Constrained Optimization in Two Variables
|
|
- Shauna Wiggins
- 5 years ago
- Views:
Transcription
1 in Two Variables James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 17, 216
2 Outline 1 2 What Does the Lagrange Multiplier Mean?
3 Let s look at the problem of finding the minimum or maimum of a function f (, y) subject to the constraint g(, y) = c where c is a constant. Let s suppose we have a point (, y ) where an etremum value occurs and assume f and g are differentiable at that point. Then, at another point (, y) which satisfies the constraint, we have g(, y) = g(, y ) + g (, y )( ) + g y (, y )(y y ) + E g (, y,, y ) But g(, y) = g(, y ) = c, so we have = g ( ) + g y (y y ) + E g (, y,, y ) where we let g = g (, y ) and g y = g y (, y ). Now given an, we assume we can find a y values so that g(, y) = c in B r () for some r >. Of course, this value need not be unique. Let φ() be the function defined by φ() = {y g(, y) = c} For eample, if g(, y) = c was the function 2 + y 2 = 25, then φ() = ± 25 2 for 5 5.
4 Clearly, we can t find a full circle B r () when = 5 or = 5, so let s assume the point (, y ) where the etremum value occurs does have such a local circle around it where we can find corresponding y values for all values in B r ( ). So locally, we have a function φ() defined around so that g(, φ()) = c for B r ( ). Then we have = g ( ) + g y (φ() φ( )) + E g (, φ(),, φ( )) Now divide through by to get ( ) φ() = g + gy φ( ) + E g (, φ(),, φ( )) Thus, g y ( ) φ() φ( ) = g E g (, φ(),, φ( )) and assuming g and g y, we can solve to find
5 φ() φ( ) = g g y 1 g y E g (, φ(),, φ( )) Eg (,φ(),,φ()) Since g is differentiable at (, φ( )), as,. Thus, we have φ is differentiable and therefore also continuous with φ ( ) = lim φ() φ( ) = g g y Now since both g and gy, the fraction g g too. This y means locally φ() is either strictly increasing or strictly decreasing; i.e. is a strictly monotone function. Let s assume φ ( ) > and so φ is increasing on some interval ( R, + R). Let s also assume the etreme value is a minimum, so we know on this interval f (, y) f (, y ) with g(, y) = c. This means
6 f (, φ()) f (, φ( ) on ( R, + R). Now do an epansion for f to get f (, φ()) = f (, φ( )) + f ( ) + f y (φ() φ( )) + E f (, φ(),, φ( )) where f = f (, φ( )) and fy = f y (, φ( )). This implies f (, φ()) f (, φ( )) = f ( ) + f y (φ() φ( )) + E f (, φ(),, φ( )) Now we have assumed f (, φ()) f (, φ( )), so we have f ( ) + f y (φ() φ( )) + E f (, φ(),, φ( )) If > we find ( ) φ() f + fy φ( ) + E f (, φ(),, φ( )) Now take the limit as + to find f + f y φ ( )
7 When <, we argue similarly to find Combining, we have f + f y φ ( ) f + f y φ ( ) which tells us at this etreme value we have the equation f + fy φ ( ) =. This implies fy = f φ ( ). Net note [ ] [ ] f f { f ( )} = = g y = f [ ] g f y f φ ( ) This says there is a scalar λ = f g f so that (f )(, y ) = λ (g)(, y ) where g(, y ) = c. The value f g is called the Lagrange Multiplier for this etremal Problem. This is the basis for the Lagrange Multiplier Technique for a constrained optimization problem. g g g y
8 The value λ above is called a Lagrange Multiplier of the etremum problem we are trying to solve. We can do a similar sort of analysis in the case the etremum is a maimum too. Our analysis assumes that the the point (, y ) where the etremum occurs is like an interior point in the {(, y) g(, y) = c}. That is, we assume for such an there is a interval B r ( ) with any B r ( ) having a corresponding value y = φ() so that g(, φ()) = c. So the argument does not handle boundary points such as the ±5 is our previous eample. To implement the Lagrange Multiplier technique, we define a new function H(, y, λ) = f (, y) + λ(g(, y) c) The critical points we see are the ones where the gradient of f is a multiple of the gradient of g for the reasons we discussed above. Note the λ we use here is the negative of the Lagrange Multiplier.
