=====Mathematics 250 - Spring 2016=====

==== Homework ====

**Section 4.3**: 6, 13, 14, 16

Also:
   - Find the natural parametrization of the curve $C$ parametrized by $F(t) = (t, \cos(2t), \sin(2t))$ for $0 \le t \le \pi$.
   - Find the natural parametrization of the curve $C$ parametrized by $F(t) = (e^t\cos(t), e^t\sin(t), e^t)$ for $0 \le t \le 2\pi$.
   - Find the curvature of the curve parametrized by $F(t) = (t, t^2, t^3)$ at $t = 1$.
   - Use Exercise 16 to find the curvature of the graph of $y = x^2$ at $x = 1$.
   - Use Exercise 16 to find the curvature of the graph of $y = \sin(x)$ at $x = \frac{\pi}{2}$.

Answers:
  - $G(\tau) = \left(\frac{\tau}{\sqrt{5}}, \cos\left(\frac{2\tau}{\sqrt{5}}\right), \sin\left(\frac{2\tau}{\sqrt{5}}\right)\right)$
  - $G(\tau) = \left(1 + \frac{\tau}{\sqrt{3}}\right)\left(\cos\left(\log\left(1 + \frac{\tau}{\sqrt{3}}\right)\right), \sin\left(\log\left(1 + \frac{\tau}{\sqrt{3}}\right)\right), 1\right)$
  - $\kappa = \frac{\sqrt{266}}{98}$
  - $\kappa = \frac{2}{5\sqrt{5}}$
  - $\kappa = 1$