Exercises

Find fx( x,y)  and fy(x,y)  for each of the following:
1.
f( x,y) = x2+y3
2.
f( x,y) = x2+2xy+y3
3.
f( x,y) = ( x+2y) 2
4.
f(x,y) = ( x2+2y) 2
5.
f( x,y) = xsin( y)
6.
f(x,y) = exln( y2+1)
7.
f( x,y) = exp( -x2-y2)
8.
f( x,y) = tan-1( xy)
9.
f( x,y) = xcos( xy)
10.
f(x,y) = xsin( xy)
11.
f( x,y) = yx
12.
f( x,y) = xy
13.
f( x,y) =  x
x2+y2
14.
f(x,y) = ó
õ 		y

x 
 sin( t2) dt

Find fxx, fxy, fyx, and fyy  for each of the following. Then show that the mixed partials are the same.
15.
f( x,y) = x2+y3
16.
f( x,y) = x2+2xy+y3
17.
f( x,y) = ( x+2y) 2
18.
f(x,y) = ( x2+2y) 2
19.
f( x,y) = xsin( y)
20.
f(x,y) = exln( y2+1)
21.
f( x,y) = xcos( xy)
22.
f(x,y) = tan-1( xy)
23.
f( x,y) = yx
24.
f( x,y) = xy

Find the indicated derivative of the given function:
25.
fxxy  for  f( x,y) = x2+y3
26.
fxxy  for  f( x,y) = x2+2xy+y3
27.
fxyx  for  f( x,y) = ( x+2y) 2
28.
fyxy  for  f( x,y) = ( x2+2y) 2
29.
fxxyy  for  f( x,y) = xsin( y)
30.
fxxxxxxxy  for  f( x,y) = exln( y2+1)
31.
fxxxy  for  f( x,y) = ycos( xy)
32.
fxyy  for  f( x,y) = sin( x) tan(xy)

       

33. A certain vibrating string has a displacement u in cm at time t in seconds and at a distance x in cm from one end which is given by
u( x,t) = 2sin( 120p( x-t) )
How fast (in units of cm per sec) is the string vibrating at a horizontal distance x = 1.2 cm from one end at time t = 2 seconds? At time t = 3 seconds?

34. Suppose that a string is attached at its endpoints x = 0 and x = l, for some number l.
and suppose that y = u( x,t) models the displacement at x in [ 0,l] of the string at time t, where
u( x,t) = Acos( at) sin æ
è 		 p
l
 x ö
ø
with A and a constant.

    1. Show that u( 0,t) = u( l,t) = 0 for all t. How does this relate to u( x,t) being a model of a string?
    2. What is the rate of change of u( x,t) at x = l/2 and for any given time?

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. How fast is the temperature decreasing at the midpoint of the rod when t = 0? When t = 1? When t = 2?

36. The function u( x,y,t) = 2sin( 3x) sin( 4y) cos( 5t) models the displacement u in cm of a vibrating rectangular membrane at time t in seconds and at a point ( x,y) on the membrane. How fast is the displacement of the membrane above the point ( 1,1) changing with respect to time at t = 1 seconds?

37. It can be shown that an ideal gas with fixed mass has an absolute temperature R, a pressure P, and a volume V that satisfies
T = kPV
where k is a constant. How fast does the temperature T change with respect to the volume V?

38. The total resistance R produced by two resistors with resistances R1 and R2, respectively, satisfies
R =  R1R2
R1+R2
What is the rate of change of the total resistance R with respect to the resistance R1?

39. If two planets with masses M and m are located at the points ( x,y,z) and ( 0,0,0) , respectively, then the potential energy of their mutual gravitational attraction is given by
f( x,y,z) = G  Mm
x2+y2+z2



where G is the universal gravitational constant. At what rate is the potential energy changing with respect to x? With respect to y?

40. A Cobb-Douglas production function is a function of the form P = bLaKb where b,a, and b are constants. What is the rate of change of P with respect to L? With respect to P?

41. If fx and fy both exist, how can the limit

lim
h® 0 
  f( x+h,y) -f( x,y+h)
h
be expressed in terms of the 1st partial derivatives of f?

42. Write to Learn: Write a short essay in which you explain why if f( x,y) continuous and third differentiable in each variable at a point ( p,q) , then theorem 3.1 implies that the following are true at every point ( p,q) :
fxyx = fxxy        and        fyxx = fxyx