Constrained Optimization: The Method of Lagrange Multipliers:


 Lambert Mosley
 2 years ago
 Views:
Transcription
1 Constrained Optimization: The Method of Lagrange Multipliers: Suppose the equation p(x,) x 60x 7 00 models profit when x represents the number of handmade chairs and is the number of handmade rockers produced per week. The optimal (maximum) situation occurs when x = 5 and =. However due to an insufficient labor force the can onl make a total of 0 chairs and rockers per week ( x + = 0). So how man chairs and how man rockers will give the realistic maximum profit? We will come back to this questions shortl but first we will look at the following example. Using Level Curves and the constraint function to determine optimal points: A Compan has a determined that its production function is the CobbDouglas function f (x,) x where x is the number of labor hours and is the number of capitol units. The budget constraint for the compan is given b 00x a) If the compan decides to spend $00, 000 on x then how much can be spent on under this budget constraint? solution: If $00000 is spent on x then 00x = and x = 000. There is $00000 left to spend on therefore 00 = and = 000. In this case total production will be f(000, 000) = (000) (000) = (08)(0) = 080 units. Suppose the amount spent on x is changed to $50,000. How will that change production? Following the same procedure as above x = 500 and = 500. The production f(500, 500) then be 89 units. Is there a wa to determine values for x and that will give the optimal production possible given this budget constraint? we will first tr to find the optimal situation using level curves with the constraint function. It turns out that the global maximum or global minimum occurs where the graph of the constraint equation is tangent to one of the level curves of the original function. In order to plot the level curves we must solve for to get f (x,) > C x x where C now represents values of f(x,). Now we will plot for various values of C. (The choice of values for C is determined mostl b common sense.) In this case we
2 will let C = 000, C = 000 and C = 000. After looking at the resulting level curves and constraint, I decide to add another value C = 00. Now I can make a good guess After considering the graph below, we can guess the optimal value will occur for x = 600 and = 400. optim al solution 5000 C=000 C=000 C=00 C=00 constraint x There is also an algebraic approach to finding the optimal solution given a certain constraint. We will use the method of Lagrange Multipliers to find the maximum situation in the problem above. In the proceeding sections ou have learned how to use partial derivatives to find the optimal situation for a multivariable equation. Now we will expand on this process to find the optimal situation with a constraint. Below is an outline of the process we will use followed examples. Procedure for Appling the Method of Lagrange Multipliers: In order to maximize or minimize the function f(x,) which is subject to the constraint g(x, ) = k we will follow the following procedure. Step # First create the LaGrange Function. This function is composed of the function to be optimized combined with the constraint function in the following wa: L(x, ) = f(x, ) [g(x, )  k]
3 Step # Now find the partial derivative with respect to each variable x, and the Lagrange multiplier of the function shown: L(x, ) = f(x, ) [g(x, )  k] Step # Set each of the partial derivatives equal to zero to get L x = 0, L = 0 and L = 0 Using L x = 0, L = 0, proceed to solve for x and solve for in terms of. Now substitute the solutions for x and so that L = 0 is in terms of onl. Now solve for and use this value to find the optimal values x and. Note: If M is the max or min value of f(x, ) subject to the constraint g(x, ) = k, then the Lagrange multiplier is the rate of change in M with respect to k. (i.e. = dm/dk and therefore approximates the change in M resulting in a one unit increase in k.) Example We will revisit the CobbDouglas function f (x,) x and the budget constraint function 00x We will use the method of Lagrange multiplier described above to find the optimal solutions. First we will create the Lagrange equation L(x,) x (00x Now find L x = 0, L = 0 and L = 0. L x x 00 L x 00 L 00x Using L x = 0, L = 0 to solve for x and Set the first derivative L x = 0 to get and solve for = solve for again to get = x 00 x 00 and then set L = 0 and We now need to use the results to turn the equation L = 0 into an equation of one variable onl. To do this we will set the equations equal to each other and solve for as shown below.
