)$!

answered Feb 09 '15 at 20:06

kryomaxim

2,882

Okay that makes sense...I do see that both components are differentiable everywhere, I was just wondering about the endpoints that appear. Interesting. Thanks! – bjd2385 Feb 09 '15 at 20:09

Continuity of $r\left(t\right)=\cos^2\left(2t\right)\hat{i}+\sin\left(3t\right)\hat{j}$ along $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$

1 Answers1