Books(current) Courses (current) Earn đŸ’° Log in(current)

Chapter 14

PARTIAL DERIVATIVES

Educators


Problem 1

In Example 2 we considered the function $W=f(T, v),$ where
$W$ is the wind-chill index, $T$ is the actual temperature, and $v$ is
the wind speed. A numerical representation is given in Table $1 .$
(a) What is the value of $f(-15,40) ?$ What is its meaning?
(b) Describe in words the meaning of the question "For what
value of $v$ is $f(-20, v)=-30 ?^{\prime \prime}$ Then answer the question.
(c) Describe in words the meaning of the question "For what
value of $T$ is $f(T, 20)=-49 ?^{\circ}$ Then answer the question.
(d) What is the meaning of the function $W=f(-5, v) ?$
Describe the behavior of this function.
(e) What is the meaning of the function $W=f(T, 50) ?$
Describe the behavior of this function.

Check back soon!

Problem 2

The temperature-humidity index $I$ (or humidex, for short) is the
perceived air temperature when the actual temperature is $T$ and
the relative humidity is $h,$ so we can write $I=f(T, h) .$ The following table of values of $I$ is an excerpt from a table compiled
by the National Oceanic $\&$ Atmospheric Administration.
(a) What is the value of $f(95,70)$ ? What is its meaning?
(b) For what value of $h$ is $f(90,$ h) $=100 ?$
(c) For what value of $T$ is $f(T, 50)=88 ?$
(d) What are the meanings of the functions $I=f(80, h)$
and $I=f(100, h) ?$ Compare the behavior of these two
functions of $h .$

Check back soon!

Problem 3

Verify for the Cobb-Douglas production function
$$P(L, K)=1.01 L^{0.75} K^{025}$$
discussed in Example 3 that the production will be doubled
if both the amount of labor and the amount of capital are
doubled. Determine whether this is also true for the general
production function
$$P(L, K)=b L^{\alpha} K^{1-\alpha}$$

Check back soon!

Problem 4

The wind-chill index $W$ discussed in Example 2 has been
modeled by the following function:
$$W T, v )=13.12+0.6215 T-11.37 v^{0.16}+0.3965 T v^{0.16}$$
Check to see how closely this model agrees with the values in
Table 1 for a few values of $T$ and $v .$

Check back soon!

Problem 5

The wave heights $h$ in the open sea depend on the speed $v$
of the wind and the length of time $t$ that the wind has been
blowing at that speed. Values of the function $h=f(v, t)$ are
recorded in feet in Table 4 .
(a) What is the value of $f(40,15) ?$ What is its meaning?
(b) What is the meaning of the function $h=f(30, t) ?$ Describe
the behavior of this function.
(c) What is the meaning of the function $h=f(v, 30) ?$ Describe
the behavior of this function.

Check back soon!

Problem 6

Let $$f(x, y)=\ln (x+y-1)$$
(a) Evaluate $f(1,1), \quad$ (b) Evaluate $f(e, 1)$
(c) Find and sketch the domain of $f$ .
(d) Find the range of $f .$

Check back soon!

Problem 7

Let $$f(x, y)=x^{2} e^{3 x y}$$
(a) Evaluate $f(2,0) . \quad$ (b) Find the domain of $f$
(c) Find the range of $f .$

Check back soon!

Problem 8

Find and sketch the domain of the function
$f(x, y)=\sqrt{1+x-y^{2}} .$ What is the range of $f ?$

Check back soon!

Problem 9

Let $$f(x, y, z)=e^{\sqrt{1-x^{2}-y^{2}}}$$
(a) Evaluate $f(2,-1,6), \quad$ (b) Find the domain of $f$
(c) Find the range of $f$

Check back soon!

Problem 10

Let $$g(x, y, z)=\ln \left(25-x^{2}-y^{2}-z^{2}\right)$$
(a) Evaluate $g(2,-2,4)$. (b) Find the domain of $g$.
(c) Find the range of $g$.

Check back soon!

Problem 11

Find and sketch the domain of the function.
$$f(x, y)=\sqrt{x+y}$$

Check back soon!

Problem 12

Find and sketch the domain of the function.
$$f(x, y)=\sqrt{x y}$$

Check back soon!

Problem 13

Find and sketch the domain of the function.
$$f(x, y)=\ln \left(9-x^{2}-9 y^{2}\right)$$

Check back soon!

Problem 14

Find and sketch the domain of the function.
$$f(x, y)=\sqrt{y-x} \ln (y+x)$$

Check back soon!

