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Section 2.2

By cssp

Please post comments if you find any errors, or if you wish to contribute answers for additional problems. You may find information on using \LaTeX here.

1. (a) \displaystyle{Df(t) = \left(3t^2, 1, 2\right)}

(c) \displaystyle{Dh(t) = \left(12t^2, \cos(t), -2e^{-2t}\right)}

2. (a) \displaystyle{A(t) = (1,12)(t - 2) + (2, 8)}

(c) \displaystyle{A(t) = \left(-\frac{\sqrt{3}}{2}, \frac{1}{2}, -\sqrt{3}\right)\left(t - \frac{\pi}{3}\right) + \left(\frac{1}{2}, \frac{\sqrt{3}}{2}, -\frac{1}{2}\right)}

7. Note that the tangent line to C at f(c) is parametrized by

\displaystyle{A(t) = (1, \varphi'(c))(t - c) + (c, \varphi(c)) = (t, \varphi'(c)(t-c) + \varphi(c))}

8. No, f is not a smooth parametrization of C since Df(0) = (0, 0) . However, \displaystyle{g(t) = \left(t, t^2\right)}, -\infty < t < \infty , is a smooth parametrization of C .

9. f is not a smooth parametrization of C since Df(0) = (0, 0) .

11. (a) \displaystyle{T(1) = \frac{1}{\sqrt{5}}(1, 2)} ; \displaystyle{N(1) = \frac{1}{\sqrt{5}}(-2, 1)}

(c) \displaystyle{T\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{5}}(-1, 2)}; \displaystyle{N\left(\frac{\pi}{4}\right) = -\frac{1}{\sqrt{5}}(2, 1)}

(e) \displaystyle{T\left(\frac{\pi}{3}\right) = \left(-\frac{1}{2}\sqrt{\frac{3}{2}}, \frac{1}{2\sqrt{2}}, \frac{1}{\sqrt{2}}\right)}; \displaystyle{N\left(\frac{\pi}{3}\right) = \left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}, 0\right)}

(g) \displaystyle{T\left(\frac{1}{2}\right) = \left(0, \frac{\pi}{\sqrt{9+\pi^2}}, \frac{3}{\sqrt{9+\pi^2}}\right)}; \displaystyle{N\left(\frac{1}{2}\right) = (-1, 0, 0)}

(i) \displaystyle{T(2) = \frac{1}{\sqrt{161}}(1, 4, 12)} ; \displaystyle{N(2) = \frac{1}{\sqrt{29141}}(-76, -143, 54)}

15. \mathbf{M}

This entry was posted on 26 June 2007 at 8:03 am and is filed under Section 2.2. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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