9 To find them, we set the partials of H equal to zero. H H y H λ = = f = λ g = = f y = λ g = = g(, y) = c The first two lines are the statement that the gradient of f is a multiple of the gradient of g and the third line is the statement that the constraint must be satisfied. As usual, solving these three nonlinear equations gives the interior point solutions and we always must also include the boundary points that solve the constraints as well.
10 Eample Etremize 2 + 3y subject to 2 + y 2 = 25. Solution Set H(, y, λ) = (2 + 3y) + λ( 2 + y 2 25). The critical point equations are then H H y H λ = 2 + λ(2) = = 3 + λ(2y) = = 2 + y 2 25 =
11 Solution Solving for λ, we find λ = 1/ = 3/(2y). Thus, = 2y/3. Using this relationship in the constraint, we find (2y/3) 2 + y 2 = 25 or 13y 2 /9 = 25. Thus, y 2 = 225/13 or y = ±15/ 13). This means = ±1/ 13. We find f (1/ 13, 15/ 13) = 45/ 13 = f ( 1/ 13, 15/ 13) = 45/ 13 = f ( 5, ) = 1 f (5, ) = 1 f (, 5) = 15 f (, 5) = 15. So the etreme values here occur at the boundary points (, ±5).
12 Eample Etremize sin(y) subject to 2 + y 2 = 1. Solution Set H(, y, λ) = (sin(y)) + λ( 2 + y 2 1). The critical point equations are then H H y H λ = y cos(y) + λ(2) = = cos(y) + λ(2y) = = 2 + y 2 1 =
13 Solution Solving for λ in the first equation, we find λ = y cos(y)/2. Now use this in the second equation to find cos(y) 2y 2 cos(y)/2 =. Thus, cos(y)( y 2 /) =. The first case is cos(y) =. This implies y = π/2 + 2nπ. These are rectangular hyperbolas and the closest one to the origin is y = π/2 = 1.57 whose closest point to the origin is ( π/2, π/2) = (1, 25, 1, 25). This point is outside of the constraint set, so this case does not matter. The second case is 2 = y 2 which, using the constraint, implies 2 = 1/2. Thus there are four possibilities: (1/ 2, 1/ 2), (1/ 2, 1/ 2), ( 1/ 2, 1/ 2) and ( 1/ 2, 1/ 2). The other possibilities are the circle s boundary points (, ±1) and (±1, ). We find sin(1/ 2, 1/ 2) = sin( 1/ 2, 1/ 2) = sin(1/2) =.48 sin(1/ 2, 1/ 2) = sin( 1/ 2, 1/ 2) = sin(1/2) =.48 sin(±1, ) =, sin(, ±1) = So the etreme values occur here at the interior points of the constraint.
14 What Does the Lagrange Multiplier Mean? Let s change our etremum problem by assuming the constraint constant is changing as a function of ; i.e. the problem is now find the etreme values of f (, y) subject to g(, y) = C(). We can use our tangent plane analysis like before. We assume the etreme value for constraint value C( ) occurs at an interior point of the constraint. We have Then = g + g y ( φ() φ( ) g(, φ()) C() = ) + ( Eg (, φ(),, φ( ) ) We assume the constraint bound function C is differentiable and so letting, we find g + g y φ ( ) = C ( ) ( ) C() C( ) Now assume ( 1, φ( 1 )) etremizes f for the constraint bound value C( 1 ).
15 What Does the Lagrange Multiplier Mean? We have f ( 1, φ( 1 )) f (, φ( )) = f ( 1 ) + f y (φ( 1 ) φ( )) +E f ( 1, φ( 1 ),, φ( )) or ( ) f (1, φ( 1 )) f (, φ( )) 1 ( ) = f + fy φ(1 ) φ( ) 1 ( ) Ef ( 1, φ( 1 ),, φ( )) + 1 Now assuming C() locally around, we can write ( )( ) f (1, φ( 1 )) f (, φ( )) C(1 ) C( ) = C( 1 ) C( ) 1 ( ) ( ) f + fy φ(1 ) φ( ) Ef ( 1, φ( 1 ),, φ( )) + 1 1
16 What Does the Lagrange Multiplier Mean? Now let Θ() be defined by Θ() = f (, φ()) when (, φ()) is the etreme value for the problem of etremizing f (, y) subject to g(, y) = C(). This is called the Optimal Value Function for this problem. Rewriting, we have ( )( ) Θ(1 ) Θ( ) C(1 ) C( ) = C( 1 ) C( ) 1 ( f + fy φ(1 ) φ( ) 1 ) + ( ) Ef ( 1, φ( 1 ),, φ( )) 1 Now let 1. We obtain ( ) Θ(1 ) Θ( ) lim C ( ) 1 C( 1 ) C( ) = f + fy φ ( ) Thus, the rate of change of the optimal value with respect to change in the constraint bound, dθ dc is well - defined and where C is the value C( ). dθ dc (C ) C ( ) = f + f y φ ( )
17 What Does the Lagrange Multiplier Mean? Assuming C ( ), we have dθ dc (C ) = f C ( ) + f y C ( ) φ ( ) But we know at the etreme value for that f y dθ dc (C ) = = f g f C ( ) + g y 1 C ( ) g f C ( ) φ ( ) = f ( g + g y φ ( ) ) = g y g f. Thus, ( C 1 + g ) y ( ) g φ ( ) But the Lagrange Multiplier λ for etremal value for is λ = f g. So ( ) dθ 1 dc (C ) = λ g + gy φ ( ) C ( ) But C ( ) = g + g y φ ( ). Hence, we find dθ dc (C ) = λ.