Problem 15

Find and sketch the domain of the function.
$$f(x, y)=\sqrt{1-x^{2}}-\sqrt{1-y^{2}}$$

Check back soon!

Problem 16

Find and sketch the domain of the function.
$$f(x, y)=\sqrt{y}+\sqrt{25-x^{2}-y^{2}}$$

Check back soon!

Problem 17

Find and sketch the domain of the function.
$$f(x, y)=\frac{\sqrt{y-x^{2}}}{1-x^{2}}$$

Check back soon!

Problem 18

Find and sketch the domain of the function.
$$f(x, y)=\arcsin \left(x^{2}+y^{2}-2\right)$$

Check back soon!

Problem 19

Find and sketch the domain of the function.
$$f(x, y, z)=\sqrt{1-x^{2}-y^{2}-z^{2}}$$

Check back soon!

Problem 20

Find and sketch the domain of the function.
$$f(x, y, z)=\ln \left(16-4 x^{2}-4 y^{2}-z^{2}\right)$$

Check back soon!

Problem 21

Sketch the graph of the function.
$$f(x, y)=3$$

Check back soon!

Problem 22

Sketch the graph of the function.
$$f(x, y)=y$$

Check back soon!

Problem 23

Sketch the graph of the function.
$$f(x, y)=10-4 x-5 y$$

Check back soon!

Problem 24

Sketch the graph of the function.
$$f(x, y)=\cos x$$

Check back soon!

Problem 25

Sketch the graph of the function.
$$f(x, y)=y^{2}+1$$

Check back soon!

Problem 26

Sketch the graph of the function.
$$f(x, y)=3-x^{2}-y^{2}$$

Check back soon!

Problem 27

Sketch the graph of the function.
$$f(x, y)=4 x^{2}+y^{2}+1$$

Check back soon!

Problem 28

Sketch the graph of the function.
$$f(x, y)=\sqrt{16-x^{2}-16 y^{2}}$$

Check back soon!

Problem 29

Sketch the graph of the function.
$$f(x, y)=\sqrt{x^{2}+y^{2}}$$

Check back soon!

Problem 30

Match the function with its graph (labeled I-VI).Give reasons
for your choices.
(a) $f(x, y)=|x|+|y|$ (b) f(x, y)=|x y|$ (c) f(x, y)=\frac{1}{1+x^{2}+y^{2}}$ (d) $f(x, y)=\left(x^{2}-y^{2}\right)^{2}$ (e) $f(x, y)=(x-y)^{2}$ (f) f(x, y)=\sin (|x|+|y|)$

Check back soon!

Problem 31

A contour map for a function $f$ is shown. Use it to estimate the
values of $f(-3,3)$ and $f(3,-2) .$ What can you say about the
shape of the graph?

Check back soon!

Problem 32

Two contour maps are shown. One is for a function $f$ whose
graph is a cone. The other is for a function $g$ whose graph is a
paraboloid. Which is which, and why?

Check back soon!

Problem 33

Locate the points $A$ and $B$ in the map of Lonesome Mountain
(Figure $12 ) .$ How would you describe the terrain near $A$ ?
Near $B$ ?

Check back soon!

Problem 34

Make a rough sketch of a contour map for the function whose
graph is shown.

Check back soon!

Problem 35

A contour map of a function is shown. Use it to make a rough sketch of the graph of $f$.

Check back soon!

Problem 36

A contour map of a function is shown. Use it to make a rough sketch of the graph of $f$.

Check back soon!

Problem 37

A contour map of a function is shown. Use it to make a rough sketch of the graph of $f .$

Check back soon!

Problem 38

A contour map of a function is shown. Use it to make a rough sketch of the graph of $f .$

Check back soon!

Problem 39

Draw a contour map of the function showing several level curves.
$$f(x, y)=(y-2 x)^{2}$$

Check back soon!

Problem 40

Draw a contour map of the function showing several level curves.
$$f(x, y)=x^{3}-y$$

Check back soon!

Problem 41

Draw a contour map of the function showing several level curves.
$$f(x, y)=x^{3}-y$$

Check back soon!

Problem 42

Draw a contour map of the function showing several level curves.
$$f(x, y)=e^{y / x}$$

Check back soon!

Problem 43

Draw a contour map of the function showing several level curves.
$$f(x, y)=y e^{x}$$

Check back soon!

Problem 44

Draw a contour map of the function showing several level curves.
$$f(x, y)=y \sec x$$

Check back soon!

Problem 45

Draw a contour map of the function showing several level curves.
$$f(x, y)=\sqrt{y^{2}-x^{2}}$$

Check back soon!

