Multivariable Calculus 9th Edition solutions

Multivariable Calculus 9
Determine whether the points \(P\) and \(Q\) lie on the given surface. \(\mathbf{r}(u, v)=\langle u+v, u-2 v, 3+u-v\rangle\) \(P(4,-5,1), Q(0,4,6)\) ________________ Equation Transcription: r?? Text Transcription: P Q r(u,v)=leftangle u+v,u-2v,3+u-v rightangle P(

Multivariable Calculus 9
Determine whether the points \(P\) and \(Q\) lie on the given surface. \(\mathbf{r}(u, v)=\left\langle 1+u-v, u+v^{2}, u^{2}-v^{2}\right\rangle\) \(P(1,2,1), Q(2,3,3)\) ________________ Equation Transcription: r?? Text Transcription: P Q r(u,v)=leftangle 1+u-v,

Multivariable Calculus 9
Identify the surface with the given vector equation. \(\mathbf{r}(u, v)=(u+v) \mathbf{i}+(3-v) \mathbf{j}+(1+4 u+5 v) \mathbf{k}\) ________________ Equation Transcription: rijk Text Transcription: r(u,v)=(u+v)i+(3-v)j+(1+4u+5v)k

Multivariable Calculus 9
Identify the surface with the given vector equation. \(\mathbf{r}(u, v)=u^{2} \mathbf{i}+u \cos v \mathbf{j}+u \sin v \mathbf{k}\) ________________ Equation Transcription: rijk Text Transcription: r(u,v)=u^2i+u cos vj+u sin vk

Multivariable Calculus 9
Identify the surface with the given vector equation. \(\mathbf{r}(s, t)=\langle 3 \cos t, s, \sin t\rangle, \quad-1 \leqslant s \leqslant 1\) ________________ Equation Transcription: r???? Text Transcription: r(s,t)=leftangle 3 cos t, sin t rightangle,-1 leqslant s leqsl

Multivariable Calculus 9
Use a computer to graph the parametric surface. Indicate on the graph which grid curves have \(u\) constant and which have \(v\) constant. \(\mathbf{r}(u, v)=\left\langle u^{2}, v^{2}, u+v\right\rangle\), \(-1 \leqslant u \leqslant 1,-1 \leqslant v \leqslant 1\) ______________

Multivariable Calculus 9
Use a computer to graph the parametric surface. Indicate on the graph which grid curves have \(u\) constant and which have \(v\) constant. \(\mathbf{r}(u, v)=\left\langle u, v^{3},-v\right\rangle\), \(-2 \leqslant u \leqslant 2,-2 \leqslant v \leqslant 2\) ________________ Equat

Multivariable Calculus 9
Use a computer to graph the parametric surface. Indicate on the graph which grid curves have \(u\) constant and which have \(v\) constant. \(\mathbf{r}(u, v)=\left\langle u^{3}, u \sin v, u \cos v\right\rangle\), \(-1 \leqslant u \leqslant 1,0 \leqslant v \leqslant 2 \pi\) _________

Multivariable Calculus 9
Use a computer to graph the parametric surface. Indicate on the graph which grid curves have \(u\) constant and which have \(v\) constant. \(\mathbf{r}(u, v)=\langle u, \sin (u+v), \sin v\rangle\), \(-\pi \leqslant u \leqslant \pi,-\pi \leqslant v \leqslant \pi\) ________________ Equation

Multivariable Calculus 9
Use a computer to graph the parametric surface. Indicate on the graph which grid curves have \(u\) constant and which have \(v\) constant. \(x=\sin v, \quad y=\cos u \sin 4 v, \quad z=\sin 2 u \sin 4 v\), \(0 \leqslant u \leqslant 2 \pi,-\pi / 2 \leqslant v \leqslant \pi / 2\) _______

Multivariable Calculus 9
Use a computer to graph the parametric surface. Indicate on the graph which grid curves have \(u constant and which have \(v constant. \(x=\cos u, \quad y=\sin u \sin v, \quad z=\cos v\), \(0 \leqslant u \leqslant 2 \pi, 0 \leqslant v \leqslant 2 \pi\) ________________ Equation Tr

