Optimus Prime
The Pirate said "rdr matey"
Double the Trouble
I'm partial to the Giants
The Dodgers suck, but Calculus applications rock!
100
The local maximum and minimum values and saddle point(s) of the function
f(x,y)=x^2+xy+y^2+y
f(1/3,-2/3) = -1/3
is a local minimum.
100
Describe the region R below using polar coordinates:
2<=r<=3
pi/2<=theta<=(3pi)/2
100
Evaluate the iterated integral:
int_0^1 int_0^y xe^(y^3) dx dy
1/6(e-1)
100
Find the first partial derivatives of the function
f(x,y)=x/y
f_x(x,y)=1/y
f_y(x,y)=-x/y^2
100
Find the equation of the tangent plane to the surface
z=e^(x-y)
at the point (2,2,1).
z=1+1(x-2)-1(y-2)
z=x-y+1
200
The local maximum and minimum values and saddle point(s) of the function
f(x,y)=2-x^4+2x^2-y^2
(0,0,2) is a saddle point.
f(1,0) = 3 and f(-1,0) = 3 are local maxima.
200
Sketch the region whose area is given by the integral and evaluate the integral.
int_(pi/4)^((3pi)/4) int_1^2 r dr d theta
200
Set up integrals for both orders of integration where
int int_D y^2 e^(xy) dA
and D is bounded by
y=x, y=4,x=0
int_0^4 int_x^4 y^2 e^(xy) dy dx
or
int_0^4 int_0^y y^2 e^(xy) dx dy
200
Find the first partial derivatives of the function
f(x,y,z)=x^3yz^2+2yz
f_x(x,y,z)=3x^2yz^2
f_y(x,y,z)=x^3z^2+2z
f_z(x,y,z)=2x^3yz+2y
200
Find the maximum rate of change of the function
f(x,y)=sin(xy)
at the point (1,0) and the direction in which it occurs.
The maximum rate of change is
|gradf(1,0)|=1
in the direction of
gradf(1,0) = <<0,1>>
300
The marketing department of a company has determined that if it spends x thousands of dollars on radio advertisements and y thousands of dollars on newspaper advertisements, the company's revenue (in thousands of dollars) will be
R(x,y)=-0.07x^2-100y^2+4x+5y+2xy
How much money should this company spend of radio advertisements and newspaper advertisements to maximize its revenue?
$33,750 on radio advertising and $362.50 on newspaper advertising will maximize revenue.
300
Set up the integral
int int_D x^2ydA
by changing to polar coordinates, where D is the top half of the disk with center origin and radius 5.
int_0^pi int_0^5 r^4 cos^2thetasinthetadrd theta
300
Evaluate the double integral
int int_D (x^2+2y)dA
, where D is bound by
y=x,y=x^3, x>=0
23/84
300
If
z=x^4+x^2y,
x=s+2t-u,
y=stu^2
Find
(delz)/(dels)
when s = 4, t = 2, and u = 1
1,582
300
Find the equation of the tangent plane to the surface
xy+yz+zx=5
at the point (1,2,1).
3(x-1)+2(y-2)+3(z-1)=0
3x+2y+3z=10
400
Use Lagrange multipliers to find teh extreme values of the function
f(x,y,z)=2x+2y+z
subject to the constraint
x^2+y^2+z^2=9
The maximum value is f(2,2,1) = 9.
The minimum value is f(-2,-2,-1) = -9
400
Use polar coordinates set-up a double integral to find the volume of the solid that lies below the cone
z=sqrt(x^2+y^2)
and above the ring
1<=x^2+y^2<=4
int_o^(2pi) int_1^2 r^2 dr d theta
400
Sketch the region of integration and change the order of integration.
int_0^2 int_(x^2)^4 f(x,y) dy dx
int_0^4 int_0^sqrt(y) f(x,y) dx dy
400
Find the gradient of
f(x,y,z)=y^2e^(xyz)
gradf(x,y,z)=<<y^3ze^(xyz),(xy^2z+2y)e^(xyz),xy^3e^(xyz)>>
500
Find the dimensions of a rectangular box of maximum volume such that the sum of the lengths of its 12 edges is 36 inches.
3 in x 3 in x 3 in
500
Convert the integral to polar coordinates:
int_0^3int_-sqrt(9-x^2)^sqrt(9-x^2)e^(x^2+y^2)dydx
int_(-pi/2)^(pi/2)int_0^3e^(r^2)r dr d theta
500
Find the volume of the solid that lies under the plane
3x+2y-z=0
and above the region enclosed by the parabolas
y=x^2
and
x=y^2
3/4
500
Find the directional derivative of the function
g(u,v)=u^2e^-v
at the point (3,0) in the direction of the vector
vecv=<<3,4>>
-18/5
500
A manufacturer has modeled its yearly production function P (the value of its entire production, in millions of dollars) as a Cobb-Douglass function:
P(L,K)=1.47L^0.65K^0.35
where L is the number of labor hours (in thousands) and K is the invested capital (in millions of dollars). Suppose that when L = 30 and K = 8, the labor force is decreasing at a rate of 2,000 labor hours per year, and capital is increasing at a rate of $500,000 per year. Find the rate of change of production.
Production is decreasing at a rate of about $596,000 per year.