4 x 00 = x 00 > x 00 = x 00 > multipl both sides b 00 and then b to get x x x x x x Now substitute the values of into L = 0 to get 00x 00( x) and solve for x = and then find to be =.. The value of x and that will maximize production is x = 666 and = and the maximum production is 6 units. Example : We will now revisit the profit equation introduced at the beginning of this section. p(x,) x 60x 7 00 where x represents the number of handmade chairs and is the number of handmade rockers produced per week. The constraint on this profit is g(x, ) = x + = 0. In order to find the optimal situation given this constraint we will follow the method given above. First we will create the LaGrange equation L(x, ) = p(x, ) [g(x, ) k] = ( x 60x 7 00) (x 0). Now find L x = 0, L = 0 and L = 0. L x 4 x 60 0 L L x 0 0 Using L x = 0, L = 0 to solve for x and to get x 5 4 and 6. Now substitute the values of x and into L = 0 to get L x 0 (5 4 ) ( 0 ) 0 ( ) and = 6.8 so x and Therefore the maximum profit will be achieved when 0 chairs and 9 rockers are produced. ( Remember that the maximum profit is x = 5 and = when there is not constraint.)
5 Notice that the value of = 6.8 and if the constraint was increased from 0 to then the maximum profit would increase b $6.80 Example : Amanda is getting a new dog. She wants to build a pen for her dog in the back ard. The pen will be rectangular using 00 feet of fence. Amanda plans to build the pen up against the wall of her house so that she will onl need three sides of fence. She wants to build a pen with the maximum amount of area. Therefore we need to maximize the area equation A = x. Since Amanda has a constraint in the amount of fencing she can use, we will use the LaGrange method and create the following equation. L(x,) x (x 00) Now find L x = 0, L = 0 and solve for x and to get that x = and =. Find L = 0 and substitute in the values for x and to put the equation in terms of. Now solve to get that = 50. Therefore the optimal values are x = 50 and = 00 feet of fencing. (Notice that if the constant 00 feet of fencing is increased b on foot to 0 feet then b marginal analsis the maximum value for the area will increase b 50 square feet. Determine what the maximum area would be for 0 feet of fencing) Answer is Example 4 The function U = f(x,) represent the utilit or customer satisfaction derived b a consumer from the consumption of a certain amount of product x and a certain amount of product. In the example below the maximum utilit is determined when the given budget constraint affects the amount of each product produced. The function f(x,) x is a utilit function with a budget constraint given to be g(x,) = x + 4 = 40. The first step is to create the LaGrange equation L(x,) x (x 4 40) L x L x x 0 and therefore = x L L x 4 0 and therefore = x The next step is to find L x and then use the partials above to help transform L into an equation with just one variable. It does not matter if it is terms of x onl, onl or onl ( as in the two examples above)
6 In this case it is easier to set the equations equal to each other and solve for either x or as seen below. set x x x x x Going back and substitute into L x 4 40 x 4( x ) 40 0 and solve to get x = 0 and then =5. ( You will also get = x = (0)(5) = 50. What is the meaning of this information? It means that if the constant is increased to 4 the maximum utilit will increase b 50 ) Practice Problems Problem ) Find the minimum value of the function f(x, ) = x +  x subject to the constraint x + =. Problem The highwa Department is planning to build a picnic area for motorist along a major highwa. It is rectangular with area of 5,000 square ards. It is to be fenced off on the three sides not adjacent to the highwa. What is the least amount of fencing needed to complete the job? Problem Using the information given in problem graph the level curves and the constraint function and indicate where the minimum point will occur. Problem 4 Two women have decided to start a cottage industr that will be run out of their homes. The are going to make two tpes of specialt shirts, rhinestone and embroider. The total gross profit is given b P(x, ) = 6x x  where x is the number of rhinestone shirts made and is the number of embroider shirts made per month. Together, the are able to make at most a total of six shirts per month. What is the number of each tpe of shirt that the should tr to make and sell in order to maximize their profit each month.? Problem 5 If the equation f(x,, z) = x + + z is subject to the constraint x  + z = 6, find the optimal point under these conditions.
7 Problem 5 A) An editor has been allotted $60000 to spend on the development and promotion of a new book. It is estimated that if x thousand dollars is spent on development and thousand on promotion, approximatel f(x, ) = 0x copies of the book will be sold. How much mone should the editor allocate to development and how much to promotion in order to maximize sales? C) Graph the level curves and the constraint function and determine the maximum point. B) Suppose the editor s allocation is increased to $6000 to spend on the development and promotion of the new book. Estimate how the additional $000 will affect the maximum sales level. Problem 6 A farmer wishes to fence off a rectangular pasture along the bank of a river. The area of the pasture is to be,00 square meters and no fencing is needed along the bank. Find the dimensions of the pasture that will require the least amount of fencing. Problem 7 A CobbDouglas production function for new compan is given b f (x,) 50x 5 5 where x represents the units of labor and represents the units of capital. Suppose units of Labor and capital cost $00 and $00 each respectivel. If the budget constraint is $0,000, find the maximum production level for this manufacture.