18 What Does the Lagrange Multiplier Mean? We see from our argument that the Lagrange Multiplier λ is the rate of change of optimal value with respect to the constraint bound. There are cases: sign(λ ) = sign( f g ) = + + = + which implies the optimal value goes up if the constraint bound is perturbed. sign(λ ) = sign( f g ) = = + which implies the optimal value goes up if the constrained bound is perturbed. sign(λ ) = sign( f g ) = + = which implies the optimal value goes down if the constrained bound is perturbed. sign(λ ) = sign( f g ) = + = which implies the optimal value goes down if the constrained bound is perturbed. Hence, we can interpret the Lagrange Multiplier as a price: the change in optimal value due to a change in constraint is essentially the loss of value eperienced due to the constraint change. For eample, if λ is very small, it says the rate of change of the optimal value with respect to constraint modification is small too. This implies the optimal value is insensitive to constraint modification.
19 What Does the Lagrange Multiplier Mean? Homework Find the etreme values of sin(y) subject to 2 + 2y 2 = 1 using the Lagrange Multiplier Technique Find the etreme values of cos(y) subject to 2 + 2y 2 = 1 using the Lagrange Multiplier Technique If a n 1 and b n 3, use an ɛ N proof to show 5a n /b n 5/ If a n 1 and b n 2, use an ɛ N proof to show 5a n b n 1.
Constrained Optimization in Two Variables
Constrained Optimization in Two Variables James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 17, 216 Outline Constrained Optimization
More information12.10 Lagrange Multipliers
.0 Lagrange Multipliers In the last two sections we were often solving problems involving maimizing or minimizing a function f subject to a 'constraint' equation g. For eample, we minimized the cost of
More informationThe First Derivative and Second Derivative Test
The First Derivative and Second Derivative Test James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 8, 2017 Outline Extremal Values The
More informationDerivatives in 2D. Outline. James K. Peterson. November 9, Derivatives in 2D! Chain Rule
Derivatives in 2D James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 9, 2016 Outline Derivatives in 2D! Chain Rule Let s go back to
More informationThe First Derivative and Second Derivative Test
The First Derivative and Second Derivative Test James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University April 9, 2018 Outline 1 Extremal Values 2
More informationOptimization Methods: Optimization using Calculus - Equality constraints 1. Module 2 Lecture Notes 4
Optimization Methods: Optimization using Calculus - Equality constraints Module Lecture Notes 4 Optimization of Functions of Multiple Variables subect to Equality Constraints Introduction In the previous
More informationGeometric Series and the Ratio and Root Test
Geometric Series and the Ratio and Root Test James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 5, 2018 Outline 1 Geometric Series
More informationCalculus of Variation An Introduction To Isoperimetric Problems
Calculus of Variation An Introduction To Isoperimetric Problems Kevin Wang The University of Sydney SSP Working Seminars, MATH2916 May 4, 2013 Contents I Lagrange Multipliers 2 1 Single Constraint Lagrange
More informationMatrix Solutions to Linear Systems of ODEs
Matrix Solutions to Linear Systems of ODEs James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 3, 216 Outline 1 Symmetric Systems of
More informationGeometric Series and the Ratio and Root Test
Geometric Series and the Ratio and Root Test James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 5, 2017 Outline Geometric Series The
More informationConsequences of Continuity
Consequences of Continuity James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 4, 2017 Outline 1 Domains of Continuous Functions 2 The
More informationConvergence of Fourier Series
MATH 454: Analysis Two James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University April, 8 MATH 454: Analysis Two Outline The Cos Family MATH 454: Analysis
More informationConvergence of Sequences
James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 5, 2018 Outline 1 2 Homework Definition Let (a n ) n k be a sequence of real numbers.
