Math 119 - Week 4

Lagrange Multiplier formula and purpose =

λ = lagrange multiplier. Optimizes a function constrained by a function g(x, y) = k ONLY.

∇f→ = λ∇g→ ONLY IF ∇g→ ≠ 0→

How to find abs max/min with lagrange multipliers =

IF INEQUALITY: Find critical points ∇f→ = 0→ and sub into f(x, y). (Must satisfy constraint)

Solve ∇f→ = λ∇g→ and g(x, y) = k to get points, sub in f(x, y)

Solve ∇g→ = 0→ and g(x, y) = k, sub in f(x, y) [IF THERE EXISTS SOLUTIONS]

If the constraint g(x, y) = k has endpoints, evaluate f(x, y) at these points

max = largest computed f

min = smallest computed f

Average value of a 2D function over a region D =

[∬_D f(x, y) dA] / [∬_D dA]

If D is difficult to integrate whole (∬_D f dA), what to do =

Split up into two non overlapping regions.

∬_D f dA = ∬_D₁ f dA + ∬_D₂ f dA

ONLY IF D₁ ∩ D₂ = ∅ and D = D₁ ∪ D₂

Plane intercept formula =

[x / a] + [y / b] + [z / c] = 1

Line intercept formula =

[x / a] + [y / b] = 1

How to calculate volume under a surface z =

Given z = h(x, y) and region D in the xy-plane

∬_D h(x, y) dx dy

ONLY IF h(x, y) ≥ 0

If m ≤ f(x, y) ≤ M on region D =

m ∬_D dA ≤ ∬_D f(x, y) dA ≤ M ∬_D dA

also: mA ≤ ∬_D f(x, y) dA ≤ MA

how to create height function for volume =

Function/curve on TOP - Function/curve on BOTTOM

ex.) f: 3z = 4 - x² - y²
g: z = x² + y²

f opens DOWN, g opens UP

• h(x, y) = f(x, y) - g(x, y)

  • f is ABOVE g

how to solve lagrange multipliers if there are more variables than equations =

solve in terms of one variable, then sub them into constraint eqn

how to reverse order of integration (double integral)

get intervals, + graph it

for bounds: look for the swapped variable:

• lower function = lower bound

• higher function = higher bound

cos²(x) half angle identity =

(1 + cos2x)/2

sin²(x) half angle identity =

(1 - cos2x)/2