4 Constrained Optimization: The Method of Lagrange Multipliers. Chapter 7 Section 4 Constrained Optimization: The Method of Lagrange Multipliers 551
Chapter 7 Section 4 Constrained Optimization: The Method of Lagrange Multipliers 551 LEVEL CURVES 2 7 2 45. f(, ) ln 46. f(, ) 6 2 12 4 16 3 47. f(, ) 2 4 4 2 (11 18) 48. Sometimes ou can classif the critical
More informationName Date. BreakEven Analysis
Name Date BreakEven Analsis In our business planning so far, have ou ever asked the questions: How much do I have to sell to reach m gross profit goal? What price should I charge to cover m costs and
More informationSolving Quadratic Equations by Graphing. Consider an equation of the form. y ax 2 bx c a 0. In an equation of the form
SECTION 11.3 Solving Quadratic Equations b Graphing 11.3 OBJECTIVES 1. Find an ais of smmetr 2. Find a verte 3. Graph a parabola 4. Solve quadratic equations b graphing 5. Solve an application involving
More information1.2 GRAPHS OF EQUATIONS
000_00.qd /5/05 : AM Page SECTION. Graphs of Equations. GRAPHS OF EQUATIONS Sketch graphs of equations b hand. Find the  and intercepts of graphs of equations. Write the standard forms of equations of
More information3 Optimizing Functions of Two Variables. Chapter 7 Section 3 Optimizing Functions of Two Variables 533
Chapter 7 Section 3 Optimizing Functions of Two Variables 533 (b) Read about the principle of diminishing returns in an economics tet. Then write a paragraph discussing the economic factors that might
More informationA Utility Maximization Example
A Utilit Maximization Example Charlie Gibbons Universit of California, Berkele September 17, 2007 Since we couldn t finish the utilit maximization problem in section, here it is solved from the beginning.
More informationEconomics 326: Marshallian Demand and Comparative Statics. Ethan Kaplan
Economics 326: Marshallian Demand and Comparative Statics Ethan Kaplan September 17, 2012 Outline 1. Utility Maximization: General Formulation 2. Marshallian Demand 3. Homogeneity of Degree Zero of Marshallian
More informationEconomic Principles Solutions to Problem Set 1
Economic Principles Solutions to Problem Set 1 Question 1. Let < be represented b u : R n +! R. Prove that u (x) is strictl quasiconcave if and onl if < is strictl convex. If part: ( strict convexit of
More informationA couple of Max Min Examples. 3.6 # 2) Find the maximum possible area of a rectangle of perimeter 200m.
A couple of Max Min Examples 3.6 # 2) Find the maximum possible area of a rectangle of perimeter 200m. Solution : As is the case with all of these problems, we need to find a function to minimize or maximize
More informationSection 3.1 Quadratic Functions and Models
Section 3.1 Quadratic Functions and Models DEFINITION: A quadratic function is a function f of the form fx) = ax 2 +bx+c where a,b, and c are real numbers and a 0. Graphing Quadratic Functions Using the
More information(a) We have x = 3 + 2t, y = 2 t, z = 6 so solving for t we get the symmetric equations. x 3 2. = 2 y, z = 6. t 2 2t + 1 = 0,
Name: Solutions to Practice Final. Consider the line r(t) = 3 + t, t, 6. (a) Find symmetric equations for this line. (b) Find the point where the first line r(t) intersects the surface z = x + y. (a) We
More informationName: ID: Discussion Section:
Math 28 Midterm 3 Spring 2009 Name: ID: Discussion Section: This exam consists of 6 questions: 4 multiple choice questions worth 5 points each 2 handgraded questions worth a total of 30 points. INSTRUCTIONS:
More informationconstraint. Let us penalize ourselves for making the constraint too big. We end up with a
Chapter 4 Constrained Optimization 4.1 Equality Constraints (Lagrangians) Suppose we have a problem: Maximize 5, (x 1, 2) 2, 2(x 2, 1) 2 subject to x 1 +4x 2 =3 If we ignore the constraint, we get the
More informationThe Graph of a Linear Equation
4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two variables 2. Graph a line from three points 3. Graph a line b the intercept method 4. Graph a line that
More informationEconomics 2020a / HBS 4010 / HKS API111 FALL 2010 Solutions to Practice Problems for Lectures 1 to 4
Economics 00a / HBS 4010 / HKS API111 FALL 010 Solutions to Practice Problems for Lectures 1 to 4 1.1. Quantity Discounts and the Budget Constraint (a) The only distinction between the budget line with
More informationProfit and Revenue Maximization
WSG7 7/7/03 4:36 PM Page 95 7 Profit and Revenue Maximization OVERVIEW The purpose of this chapter is to develop a general framework for finding optimal solutions to managerial decisionmaking problems.
