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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, | ||
+ | |||
+ | (b) $ \displaystyle{\lim_{(x, | ||
+ | |||
+ | %%(c)%% $ \displaystyle{\lim_{(x, | ||
+ | |||
+ | (d) $ \displaystyle{\lim_{(x, | ||
+ | |||
+ | 4. (a) $ \displaystyle{\lim_{(x, | ||
+ | |||
+ | %%(c)%% The limit does not exist: | ||
+ | \[ | ||
+ | f(x,y) = \frac{x}{x+y^2}, | ||
+ | \] | ||
+ | $\alpha(t) = (0, t)$, 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, | ||
+ | |||
+ | 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, | ||
+ | |||
+ | 2. (a) $ \displaystyle{f_x(x, | ||
+ | |||
+ | %%(c)%% $ \displaystyle{f_x(x, | ||
+ | |||
+ | $ \displaystyle{f_y(x, | ||
+ | |||
+ | $ \displaystyle{f_z(x, | ||
+ | |||
+ | (e) $ \displaystyle{g_w(w, | ||
+ | |||
+ | $ \displaystyle{g_x(w, | ||
+ | |||
+ | $ \displaystyle{g_y(w, | ||
+ | |||
+ | $ \displaystyle{g_z(w, | ||
+ | |||
+ | 3. (a) $ \displaystyle{\nabla f(x,y,z) = \frac{1}{\sqrt{x^2+y^2+z^2}}(x, | ||
+ | |||
+ | %%(c)%% $ \displaystyle{\nabla f(w,x,y,z) = \frac{1}{1 + (4w+3x+5y+z)^2}(4, | ||
+ | |||
+ | 4. (a) $ \displaystyle{D_{\mathbf{u}}(-2, | ||
+ | |||
+ | %%(c)%% $ \displaystyle{D_{\mathbf{u}}(-2, | ||
+ | |||
+ | 5. (a) $ \displaystyle{D_{\mathbf{u}}f(-2, | ||
+ | |||
+ | %%(c)%% | ||
+ | |||
+ | 6. (a) $ \displaystyle{-20\sqrt{10}e^{-1}}$ | ||
+ | |||
+ | (b) Direction: $ \displaystyle{\frac{1}{\sqrt{5}}(-1, | ||
+ | |||
+ | Rate of change: $ \displaystyle{40\sqrt{5}e^{-1}}$ | ||
+ | |||
+ | %%(c)%% Direction: $ \displaystyle{\frac{1}{\sqrt{5}}(1, | ||
+ | |||
+ | 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)%% | ||
+ | |||
+ | 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}}}$; | ||
+ | |||
+ | 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, | ||
+ | |||
+ | 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, | ||
+ | |||
+ | (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, | ||
+ | |||
+ | (b) $ \displaystyle{Hf(x, | ||
+ | |||
+ | 4. (a) $ \displaystyle{P_2(x, | ||
+ | |||
+ | %%(c)%% $ \displaystyle{P_2(x, | ||
+ | |||
+ | $ \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}}, | ||
+ | |||
+ | 3. Maximum value of $ 10 $ at $ \left(\sqrt{2}, | ||
+ | |||
+ | 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}}, | ||
+ | |||
+ | 15. Minimum distance of $ \dfrac{2\sqrt{21}}{7} $ at $ \displaystyle{\frac{2}{7}(2, | ||
+ | |||
+ | 17. Hottest point: $ 98.28^\circ $ at $ \left(\sqrt{2}, | ||
+ | |||
+ | 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, | ||
+ | |||
+ | (b) Mass: $ \dfrac{16}{3} $; center of mass: $ \displaystyle{\left(\frac{4}{5}, | ||
+ | |||
+ | 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}, | ||
+ | |||
+ | %%(c)%% $ \left(\sqrt{10}, | ||
+ | |||
+ | 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}, | ||
+ | |||
+ | 15. $ \displaystyle{\left(-\sqrt{\frac{3}{2}}, | ||
+ | |||
+ | 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} $ | ||
+ | |||
+ | ---- |