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 — cfsv:chapter-2 [2012/06/25 12:42] (current)dcs created 2012/06/25 12:42 dcs created 2012/06/25 12:42 dcs created Line 1: Line 1: + ====The Calculus of Functions of Several Variables==== + ===Answers for selected problems=== + + **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. $\displaystyle{\mathbf{x}(t) = \left(2-\cos(t),​ 2+2t-\sin(2t),​t\right)}$ + + ----