More informationSolving Systems of Equations
Solving Sstems of Equations When we have or more equations and or more unknowns, we use a sstem of equations to find the solution. Definition: A solution of a sstem of equations is an ordered pair that
More informationLecture 4: Equality Constrained Optimization. Tianxi Wang
Lecture 4: Equality Constrained Optimization Tianxi Wang wangt@essex.ac.uk 2.1 Lagrange Multiplier Technique (a) Classical Programming max f(x 1, x 2,..., x n ) objective function where x 1, x 2,..., x
More informationCONSUMER PREFERENCES THE THEORY OF THE CONSUMER
CONSUMER PREFERENCES The underlying foundation of demand, therefore, is a model of how consumers behave. The individual consumer has a set of preferences and values whose determination are outside the
More information8 Polynomials Worksheet
8 Polynomials Worksheet Concepts: Quadratic Functions The Definition of a Quadratic Function Graphs of Quadratic Functions  Parabolas Vertex Absolute Maximum or Absolute Minimum Transforming the Graph
More informationSECTION 2.2. Distance and Midpoint Formulas; Circles
SECTION. Objectives. Find the distance between two points.. Find the midpoint of a line segment.. Write the standard form of a circle s equation.. Give the center and radius of a circle whose equation
More information6. The given function is only drawn for x > 0. Complete the function for x < 0 with the following conditions:
Precalculus Worksheet 1. Da 1 1. The relation described b the set of points {(, 5 ),( 0, 5 ),(,8 ),(, 9) } is NOT a function. Eplain wh. For questions  4, use the graph at the right.. Eplain wh the graph
More informationChapter 6 Quadratic Functions
Chapter 6 Quadratic Functions Determine the characteristics of quadratic functions Sketch Quadratics Solve problems modelled b Quadratics 6.1Quadratic Functions A quadratic function is of the form where
More informationZeros of Polynomial Functions. The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra. zero in the complex number system.
_.qd /7/ 9:6 AM Page 69 Section. Zeros of Polnomial Functions 69. Zeros of Polnomial Functions What ou should learn Use the Fundamental Theorem of Algebra to determine the number of zeros of polnomial
More informationWe start with the basic operations on polynomials, that is adding, subtracting, and multiplying.
R. Polnomials In this section we want to review all that we know about polnomials. We start with the basic operations on polnomials, that is adding, subtracting, and multipling. Recall, to add subtract
More information4.1 Ordinal versus cardinal utility
Microeconomics I. Antonio Zabalza. Universit of Valencia 1 Micro I. Lesson 4. Utilit In the previous lesson we have developed a method to rank consistentl all bundles in the (,) space and we have introduced
More informationTo Be or Not To Be a Linear Equation: That Is the Question
To Be or Not To Be a Linear Equation: That Is the Question Linear Equation in Two Variables A linear equation in two variables is an equation that can be written in the form A + B C where A and B are not
More informationSECTION 51 Exponential Functions
354 5 Eponential and Logarithmic Functions Most of the functions we have considered so far have been polnomial and rational functions, with a few others involving roots or powers of polnomial or rational
More informationProt Maximization and Cost Minimization
Simon Fraser University Prof. Karaivanov Department of Economics Econ 0 COST MINIMIZATION Prot Maximization and Cost Minimization Remember that the rm's problem is maximizing prots by choosing the optimal
More informationMA107 Precalculus Algebra Exam 2 Review Solutions
MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write
More informationALGEBRA I (Common Core)
The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA I (Common Core) Wednesday, June 17, 2015 1:15 to 4:15 p.m. MODEL RESPONSE SET Table of Contents Question 25..................
More information53 Polynomial Functions. not in one variable because there are two variables, x. and y
y. 53 Polynomial Functions State the degree and leading coefficient of each polynomial in one variable. If it is not a polynomial in one variable, explain why. 1. 11x 6 5x 5 + 4x 2 coefficient of the
More informationDeriving MRS from Utility Function, Budget Constraints, and Interior Solution of Optimization
Utilit Function, Deriving MRS. Principles of Microeconomics, Fall ChiaHui Chen September, Lecture Deriving MRS from Utilit Function, Budget Constraints, and Interior Solution of Optimization Outline.
