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[email protected] • MTH16_Lec-08_sec_7-5_LaGrange_Multipliers.pptx 1
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
§7.5 LaGrangeMultipliers
[email protected] • MTH16_Lec-08_sec_7-5_LaGrange_Multipliers.pptx 2
Bruce Mayer, PE Chabot College Mathematics
Review §
Any QUESTIONS About• §7.4 → Least Squares Linear
Regression
Any QUESTIONS About HomeWork• §7.4 → HW-07
7.4
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Bruce Mayer, PE Chabot College Mathematics
§7.5 Learning Goals
Study the method of Lagrange multipliers as a procedure for locating points on a graph where constrained optimization can occur
Use the method of Lagrange multipliers in a number of applied problems including utility and allocation of resources
Discuss the significance of the Lagrange multiplier λ
[email protected] • MTH16_Lec-08_sec_7-5_LaGrange_Multipliers.pptx 4
Bruce Mayer, PE Chabot College Mathematics
Lagrange Multipliers Often the Domain of
an Optimization is CONSTRAINED for some Reason; that is,
• k a CONSTANT
The constraint Eqn could be solved for, say y:
In other words, the Constraint fcn describes a LINE in the xy-Plane Domain surface kyxg ,
kxhy ,Constrained Domain LINE
[email protected] • MTH16_Lec-08_sec_7-5_LaGrange_Multipliers.pptx 5
Bruce Mayer, PE Chabot College Mathematics
Lagrange Multipliers The Constrained
DOMAIN Line is then Projected Up or Down by the fcn
Functional projection produces a LINE on the Range Surface
It can be shown than any extremum on the range line must be a C.P. of
yxfz ,
Constrained Range LINE
kyxgyxfyxF ,,,
[email protected] • MTH16_Lec-08_sec_7-5_LaGrange_Multipliers.pptx 6
Bruce Mayer, PE Chabot College Mathematics
Lagrange Multipliers Where λ is a new
independent variable
To Find max/min for F(x,y) take
Solving the 3 eqns:
From the above equations determine the Critical Point (C.P.) Location:
Then
kyxgyxF
y
yxg
y
yxf
y
yxF
x
yxg
x
yxf
x
yxF
,0,
,,0
,
,,0
,
kgy
g
y
f
x
g
x
f
byax
bafz ,minmax
[email protected] • MTH16_Lec-08_sec_7-5_LaGrange_Multipliers.pptx 7
Bruce Mayer, PE Chabot College Mathematics
Lagrange Multiplier Method
[email protected] • MTH16_Lec-08_sec_7-5_LaGrange_Multipliers.pptx 8
Bruce Mayer, PE Chabot College Mathematics
Example Lagrange Multipliers
Use the method of Lagrange multipliers to find the maximum value of
Subject to the Constraint of
yxxyyxfz 22),(
10 yx
[email protected] • MTH16_Lec-08_sec_7-5_LaGrange_Multipliers.pptx 9
Bruce Mayer, PE Chabot College Mathematics
Example Lagrange Multipliers
SOLUTION First find the partial derivatives of f & g:
And set each equal to the Lagrange multiplier, λ, times the partials of the left side of the constraint equation:
11and22 22
y
g
x
gxxy
y
zxyy
x
z
yxgyx ,10
12and12 22 xxyxyy
[email protected] • MTH16_Lec-08_sec_7-5_LaGrange_Multipliers.pptx 10
Bruce Mayer, PE Chabot College Mathematics
Example Lagrange Multipliers
Solving the first two equations for λ:
By the Last Eqn: Now use the Constraint Eqn:
The ONLY Soln to the last eqn:
xyxxyy 2and2 22
xy
xxyxyx 10so1010
5x
[email protected] • MTH16_Lec-08_sec_7-5_LaGrange_Multipliers.pptx 11
Bruce Mayer, PE Chabot College Mathematics
Example Lagrange Multipliers
Recall eqn for y(x):
Thus have Two Critical Points
Check max/min by functional evaluation
Thus the MAX value of 250 occurs at (5,−5)
55 yyxy
5,5and5,5 2,21,1 baba
0)5()5()5(5)5,5( 22 f
250)5()5()5(5)5,5( 22 f
[email protected] • MTH16_Lec-08_sec_7-5_LaGrange_Multipliers.pptx 12
Bruce Mayer, PE Chabot College Mathematics
Example Find 2Var Domain
A seller’s assigned area is the six-mile radius surrounding the center of a city.
