Calculus Volume 3

Gilbert Strang

Chapter 4

Differentiation of Functions of Several Variables - all with Video Answers

Educators

Section 1

Functions of Several Variables

Problem 1

For the following exercises, evaluate each function at the indicated values.
$$
W(x, y)=4 x^{2}+y^{2} . \text { Find } W(2,-1), \quad W(-3,6)
$$

Amy Jiang

Amy Jiang

Numerade Educator

Problem 2

For the following exercises, evaluate each function at the indicated values.
$$
W(x, y)=4 x^{2}+y^{2} . \text { Find } W(2+h, 3+h)
$$

Amy Jiang

Numerade Educator

Problem 3

The volume of a right circular cylinder is calculated by a function of two variables, $V(x, y)=\pi x^{2} y,$ where $x$ is theradius of the right circular-cylinder and $y$ represents the height of the cylinder. Evaluate $V(2,5)$ and explain what this means.

Andrew Bassila

Numerade Educator

Problem 4

An oxygen tank is constructed of a right cylinder of height $y$ and radius $x$ with two hemispheres of radius $x$ mounted on the top and bottom of the cylinder. Express the volume of the cylinder as a function of two variables, $x$ and $y,$ find $V(10,2),$ and explain what this means.

Amy Jiang

Numerade Educator

Problem 5

For the following exercises, find the domain of the function.
$$
V(x, y)=4 x^{2}+y^{2}
$$

Oswaldo Jiménez

Oswaldo Jiménez

Numerade Educator

Problem 6

For the following exercises, find the domain of the function.
$$
f(x, y)=\sqrt{x^{2}+y^{2}-4}
$$

Amy Jiang

Numerade Educator

Problem 7

For the following exercises, find the domain of the function.
$$
f(x, y)=4 \ln \left(y^{2}-x\right)
$$

Amy Jiang

Numerade Educator

Problem 8

For the following exercises, find the domain of the function.
$$
g(x, y)=\sqrt{16-4 x^{2}-y^{2}}
$$

Amy Jiang

Numerade Educator

Problem 9

For the following exercises, find the domain of the function.
$$
z(x, y)=y^{2}-x^{2}
$$

Carson Merrill

Numerade Educator

Problem 10

For the following exercises, find the domain of the function.
$$
f(x, y)=\frac{y+2}{x^{2}}
$$

Carson Merrill

Numerade Educator

Problem 11

Find the range of the functions.
$$
g(x, y)=\sqrt{16-4 x^{2}-y^{2}}
$$

Amy Jiang

Numerade Educator

Problem 12

Find the range of the functions.
$$
V(x, y)=4 x^{2}+y^{2}
$$

Amy Jiang

Numerade Educator

Problem 13

Find the range of the functions.
$$
z=y^{2}-x^{2}
$$

Amy Jiang

Numerade Educator

Problem 14

For the following exercises, find the level curves of each function at the indicated value of $c$ to visualize the given function.
$$
z(x, y)=y^{2}-x^{2}, \quad c=1
$$

Amy Jiang

Numerade Educator

Problem 15

For the following exercises, find the level curves of each function at the indicated value of $c$ to visualize the given function.
$$
z(x, y)=y^{2}-x^{2}, \quad c=4
$$

Amy Jiang

Numerade Educator

Problem 16

For the following exercises, find the level curves of each function at the indicated value of $c$ to visualize the given function.
$$
g(x, y)=x^{2}+y^{2} ; c=4, c=9
$$

Amy Jiang

Numerade Educator

Problem 17

For the following exercises, find the level curves of each function at the indicated value of $c$ to visualize the given function.
$$
g(x, y)=4-x-y ; c=0,4
$$

Amy Jiang

Numerade Educator

Problem 18

For the following exercises, find the level curves of each function at the indicated value of $c$ to visualize the given function.
$$
f(x, y)=x y ; c=1 ; c=-1
$$

Amy Jiang

Numerade Educator

Problem 19

For the following exercises, find the level curves of each function at the indicated value of $c$ to visualize the given function.
$$
h(x, y)=2 x-y ; c=0,-2,2
$$

Amy Jiang

Numerade Educator

Problem 20

For the following exercises, find the level curves of each function at the indicated value of $c$ to visualize the given function.
$$
f(x, y)=x^{2}-y ; c=1,2
$$

