Consider the ellipse $ r'(t) = \langle3\cos(t),4\sin(t)\rangle$ for $ 0\le t \le 2. $
(a) At what points $\|r'\|$ have maximum and minimum values?
(b) At what points does the curvature have maximum and minimum values?
I'm trying to figure this out, and I think I'm overthinking it. I calculated $r'$ and $r''$, and got
$$ r' = \langle -3\sin(t), 4\cos(t) \rangle $$ $$ r'' = \langle -3\cos(t), -4\sin(t) \rangle $$
I would think that in order to find where $|r'|$ is at max and min, I should just set $|r''|$ to zero and solve for that, but if I do that I get that $0=9\cos^2(t) + 16\sin^2(t)$, which has no solution. What am I doing wrong here?
If $r'' = \langle -3\cos(t), -4\sin(t) \rangle$ then $\|r''\| = \sqrt{(-3\cos t)^2 +(-4\sin t)^2} $.
But what you need is not $\|r''\|$, but $\|r'\|'$. If $r' = \langle -3\sin(t), 4\cos(t) \rangle$ then $\|r'\| = \sqrt{(-3\sin t)^2+(4\cos t)^2}$, and then you would take the derivative of that with respect to $t$.
But that's the hard way. The easier way is this: to find the values of $t$ that maximize or minimize $\|r'\|$, just find the values of $t$ that maximize or minimize $\|r'\|^2$. Differentiating that is less complicated. The results will be the same because for positive numbers $a$, the square $a^2$ is an increasing function of $a$.