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

Question 1

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

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

Answer 1

sphere

Question 2

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

A) $5 x^{3} y^{4}+x \cos x y$
B) $3 x^{2} y^{5}+y \cos x y$
C) $3 x^{2} y^{5}+\cos x y$
D) $3 x^{2} y^{5}+x \cos x y$

Answer 2

$3 x^{2} y^{5}+y \cos x y$

Question 3

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

A) $-\frac{V^{2}}{R^{2}}$
B) $-\frac{2 V}{R^{2}}$
C) $\frac{2 \mathrm{~V}}{\mathrm{R}^{2}}$
D) $\frac{V^{2}}{R^{2}}$

Answer 3

$-\frac{V^{2}}{R^{2}}$

Question 4

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

A) $d z=(2 x y \cos y) d x+\left(x^{2}+\cos y\right) d y$
B) $d z=2 x y d x+\left(x^{2}+\cos y\right) d y$
C) $d z=2 x y d x+(2 x+\cos y) d y$
D) $d z=(2 x y+\cos y) d x+x^{2} d y$

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 $3.98 \mathrm{~cm}$ .

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

Question 6

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

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

Question 7

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

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

Question 8

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

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

Question 9

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

Question 10

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

Question 11

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

Question 12

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 $(-1,1,61)$ .

Question 13

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

Question 14

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.

Question 15

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

Question 16

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

Question 17

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