Chapter 2

Section 2.1

4. The curve parametrized by $ f $ is the graph of $ g $.

5. (a) $ \displaystyle{\lim_{n \to \infty}\mathbf{x}_n = \left(\frac{1}{2}, 3\right)}$

(b) $ \displaystyle{\lim_{n \to \infty}\mathbf{x}_n = \left(\sin(1), \cos(1), 1\right)}$

(c) $ \displaystyle{\lim_{n \to \infty}\mathbf{x}_n = \left(0, 3, 4, 0\right)}$

6. (a) $ \displaystyle{\lim_{t \to \pi}f(t) = \left(0, -1, 3\pi^2\right)}$

(b) $ \displaystyle{\lim_{t \to 1}f(t) = \left(\sin(1), \cos(1), 3\right)}$

(c) $ \displaystyle{\lim_{t \to 0}f(t) = \left(1, 1, 0\right)}$

8. $ \displaystyle{\lim_{h \to 0}\frac{f(t+h)-f(t)}{h} = \left(2t, 3, 2\right)}$

Section 2.2

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} $

Section 2.3

1. (a) $ \mathbf{v} = (2t, \cos(t)) $; $ \mathbf{a} = (2, -\sin(t) $

(c) $ \displaystyle{\mathbf{v} = \left(-6t\sin(3t^2), 6t\cos(3t^2)\right)} $;

$ \displaystyle{\mathbf{a} = \left(-36t^2\cos(3t^2)-6\sin(3t^2), -36t^2\sin(3t^2)+6\cos(3t^2)\right)} $

2. (a) $ \dfrac{2}{5\sqrt{5}}$

(c) $ \dfrac{1}{2} $

4. (a) $ a_T = 0 $; $ a_N = 1 $; $ \displaystyle{a\left(\frac{\pi}{3}\right) = N\left(\frac{\pi}{3}\right)} $

(c) $ \displaystyle{a_T = \frac{4}{\sqrt{5}}}$; $ \displaystyle{a_N = \frac{2}{\sqrt{5}}}$; $ \displaystyle{a(1) = \frac{4}{\sqrt{5}}T(1) + \frac{2}{\sqrt{5}}N(1)}$

6. $ \dfrac{1}{2} $

7. $ \dfrac{\sqrt{74}}{11\sqrt{11}}$

9. $ \dfrac{2}{5\sqrt{5}} $ and $ \dfrac{2}{17\sqrt{17}} $

11. $ 1 $ and $ \dfrac{2}{3\sqrt{3}} $

12. (a) $ 13.3649 $

(c) $ \displaystyle{\sqrt{17} + \frac{1}{4}\sinh^{-1}(4) \approx 4.64678} $

(e) $ \displaystyle{\frac{1}{2}\sqrt{37+4\pi^2} + \frac{1}{12}\left(1+4\pi^2\right)\sinh^{-1}\left(\frac{6}{\sqrt{1+4\pi^2}}\right) \approx 7.20788} $

(g) $ 32.2744 $

14. $ 2.4221 $

15. $ 6 $

17. $ 3.8202 $

20. $ \displaystyle{\mathbf{x}(t) = \left(\frac{1}{2}\sin(2t), \frac{3}{2} - \frac{1}{2}\cos(2t), \frac{3}{2}t^2\right)} $

21. $ \displaystyle{\mathbf{x}(t) = \left(2-\cos(t), 2+2t-\sin(2t),t\right)} $