Question 1

Find the distance between the points $(4,-5,8)$ and (3, 7 - 2).

Accepted Answer

$7 \sqrt{5}$

Question 2

Find the volume of the solid bounded by the plane $5 x+2 y+z=10$ and the coordinate planes.

Accepted Answer

$\mathrm{V}=16 \frac{2}{3}$

Question 3

Name and sketch the graph of $9 x^{2}+y^{2}+4 z^{2}=36$.

Accepted Answer

$9 x^{2}+y^{2}+4 z^{2}=36$
Ellipsoid

Question 4

The total resistance $R$ of two resistors $R_{1}$ and $R_{2}$, connected in parallel, is $\frac{R_{1} R_{2}}{R_{1}+R_{2}}$. $R_{1}$ measures $525 \Omega$ with a maximum error of $35 \Omega$. $\mathrm{R}_{2}$ measures $375 \Omega$ with a maximum error of $24 \Omega$. Use a differential to approximate the change in $\mathrm{R}$ to two significant digits.

Accepted Answer

The answer of The total resistance $R$ of two resistors...

Question 5

Use the total differential to estimate the change in the volume of a cylinder, $V=\pi r^{2} h$, if its radius changed from $5.00 \mathrm{~cm}$ to $5.03 \mathrm{~cm}$ and its height changed from $4.00 \mathrm{~cm}$ to $3.98 \mathrm{~cm}$.

Accepted Answer

A)    dV=1.96 cm2\mathrm{dV}=1.96 \mathrm{~cm}^{2}dV=1.96 cm2 
B)    dV=2.20 cm2\mathrm{dV}=2.20 \mathrm{~cm}^{2}dV=2.20 cm2 
C)    dV=2.07 cm2\mathrm{dV}=2.07 \mathrm{~cm}^{2}dV=2.07 cm2 
D)    dV=5.34 cm2\mathrm{dV}=5.34 \mathrm{~cm}^{2}dV=5.34 cm2

Question 6

Find the partial derivative $\frac{\partial z}{\partial x}$ if $z=x^{3} y^{5}+\sin x y$.

Accepted Answer

A)    5x3y4+xcos⁡xy5 x^{3} y^{4}+x \cos x y5x3y4+xcosxy 
B)    3x2y5+ycos⁡xy3 x^{2} y^{5}+y \cos x y3x2y5+ycosxy 
C)    3x2y5+cos⁡xy3 x^{2} y^{5}+\cos x y3x2y5+cosxy 
D)    3x2y5+xcos⁡xy3 x^{2} y^{5}+x \cos x y3x2y5+xcosxy

Question 7

Find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$ for $z=e^{4 x} \cos x y$.

Accepted Answer

The answer of Find $\frac{\partial z}{\partial x}$ and \(\frac{\partial z}{\partial...

Question 8

The three- space equation $(x-2)^{2}+\left(y^{2}+5\right)^{2}+(z-1)^{2}=16$ represents $a$ :

Accepted Answer

A)   parabolic cylinder
B)   sphere
C)   hyperboloid of one sheet
D)   elliptic cone

Question 9

Find the total differential of the function $z=x^{2} y+\sin y$.

Accepted Answer

A)    dz=(2xycos⁡y) dx+(x2+cos⁡y) dyd z=(2 x y \cos y)  d x+\left(x^{2}+\cos y\right)  d ydz=(2xycosy) dx+(x2+cosy) dy 
B)    dz=2xydx+(x2+cos⁡y) dyd z=2 x y d x+\left(x^{2}+\cos y\right)  d ydz=2xydx+(x2+cosy) dy 
C)    dz=2xydx+(2x+cos⁡y) dyd z=2 x y d x+(2 x+\cos y)  d ydz=2xydx+(2x+cosy) dy 
D)    dz=(2xy+cos⁡y) dx+x2dyd z=(2 x y+\cos y)  d x+x^{2} d ydz=(2xy+cosy) dx+x2dy

Question 10

Find all critical points and any relative maximum or minimum points or saddle points for the function: $z=2 x^{2}+5 y^{2}-8 x y+6 x$

Accepted Answer

The answer of Find all critical points and any relative...

Question 11

Find any relative maximum or minimum points or saddle points of the function $z=f(x, y)=x^{2}+y^{2}+2 x-6 y+3$.

Accepted Answer

A)   saddle point at  (1,3,−3) (1,3,-3) (1,3,−3)  
B)   relative minimum at (-  1,3,−7) 1,3,-7) 1,3,−7)  
C)   saddle point at (  −1,−3,29-1,-3,29−1,−3,29  ) 
D)   relative maximum at  (1,−3,33) (1,-3,33) (1,−3,33)

Question 12

Find the volume of the solid bounded by the surface $3 x+4 y+2 z=12$ and the coordinate planes.

