MTH16_Lec-08_sec_7-5_LaGrange_Multipliers.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics §7.5 LaGrange Multipliers

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

MTH16_Lec-08_sec_7-5_LaGrange_Multipliers.pptx 3 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

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 Constrained Domain LINE

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 Constrained Range LINE

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

MTH16_Lec-08_sec_7-5_LaGrange_Multipliers.pptx 7 Bruce Mayer, PE Chabot College Mathematics Lagrange Multiplier Method

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

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:

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:

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)

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

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:

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

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 Zero Specify x ≠ 0  Use the above result in eqn (2)  Solving the Above

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:

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)

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

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

MTH16_Lec-08_sec_7-5_LaGrange_Multipliers.pptx 20 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §7.5 P → Constant Elasticity of Substitution (CES) Production Function

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 Joseph Louis Lagrange

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 Joseph Louis Lagrange

MTH16_Lec-08_sec_7-5_LaGrange_Multipliers.pptx 23 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics Appendix –

MTH16_Lec-08_sec_7-5_LaGrange_Multipliers.pptx 24 Bruce Mayer, PE Chabot College Mathematics

MTH16_Lec-08_sec_7-5_LaGrange_Multipliers.pptx 25 Bruce Mayer, PE Chabot College Mathematics

MTH16_Lec-08_sec_7-5_LaGrange_Multipliers.pptx 26 Bruce Mayer, PE Chabot College Mathematics

MTH16_Lec-08_sec_7-5_LaGrange_Multipliers.pptx 27 Bruce Mayer, PE Chabot College Mathematics

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)

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)