History indicates that x miles east and y miles north of city center, his/her sales competition by other businesses is Modeled by
Find• the location(s) for minimum competition • The minimum level of competition
[email protected] • MTH16_Lec-08_sec_7-5_LaGrange_Multipliers.pptx 13
Bruce Mayer, PE Chabot College Mathematics
Example Find 2Var Domain
SOLUTION The constraint for this function is the
circle of radius six miles centered about the middle of the city. Such a circle can be described by the points (x,y) satisfying the equation:
Taking the partials of the competition function find:
222 6 yx
21.0and2.0 xy
Cxy
x
C
[email protected] • MTH16_Lec-08_sec_7-5_LaGrange_Multipliers.pptx 14
Bruce Mayer, PE Chabot College Mathematics
Example Find 2Var Domain
In this case g(x,y) = k →
ReCall the Lagrange Equation:
Then the Lagrange Multiplier Minimum System
kyxyxg 222 6,
kyxgyxfyxF ,,,
336
221.0
122.0
22
2
yxkg
yxy
g
y
f
xxyx
g
x
f
[email protected] • MTH16_Lec-08_sec_7-5_LaGrange_Multipliers.pptx 15
Bruce Mayer, PE Chabot College Mathematics
Example Find 2Var Domain
Using eqn (1) to Solve for y• To prevent
Division by ZeroSpecify x ≠ 0
Use the above result in eqn (2)
SolvingtheAbove
xxy 22.0
1021.021.0 22 xyx
2222 200201.0 xx
210200200222 xx
[email protected] • MTH16_Lec-08_sec_7-5_LaGrange_Multipliers.pptx 16
Bruce Mayer, PE Chabot College Mathematics
Example Find 2Var Domain
Combining this result with the solution for y in terms of λ and the constraint equation to solve for λ:
3610210 22
346.0300
36
[email protected] • MTH16_Lec-08_sec_7-5_LaGrange_Multipliers.pptx 17
Bruce Mayer, PE Chabot College Mathematics
Example Find 2Var Domain
Finally, use the value of λ to determine values of x & y for minimum competition:
Testing the Four (x,y) Pairs find:
Thus the minimum of 1.69 businesses occurs 3.46 miles north and 4.90 miles either east/west of the center of the city
(x,y) (−4.90, −3.46) (−4.90,3.46) (4.90,−3.46) (4.90,3.46)
C(x,y) 18.31 1.69 18.31 1.69
[email protected] • MTH16_Lec-08_sec_7-5_LaGrange_Multipliers.pptx 18
Bruce Mayer, PE Chabot College Mathematics
Lagrange Multiplier as a Rate
Thus λ is a Marginal Rate for the max or min with respect to a change in the constraint value
[email protected] • MTH16_Lec-08_sec_7-5_LaGrange_Multipliers.pptx 19
Bruce Mayer, PE Chabot College Mathematics
Example Lagrange as Rate
In the Previous the minimum value was M=1.69 Businesses, with k = 36 sq-miles
If k increased by 1 sq-mi (in context this would be increasing the radius of the seller’s route), the approximate change in the minimum value:
The min no. of competing businesses would INcrease by about 0.346
346.0dk
dM
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Bruce Mayer, PE Chabot College Mathematics
WhiteBoard Work
Problems From §7.5• P7.5-32 →
ConstantElasticityof Substitution(CES)ProductionFunction
[email protected] • MTH16_Lec-08_sec_7-5_LaGrange_Multipliers.pptx 21
Bruce Mayer, PE Chabot College Mathematics
All Done for Today
Born: 25 January 1736 Died: 10 April 1813 (aged 77)
Professorship • École Polytechnique
Academic advisors• Leonhard Euler• Giovanni Beccaria
Doctoral students• Joseph Fourier• Giovanni Plana• Siméon Poisson
JosephLouis
Lagrange
[email protected] • MTH16_Lec-08_sec_7-5_LaGrange_Multipliers.pptx 22
Bruce Mayer, PE Chabot College Mathematics
All Done for Today
Born: 25 January 1736 Died: 10 April 1813 (aged 77)
Professorship • École Polytechnique
Academic advisors• Leonhard Euler• Giovanni Beccaria
Doctoral students• Joseph Fourier• Giovanni Plana• Siméon Poisson
JosephLouis
Lagrange
[email protected] • MTH16_Lec-08_sec_7-5_LaGrange_Multipliers.pptx 23
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
Appendix
–
srsrsr 22
a2 b2
[email protected] • MTH16_Lec-08_sec_7-5_LaGrange_Multipliers.pptx 24
Bruce Mayer, PE Chabot College Mathematics
[email protected] • MTH16_Lec-08_sec_7-5_LaGrange_Multipliers.pptx 25
Bruce Mayer, PE Chabot College Mathematics
[email protected] • MTH16_Lec-08_sec_7-5_LaGrange_Multipliers.pptx 26
Bruce Mayer, PE Chabot College Mathematics
[email protected] • MTH16_Lec-08_sec_7-5_LaGrange_Multipliers.pptx 27
Bruce Mayer, PE Chabot College Mathematics
[email protected] • MTH16_Lec-08_sec_7-5_LaGrange_Multipliers.pptx 28
Bruce Mayer, PE Chabot College Mathematics
Q := 50*(0.3*K^(-1/5) + 0.7*L^(-1/5))^-5
dQdK = diff(Q, K)
dQdL = diff(Q, L)
K := 140/(5+2*(35/6)^(5/6))
Kn := float(K)
L := K*(35/6)^(5/6)
Ln := float(L)
[email protected] • MTH16_Lec-08_sec_7-5_LaGrange_Multipliers.pptx 29
Bruce Mayer, PE Chabot College Mathematics
Qmax = subs(Q, K = Kn, L = Ln)
Qmax = subs(Q, K = K, L = L)