Amy Jiang

Numerade Educator

Problem 21

For the following exercises, find the level curves of each function at the indicated value of $c$ to visualize the given function.
$$
g(x, y)=\frac{x}{x+y} ; c=-1,0,2
$$

Amy Jiang

Numerade Educator

Problem 22

For the following exercises, find the level curves of each function at the indicated value of $c$ to visualize the given function.
$$
g(x, y)=x^{3}-y ; c=-1,0,2
$$

Amy Jiang

Numerade Educator

Problem 23

For the following exercises, find the level curves of each function at the indicated value of $c$ to visualize the given function.
$$
g(x, y)=e^{x y} ; c=\frac{1}{2}, 3
$$

Amy Jiang

Numerade Educator

Problem 24

For the following exercises, find the level curves of each function at the indicated value of $c$ to visualize the given function.
$$
f(x, y)=x^{2} ; c=4,9
$$

Amy Jiang

Numerade Educator

Problem 25

For the following exercises, find the level curves of each function at the indicated value of $c$ to visualize the given function.
$$
f(x, y)=x y-x ; c=-2,0,2
$$

Amy Jiang

Numerade Educator

Problem 26

For the following exercises, find the level curves of each function at the indicated value of $c$ to visualize the given function.
$$
h(x, y)=\ln \left(x^{2}+y^{2}\right) ; c=-1,0,1
$$

Amy Jiang

Numerade Educator

Problem 27

For the following exercises, find the level curves of each function at the indicated value of $c$ to visualize the given function.
$$
g(x, y)=\ln \left(\frac{y}{x^{2}}\right) ; c=-2,0,2
$$

Amy Jiang

Numerade Educator

Problem 28

For the following exercises, find the level curves of each function at the indicated value of $c$ to visualize the given function.
$$
z=f(x, y)=\sqrt{x^{2}+y^{2}}, \quad c=3
$$

Amy Jiang

Numerade Educator

Problem 29

For the following exercises, find the level curves of each function at the indicated value of $c$ to visualize the given function.
$$f(x, y)=\frac{y+2}{x^{2}}, \quad c= any\quad constant$$

Amy Jiang

Numerade Educator

Problem 30

For the following exercises, find the vertical traces of the functions at the indicated values of $x$ and $y,$ and plot the traces.
$$
z=4-x-y ; x=2
$$

Amy Jiang

Numerade Educator

Problem 31

For the following exercises, find the vertical traces of the functions at the indicated values of $x$ and $y,$ and plot the traces.
$$
f(x, y)=3 x+y^{3}, x=1
$$

Melissa Munoz

Numerade Educator

Problem 32

For the following exercises, find the vertical traces of the functions at the indicated values of $x$ and $y,$ and plot the traces.
$$
z=\cos \sqrt{x^{2}+y^{2}} \quad x=1
$$

Amy Jiang

Numerade Educator

Problem 33

Find the domain of the following functions.
$$
z=\sqrt{100-4 x^{2}-25 y^{2}}
$$

Carson Merrill

Numerade Educator

Problem 34

Find the domain of the following functions.
$$
z=\ln \left(x-y^{2}\right)
$$

Amy Jiang

Numerade Educator

Problem 35

Find the domain of the following functions.
$$
f(x, y, z)=\frac{1}{\sqrt{36-4 x^{2}-9 y^{2}-z^{2}}}
$$

Amy Jiang

Numerade Educator

Problem 36

Find the domain of the following functions.
$$
f(x, y, z)=\sqrt{49-x^{2}-y^{2}-z^{2}}
$$

Amy Jiang

Numerade Educator

Problem 37

Find the domain of the following functions.
$$
f(x, y, z)=\sqrt[3]{16-x^{2}-y^{2}-z^{2}}
$$

Vikash Ranjan

Numerade Educator

Problem 38

Find the domain of the following functions.
$$
f(x, y)=\cos \sqrt{x^{2}+y^{2}}
$$

Amy Jiang

Numerade Educator

Problem 39

For the following exercises, plot a graph of the function.
$$
z=f(x, y)=\sqrt{x^{2}+y^{2}}
$$

