User Tools


Differences

This shows you the differences between two versions of the page.

Link to this comparison view

cfsv:chapter-2 [2012/06/25 12:42] (current)
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)} $
 +
 +----