Accepted Answer

A)   12
B)   24
C)   14
D)   16

Question 13

Find the total differential for $z=4 x^{2}+2 x y+3 y^{3}$.

Accepted Answer

The answer of Find the total differential for \(z=4 x^{2}+2...

Question 14

Evaluate $\int_{1}^{3} \int_{0}^{x^{2}}\left(x^{2}-5 x y\right) d y d x$.

Accepted Answer

The answer of Evaluate \(\int_{1}^{3} \int_{0}^{x^{2}}\left(x^{2}-5 x y\right) d y...

Question 15

Evaluate the double integral $\int_{0}^{1} \int_{x^{2}}^{3 x+1} 4 x d y d x$

Accepted Answer

A)   6
B)   3
C)   4
D)   5

Question 16

Find $\frac{\partial P}{\partial R}$ if $P=\frac{V^{2}}{R}$.

Accepted Answer

A)    −V2R2-\frac{V^{2}}{R^{2}}−R2V2​ 
B)    −2VR2-\frac{2 V}{R^{2}}−R22V​ 
C)    2 VR2\frac{2 \mathrm{~V}}{\mathrm{R}^{2}}R22 V​ 
D)    V2R2\frac{V^{2}}{R^{2}}R2V2​

Question 17

Given the surface $z=36 x^{2}+25 y^{2}$, find the slope of the tangent line parallel to the $x z$ - plane and through the point $(-1,1,61)$.

Accepted Answer

The answer of Given the surface $z=36 x^{2}+25 y^{2}$, find...

Exam 8: Three-Space: Partial Derivatives and Double Integrals

Find the distance between the points $(4,-5,8)$ and (3, 7 - 2).

Find the volume of the solid bounded by the plane $5 x+2 y+z=10$ and the coordinate planes.

Name and sketch the graph of $9 x^{2}+y^{2}+4 z^{2}=36$ .

Use the total differential to estimate the change in the volume of a cylinder, $V=\pi r^{2} h$ , if its radius changed from $5.00 \mathrm{~cm}$ to $5.03 \mathrm{~cm}$ and its height changed from $4.00 \mathrm{~cm}$ to $3.98 \mathrm{~cm}$ .

Find the partial derivative $\frac{\partial z}{\partial x}$ if $z=x^{3} y^{5}+\sin x y$ .

Find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$ for $z=e^{4 x} \cos x y$ .

The three- space equation $(x-2) ^{2}+\left(y^{2}+5\right) ^{2}+(z-1) ^{2}=16$ represents $a$ :

Find the total differential of the function $z=x^{2} y+\sin y$ .

Find all critical points and any relative maximum or minimum points or saddle points for the function: $z=2 x^{2}+5 y^{2}-8 x y+6 x$

Find any relative maximum or minimum points or saddle points of the function $z=f(x, y) =x^{2}+y^{2}+2 x-6 y+3$ .

Find the volume of the solid bounded by the surface $3 x+4 y+2 z=12$ and the coordinate planes.

Find the total differential for $z=4 x^{2}+2 x y+3 y^{3}$ .

Evaluate $\int_{1}^{3} \int_{0}^{x^{2}}\left(x^{2}-5 x y\right) d y d x$ .

Evaluate the double integral $\int_{0}^{1} \int_{x^{2}}^{3 x+1} 4 x d y d x$

Find $\frac{\partial P}{\partial R}$ if $P=\frac{V^{2}}{R}$ .

Given the surface $z=36 x^{2}+25 y^{2}$ , find the slope of the tangent line parallel to the $x z$ - plane and through the point $(-1,1,61)$ .

Exam 8: Three-Space: Partial Derivatives and Double Integrals

Find the distance between the points (4,−5,8)(4,-5,8)(4,−5,8) and (3, 7 - 2).

Find the volume of the solid bounded by the plane 5x+2y+z=105 x+2 y+z=105x+2y+z=10 and the coordinate planes.

Name and sketch the graph of 9x2+y2+4z2=369 x^{2}+y^{2}+4 z^{2}=369x2+y2+4z2=36 .

Use the total differential to estimate the change in the volume of a cylinder, V=πr2hV=\pi r^{2} hV=πr2h , if its radius changed from 5.00 cm5.00 \mathrm{~cm}5.00 cm to 5.03 cm5.03 \mathrm{~cm}5.03 cm and its height changed from 4.00 cm4.00 \mathrm{~cm}4.00 cm to 3.98 cm3.98 \mathrm{~cm}3.98 cm .