More informationCable Convergence. James K. Peterson. May 7, Department of Biological Sciences and Department of Mathematical Sciences Clemson University
Cable Convergence James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University May 7, 2018 Outline 1 Fourier Series Convergence Redux 2 Fourier Series
More information32. Method of Lagrange Multipliers
32. Method of Lagrange Multipliers The Method of Lagrange Multipliers is a generalized approach to solving constrained optimization problems. Assume that we are seeking to optimize a function z = f(, y)
More informationA Basic Course in Real Analysis Prof. P. D. Srivastava Department of Mathematics Indian Institute of Technology, Kharagpur
A Basic Course in Real Analysis Prof. P. D. Srivastava Department of Mathematics Indian Institute of Technology, Kharagpur Lecture - 36 Application of MVT, Darbou Theorem, L Hospital Rule (Refer Slide
More informationMultivariable Calculus and Matrix Algebra-Summer 2017
Multivariable Calculus and Matrix Algebra-Summer 017 Homework 4 Solutions Note that the solutions below are for the latest version of the problems posted. For those of you who worked on an earlier version
More informationConvergence of Sequences
Convergence of Sequences James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University February 12, 2018 Outline Convergence of Sequences Definition Let
More informationConsequences of Continuity
Consequences of Continuity James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 4, 2017 Outline Domains of Continuous Functions The Intermediate
More informationFourier Sin and Cos Series and Least Squares Convergence
Fourier and east Squares Convergence James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University May 7, 28 Outline et s look at the original Fourier sin
More informationMatrices and Vectors
Matrices and Vectors James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 11, 2013 Outline 1 Matrices and Vectors 2 Vector Details 3 Matrix
More informationand ( x, y) in a domain D R a unique real number denoted x y and b) = x y = {(, ) + 36} that is all points inside and on
Mat 7 Calculus III Updated on 10/4/07 Dr. Firoz Chapter 14 Partial Derivatives Section 14.1 Functions o Several Variables Deinition: A unction o two variables is a rule that assigns to each ordered pair
More informationMore Series Convergence
More Series Convergence James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University December 4, 218 Outline Convergence Analysis for Fourier Series Revisited
More informationI know this time s homework is hard... So I ll do as much detail as I can.
I know this time s homework is hard... So I ll do as much detail as I can. Theorem (Corollary 7). If f () g () in (a, b), then f g is a constant function on (a, b). Proof. Call h() f() g(). Then, we try
More informationMath 212-Lecture Interior critical points of functions of two variables
Math 212-Lecture 24 13.10. Interior critical points of functions of two variables Previously, we have concluded that if f has derivatives, all interior local min or local max should be critical points.
More informationUniform Convergence Examples
Uniform Convergence Examples James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 13, 2017 Outline 1 Example Let (x n ) be the sequence
More informationReview: critical point or equivalently f a,
Review: a b f f a b f a b critical point or equivalentl f a, b A point, is called a of if,, 0 A local ma or local min must be a critical point (but not conversel) 0 D iscriminant (or Hessian) f f D f f
More informationSolutions of Math 53 Midterm Exam I
Solutions of Math 53 Midterm Exam I Problem 1: (1) [8 points] Draw a direction field for the given differential equation y 0 = t + y. (2) [8 points] Based on the direction field, determine the behavior
More informationf x, y x 2 y 2 2x 6y 14. Then
SECTION 11.7 MAXIMUM AND MINIMUM VALUES 645 absolute minimum FIGURE 1 local maimum local minimum absolute maimum Look at the hills and valles in the graph of f shown in Figure 1. There are two points a,
More informationProject One: C Bump functions
Project One: C Bump functions James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 2, 2018 Outline 1 2 The Project Let s recall what the
More informationSolutions to old Exam 3 problems
Solutions to old Exam 3 problems Hi students! I am putting this version of my review for the Final exam review here on the web site, place and time to be announced. Enjoy!! Best, Bill Meeks PS. There are
More informationDirchlet s Function and Limit and Continuity Arguments
Dirchlet s Function and Limit and Continuity Arguments James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University February 23, 2018 Outline 1 Dirichlet
More information4.3 How derivatives affect the shape of a graph. The first derivative test and the second derivative test.