Multivariable Calculus 9
Match the equations with the graphs labeled I-VI and give reasons for your answers. Determine which families of grid curves have \(u\) constant and which have \(v\) constant. \(\mathbf{r}(u, v)=u \cos v \mathbf{i}+u \sin v \mathbf{j}+v \mathbf{k}\) __________

Multivariable Calculus 9
Match the equations with the graphs labeled IVI and give reasons for your answers. Determine which families of grid curves have \(u\) constant and which have \(v\) constant. \(\mathbf{r}(u, v)=u v^{2} \mathbf{i}+u^{2} v \mathbf{j}+\left(u^{2}-v^{2}\right) \mathbf{k}\)

Multivariable Calculus 9
Match the equations with the graphs labeled IVI and give reasons for your answers. Determine which families of grid curves have \(u\) constant and which have \(v\) constant. \(\mathbf{r}(u, v)=\left(u^{3}-u\right) \mathbf{i}+v^{2} \mathbf{j}+u^{2} \mathbf{k}\)

Multivariable Calculus 9
Match the equations with the graphs labeled IVI and give reasons for your answers. Determine which families of grid curves have \(u\) constant and which have \(v\) constant. \(x=(1-u)(3+\cos v) \cos 4 \pi u\)

Multivariable Calculus 9
Match the equations with the graphs labeled IVI and give reasons for your answers. Determine which families of grid curves have \(u\) constant and which have \(v\) constant. \(x=\cos ^{3} u \cos ^{3} v, \quad y=\sin ^{3} u \cos ^{3} v, \quad z=\sin ^{3} v\)

Multivariable Calculus 9
Match the equations with the graphs labeled IVI and give reasons for your answers. Determine which families of grid curves have \(u\) constant and which have \(v\) constant. \(x=\sin u, \quad y=\cos u \sin v, \quad z=\sin v\)

Multivariable Calculus 9
Find a parametric representation for the surface. The plane through the origin that contains the vectors \(\mathbf{i}-\mathbf{j}\) and \(\mathbf{j}-\mathbf{k}\) ________________ Equation Transcription: ij jk Text Transcription: i-j j-k

Multivariable Calculus 9
Find a parametric representation for the surface. The plane that passes through the point \((0,-1,5)\) and contains the vectors \(\langle 2,1,4\rangle\) and \(\langle-3,2,5\rangle\). ________________ Equation Transcription: ?? ?? Text Transcription: (0,-1,5) leftangle 2,1

Multivariable Calculus 9
Find a parametric representation for the surface. The part of the hyperboloid \(4 x^{2}-4 y^{2}-z^{2}=4\) that lies in front of the \(y z\)-plane ________________ Equation Transcription: Text Transcription: 4x^2-4y^2-z^2=4 yz-plane

Multivariable Calculus 9
Find a parametric representation for the surface. The part of the ellipsoid \(x^{2}+2 y^{2}+3 z^{2}=1\) that lies to the left of the \(x z\)-plane ________________ Equation Transcription: Text Transcription: x^2+2y^2+3z^2=1 xz-plane

Multivariable Calculus 9
Find a parametric representation for the surface. The part of the sphere \(x^{2}+y^{2}+z^{2}=4\) that lies above the cone \(z=\sqrt{x^{2}+y^{2}}\). ________________ Equation Transcription: Text Transcription: x^2+y^2+z^2=4 z=sqrt x^2+y^2

Multivariable Calculus 9
Find a parametric representation for the surface. The part of the cylinder \(x^{2}+z^{2}=9\) that lies above the \(x y\)-plane and between the planes \(y=-4\) and \(y=4\). ________________ Equation Transcription: Text Transcription: x^2+z^2=9 xy-plane y=-4

Multivariable Calculus 9
Find a parametric representation for the surface. The part of the sphere \(x^{2}+y^{2}+z^{2}=36\) that lies between the planes \(z=0\) and \(z=3 \sqrt{3}\). ________________ Equation Transcription: Text Transcription: x^2+y^2+z^2=36 z=0 z=3 sqrt 3

Multivariable Calculus 9
Find a parametric representation for the surface. The part of the plane \(z=x+3\) that lies inside the cylinder \(x^{2}+y^{2}=1\) ________________ Equation Transcription: Text Transcription: z=x+3 x^2+y^2=1

Multivariable Calculus 9
Use a computer to produce a graph that looks like the given one.

