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Mathematics 250 - Spring 2016

Homework

Section 4.3: 6, 13, 14, 16

Also:

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

Answers:

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