Chapter 4: Applications of Differentiation In this chapter we will cover: 41 Maimum and minimum values The critical points method for finding etrema 43 How derivatives affect the shape of a graph The first
More informationMat 267 Engineering Calculus III Updated on 9/19/2010
Chapter 11 Partial Derivatives Section 11.1 Functions o Several Variables Deinition: A unction o two variables is a rule that assigns to each ordered pair o real numbers (, ) in a set D a unique real number
More informationDifferentiation - Important Theorems
Differentiation - Important Theorems Philippe B Laval KSU Spring 2012 Philippe B Laval (KSU) Differentiation - Important Theorems Spring 2012 1 / 10 Introduction We study several important theorems related
More informationDirectional derivatives and gradient vectors (Sect. 14.5). Directional derivative of functions of two variables.
Directional derivatives and gradient vectors (Sect. 14.5). Directional derivative of functions of two variables. Partial derivatives and directional derivatives. Directional derivative of functions of
More informationCHAPTER 1-2: SHADOW PRICES
Essential Microeconomics -- CHAPTER -: SHADOW PRICES An intuitive approach: profit maimizing firm with a fied supply of an input Shadow prices 5 Concave maimization problem 7 Constraint qualifications
More informationFunction Composition and Chain Rules
Function Composition and s James K. Peterson Department of Biological Sciences and Department of Matematical Sciences Clemson University Marc 8, 2017 Outline 1 Function Composition and Continuity 2 Function
More informationReview sheet Final Exam Math 140 Calculus I Fall 2015 UMass Boston
Review sheet Final Eam Math Calculus I Fall 5 UMass Boston The eam is closed tetbook NO CALCULATORS OR ELECTRONIC DEVICES ARE ALLOWED DURING THE EXAM The final eam will contain problems of types similar
More informationLower semicontinuous and Convex Functions
Lower semicontinuous and Convex Functions James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 6, 2017 Outline Lower Semicontinuous Functions
More information3.1 ANALYSIS OF FUNCTIONS I INCREASE, DECREASE, AND CONCAVITY
MATH00 (Calculus).1 ANALYSIS OF FUNCTIONS I INCREASE, DECREASE, AND CONCAVITY Name Group No. KEYWORD: increasing, decreasing, constant, concave up, concave down, and inflection point Eample 1. Match the
More informationThis theorem guarantees solutions to many problems you will encounter. exists, then f ( c)
Maimum and Minimum Values Etreme Value Theorem If f () is continuous on the closed interval [a, b], then f () achieves both a global (absolute) maimum and global minimum at some numbers c and d in [a,
More information4.3 - How Derivatives Affect the Shape of a Graph
4.3 - How Derivatives Affect the Shape of a Graph 1. Increasing and Decreasing Functions Definition: A function f is (strictly) increasing on an interval I if for every 1, in I with 1, f 1 f. A function
More informationUniform Convergence Examples
Uniform Convergence Examples James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 13, 2017 Outline More Uniform Convergence Examples Example
More informationFirst Variation of a Functional
First Variation of a Functional The derivative of a function being zero is a necessary condition for the etremum of that function in ordinary calculus. Let us now consider the equivalent of a derivative
More informationPRACTICE PROBLEMS FOR MIDTERM I
Problem. Find the limits or explain why they do not exist (i) lim x,y 0 x +y 6 x 6 +y ; (ii) lim x,y,z 0 x 6 +y 6 +z 6 x +y +z. (iii) lim x,y 0 sin(x +y ) x +y Problem. PRACTICE PROBLEMS FOR MIDTERM I
More informationHOMEWORK SOLUTIONS MATH 1910 Sections 6.1, 6.2, 6.3 Fall 2016
HOMEWORK SOLUTIONS MATH 191 Sections.1,.,. Fall 1 Problem.1.19 Find the area of the shaded region. SOLUTION. The equation of the line passing through ( π, is given by y 1() = π, and the equation of the
More informationPhysics 6010, Fall 2016 Constraints and Lagrange Multipliers. Relevant Sections in Text:
Physics 6010, Fall 2016 Constraints and Lagrange Multipliers. Relevant Sections in Text: 1.3 1.6 Constraints Often times we consider dynamical systems which are defined using some kind of restrictions
More informationMath 210, Final Exam, Spring 2012 Problem 1 Solution. (a) Find an equation of the plane passing through the tips of u, v, and w.