More informationMath 120 Final Exam Practice Problems, Form: A
Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,
More information2.7 Applications of Derivatives to Business
80 CHAPTER 2 Applications of the Derivative 2.7 Applications of Derivatives to Business and Economics Cost = C() In recent ears, economic decision making has become more and more mathematicall oriented.
More information2.4 Inequalities with Absolute Value and Quadratic Functions
08 Linear and Quadratic Functions. Inequalities with Absolute Value and Quadratic Functions In this section, not onl do we develop techniques for solving various classes of inequalities analticall, we
More informationSOLUTIONS. f x = 6x 2 6xy 24x, f y = 3x 2 6y. To find the critical points, we solve
SOLUTIONS Problem. Find the critical points of the function f(x, y = 2x 3 3x 2 y 2x 2 3y 2 and determine their type i.e. local min/local max/saddle point. Are there any global min/max? Partial derivatives
More informationSystems of Equations. from Campus to Careers Fashion Designer
Sstems of Equations from Campus to Careers Fashion Designer Radius Images/Alam. Solving Sstems of Equations b Graphing. Solving Sstems of Equations Algebraicall. Problem Solving Using Sstems of Two Equations.
More informationMATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60
MATH REVIEW SHEETS BEGINNING ALGEBRA MATH 60 A Summar of Concepts Needed to be Successful in Mathematics The following sheets list the ke concepts which are taught in the specified math course. The sheets
More informationLESSON EIII.E EXPONENTS AND LOGARITHMS
LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS OVERVIEW Here s what ou ll learn in this lesson: Eponential Functions a. Graphing eponential functions b. Applications of eponential
More informationSection 9: Applied Optimization
Chapter The Derivative Applied Calculus 148 Section 9: Applied Optimization We have used derivatives to help find the maximums and minimums of some functions given by equations, but it is very unlikely
More informationLinear Inequality in Two Variables
90 (7) Chapter 7 Sstems of Linear Equations and Inequalities In this section 7.4 GRAPHING LINEAR INEQUALITIES IN TWO VARIABLES You studied linear equations and inequalities in one variable in Chapter.
More informationThe variable λ is a dummy variable called a Lagrange multiplier ; we only really care about the values of x, y, and z.
Math a Lagrange Multipliers Spring, 009 The method of Lagrange multipliers allows us to maximize or minimize functions with the constraint that we only consider points on a certain surface To find critical
More informationALGEBRA I (Common Core)
The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA I (Common Core) Wednesday, August 13, 2014 8:30 a.m. MODEL RESPONSE SET Table of Contents Question 25...................
More informationMath 53 Worksheet Solutions Minmax and Lagrange
Math 5 Worksheet Solutions Minmax and Lagrange. Find the local maximum and minimum values as well as the saddle point(s) of the function f(x, y) = e y (y x ). Solution. First we calculate the partial
More informationSolving Quadratic Equations
9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation
More informationLearning Objectives for Section 1.2 Graphs and Lines. Linear Equations in Two Variables. Linear Equations
Learning Objectives for Section 1.2 Graphs and Lines After this lecture and the assigned homework, ou should be able to calculate the slope of a line. identif and work with the Cartesian coordinate sstem.
More informationAnalyzing the Graph of a Function
SECTION A Summar of Curve Sketching 09 0 00 Section 0 0 00 0 Different viewing windows for the graph of f 5 7 0 Figure 5 A Summar of Curve Sketching Analze and sketch the graph of a function Analzing the
More informationax 2 by 2 cxy dx ey f 0 The Distance Formula The distance d between two points (x 1, y 1 ) and (x 2, y 2 ) is given by d (x 2 x 1 )
SECTION 1. The Circle 1. OBJECTIVES The second conic section we look at is the circle. The circle can be described b using the standard form for a conic section, 1. Identif the graph of an equation as
More informationConstrained optimization.
