# 3 The Utility Maximization Problem

Size: px
Start display at page:

Transcription

1 3 The Utility Mxiiztion Proble We hve now discussed how to describe preferences in ters of utility functions nd how to forulte siple budget sets. The rtionl choice ssuption, tht consuers pick the best ordble bundle, cn then be described s n optiiztion proble. The proble is to nd bundle (x 1; x ); which is in the budget set, ening tht x 1 + x nd x 1 0; x 0; which is such tht u(x 1; x ) u(x 1 ; x ) for ll (x 1 ; x ) in the budget set. It is convenient to introduce soe nottion for this type of probles. Using rther stndrd conventions I will write this s subj. to x u (x 1 ; x ) x 1 ;x x 1 + x x 1 0 x 0 ; which is n exple of proble of constrined optiiztion. A coon tendency of students is to skip the step where the proble is written down. This is bd ide. The reson is tht we will often study vrints of optiiztion probles tht di er in wht sees to be sll detils. Indeed, often ties the di cult step when thinking bout proble is to forulte the right optiiztion proble. For this reson I wnt you to 1. Write out the x in front of the utility function (the xind, or, objective function). This clri es tht the consuer is supposed to solve n optiiztion proble.. Below the x, it is good ide to indicte wht the choice vribles re for the consuer (x 1 nd x in this exple). This is to clrify the di erence between the vribles tht re under control of the decision ker nd vribles tht the decision ker hs no control over, which re referred to s preters. In the ppliction bove ; nd re preters. 37

2 3. Finlly, it is iportnt tht it is cler wht the constrints to the proble re. I good hbit is to write subject to or, ore concisely, s.t. nd then list whtever constrints there re, s in the proble bove. 3.1 Solving the Utility Mxiiztion Proble Optionl Reding You y wnt to look t the ppendix to chpter 5 in Vrin (pges in the ost recent Edition). We seek to solve the consuer proble x u (x 1 ; x ) x 1 ;x subj. to x 1 + x x 1 0 x 0 Given tht we re willing to ssue tht preferences re onotonic (which we re) we rst ke the siple observtion tht we y replce the inequlity x 1 + x with x 1 + x = To understnd this one just needs to drw grph. Suppose we would be ble to nd sine optil solution x = (x 1; x ) to the consuer proble such tht x 1 + x < nd tht preferences re onotonic. Then we observe tht we cn increse good 1 (or good or both) little bit without violting the budget constrint. But, by the onotonicity, this increses the utility of the consuer, which ens tht x wsn t optil. Hence we conclude tht no optil solution cn be interior in the budget set. 38

3 x 6 Better Bundles s - x 1 Figure 1 Interior Bundles Cn t be Optil with Monotonic Preferences Thus, we cn rule out ll interior points in the budget set nd solve x u (x 1 ; x ) x 1 ;x subj. to x 1 + x = x 1 0 x 0 But since the constrint ust hold with equlity we know tht in ny optil solution it ust be the cse tht x = x 1 We cn thus siply plug in the constrint into the objective function nd solve the sipler proble x u x 1 ; p 1x 1 ; 0x 1 WHICH IS A MAXIMIZATION PROBLEM WITH A SINGLE VARIABLE (exctly on the for x x f (x) s.t. x b which we hd in our discussion on xi). Ignoring for now the possibility of corner solutions we cn just di erentite this nd set the derivtive to zero to get the rst order condition. 39

4 3.1.1 Optility Conditions for Consuer Choice Proble Using the chin rule, di erentite the utility function with respect to the choice vrible x 1 to get u x 1; x 1 + x 1 u x 1; x 1 x p1 = 0 Since the budget constrint is stis ed with equlity we cn now substitute bck x = x 1 ; which leds to the condition u(x 1 ;x ) x 1 = u(x 1 ;x ) x This condition, which we interpret s tngency condition below, is necessry condition for n interior optiu. 3. Interprettion of The Optility Condition The optility condition hs useful geoetric interprettion s tngency condition between the indi erence curve nd the budget line nd if often referred to s sying tht slope of indi erence curve=mrs=slope of budget line This geoetric interprettion is useful since it will llow us to go bck nd forth between pictures nd th. The rst thing to relize is tht u(x 0 1 ;x0 ) x 1 u(x 0 1 ;x0 ) x is the slope of the indi erence curve for ny point (x 0 1; x 0 )(Rerk bout rguents & nottionl sloppiness). To see this, pick point (x 0 1; x 0 ) nd look for ll (x 1 ; x ) such tht the consuer is indi erent between these points nd (x 0 1; x 0 ) Tht is u (x 1 ; x ) = u (x 0 1; x 0 ) k {z } just nuber- k for short 40