Multivariable Calculus 9
Use a computer to produce a graph that looks like the given one.

Multivariable Calculus 9
Find parametric equations for the surface obtained by rotating the curve \(y=1 /\left(1+x^{2}\right),\ -2 \leqslant x \leqslant 2\), about the \(x\)-axis and use them to graph the surface. Equation Transcription: Text Transcription: y = 1/(1 + x^2),

Multivariable Calculus 9
Find parametric equations for the surface obtained by rotating the curve \(x=1 / y, y \geqslant 1\), about the \(y\)-axis an use them to graph the surface. Equation Transcription: Text Transcription: x=1/y, y > or =1 y

Multivariable Calculus 9
(a) What happens to the spiral tube in Example 2 (sed ure 5) if we replace \(\cos u\) by \(\sin u\) and \(\sin u\)by \(\cos u\)? (b) What happens if we replace \(\cos u\) by \(\cos 2 u\) and \(\sin u\) by \(\sin 2 u ?\) Equation Transcription: Tex

Multivariable Calculus 9
The surface with parametric equations \(x=2 \cos \theta+r \cos (\theta / 2) \\\) \(y=2 \sin \theta+r \cos (\theta / 2) \\\) \(z=r \sin (\theta / 2)\) Where \(-\frac{1}{2} \leqslant r \leqslant \frac{1}{2}\) and \(0 \leqslant \theta \leqslant 2 \pi\), is called a Mbius strip. Graph this sur

Multivariable Calculus 9
Find an equation of the tangent plane to the given parametric surface at the specified point. \(x=u+v, \quad y=3 u^{2}, \quad z=u-v ; \quad(2,3,0)\) Equation Transcription: Text Transcription: x = u + v, y = 3u^2, z = u - v; (2, 3, 0)

Multivariable Calculus 9
Find an equation of the tangent plane to the given parametric surface at the specified point. \(x=u^{2}+1, \quad y=v^{3}+1, \quad z=u+v ; \quad(5,2,3)\) Equation Transcription: Text Transcription: x=u^2+1, y=v^3+1, z=u+v; (5,2,3)

Multivariable Calculus 9
Find an equation of the tangent plane to the given parametric surface at the specified point. \(\mathbf{r}(u, v)=u \cos v \mathbf{i}+u \sin v \mathbf{j}+v \mathbf{k} ; \quad u=1, v=\pi / 3\) Equation Transcription: Text Transcription: r(u, v) = u

Multivariable Calculus 9
Find an equation of the tangent plane to the given parametric surface at the specified point. \(\mathbf{r}(u, v)=\sin u \mathbf{i}+\cos u \sin v \mathbf{j}+\sin v \mathbf{k} ;\) \(u=\pi / 6, v=\pi / 6\) Equation Transcription: Text Transcription:

Multivariable Calculus 9
Find an equation of the tangent plane to the given parametric surface at the specified point. Graph the surface and the tangent plane. \(\mathbf{r}(u, v)=u^{2} \mathbf{i}+2 u \sin v \mathbf{j}+u \cos v \mathbf{k} ; \quad u=1, v=0\) Equation Transcription:

Multivariable Calculus 9
Find an equation of the tangent plane to the given parametric surface at the specified point. Graph the surface and the tangent plane. \(\mathbf{r}(u, v)=\left(1-u^{2}-v^{2}\right) \mathbf{i}-v \mathbf{j}-u \mathbf{k} ; \quad(-1,-1,-1)\) Equation Transcription:

Multivariable Calculus 9
Find the area of the surface. The part of the plane \(3 x+2 y+z=6\) that lies in the first octant Equation Transcription: Text Transcription: 3 x+2 y+z=6

Multivariable Calculus 9
Find the area of the surface. The part of the plane with vector equation \(\mathbf{r}(u, v)=\langle u+v, 2-3 u, 1+u-v\rangle\) that is given by \(0 \leqslant u \leqslant 2,-1 \leqslant v \leqslant 1\) Equation Transcription: Text Transcriptio