Math, Final Exam, Spring Problem Solution. Consider three position vectors (tails are the origin): u,, v 4,, w,, (a) Find an equation of the plane passing through the tips of u, v, and w. (b) Find an equation
More informationDirchlet s Function and Limit and Continuity Arguments
Dirchlet s Function and Limit and Continuity Arguments James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 2, 2018 Outline Dirichlet
More informationA. Correct! These are the corresponding rectangular coordinates.
Precalculus - Problem Drill 20: Polar Coordinates No. 1 of 10 1. Find the rectangular coordinates given the point (0, π) in polar (A) (0, 0) (B) (2, 0) (C) (0, 2) (D) (2, 2) (E) (0, -2) A. Correct! These
More informationThe Limit Inferior and Limit Superior of a Sequence
The Limit Inferior and Limit Superior of a Sequence James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University February 13, 2018 Outline The Limit Inferior
More informationConstrained Maxima and Minima EXAMPLE 1 Finding a Minimum with Constraint
1038 Chapter 14: Partial Derivatives 14.8 Lagrange Multipliers HISTORICAL BIOGRAPHY Joseph Louis Lagrange (1736 1813) Sometimes we need to find the etreme values of a function whose domain is constrained
More information1 Exponential Functions Limit Derivative Integral... 5
Contents Eponential Functions 3. Limit................................................. 3. Derivative.............................................. 4.3 Integral................................................
More informationSTATIC LECTURE 4: CONSTRAINED OPTIMIZATION II - KUHN TUCKER THEORY
STATIC LECTURE 4: CONSTRAINED OPTIMIZATION II - KUHN TUCKER THEORY UNIVERSITY OF MARYLAND: ECON 600 1. Some Eamples 1 A general problem that arises countless times in economics takes the form: (Verbally):
More informationFunction Composition and Chain Rules
Function Composition an Cain Rules James K. Peterson Department of Biological Sciences an Department of Matematical Sciences Clemson University November 2, 2018 Outline Function Composition an Continuity
More informationMath Maximum and Minimum Values, I
Math 213 - Maximum and Minimum Values, I Peter A. Perry University of Kentucky October 8, 218 Homework Re-read section 14.7, pp. 959 965; read carefully pp. 965 967 Begin homework on section 14.7, problems
More informationThe Kuhn-Tucker and Envelope Theorems
The Kuhn-Tucker and Envelope Theorems Peter Ireland EC720.01 - Math for Economists Boston College, Department of Economics Fall 2010 The Kuhn-Tucker and envelope theorems can be used to characterize the
More informationDerivatives and the Product Rule
Derivatives and the Product Rule James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University January 28, 2014 Outline 1 Differentiability 2 Simple Derivatives
More informationMath Lagrange Multipliers
Math 213 - Lagrange Multipliers Peter A. Perr Universit of Kentuck October 12, 2018 Homework Re-read section 14.8 Begin practice homework on section 14.8, problems 3-11 (odd), 15, 21, 23 Begin (or continue!)
More informationPredator - Prey Model Trajectories are periodic
Predator - Prey Model Trajectories are periodic James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 4, 2013 Outline 1 Showing The PP
More information3.3.1 Linear functions yet again and dot product In 2D, a homogenous linear scalar function takes the general form:
3.3 Gradient Vector and Jacobian Matri 3 3.3 Gradient Vector and Jacobian Matri Overview: Differentiable functions have a local linear approimation. Near a given point, local changes are determined by
More information2.10 Saddles, Nodes, Foci and Centers
2.10 Saddles, Nodes, Foci and Centers In Section 1.5, a linear system (1 where x R 2 was said to have a saddle, node, focus or center at the origin if its phase portrait was linearly equivalent to one
More informationMATHEMATICS 200 April 2010 Final Exam Solutions
MATHEMATICS April Final Eam Solutions. (a) A surface z(, y) is defined by zy y + ln(yz). (i) Compute z, z y (ii) Evaluate z and z y in terms of, y, z. at (, y, z) (,, /). (b) A surface z f(, y) has derivatives
More informationMATH section 3.1 Maximum and Minimum Values Page 1 of 7
MATH section. Maimum and Minimum Values Page of 7 Definition : Let c be a number in the domain D of a function f. Then c ) is the Absolute maimum value of f on D if ) c f() for all in D. Absolute minimum