Problem 46

Draw a contour map of the function showing several level curves.
$$f(x, y)=y /\left(x^{2}+y^{2}\right)$$

Check back soon!

Problem 47

Sketch both a contour map and a graph of the function and compare them.
$$f(x, y)=x^{2}+9 y^{2}$$

Check back soon!

Problem 48

Sketch both a contour map and a graph of the function and compare them.
$$f(x, y)=\sqrt{36-9 x^{2}-4 y^{2}}$$

Check back soon!

Problem 49

A thin metal plate, located in the $x y$ -plane, has temperature $T(x, y)$ at the point $(x, y) .$ The level curves of $T$ are called isothermals because at all points on an isothermal the temperature is the same. Sketch some isothermals if the temperature function is given by
$$T(x, y)=100 /\left(1+x^{2}+2 y^{2}\right)$$

Check back soon!

Problem 50

If $V(x, y)$ is the electric potential at a point $(x, y)$ in the xy-plane, then the level curves of $V$ are called equipotential curves because at all points on such a curve the electric potential is the same. Sketch some equipotential curves if $V(x, y)=c / \sqrt{r^{2}-x^{2}-y^{2}},$ where $c$ is a positive constant.

Check back soon!

Problem 51

Use a computer to graph the function using various domains and viewpoints. Get a printout of one that, in your opinion, gives a good view. If your software also produces level curves, then plot some contour lines of the same function and compare with the graph.
$$f(x, y)=e^{-x^{2}}+e^{-2 y^{2}}$$

Check back soon!

Problem 52

Use a computer to graph the function using various domains and viewpoints. Get a printout of one that, in your opinion, gives a good view. If your software also produces level curves, then plot some contour lines of the same function and compare with the graph.
$$f(x, y)=\left(1-3 x^{2}+y^{2}\right) e^{1-x^{2}-y^{2}}$$

Check back soon!

Problem 53

Use a computer to graph the function using various domains and viewpoints. Get a printout of one that, in your opinion, gives a good view. If your software also produces level curves, then plot some contour lines of the same function and compare with the graph.
$f(x, y)=x y^{2}-x^{3} \quad $(monkey saddle)

Check back soon!

Problem 54

Use a computer to graph the function using various domains and viewpoints. Get a printout of one that, in your opinion, gives a good view. If your software also produces level curves, then plot some contour lines of the same function and compare with the graph.
$f(x, y)=x y^{3}-y x^{3} \quad($ dog saddle $)$

Check back soon!

Problem 55

Match the function (a) with its graph (labeled $A-F$ on page 869 ) and (b) with its contour map (labeled I-VI). Give reasons for your choices.
$$z=\sin (x y)$$

Check back soon!

Problem 56

Match the function (a) with its graph (labeled $A-F$ on page 869 ) and (b) with its contour map (labeled I-VI). Give reasons for your choices.
$$z=e^{x} \cos y$$

Check back soon!

Problem 56

Match the function (a) with its graph (labeled $A-F$ on page 869 ) and (b) with its contour map (labeled I-VI). Give reasons for your choices.
$$z=e^{x} \cos y$$

Check back soon!

Problem 57

Match the function (a) with its graph (labeled $A-F$ on page 869 ) and (b) with its contour map (labeled I-VI). Give reasons for your choices.
$$z=\sin (x-y)$$

Check back soon!

Problem 58

Match the function (a) with its graph (labeled $A-F$ on page 869 ) and (b) with its contour map (labeled I-VI). Give reasons for your choices.
$$z=\sin x-\sin y$$

Check back soon!

Problem 59

Match the function (a) with its graph (labeled $A-F$ on page 869 ) and (b) with its contour map (labeled I-VI). Give reasons for your choices.
$$z=\left(1-x^{2}\right)\left(1-y^{2}\right)$$

Check back soon!

Problem 60

Match the function (a) with its graph (labeled $A-F$ on page 869 ) and (b) with its contour map (labeled I-VI). Give reasons for your choices.
$$z=\frac{x-y}{1+x^{2}+y^{2}}$$

Check back soon!

Problem 61

Describe the level surfaces of the function.
$$f(x, y, z)=x+3 y+5 z$$

Check back soon!

Problem 62

Describe the level surfaces of the function.
$$f(x, y, z)=x^{2}+3 y^{2}+5 z^{2}$$

Check back soon!

Problem 63

Describe the level surfaces of the function.
$$f(x, y, z)=x^{2}-y^{2}+z^{2}$$

Check back soon!

