MATH 16B Lecture Notes - Lecture 8: Minimax, Multiple Integral

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21 Apr 2017
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Three-variable equations
Max/min of f(x, y, z) such that g(x, y, z) = 0
F(x, y, z, lambda) = f(x, y, z) + (lambda)g(x, y, z)
Calculate PD(x), PD(y), PD(z), PD(lambda)
Solve all equations when equal to 0
(a, b, c, d) solution --> (a, b, c) potential max/min
Example: Determine the maximum possible volume of a cuboid such that surface area is 6.
Volume = xyz; surface area = 2xy + 2xz + 2yz = 6 (assume max volume exists and x, y, z > 0)
Xyz = f(x, y, z); 2xy + 2xz + 2yz = 6 --> xy + xz + yz -3 = 0 = g(x, y, z)
F(x, y, z, lambda) = xyz + lambda(xy) + lambda(xz) + lambda (yz) - lambda(3)
PD(x) = yz + lambda(y) + lambda(z), PD(y) = xz + lambda(x) + lambda(z), PD(z) =
xy + lambda(x) + lambda(y), PD(lambda) = xy + xz + yz - 3
Yz + lambda(y) + lambda(z) = 0, xz + lambda(x) + lambda(z) = 0, xy + lambda(x) +
lambda(y) = 0, xy + xz + yz - 3 = 0
Lambda = -yz/(y+z), lambda = -xz/(x + z), lambda = -xy/(x + y) -->
-yz/(y + z) = -xz/(x + z) and -xz/(x + z) = -xy/(x + y)
-y/(y + z) = -x/(x + z) and -z/(x + z) = -y/(x + y) --> -y(x + z) = -x(y + z) and -z(x + y) = -y(x + z) --> -yz
= -xz and -xz = -xy -->
-yz = -xz and -xz = -xy --> y = x and y = z
Xy + xz + yz - 3 = 0 --> x^2 + x^2 + x^2 - 3 = 0 --> 3x^2 - 3 = 0 --> x^2 = 1 --> x = 1
Y = 1 and z = 1 and lambda = -1/2
(1, 1, 1, -1/2) is the only solution (x, y, z > 0)
Max must be at (1, 1, 1) so the max volume is 1
Double Integrals
f(x): single-variable function, continuous on [a, b]
Definite integral of f(x) = area above x-axis bounded by y = f(x) between a and b minus
area below x-axis bounded by y = f(x) between a and b
Fundamental Theorem: definite integral of f(x) = F(b) - F(a)
F'(x) = f(x)
Aim: Generalize to 2-variable functions
Replace [a, b] with rectangle R in xy-plane
Replace f(x) with f(x, y) continuous 2-variable function on R
Definition: Double integral of f(x, y) on R = volume above xy-plane bounded by z
= f(x, y) and R minus volume below xy-plane bounded by z = f(x, y) and R
How do we calculate the double integral of f(x, y) on R?
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