ams/econ 11b supplementary notes ucsc Constrained optimization. c 2010, Yonatan Katznelson 1. Constraints In many of the optimization problems that arise in economics, there are restrictions on the values
More informationP1. Plot the following points on the real. P2. Determine which of the following are solutions
Section 1.5 Rectangular Coordinates and Graphs of Equations 9 PART II: LINEAR EQUATIONS AND INEQUALITIES IN TWO VARIABLES 1.5 Rectangular Coordinates and Graphs of Equations OBJECTIVES 1 Plot Points in
More informationA Detailed Price Discrimination Example
A Detailed Price Discrimination Example Suppose that there are two different types of customers for a monopolist s product. Customers of type 1 have demand curves as follows. These demand curves include
More informationA Summary of Curve Sketching. Analyzing the Graph of a Function
0_00.qd //0 :5 PM Page 09 SECTION. A Summar of Curve Sketching 09 0 00 Section. 0 0 00 0 Different viewing windows for the graph of f 5 7 0 Figure. 5 A Summar of Curve Sketching Analze and sketch the graph
More informationQuadratic Modeling Business 10 Profits
Quadratic Modeling Business 10 Profits In this activity, we are going to look at modeling business profits. We will allow q to represent the number of items manufactured and assume that all items that
More information1.3. Maximum or Minimum of a Quadratic Function. Investigate A
< P16 photo of a large arched bridge, similar to the one on page 292 or p 360361of the fish book> Maximum or Minimum of a Quadratic Function 1.3 Some bridge arches are defined by quadratic functions.
More informationa b c d e You have two hours to do this exam. Please write your name on this page, and at the top of page three. GOOD LUCK! 3. a b c d e 12.
MA123 Elem. Calculus Fall 2015 Exam 2 20151022 Name: Sec.: Do not remove this answer page you will turn in the entire exam. No books or notes may be used. You may use an ACTapproved calculator during
More information), 35% use extra unleaded gas ( A
. At a certain gas station, 4% of the customers use regular unleaded gas ( A ), % use extra unleaded gas ( A ), and % use premium unleaded gas ( A ). Of those customers using regular gas, onl % fill their
More informationDERIVATIVES AS MATRICES; CHAIN RULE
DERIVATIVES AS MATRICES; CHAIN RULE 1. Derivatives of Realvalued Functions Let s first consider functions f : R 2 R. Recall that if the partial derivatives of f exist at the point (x 0, y 0 ), then we
More informationIn this section, we will consider techniques for solving problems of this type.
Constrained optimisation roblems in economics typically involve maximising some quantity, such as utility or profit, subject to a constraint for example income. We shall therefore need techniques for solving
More informationFlorida Algebra I EOC Online Practice Test
Florida Algebra I EOC Online Practice Test 1 Directions: This practice test contains 65 multiplechoice questions. Choose the best answer for each question. Detailed answer eplanations appear at the end
More informationTo give it a definition, an implicit function of x and y is simply any relationship that takes the form:
2 Implicit function theorems and applications 21 Implicit functions The implicit function theorem is one of the most useful single tools you ll meet this year After a while, it will be second nature to
More informationMicroeconomics Sept. 16, 2010 NOTES ON CALCULUS AND UTILITY FUNCTIONS
DUSP 11.203 Frank Levy Microeconomics Sept. 16, 2010 NOTES ON CALCULUS AND UTILITY FUNCTIONS These notes have three purposes: 1) To explain why some simple calculus formulae are useful in understanding
More informationALGEBRA I (Common Core) Wednesday, August 13, 2014 8:30 to 11:30 a.m., only
ALGEBRA I (COMMON CORE) The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA I (Common Core) Wednesday, August 13, 2014 8:30 to 11:30 a.m., only Student Name: School Name: The
More information15.1. Exact Differential Equations. Exact FirstOrder Equations. Exact Differential Equations Integrating Factors
SECTION 5. Eact FirstOrder Equations 09 SECTION 5. Eact FirstOrder Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Section 5.6, ou studied applications of differential
More informationMICROECONOMICS AND POLICY ANALYSIS  U8213 Professor Rajeev H. Dehejia Class Notes  Spring 2001
MICROECONOMICS AND POLICY ANALYSIS  U8213 Professor Rajeev H. Dehejia Class Notes  Spring 2001 General Equilibrium and welfare with production Wednesday, January 24 th and Monday, January 29 th Reading:
More informationAlgebra II Semester Exam Review Sheet
Name: Class: Date: ID: A Algebra II Semester Exam Review Sheet 1. Translate the point (2, 3) left 2 units and up 3 units. Give the coordinates of the translated point. 2. Use a table to translate the graph
More informationThe composition g f of the functions f and g is the function (g f)(x) = g(f(x)). This means, "do the function f to x, then do g to the result.
30 5.6 The chain rule The composition g f of the functions f and g is the function (g f)(x) = g(f(x)). This means, "do the function f to x, then do g to the result." Example. g(x) = x 2 and f(x) = (3x+1).
