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cfsv:chapter-3 [2012/06/25 12:43] (current)
dcs created
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 +====The Calculus of Functions of Several Variables====
  
 +===Answers for selected problems===
 +
 +**Chapter 3**
 +
 +----
 +
 +**Section 3.1**
 +
 +3. (a) $ \displaystyle{\lim_{(x,​y) \to (2,​1)}\left(3xy + x^2y + 4\right) = 14}$
 +
 +(b) $ \displaystyle{\lim_{(x,​y,​z) \to (1,​2,​1)}\frac{3xyz}{2xy^2+4z} = \frac{1}{2}}$
 +
 +%%(c)%% $ \displaystyle{\lim_{(x,​y) \to (2,​0)}\frac{\cos(3xy)}{\sqrt{x^2 + 1}} = \frac{1}{\sqrt{5}}}$
 +
 +(d) $ \displaystyle{\lim_{(x,​y,​z) \to (2,​1,​3)}ye^{2x-3y+z} = e^4}$
 +
 +4. (a) $ \displaystyle{\lim_{(x,​y) \to (0,​0)}\frac{xy^2}{x^2+y^2} = 0}$
 +
 +%%(c)%% The limit does not exist: ​ For example, if we let
 +\[
 +f(x,y) = \frac{x}{x+y^2},​
 +\]
 +$\alpha(t) = (0, t)$, and $ \beta(t) = (t, 0) $, then
 +\[
 +\lim_{t \to 0}f(\alpha(t)) = 0,
 +\]
 +while
 +\[
 +\lim_{t \to 0}f(\beta(t)) = 1.
 +\]
 +
 +(e) $ \displaystyle{\lim_{(x,​y) \to (0,​0)}\frac{1- e^{-(x^2 + y^2)}}{x^2 + y^2} = 1}$
 +
 +8. (a)  Open
 +
 +(b) Closed
 +
 +%%(c)%% Open
 +
 +(d) Open
 +
 +(e) Closed
 +
 +(f) Closed
 +
 +(g) Open
 +
 +(h) Neither open nor closed
 +
 +----
 +
 +**Section 3.2**
 +
 +1. $ \displaystyle{D_{\mathbf{u}}f(3,​ 1) = \frac{26}{\sqrt{5}}}$
 +
 +2. (a) $ \displaystyle{f_x(x,​y) = \frac{4y^2-4x^2}{(x^2+y^2)^2}}$; ​ $ \displaystyle{f_y(x,​y) = -\frac{8xy}{(x^2+y^2)^2}}$
 +
 +%%(c)%% $ \displaystyle{f_x(x,​y,​z) = 6xy^3z^4-26xy}$
 +
 +$ \displaystyle{f_y(x,​y,​z) = 9x^2y^2z^4-13x^2}$
 +
 +$ \displaystyle{f_z(x,​y,​z) = 12x^2y^3z^3}$
 +
 +(e) $ \displaystyle{g_w(w,​x,​y,​z) = \frac{w\cos\left(\sqrt{w^2+x^2+2y^2+3z^2}\right)}{\sqrt{w^2+x^2+2y^2+3z^2}}}$
 +
 +$ \displaystyle{g_x(w,​x,​y,​z) = \frac{x\cos\left(\sqrt{w^2+x^2+2y^2+3z^2}\right)}{\sqrt{w^2+x^2+2y^2+3z^2}}}$
 +
 +$ \displaystyle{g_y(w,​x,​y,​z) = \frac{2y\cos\left(\sqrt{w^2+x^2+2y^2+3z^2}\right)}{\sqrt{w^2+x^2+2y^2+3z^2}}}$
 +
 +$ \displaystyle{g_z(w,​x,​y,​z) = \frac{3z\cos\left(\sqrt{w^2+x^2+2y^2+3z^2}\right)}{\sqrt{w^2+x^2+2y^2+3z^2}}}$
 +
 +3. (a) $ \displaystyle{\nabla f(x,y,z) = \frac{1}{\sqrt{x^2+y^2+z^2}}(x,​y,​z)}$
 +
 +%%(c)%% $ \displaystyle{\nabla f(w,x,y,z) = \frac{1}{1 + (4w+3x+5y+z)^2}(4,​3,​5,​1)}$
 +
 +4. (a) $ \displaystyle{D_{\mathbf{u}}(-2,​1) = -\frac{56}{\sqrt{13}}}$