Amy Jiang

Numerade Educator

Problem 40

For the following exercises, plot a graph of the function.
$$
z=x^{2}+y^{2}
$$

Amy Jiang

Numerade Educator

Problem 41

For the following exercises, plot a graph of the function.
Use technology to graph $z=x^{2} y$

Amy Jiang

Numerade Educator

Problem 42

Sketch the following by finding the level curves. Verify the graph using technology.
$$
f(x, y)=\sqrt{4-x^{2}-y^{2}}
$$

Amy Jiang

Numerade Educator

Problem 43

Sketch the following by finding the level curves. Verify the graph using technology.
$$
f(x, y)=2-\sqrt{x^{2}+y^{2}}
$$

Amy Jiang

Numerade Educator

Problem 44

Sketch the following by finding the level curves. Verify the graph using technology.
$$
z=1+e^{-x^{2}-y^{2}}
$$

Amy Jiang

Numerade Educator

Problem 45

Sketch the following by finding the level curves. Verify the graph using technology.
$$
z=\cos \sqrt{x^{2}+y^{2}}
$$

Amy Jiang

Numerade Educator

Problem 46

Sketch the following by finding the level curves. Verify the graph using technology.
$$
z=y^{2}-x^{2}
$$

Amy Jiang

Numerade Educator

Problem 47

Sketch the following by finding the level curves. Verify the graph using technology.
Describe the contour lines for several values of $c$ for $z=x^{2}+y^{2}-2 x-2 y$

Amy Jiang

Numerade Educator

Problem 48

Find the level surface for the functions of three variables and describe it.
$$
w(x, y, z)=x-2 y+z, c=4
$$

Amy Jiang

Numerade Educator

Problem 49

Find the level surface for the functions of three variables and describe it.
$$
w(x, y, z)=x^{2}+y^{2}+z^{2}, c=9
$$

Amy Jiang

Numerade Educator

Problem 50

Find the level surface for the functions of three variables and describe it.
$$
w(x, y, z)=x^{2}+y^{2}-z^{2}, c=-4
$$

Amy Jiang

Numerade Educator

Problem 51

Find the level surface for the functions of three variables and describe it.
$$
w(x, y, z)=x^{2}+y^{2}-z^{2}, c=4
$$

Amy Jiang

Numerade Educator

Problem 52

Find the level surface for the functions of three variables and describe it.
$$
w(x, y, z)=9 x^{2}-4 y^{2}+36 z^{2}, c=0
$$

Amy Jiang

Numerade Educator

Problem 53

For the following exercises, find an equation of the level curve of $f$ that contains the point $P .$
$$
f(x, y)=1-4 x^{2}-y^{2}, P(0,1)
$$

Amy Jiang

Numerade Educator

Problem 54

For the following exercises, find an equation of the level curve of $f$ that contains the point $P .$
$$
g(x, y)=y^{2} \arctan x, P(1,2)
$$

Amy Jiang

Numerade Educator

Problem 55

For the following exercises, find an equation of the level curve of $f$ that contains the point $P .$
$$
g(x, y)=e^{x y}\left(x^{2}+y^{2}\right), P(1,0)
$$

Amy Jiang

Numerade Educator

Problem 56

The strength $E$ of an electric field at point $(x, y, z)$ resulting from an infinitely long charged wire lying along the $y$ -axis is given by $E(x, y, z)=k / \sqrt{x^{2}+y^{2}},$ where $k$ is a positive constant. For simplicity, let $k=1$ and find the equations of the level surfaces for $E=10$ and $E=100$ .

Amy Jiang

Numerade Educator

Problem 57

A thin plate made of iron is located in the $x y$ -plane. The temperature $T$ in degrees Celsius at a point $P(x, y)$ is inversely proportional to the square of its distance from the origin. Express $T$ as a function of $x$ and $y$

Suzanne W.

Numerade Educator

Problem 58

Refer to the preceding problem. Using the temperature function found there, determine the proportionality constant if the temperature at point $P(1,2)$ is $50^{\circ} \mathrm{C} .$ Use this constant to determine the temperature at point $Q(3,4)$

Amy Jiang

Numerade Educator

Problem 59

Refer to the preceding problem. Find the level curves for $T=40^{\circ} \mathrm{C}$ and $T=100^{\circ} \mathrm{C},$ and describe what the level curves represent.

Abhijith V

Numerade Educator