Optimization & Lagrange
Find the critical points of the function
f(x,y) = x^2 - 4xy + y^2 + 6y + 2
The critical point is
(1, -4)
Evaluate the iterated integral: int_0^2 int_1^3 6x^2 y \ dy \ dx
It is a sphere. Spherical form: rho = 4 cos(phi).
Evaluate the iterated triple integral: int_0^1 int_0^2 int_0^3 xyz \ dz \ dy \ dx
9/2
Convert Cartesian coordinates (-sqrt(3), 1, 5) to cylindrical coordinates (r, theta, z).
(2, (5pi)/6, 5)
Find the gradient vector field grad f of the scalar function f(x,y,z) = x^2 y sin(z).
grad f = << 2xy sin(z), x^2 sin(z), x^2 y cos(z) >>
Use the Second Derivative Test to classify the critical point. At critical point (a,b),
f_{xx} = 4, f_{yy} = 2, f_{xy} = 3
It is a saddle point.
D = f_{xx}f_{yy} - (f_{xy})^2 = 8 - 9 = -1 < 0
Sketch the region and reverse the order of integration: int_0^4 int_{sqrt(x)}^2 f(x,y) \ dy \ dx
int_0^2 int_0^{y^2} f(x,y) \ dx \ dy
Set up the triple integral to find the volume of the tetrahedron bounded by the coordinate planes and the plane 2x + y + z = 4.
int_0^2 int_0^{4-2x} int_0^{4-2x-y} dz \ dy \ dx
Identify the surface and write its equation in spherical coordinates: x^2 + y^2 + z^2 = 4z.
It is a sphere. Spherical form: rho = 4 cos(phi).
Sketch the 2D vector field bb F(x,y) = << y, -x >> by plotting at least four representative vectors.
Vectors rotate clockwise. E.g.: at (1,0): <<0,-1>>; at (0,1): <<1,0>>; at (-1,0): <<0,1>>; at (0,-1): <<-1,0>>.
Set up (do NOT solve) the Lagrange multiplier system for: maximize/minimize f(x,y,z) = 2x + 3y - z subject to the constraint x^2 + y^2 + z^2 = 16.
2 = lambda(2x), 3 = lambda(2y), -1 = lambda(2z), and x^2 + y^2 + z^2 = 16
Evaluate int int_D (x+y) \ dA, where D is bounded by the parabola y = x^2 and the line y = x.
3/8
Express as an iterated triple integral: the volume of the solid bounded by y = x^2, z=0, z=3, and y=4.
int_{-2}^2 int_{x^2}^4 int_0^3 dz \ dy \ dx
Set up in cylindrical coordinates to evaluate int int int_E x^2 \ dV, where E lies within x^2+y^2=4 between z=0 and z=5.
int_0^{2pi} int_0^2 int_0^5 r^2 cos^2(theta) \ r \ dz \ dr \ d theta
Evaluate the line integral int_C x \ ds, where C is the line segment from (0,0) to (3,4).
15/2
Find the absolute maximum and minimum values of f(x,y) = x^2 + 2y^2 - 4x on the closed region D = {(x,y) | 0 <= x <= 3, 0 <= y <= 2}.
Abs. max = 7 at (3,2) and (0,2); Abs. min = -4 at (2,0).
Convert to polar coordinates (do NOT evaluate): int_0^3 int_0^{sqrt(9-x^2)} (x^2+y^2)^{3//2} \ dy \ dx
int_0^{pi//2} int_0^3 r^4 \ dr \ d theta
Evaluate int int int_E z \ dV, where E is bounded by x=0, y=0, z=0, and x+y+z=2.
1/6
Set up in spherical coordinates: int int int_E (x^2+y^2+z^2) \ dV, where E is the unit ball x^2+y^2+z^2 <= 1.
int_0^{2pi} int_0^{pi} int_0^1 rho^2 \ rho^2 sin(phi) \ d rho \ d phi \ d theta
Evaluate int_C bb F * d bb r, where bb F(x,y) = << xy, x^2 >> and C is given by bb r(t) = << t, t^2 >> for 0 <= t <= 1.
5/12
Use Lagrange multipliers to find the max and min of f(x,y) = x^2 y subject to x^2 + 2y^2 = 6.
Max = 4 at (+-2, 1); Min = -4 at (+-2, -1).
Use polar coordinates to find the volume of the solid bounded by the paraboloid z = 18 - 2x^2 - 2y^2 and the xy-plane.
V = 81 pi
Rewrite int_0^1 int_0^{1-x} int_0^{1-x-y} f(x,y,z) \ dz \ dy \ dx in the order dy \ dx \ dz.
int_0^1 int_0^{1-z} int_0^{1-x-z} f \ dy \ dx \ dz
Use spherical coordinates to set up the volume of the solid above the cone z = sqrt(x^2+y^2) and below the sphere x^2+y^2+z^2 = 8.
int_0^{2pi} int_0^{pi//4} int_0^{2sqrt(2)} rho^2 sin(phi) \ d rho \ d phi \ d theta
Evaluate the line integral int_C x^2 z \ ds, where C is the line segment in 3D from (0,6,-1) to (4,1,5).
(1984 sqrt(77))/21 * 1/77 = (1984)/(21 sqrt(77)). Parameterize C: x=4t, y=6-5t, z=-1+6t, |r'| = sqrt(77), then integrate from 0 to 1.