 +
 +%%(c)%% $ \displaystyle{D_{\mathbf{u}}(-2,​2,​1) = -\frac{1}{9\sqrt{6}}}$
 +
 +5. (a) $ \displaystyle{D_{\mathbf{u}}f(-2,​1) = \frac{12}{\sqrt{13}}}$,​ where $ \displaystyle{\mathbf{u} = \frac{1}{\sqrt{13}}(2,​3)}$
 +
 +%%(c)%% ​ $ \displaystyle{D_{\mathbf{u}}f(2,​1,​-1,​2) = \frac{4}{\sqrt{15}}}$,​ where $ \displaystyle{\mathbf{u} = \frac{1}{\sqrt{15}}(1,​-1,​2,​3)}$
 +
 +6. (a) $ \displaystyle{-20\sqrt{10}e^{-1}}$
 +
 +(b) Direction: $ \displaystyle{\frac{1}{\sqrt{5}}(-1,​ 2)}$
 +
 +Rate of change: $ \displaystyle{40\sqrt{5}e^{-1}}$
 +
 +%%(c)%% Direction: $ \displaystyle{\frac{1}{\sqrt{5}}(1,​ -2)}$
 +
 +Rate of change: $ \displaystyle{-40\sqrt{5}e^{-1}}$
 +
 +9. (b) $ \displaystyle{\frac{\partial}{\partial y}d_A(x,y) > 0}$, $ \displaystyle{\frac{\partial}{\partial x}d_B(x,y) > 0}$
 +
 +%%(c)%% ​ $ \displaystyle{\frac{\partial}{\partial y}d_A(x,y) < 0}$, $ \displaystyle{\frac{\partial}{\partial x}d_B(x,y) < 0}$
 +
 +11. (a) $ \displaystyle{D_{\mathbf{-u}}f(\mathbf{c}) = -D_{\mathbf{u}}f(\mathbf{c})}$
 +
 +(b) No
 +
 +----
 +
 +**Section 3.3**
 +
 +1. (a) $ A(x,y) = 12x + 4y - 12 $
 +
 +%%(c)%% $ A(x,y) = 0 $
 +
 +(e) $ A(w,x,y,z) = 2w + 4x - 12y - 4z -19 $
 +
 +2. (a) $ 8x - 2y - z = 5 $
 +
 +%%(c)%% $ 4x - 4y + z = 17 $
 +
 +4. (a) $ A(x,y,z) = 8x + 12y + 6z - 48 $
 +
 +(b) $ 26h $
 +
 +5. $ \displaystyle{\left. \frac{dT}{dt}\right|_{t=0} = 0} $; $ \displaystyle{\left. \frac{dT}{dt}\right|_{t=\frac{\pi}{4}} = 140e^{-\frac{1}{2}}}$;​ $ \displaystyle{\left. \frac{dT}{dt}\right|_{t=\frac{\pi}{2}} = 0}$
 +
 +7. $ \displaystyle{\left. \frac{dw}{dt}\right|_{t=\frac{\pi}{3}} = -\frac{96\pi}{16\pi^2+9} \approx -1.807}$
 +
 +9. (a) $ 2x + y = 5 $
 +
 +%%(c)%% $ x + 4y = -4 $
 +
 +10. (a)  $ 2x + y - 3z = 14 $
 +
 +%%(c)%% $ x + 2y - 2z = 1 $
 +
 +----
 +
 +**Section 3.4**
 +
 +1. (a) $ \displaystyle{\frac{\partial^2}{\partial x \partial y}f(x,y) = 6x^2 + 24e^{-3y}}$
 +
 +%%(c)%% $ \displaystyle{\frac{\partial^2}{\partial x^2}f(x,y) = 6xy^2 - 8e^{-3y}}$
 +
 +(e) $ \displaystyle{\frac{\partial^3}{\partial x \partial y^2}f(x,y) = 6x^2 - 72xe^{-3y}}$
 +
 +(g) $ \displaystyle{f_{yy}(x,​y) = x^3 - 36x^2e^{-3y}}$
 +
 +2. (a) $ \displaystyle{\frac{\partial^2}{\partial z \partial x}f(x,y,z) = \frac{2yz(3x^2 - y^2 - z^2)}{(x^2 + y^2 + z^2)^3}}$
 +