Multivariable Calculus 9
Find the area of the surface. The part of the plane \(x+2 y+3 z=1\) that lies inside the cylinder \(x^{2}+y^{2}=3\) Equation Transcription: Text Transcription: x + 2y + 3z = 1 x^2 + y^2 = 3

Multivariable Calculus 9
Find the area of the surface. The part of the cone \(z=\sqrt{x^{2}+y^{2}}\) that lies between the plane \(y=x\) and the cylinder \(y=x^{2}\) Equation Transcription: Text Transcription: z = sqrt x^2 + y^2 y = x y = x^2

Multivariable Calculus 9
Find the area of the surface. The surface \(z=\frac{2}{3}\left(x^{3 / 2}+y^{3 / 2}\right), 0 \leqslant \ x \leqslant \ 1,\ 0 \leqslant \ y \leqslant 1\) Equation Transcription: Text Transcription: z = 23(x^3/2 + y^3/2), 0 < or = x < or = 1, 0 < or = y

Multivariable Calculus 9
Find the area of the surface. The part of the surface \(z=4-2 x^{2}+y\) that lies above the triangle with vertices \((0,0),(1,0)\), and \((1,1)\) Equation Transcription: Text Transcription: z=4-2x^2+y (0,0),(1,0), (1,1)

Multivariable Calculus 9
Find the area of the surface. The part of the surface \(z=x y\) that lies within the cylinder \(x^{2}+y^{2}=1\) Equation Transcription: Text Transcription: z = xy x^2 + y^2 = 1

Multivariable Calculus 9
Find the area of the surface. The part of the surface \(x=z^{2}+y\) that lies between the planes \(y=0, \ y=2, \ z=0\), and \(z=2\) Equation Transcription: , Text Transcription: x = z^2 + y y = 0, y = 2, z = 0 z = 2

Multivariable Calculus 9
Find the area of the surface. The part of the paraboloid \(y=x^{2}+z^{2}\) that lies within the cylinder \(x^{2}+z^{2}=16\) Equation Transcription: Text Transcription: y = x^2 + z^2 x^2 + z^2 = 16

Multivariable Calculus 9
Find the area of the surface. The helicoid (or spiral ramp) with vector equation \(\mathbf{r}(u, v)=u \cos v \mathbf{i}+u \sin v \mathbf{j}+v \mathbf{k}, 0 \leqslant u \leqslant 1\), \(0 \leqslant v \leqslant \pi\) Equation Transcription: Text Transcriptio

Multivariable Calculus 9
Find the area of the surface. The surface with parametric equations \(x=u^{2}, y=u v\), \(z=\frac{1}{2} v^{2}, 0 \leqslant u \leqslant 1,0 \leqslant v \leqslant 2\) Equation Transcription: Text Transcription: x = u^2, y = uv, z= v^2, 0 < or = u < or

Multivariable Calculus 9
Find the area of the surface. The part of the sphere \(x^{2}+y^{2}+z^{2}=b^{2}\) that lies inside the cylinder \(x^{2}+y^{2}=a^{2}\), where \(0<a<b\) Equation Transcription: Text Transcription: x^2 + y^2 + z^2 = b^2 x^2 + y^2 = a^2 0 < a < b

Multivariable Calculus 9
If the equation of a surface \(S\) is \(z=f(x, y)\), where \(x^{2}+y^{2} \leqslant R^{2}\), and you know that \(\left|f_{x}\right| \leqslant 1\) and \(\left|f_{y}\right| \leqslant 1\) what can you say about \(A(S) ?\) Equation Transcription:

Multivariable Calculus 9
Find the area of the surface correct to four decimal places by first simplifying an expression for area to one in terms of a single integral and then evaluating the integral numerically. The part of the surface \(z=\cos \left(x^{2}+y^{2}\right)\) that lies inside the cylinder \(x^{2}+y^{2}=1\)

Multivariable Calculus 9
Find the area of the surface correct to four decimal places by first simplifying an expression for area to one in terms of a single integral and then evaluating the integral numerically. The part of the surface \(z=\ln \left(x^{2}+y^{2}+2\right)\) that lies above the disk \(x^{2}+y^{2} \leqslant 1\