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA. Tuesday, January 22, 2013 9:15 a.m. SAMPLE RESPONSE SET
The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA Tuesday, January 22, 2013 9:15 a.m. SAMPLE RESPONSE SET Table of Contents Practice Papers Question 31.......................
More informationMicroeconomic Theory: Basic Math Concepts
Microeconomic Theory: Basic Math Concepts Matt Van Essen University of Alabama Van Essen (U of A) Basic Math Concepts 1 / 66 Basic Math Concepts In this lecture we will review some basic mathematical concepts
More informationIncreasing for all. Convex for all. ( ) Increasing for all (remember that the log function is only defined for ). ( ) Concave for all.
1. Differentiation The first derivative of a function measures by how much changes in reaction to an infinitesimal shift in its argument. The largest the derivative (in absolute value), the faster is evolving.
More informationIntroduction to Calculus
Introduction to Calculus Contents 1 Introduction to Calculus 3 11 Introduction 3 111 Origin of Calculus 3 112 The Two Branches of Calculus 4 12 Secant and Tangent Lines 5 13 Limits 10 14 The Derivative
More informationEQUATIONS OF LINES IN SLOPE INTERCEPT AND STANDARD FORM
. Equations of Lines in SlopeIntercept and Standard Form ( ) 8 In this SlopeIntercept Form Standard Form section Using SlopeIntercept Form for Graphing Writing the Equation for a Line Applications (0,
More informationDividing Polynomials Quiz
Dividing Polynomials Quiz Melanie Bellido Puna is reviewing answers for this assessment. Points scored may differ from the grading guidelines because of teacher review. Contact your program teacher if
More information1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered
Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,
More informationGraphing Linear Equations
6.3 Graphing Linear Equations 6.3 OBJECTIVES 1. Graph a linear equation b plotting points 2. Graph a linear equation b the intercept method 3. Graph a linear equation b solving the equation for We are
More informationName Eco200: Practice Test 3A Covering Chapters 16, 1821
Name Eco200: Practice Test 3A Covering Chapters 16, 1821 1. The following information describes the demand schedule for a unique type of apple. This type of apple can only be produced by two firms because
More informationLet (x 1, y 1 ) (0, 1) and (x 2, y 2 ) (x, y). x 0. y 1. y 1 2. x x Multiply each side by x. y 1 x. y x 1 Add 1 to each side. SlopeIntercept Form
8 () Chapter Linear Equations in Two Variables and Their Graphs In this section SlopeIntercept Form Standard Form Using SlopeIntercept Form for Graphing Writing the Equation for a Line Applications
More informationCritical points of once continuously differentiable functions are important because they are the only points that can be local maxima or minima.
Lecture 0: Convexity and Optimization We say that if f is a once continuously differentiable function on an interval I, and x is a point in the interior of I that x is a critical point of f if f (x) =
More informationPreferences. Preferences. Budget Constraint and Line 9/10/2014
9/0/04 references Chapter 4 At this point, we know a lot about preferences and their representation with utilit functions references tell us how a consumer ranks a given bundle compared to another bundle,
More informationIntroduction. Introduction
Introduction Solving Sstems of Equations Let s start with an eample. Recall the application of sales forecasting from the Working with Linear Equations module. We used historical data to derive the equation
More information6.3 Parametric Equations and Motion
SECTION 6.3 Parametric Equations and Motion 475 What ou ll learn about Parametric Equations Parametric Curves Eliminating the Parameter Lines and Line Segments Simulating Motion with a Grapher... and wh
More informationNAME DATE PERIOD. 11. Is the relation (year, percent of women) a function? Explain. Yes; each year is
 NAME DATE PERID Functions Determine whether each relation is a function. Eplain.. {(, ), (0, 9), (, 0), (7, 0)} Yes; each value is paired with onl one value.. {(, ), (, ), (, ), (, ), (, )}. No; in the
More informationManagerial Economics Prof. Trupti Mishra S.J.M. School of Management Indian Institute of Technology, Bombay. Lecture  13 Consumer Behaviour (Contd )
(Refer Slide Time: 00:28) Managerial Economics Prof. Trupti Mishra S.J.M. School of Management Indian Institute of Technology, Bombay Lecture  13 Consumer Behaviour (Contd ) We will continue our discussion
More informationSECOND DERIVATIVE TEST FOR CONSTRAINED EXTREMA
SECOND DERIVATIVE TEST FOR CONSTRAINED EXTREMA This handout presents the second derivative test for a local extrema of a Lagrange multiplier problem. The Section 1 presents a geometric motivation for the
More informationFind the Relationship: An Exercise in Graphing Analysis
Find the Relationship: An Eercise in Graphing Analsis Computer 5 In several laborator investigations ou do this ear, a primar purpose will be to find the mathematical relationship between two variables.