More information6.5 Trigonometric Equations
6. Trigonometric Equations In this section, we discuss conditional trigonometric equations, that is, equations involving trigonometric functions that are satisfied only by some values of the variable (or
More informationMcGill University April Calculus 3. Tuesday April 29, 2014 Solutions
McGill University April 4 Faculty of Science Final Examination Calculus 3 Math Tuesday April 9, 4 Solutions Problem (6 points) Let r(t) = (t, cos t, sin t). i. Find the velocity r (t) and the acceleration
More informationSolving systems of ODEs with Matlab
Solving systems of ODEs with Matlab James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 20, 2013 Outline 1 Systems of ODEs 2 Setting Up
More informationReview for the First Midterm Exam
Review for the First Midterm Exam Thomas Morrell 5 pm, Sunday, 4 April 9 B9 Van Vleck Hall For the purpose of creating questions for this review session, I did not make an effort to make any of the numbers
More informationMaximum and Minimum Values - 3.3
Maimum and Minimum Values - 3.3. Critical Numbers Definition A point c in the domain of f is called a critical number offiff c or f c is not defined. Eample a. The graph of f is given below. Find all possible
More informationTaylor Polynomials. James K. Peterson. Department of Biological Sciences and Department of Mathematical Sciences Clemson University
James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 24, 2013 Outline 1 First Order Approximation s Second Order Approximations 2 Approximation
More informationMATH 228: Calculus III (FALL 2016) Sample Problems for FINAL EXAM SOLUTIONS
MATH 228: Calculus III (FALL 216) Sample Problems for FINAL EXAM SOLUTIONS MATH 228 Page 2 Problem 1. (2pts) Evaluate the line integral C xy dx + (x + y) dy along the parabola y x2 from ( 1, 1) to (2,
More informationMathematics 205 HWK 18 Solutions Section 15.3 p725
Mathematics 205 HWK 18 Solutions Section 15.3 p725 Problem 3, 15.3, p725. Use Lagrange multipliers to find the maximum and minimum values, if they exist, of f(x, y) = 3x 2y subject to the constraint x
More informationGeneral Power Series
General Power Series James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University March 29, 2018 Outline Power Series Consequences With all these preliminaries
More informationCALCULUS 4 QUIZ #3 REVIEW Part 2 / SPRING 09
CACUUS QUIZ #3 REVIEW Part / SPRING 09 (.) Determine the following about maima & minima of functions of variables. (a.) Complete the square for f( ) = + and locate all absolute maima & minima.. ( ) ( )
More informationReview Test 2. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. D) ds dt = 4t3 sec 2 t -
Review Test MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the derivative. ) = 7 + 0 sec ) A) = - 7 + 0 tan B) = - 7-0 csc C) = 7-0 sec tan
More informationMath 171 Calculus I Spring, 2019 Practice Questions for Exam II 1
Math 171 Calculus I Spring, 2019 Practice Questions for Eam II 1 You can check your answers in WebWork. Full solutions in WW available Sunday evening. Problem 1. Find the average rate of change of the
More informationAssignment 16 Assigned Weds Oct 11
Assignment 6 Assigned Weds Oct Section 8, Problem 3 a, a 3, a 3 5, a 4 7 Section 8, Problem 4 a, a 3, a 3, a 4 3 Section 8, Problem 9 a, a, a 3, a 4 4, a 5 8, a 6 6, a 7 3, a 8 64, a 9 8, a 0 56 Section
More informationExercises of Mathematical analysis II
Eercises of Mathematical analysis II In eercises. - 8. represent the domain of the function by the inequalities and make a sketch showing the domain in y-plane.. z = y.. z = arcsin y + + ln y. 3. z = sin
More informationSections 5.1: Areas and Distances
Sections.: Areas and Distances In this section we shall consider problems closely related to the problems we considered at the beginning of the semester (the tangent and velocity problems). Specifically,
More informationis the intuition: the derivative tells us the change in output y (from f(b)) in response to a change of input x at x = b.
Uses of differentials to estimate errors. Recall the derivative notation df d is the intuition: the derivative tells us the change in output y (from f(b)) in response to a change of input at = b. Eamples.