 +%%(c)%% $ \displaystyle{\frac{\partial^2}{\partial z^2}f(x,​y,​z) = \frac{2xy(3z^2 - x^2 - y^2)}{(x^2 + y^2 + z^2)^3}}$
 +
 +(e) $ \displaystyle{\frac{\partial^3}{\partial z \partial y \partial x}f(x,y,z) =\frac{2z(3x^4-18x^2y^2+3y^4+2x^2z^2+2y^2z^2-z^4)} {(x^2+y^2+z^2)^4}}$
 +
 +3. (a)  $ \displaystyle{Hf(x,​y) = \begin{bmatrix}6y & 6x - 12y^2 \\ 6x - 12y^2 & -24xy \end{bmatrix}}$
 +
 +(b) $ \displaystyle{Hf(x,​y,​z) = \begin{bmatrix}0 & 8yz^3 & 12y^2z^2 \\ 8yz^3 & 8xz^3 & 24xyz^2\\ 12y^2z^2 & 24xyz^2 & 24xy^2z\end{bmatrix}}$
 +
 +4. (a) $ \displaystyle{P_2(x,​y) = x - xy}$
 +
 +%%(c)%% $ \displaystyle{P_2(x,​y) = \frac{1}{2} - \frac{1}{4}(x - 1) - \frac{1}{4}(y - 1)}$
 +
 +$ \displaystyle{+ \frac{1}{8}(x-1)^2 + \frac{1}{4}(x-1)(y-1) + \frac{1}{8}(y-1)^2}$
 +
 +5. (a) Positive definite
 +
 +%%(c)%% Negative definite
 +
 +(e) Positive definite
 +
 +----
 +
 +**Section 3.5**
 +
 +1. Maximum value of $ \dfrac{1}{2} $ at $ \displaystyle{\left(\frac{1}{\sqrt{2}},​ \frac{1}{\sqrt{2}}\right)}$ and $ \displaystyle{\left(-\frac{1}{\sqrt{2}},​ -\frac{1}{\sqrt{2}}\right)}$;​ minimum value of $ -\dfrac{1}{2} $ at $ \displaystyle{\left(-\frac{1}{\sqrt{2}},​ \frac{1}{\sqrt{2}}\right)}$ and $ \displaystyle{\left(\frac{1}{\sqrt{2}},​ -\frac{1}{\sqrt{2}}\right)}$
 +
 +3. Maximum value of $ 10 $ at $ \left(\sqrt{2},​ \sqrt{2}\right)$ and  $ \left(-\sqrt{2},​ -\sqrt{2}\right) $; minimum value of $ -2 $ at $ \left(-\sqrt{2},​ \sqrt{2}\right)$ and $ \left(\sqrt{2},​ -\sqrt{2}\right) $
 +
 +5. Local minimum of $ \phantom{}0 $ at all points of the form $ (0, y) $, $ -\infty < y < \infty $; local maximum of $ e^{-1} $ at $ (1, 0) $ and $ (-1, 0) $
 +
 +7. Local maximum of $ 1 $ at $ (1, 1) $ and $ (-1, -1) $; saddle point at $ (0, 0) $
 +
 +9. Local minimum of $ \phantom{}0 $ at $ (0, 0, 0) $
 +
 +11. $ 10\hbox{ meters} \times 10\hbox{ meters} \times 10\hbox{ meters} $
 +
 +12.  $ 8.43\hbox{ meters} \times 8.43\hbox{ meters} \times 8.43\hbox{ meters} $
 +
 +13. Maximum value of $ \sqrt{3} $ at $ \displaystyle{\left(\frac{1}{\sqrt{3}},​ \frac{1}{\sqrt{3}},​ \frac{1}{\sqrt{3}}\right)} $; minimum value of $ -\sqrt{3} $ at  $ \displaystyle{\left(-\frac{1}{\sqrt{3}},​ -\frac{1}{\sqrt{3}},​ -\frac{1}{\sqrt{3}}\right)}$
 +
 +15. Minimum distance of $ \dfrac{2\sqrt{21}}{7} $ at $ \displaystyle{\frac{2}{7}(2,​ 4, 1)} $
 +
