### The Calculus of Functions of Several Variables

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. $\dfrac{16}{3}$