Exercises

Find the domains of the following functions. Sketch a graph of the domain, and then determine whether is it open or closed, bounded or unbounded, connected or not connected.
1.
f( x,y) = ( 1-2x+y) 1/2
2.
f(x,y) = ( 1-3x+2y) 1/2
3.
f( x,y) = 

x
+
y
4.
f( x,y) = 

xy - 1
5.
f( x,y) = 

1-x2-y2
x2+y2
6.
f(x,y) = 
x-y
x+y
7.
f( x,y) = 
ln(x2y)
ln(4-x2-y2)
8.
f(x,y) = ln 
x2y
1-x2-y2
9.
f( x,y) = yx
10.
f( x,y) = ytan( x)
11.
f( x,y) =  

x
ln( 1-y)
12.
f( x,y) = ln( 1-x2-y2)
13.
f( x,y) = ln| x2-y2|
14.
f( x,y) = ln| x-y|
15.
f( x,y) = sin-1( x-y)
16.
f(x,y) = sin-1( xy)

Use a computer algebra system to sketch the graph of the given function
17.
f( x,y) = y
18.
f( x,y) = 3x+2y+1
19.
f( x,y) = 9-x2-y2
20.
f( x,y) = 1-xy
21.
f( x,y) = sin( x2+y2)
22.
f( x,y) = sin( x) cos( y)
23.
f( x,y) = 
sin(x)
1+x2
24.
f( x,y) = 
e-y
1+y2
25.
f( x,y) =
x
x2+y2
26.
f(x,y) =
x+y
x2+y2

       

In 27-30, u( x,t)  is the displacement of a vibrating string at a point x in [ 0,p] and at time t. Sketch the graph of the string at each of the given times.
27.
u( x,t) = cos
p
6
sin(x)
28.
u( x,t) = cos( pt) cos( 2x)
t = 0,1,2,3
t = 0,1,2,3
29.
u( x,t) = sin
p
4
cos(2x)
30.
u( x,t) = e-t/5sin
p
6
sin( x)
t = 0,1,2,3
t = 0,1,2,3

       

31. Explain why the set { ( x,y)    |    y ¹ 2x} is the same as
{ ( x,y)   |  y < 2x} È { ( x,y)  |  y > 2x}

32. What is the domain of the function
f( x,y) = 
1-2cosh( xy)
(1)
Does it contain any points at all? Can (1) even be considered a function?

33. The function u( x,t) = sin(x-t) +sin( x+t) models the shape of a vibrating string which is fixed at x = 0 and x = p. Graph u( x,t) over x in [ 0,p] for several different values of t (e.g., t = 0, 1, 2, etc.). What does the vibration of the string look like? How long does it take until it returns to its original shape?

34. The function
u( x,t) = ( x-t) e-( x-t) 2
is an example of a traveling wave. Graph u( x,t) for several different values of t (e.g., t = 0, 1, 2, etc.). Why might we call this a traveling wave?

35. The function u( x,t) = e-tsin2( px) +32 models the temperature in °F of a 1 foot long thin rod in which both ends are held at the freezing point at all times t. Graph u( x,t) over x in [ 0,1] for several different values of t (e.g., t = 0, 1, 2, etc.). What happens to the temperature of the rod as t increases?

36. What happens to the temperature of the 1 foot long thin rod in exercise 35 as t approaches ¥.

37. Suppose that a function of 1 variable, g(x) , has a domain of [ a,b] . If we define a function of 2 variables f( x,y) = g( x) , then what is dom(f) ? Is it bounded? Is it open or closed?

38. Suppose that g( x) has a domain of [ a,b] and that h( y) has a domain of [c,d] . Then what is the domain of f( x,y) = g(x) h( y) ? Is it bounded? Is it open or closed?

39. Write to Learn: Suppose that f( x,y) and g( x,y) are defined everywhere. What is the domain of the function
h( x,y) =
f(x,y)
g(x,y)
Consider the example f( x,y) = x2y and g( x,y) = x-y+1, and include this example in a short essay describing the domain of h.

40. Write to Learn: Suppose that the domain of f(x,y) is dom( f) and the domain of g( x,y) is dom( g) . Write a short essay which explains why if h(x,y) = f( x,y) +g( x,y) , then the domain of h( x,y) is
dom( h) = dom( f) Ç dom( g)