Find the partial derivative ∂z∂x\frac{\partial z}{\partial x}∂x∂z​ if z=x3y5+sin⁡xyz=x^{3} y^{5}+\sin x yz=x3y5+sinxy .

Find ∂z∂x\frac{\partial z}{\partial x}∂x∂z​ and ∂z∂y\frac{\partial z}{\partial y}∂y∂z​ for z=e4xcos⁡xyz=e^{4 x} \cos x yz=e4xcosxy .

The three- space equation (x−2) 2+(y2+5) 2+(z−1) 2=16(x-2) ^{2}+\left(y^{2}+5\right) ^{2}+(z-1) ^{2}=16(x−2) 2+(y2+5) 2+(z−1) 2=16 represents aaa :

Find the total differential of the function z=x2y+sin⁡yz=x^{2} y+\sin yz=x2y+siny .

Find all critical points and any relative maximum or minimum points or saddle points for the function: z=2x2+5y2−8xy+6xz=2 x^{2}+5 y^{2}-8 x y+6 xz=2x2+5y2−8xy+6x

Find any relative maximum or minimum points or saddle points of the function z=f(x,y) =x2+y2+2x−6y+3z=f(x, y) =x^{2}+y^{2}+2 x-6 y+3z=f(x,y) =x2+y2+2x−6y+3 .

Find the volume of the solid bounded by the surface 3x+4y+2z=123 x+4 y+2 z=123x+4y+2z=12 and the coordinate planes.

Find the total differential for z=4x2+2xy+3y3z=4 x^{2}+2 x y+3 y^{3}z=4x2+2xy+3y3 .

Evaluate ∫13∫0x2(x2−5xy)dydx\int_{1}^{3} \int_{0}^{x^{2}}\left(x^{2}-5 x y\right) d y d x∫13​∫0x2​(x2−5xy)dydx .

Evaluate the double integral ∫01∫x23x+14xdydx\int_{0}^{1} \int_{x^{2}}^{3 x+1} 4 x d y d x∫01​∫x23x+1​4xdydx

Find ∂P∂R\frac{\partial P}{\partial R}∂R∂P​ if P=V2RP=\frac{V^{2}}{R}P=RV2​ .

Given the surface z=36x2+25y2z=36 x^{2}+25 y^{2}z=36x2+25y2 , find the slope of the tangent line parallel to the xzx zxz - plane and through the point (−1,1,61)(-1,1,61)(−1,1,61) .

Find the distance between the points $(4,-5,8)$ and (3, 7 - 2).

Find the volume of the solid bounded by the plane $5 x+2 y+z=10$ and the coordinate planes.

Name and sketch the graph of $9 x^{2}+y^{2}+4 z^{2}=36$ .

Use the total differential to estimate the change in the volume of a cylinder, $V=\pi r^{2} h$ , if its radius changed from $5.00 \mathrm{~cm}$ to $5.03 \mathrm{~cm}$ and its height changed from $4.00 \mathrm{~cm}$ to $3.98 \mathrm{~cm}$ .

Find the partial derivative $\frac{\partial z}{\partial x}$ if $z=x^{3} y^{5}+\sin x y$ .

Find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$ for $z=e^{4 x} \cos x y$ .

The three- space equation $(x-2) ^{2}+\left(y^{2}+5\right) ^{2}+(z-1) ^{2}=16$ represents $a$ :

Find the total differential of the function $z=x^{2} y+\sin y$ .

Find all critical points and any relative maximum or minimum points or saddle points for the function: $z=2 x^{2}+5 y^{2}-8 x y+6 x$

Find any relative maximum or minimum points or saddle points of the function $z=f(x, y) =x^{2}+y^{2}+2 x-6 y+3$ .

Find the volume of the solid bounded by the surface $3 x+4 y+2 z=12$ and the coordinate planes.

Find the total differential for $z=4 x^{2}+2 x y+3 y^{3}$ .

Evaluate $\int_{1}^{3} \int_{0}^{x^{2}}\left(x^{2}-5 x y\right) d y d x$ .

Evaluate the double integral $\int_{0}^{1} \int_{x^{2}}^{3 x+1} 4 x d y d x$

Find $\frac{\partial P}{\partial R}$ if $P=\frac{V^{2}}{R}$ .

Given the surface $z=36 x^{2}+25 y^{2}$ , find the slope of the tangent line parallel to the $x z$ - plane and through the point $(-1,1,61)$ .