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 — cfsv:chapter-3 [2012/06/25 12:43] (current)dcs created 2012/06/25 12:43 dcs created 2012/06/25 12:43 dcs created Line 1: Line 1: + ====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. $\dfrac{16}{3}$ + + ----