More informationFactor and Solve Polynomial Equations. In Chapter 4, you learned how to factor the following types of quadratic expressions.
5.4 Factor and Solve Polynomial Equations Before You factored and solved quadratic equations. Now You will factor and solve other polynomial equations. Why? So you can find dimensions of archaeological
More informationAlgebra 2 Chapter 5 Practice Test (Review)
Name: Class: Date: Algebra 2 Chapter 5 Practice Test (Review) Multiple Choice Identify the choice that best completes the statement or answers the question. Determine whether the function is linear or
More informationThe Envelope Theorem 1
John Nachbar Washington University April 2, 2015 1 Introduction. The Envelope Theorem 1 The Envelope theorem is a corollary of the KarushKuhnTucker theorem (KKT) that characterizes changes in the value
More information5.3 Graphing Cubic Functions
Name Class Date 5.3 Graphing Cubic Functions Essential Question: How are the graphs of f () = a (  h) 3 + k and f () = ( 1_ related to the graph of f () = 3? b (  h) 3 ) + k Resource Locker Eplore 1
More informationAPPLICATIONS AND MODELING WITH QUADRATIC EQUATIONS
APPLICATIONS AND MODELING WITH QUADRATIC EQUATIONS Now that we are starting to feel comfortable with the factoring process, the question becomes what do we use factoring to do? There are a variety of classic
More information6.3 PARTIAL FRACTIONS AND LOGISTIC GROWTH
6 CHAPTER 6 Techniques of Integration 6. PARTIAL FRACTIONS AND LOGISTIC GROWTH Use partial fractions to find indefinite integrals. Use logistic growth functions to model reallife situations. Partial Fractions
More information1 Calculus of Several Variables
1 Calculus of Several Variables Reading: [Simon], Chapter 14, p. 30031. 1.1 Partial Derivatives Let f : R n R. Then for each x i at each point x 0 = (x 0 1,..., x 0 n) the ith partial derivative is defined
More informationUNIVERSITY OF WISCONSIN SYSTEM
Name UNIVERSITY OF WISCONSIN SYSTEM MATHEMATICS PRACTICE EXAM Check us out at our website: http://www.testing.wisc.edu/center.html GENERAL INSTRUCTIONS: You will have 90 minutes to complete the mathematics
More informationCHAPTER 4 Consumer Choice
CHAPTER 4 Consumer Choice CHAPTER OUTLINE 4.1 Preferences Properties of Consumer Preferences Preference Maps 4.2 Utility Utility Function Ordinal Preference Utility and Indifference Curves Utility and
More informationALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form
ALGEBRA 2: 4.1 Graph Quadratic Functions in Standard Form Goal Graph quadratic functions. VOCABULARY Quadratic function A function that can be written in the standard form y = ax 2 + bx+ c where a 0 Parabola
More informationLecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization
Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization 2.1. Introduction Suppose that an economic relationship can be described by a realvalued
More informationThe Method of Lagrange Multipliers
The Method of Lagrange Multipliers S. Sawyer October 25, 2002 1. Lagrange s Theorem. Suppose that we want to maximize (or imize a function of n variables f(x = f(x 1, x 2,..., x n for x = (x 1, x 2,...,
More informationMATH 34A REVIEW FOR MIDTERM 2, WINTER 2012. 1. Lines. (1) Find the equation of the line passing through (2,1) and (2,9). y = 5
MATH 34A REVIEW FOR MIDTERM 2, WINTER 2012 ANSWERS 1. Lines (1) Find the equation of the line passing through (2,1) and (2,9). y = 5 2 x + 4. (2) Find the equation of the line which meets the xaxis
More informationCalculus 1st Semester Final Review
Calculus st Semester Final Review Use the graph to find lim f ( ) (if it eists) 0 9 Determine the value of c so that f() is continuous on the entire real line if f ( ) R S T, c /, > 0 Find the limit: lim
More information2011 Pearson Education. Elasticities of Demand and Supply: Today add elasticity and slope, cross elasticities
2011 Pearson Education Elasticities of Demand and Supply: Today add elasticity and slope, cross elasticities What Determines Elasticity? Influences on the price elasticity of demand fall into two categories:
More information