Problem 64

Describe the level surfaces of the function.
$$f(x, y, z)=x^{2}-y^{2}$$

Check back soon!

Problem 65

Describe how the graph of $g$ is obtained from the graph of $f .$
$$g(x, y)=f(x, y)+2 \quad \text { (b) } g(x, y)=2 f(x, y)$$
$$g(x, y)=-f(x, y) \quad \text { (d) } g(x, y)=2-f(x, y)$$

Check back soon!

Problem 66

Describe how the graph of $g$ is obtained from the graph of $f .$
(a) $g(x, y)=f(x-2, y) \quad$ (b) $g(x, y)=f(x, y+2)$
(c)$g(x, y)=f(x+3, y-4)$

Check back soon!

Problem 66

Describe how the graph of $g$ is obtained from the graph of $f .$
(a) $g(x, y)=f(x-2, y) \quad$ (b) $g(x, y)=f(x, y+2)$
(c)$g(x, y)=f(x+3, y-4)$

Check back soon!

Problem 67

Use a computer to graph the function using various domains and viewpoints. Get a printout that gives a good view of the "peaks and valleys." Would you say the function has a maximum value? Can you identify any points on the graph that you might consider to be "local maximum points"? What about "local minimum points"?
$$f(x, y)=3 x-x^{4}-4 y^{2}-10 x y$$

Check back soon!

Problem 68

Use a computer to graph the function using various domains and viewpoints. Get a printout that gives a good view of the "peaks and valleys." Would you say the function has a maximum value? Can you identify any points on the graph that you might consider to be "local maximum points"?What about "local minimum points"?
$$f(x, y)=x y e^{-x^{2}-y^{2}}$$

Check back soon!

Problem 69

Use a computer to graph the function using various domains and viewpoints. Comment on the limiting behavior of the function. What happens as both $x$ and $y$ become large? What happens as $(x, y)$ approaches the origin?
$$f(x, y)=\frac{x+y}{x^{2}+y^{2}}$$

Check back soon!

Problem 70

Use a computer to graph the function using various domains and viewpoints. Comment on the limiting behavior of the function. What happens as both $x$ and $y$ become large? What happens as $(x, y)$ approaches the origin?
$$f(x, y)=\frac{x y}{x^{2}+y^{2}}$$

Check back soon!

Problem 71

Use a computer to investigate the family of functions $f(x, y)=e^{c x^{2}+y^{2}} .$ How does the shape of the graph depend on $c ?$

Check back soon!

Problem 72

Use a computer to investigate the family of surfaces
$$z=\left(a x^{2}+b y^{2}\right) e^{-x^{2}-y^{2}}$$
How does the shape of the graph depend on the numbers a and $b ?$

Check back soon!

Problem 73

Use a computer to investigate the family of surfaces $z=x^{2}+y^{2}+c x y .$ In particular, you should determine the transitional values of $c$ for which the surface changes from one type of quadric surface to another.

Check back soon!

Problem 74

Graph the functions
Use a computer to investigate the family of surfaces $z=x^{2}+y^{2}+c x y .$ In particular, you should determine the transitional values of $c$ for which the surface changes from one type of quadric surface to another.
$$f(x, y)=\sqrt{x^{2}+y^{2}} \quad f(x, y)=e^{\sqrt{x^{2}+y^{2}}}$$
$$f(x, y)=\ln \sqrt{x^{2}+y^{2}} \quad f(x, y)=\sin \left(\sqrt{x^{2}+y^{2}}\right)$$
and $$f(x, y)=\frac{1}{\sqrt{x^{2}+y^{2}}}$$
In general, if $g$ is a function of one variable, how is the graph of
$$f(x, y)=g\left(\sqrt{x^{2}+y^{2}}\right)$$
obtained from the graph of $g ?$

Check back soon!

Problem 75

(a) Show that, by taking logarithms, the general Cobb- Douglas function $P=b L^{\alpha} K^{1-\alpha}$ can be expressed as
$$\ln \frac{P}{K}=\ln b+\alpha \ln \frac{L}{K}$$
(b) If we let $x=\ln (L / K)$ and $y=\ln (P / K),$ the equation in part (a) becomes the linear equation $y=\alpha X+\ln b .$ Use Table 2 (in Example 3 ) to make a table of values of $\ln (L / K)$ and $\ln (P / K)$ for the years $1899-1922 .$ Then use a graphing calculator or computer to find the least squares
regression line through the points $(\ln (L / K), \ln (P / K))$
(c) Deduce that the Cobb-Douglas production function is $P=1.01 L^{0.75} K^{0.5}$

Check back soon!