Multivariable Calculus 9
Use a computer algebra system to find, to four decimal places, the area of the part of the surface \(z=\left(1+x^{2}\right) /\left(1+y^{2}\right)\) that lies above the square \(|x|+|y| \leqslant 1\). Illustrate by graphing this part of the surface. Equation Transcription:

Multivariable Calculus 9
(a) Use the Midpoint Rule for double integrals (see Section 15.1) with six squares to estimate the area of the surface \(z=1 /\left(1+x^{2}+y^{2}\right), 0 \leqslant x \leqslant 6,0 \leqslant y \leqslant 4\) (b) Use a computer algebra system to approximate the surface area in part (a) to four decim

Multivariable Calculus 9
Use a computer algebra system to find the area of the surface with vector equation \(\mathbf{r}(u, v)=\left\langle\cos ^{3} u \cos ^{3} v, \sin ^{3} u \cos ^{3} v, \sin ^{3} v\right\rangle\) \(0 \leqslant u \leqslant \pi, 0 \leqslant v \leqslant 2 \pi\). State your answer correct to four decimal

Multivariable Calculus 9
Use a computer algebra system to find the exact area of the surface \(z=1+2 x+3 y+4 y^{2}, 1 \leqslant x \leqslant 4,0 \leqslant y \leqslant 1\). Equation Transcription: Text Transcription: z=1+2x+ 3y + 4y^2, 1< or = x < or = 4, 0< or = y < or =1

Multivariable Calculus 9
(a) Set up, but do not evaluate, a double integral for the area of the surface with parametric equations \(x=a u \ \cos \ v, y=b u \ \sin v, z=u^{2}, 0 \leqslant u \leqslant 2\) \(0 \leqslant v \leqslant 2 \pi\) (b) Eliminate the parameters to show that the surface is an elliptic paraboloid and set

Multivariable Calculus 9
(a) Show that the parametric equations \(x=a \ \sin \ u \ \cos \ v\), \(y=b \ \sin \ u \ \sin \ v, z=c \cos u, 0 \leqslant u \leqslant \pi\) \(0 \leqslant v \leqslant 2 \pi\), represent an ellipsoid. (b) Use the parametric equations in part (a) to graph the ellipsoid for the case \(a=1, b=2, c=3\).

Multivariable Calculus 9
(a) Show that the parametric equations \(x=a \ \cosh \ u \ \cos \ v\), \(y=b \cosh \ u \sin \ v, z=c \sinh u\), represent a hyperboloid of one sheet. (b) Use the parametric equations in part (a) to graph the hyperboloid for the case \(a=1, b=2, c=3\). (c) Set up, but do not evaluate, a double int

Multivariable Calculus 9
Find the area of the part of the sphere \(x^{2}+y^{2}+z^{2}=4 z\) that lies inside the paraboloid \(z=x^{2}+y^{2}\). Equation Transcription: Text Transcription: x^2+y^2+z^2=4z z=x^2+y^2

Multivariable Calculus 9
The figure shows the surface created when the cylinder \(y^{2}+z^{2}=1\) intersects the cylinder \(x^{2}+z^{2}=1\). Find the area of this surface. Equation Transcription: Text Transcription: y^2+z^2=1 x^2+z^2=1

Multivariable Calculus 9
Find the area of the part of the sphere \(x^{2}+y^{2}+z^{2}=a^{2}\) that lies inside the cylinder \(x^{2}+y^{2}=a x\). Equation Transcription: Text Transcription: x^2+y^2+z^2=a^2 x^2+y^2=ax

Multivariable Calculus 9
(a) Find a parametric representation for the torus obtained by rotating about the \(z\)-axis the circle in the \(x z\)-plane with center \((b, 0,0)\) and radius \(a<b\). [Hint: Take as parameters the angles \(\theta\) and \(\alpha\) shown in the figure.] (b) Use the parametric equations found in pa

Multivariable Calculus 9
Identify the surface with the given vector equation. \(\mathbf{r}(s, \ t)=\langle s \ \cos \ t, s \ \sin \ t, \ s\rangle\) Equation Transcription: ? ? Text Transcription: r=(s, t)=?s cos t, s sin t, s?