 +17. Hottest point: $ 98.28^\circ $ at $ \left(\sqrt{2},​ 0, -\sqrt{2}\right) $; coldest point: $ 41.72^\circ $ at $ \left(-\sqrt{2},​ 0, \sqrt{2}\right) $
 +
 +20. Local minimum of $ \phantom{}0 $ at all points of the form $ (x, x) $, $ -\infty < x < \infty $
 +
 +24. $ y = 9.23x + 114.72 $
 +
 +----
 +
 +**Section 3.6**
 +
 +1. (a) $ \displaystyle{\int_1^3 \int_0^2 3xy^2dydx = 32}$
 +
 +%%(c)%% $ \displaystyle{\int_{-2}^2 \int_{-1}^1 (4 - x^2y^2)dxdy = \frac{256}{9}}$
 +
 +2. (a) $ \displaystyle{\int\int_D (y^2 - 2xy)dxdy = -\frac{4}{3}}$
 +
 +%%(c)%% $ \displaystyle{\int\int_D ye^{-x}dxdy = 2(1 - e^{-1}}$
 +
 +3. (a) $ \displaystyle{\int_0^2 \int_0^y (xy^2 - x^2)dxdy = \frac{16}{15}}$
 +
 +%%(c)%% $ \displaystyle{\int_0^2 \int_0^{\sqrt{4-x^2}} (4 - x^2 - y^2)dydx = 2\pi}$
 +
 +5. $ \dfrac{32}{3} $
 +
 +7. $ \displaystyle{\int\int_D e^{-x^2}dxdy = \frac{1}{2}(1 - e^{-1})}$
 +
 +9. $ 56\pi $
 +
 +11. (a) $ \displaystyle{\int_1^2 \int_0^3 \int_{-2}^2 (4 - x^2 - z^2)dydxdz = -16}$
 +
 +%%(c)%% $ \displaystyle{\int_0^4 \int_0^x \int_0^{x+y} (x^2 - yz)dzdydx = \frac{2432}{15}}$
 +
 +12. $ 16\pi $
 +
 +14. (a) Mass: $ \dfrac{16}{3} $; center of mass: $ \displaystyle{\left(1,​ \frac{1}{2},​ 1\right)}$
 +
 +(b) Mass: $ \dfrac{16}{3} $; center of mass: $ \displaystyle{\left(\frac{4}{5},​ \frac{2}{5},​ \frac{8}{5}\right)}$
 +
 +16. (a) $ \dfrac{1}{6} $
 +
 +(b) $ \dfrac{1}{6} $
 +
 +----
 +
 +**Section 3.7**
 +
 +1. $ 2\pi $
 +
 +3. $ \dfrac{40\pi}{3} $
 +
 +5. (a) $ \displaystyle{\left(\sqrt{2},​ \frac{\pi}{4}\right)}$
 +
 +%%(c)%% $ \left(\sqrt{10},​ 4.3906\right) $
 +
 +6. (a) $ (3, 0) $
 +
 +%%(c)%% $ (-5, 0) $
 +
 +7. $ \displaystyle{\int\int_D (x^2 + y^2)dxdy = 8\pi}$
 +
 +9. $ \displaystyle{\int\int_D \frac{1}{x^2 + y^2}dxdy = \pi\log(2)}$
 +
 +10. $ \displaystyle{\int\int_D \log(x^2 + y^2)dxdy = \pi(8\log(2) - 3)}$
 +
 +13. $ \displaystyle{\left(\sqrt{6},​ \frac{3\pi}{4},​ 0.6155\right)}$
 +
 +15. $ \displaystyle{\left(-\sqrt{\frac{3}{2}},​ \sqrt{\frac{3}{2}},​ -1\right)}$
 +
 +17. $ \displaystyle{\int\int\int_D (x^2 + y^2 + z^2)dxdydz = \frac{128\pi}{5}}$
 +
 +19. $ \displaystyle{\int\int\int_D \sin(\sqrt{x^2+y^2+z^2}dxdydz = \frac{\pi}{2}(2\sin(1) + \cos(1) - 2) \approx 0.3506}$
 +
 +21. (b) $ \dfrac{\pi}{3}(2 - \sqrt{2}) $
 +
 +23. $ \displaystyle{\int\int\int_D \sqrt{x^2 + y^2}dxdydz = \frac{70\pi}{3}}$
 +
 +25. $ \dfrac{16}{3} $
 +
 +----