More information2017 HSC Mathematics Extension 2 Marking Guidelines
07 HSC Mathematics Etension Marking Guidelines Section I Multiple-choice Answer Key Question Answer C B 3 D 4 C 5 B 6 A 7 A 8 B 9 C 0 B NESA 07 HSC Mathematics Etension Sample Answers Section II Question
More informationM151B Practice Problems for Exam 1
M151B Practice Problems for Eam 1 Calculators will not be allowed on the eam. Unjustified answers will not receive credit. 1. Compute each of the following its: 1a. 1b. 1c. 1d. 1e. 1 3 4. 3. sin 7 0. +
More information4.3 Mean-Value Theorem and Monotonicity
.3 Mean-Value Theorem and Monotonicit 1. Mean Value Theorem Theorem: Suppose that f is continuous on the interval a, b and differentiable on the interval a, b. Then there eists a number c in a, b such
More information(d by dx notation aka Leibniz notation)
n Prerequisites: Differentiating, sin and cos ; sum/difference and chain rules; finding ma./min.; finding tangents to curves; finding stationary points and their nature; optimising a function. Maths Applications:
More informationPredator - Prey Model Trajectories are periodic
Predator - Prey Model Trajectories are periodic James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 4, 2013 Outline Showing The PP Trajectories
More informationIntro to Nonlinear Optimization
Intro to Nonlinear Optimization We now rela the proportionality and additivity assumptions of LP What are the challenges of nonlinear programs NLP s? Objectives and constraints can use any function: ma
More informationPartial Derivatives. w = f(x, y, z).
Partial Derivatives 1 Functions of Several Variables So far we have focused our attention of functions of one variable. These functions model situations in which a variable depends on another independent
More information17 Solution of Nonlinear Systems
17 Solution of Nonlinear Systems We now discuss the solution of systems of nonlinear equations. An important ingredient will be the multivariate Taylor theorem. Theorem 17.1 Let D = {x 1, x 2,..., x m
More informationPreface.
This document was written and copyrighted by Paul Dawkins. Use of this document and its online version is governed by the Terms and Conditions of Use located at. The online version of this document is
More informationThe Explicit Form of a Function
Section 3 5 Implicit Differentiation The Eplicit Form of a Function The normal way we see function notation has f () on one sie of an equation an an epression in terms of on the other sie. We know the
More informationDIFFERENTIATION. 3.1 Approximate Value and Error (page 151)
CHAPTER APPLICATIONS OF DIFFERENTIATION.1 Approimate Value and Error (page 151) f '( lim 0 f ( f ( f ( f ( f '( or f ( f ( f '( f ( f ( f '( (.) f ( f '( (.) where f ( f ( f ( Eample.1 (page 15): Find
More informationLESSON 25: LAGRANGE MULTIPLIERS OCTOBER 30, 2017
LESSON 5: LAGRANGE MULTIPLIERS OCTOBER 30, 017 Lagrange multipliers is another method of finding minima and maxima of functions of more than one variable. In fact, many of the problems from the last homework
More informationHOMEWORK 7 SOLUTIONS
HOMEWORK 7 SOLUTIONS MA11: ADVANCED CALCULUS, HILARY 17 (1) Using the method of Lagrange multipliers, find the largest and smallest values of the function f(x, y) xy on the ellipse x + y 1. Solution: The
More informationMath Midterm Solutions
Math 50 - Midterm Solutions November 4, 009. a) If f ) > 0 for all in a, b), then the graph of f is concave upward on a, b). If f ) < 0 for all in a, b), then the graph of f is downward on a, b). This
More informationPredator - Prey Model Trajectories and the nonlinear conservation law
Predator - Prey Model Trajectories and the nonlinear conservation law James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 28, 2013 Outline
More informationIt is convenient to introduce some notation for this type of problems. I will write this as. max u (x 1 ; x 2 ) subj. to. p 1 x 1 + p 2 x 2 m ;
4 Calculus Review 4.1 The Utility Maimization Problem As a motivating eample, consider the problem facing a consumer that needs to allocate a given budget over two commodities sold at (linear) prices p
More informationComplex Differentials and the Stokes, Goursat and Cauchy Theorems
Complex Differentials and the Stokes, Goursat and Cauchy Theorems Benjamin McKay June 21, 2001 1 Stokes theorem Theorem 1 (Stokes) f(x, y) dx + g(x, y) dy = U ( g y f ) dx dy x where U is a region of the
More informationUniversity Mathematics 2
University Matematics 2 1 Differentiability In tis section, we discuss te differentiability of functions. Definition 1.1 Differentiable function). Let f) be a function. We say tat f is differentiable at
More informationProofs Not Based On POMI
s Not Based On POMI James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University February 12, 2018 Outline 1 Non POMI Based s 2 Some Contradiction s 3
More information