5 In principle, this cn be solved for x s function of x 1 (s when we solved exples in clss & hoework). This solution is soe reltion x = y (x 1 ) stisfying u (x 1 ; y (x 1 )) = k [I cheting here in tht I ignoring soe deep th...there is question of whether u (x 1 ; x ) = k cn be solved for x s function of x 1 ] Once you ve tken on fith tht we cn solve for function y (x 1 ) we hve tht the condition bove is n identity ( holds for ll vlues of x 1 ). Hence, we cn di erentite the identity with respect to x 1 to get u (x 1 ; y (x 1 )) x 1 + u (x 1; y (x 1 )) x dy (x 1 ) dx 1 = d dx 1 k = 0 dy (x 1 ) dx 1 {z } Slope = u(x 1 ;y(x 1 )) x 1 u(x 1 ;y(x 1 )) x 0 1 = MRS x 1 ; y (x 1 ) A {z } =x Now y (x 1 ) is constructed so tht for ech x 1 you plug in you get the se utility level (it is n indi erence curve). So the interprettion of the slope of y (x 1 ) is tht it tells you how uch of good you need to tke wy if you increse the consuption of good 1 slightly in order to keep the consuer indi erent. x 6 y s slope + u(x 1 ;y(x 1 )) x 1 u(x 1 ;y(x 1 )) x x - x 1 Figure The Mrginl Rte of Substitution For discrete chnges, the rginl rte of substitution will give you slightly wrong nswer to the question, but for su ciently sll chnges the error will be negligible. So 41

6 the optility condition cn be interpreted s (fter ultiplying the optility condition by 1) MRS (x 1; x ) = u(x 1 ;x ) x 1 = u(x 1 ;x ) p x {z {z} } rte t which the rket rte t which the consuer is willing to exchnge goods is willing to exchnge goods 3..1 Exple Cobb-Dougls Utility u (x 1 ; x ) = x 1x b Utility xiiztion proble subj to x 1 + x x x 1 ;x x 1x b Budget constrint ust bind)solve out x fro constrint to get proble. b x x p1 x 1 0x 1 1 Assuing tht the solution is not t the boundry points, the rst order condition needs to be stis ed. Tht is x 1 1 b b 1 p1 x 1 + x p1 x 1 p1 1b = 0 At this point only soe lgebr reins. This cn be done in nuber of di erent wys, but here is one clcultion p1 x 1 x 1 1, b + x 1b ultiply with 1 x 1 b 1 p1 x 1 p1 x 1 bp x 1 x 1 b > 0, = 0 p1 = 0 ( x 1 ) b x 1 = 0 4, x 1 ( + b) =, x 1 = + b

7 Now we hve the cndidte solution for good one. Plugging bck into budget constrint gives = x 1 + x = + x + b + b + b = + b, x = b + b = b + b = x Hence, the cndidte solution is x 1 = x = + b b + b Now, We know tht solution ust either stisfy the rst order condition (the point (x 1; x ) is the only point on budget line which does) or be t boundry point. In this exple, ny bundle with either x 1 = 0 or x = 0 gives utility u (x 1 ; x ) = 0; wheres u (x 1; x ) = (x 1) (x ) b > 0 since x 1 > 0 nd x > 0 Cobining these two fcts we know tht the bundle (x 1; x ) = the solution to the consuer choice proble. b +b ; +b is indeed 3.. Finl Rerk About Ordinlity nd Monotone Trnsfortions Let c = Plug in c insted of nd 1 + b ) 1 c = 1 + b = + b + b = b + b c insted of b bove nd you ieditely get tht x 1 = c = + b x (1 c) = = b ; + b 43

8 so the solution does not chnge. This is becuse f (u) = u 1 +b is n incresing function of u nd x 1x b 1 +b = x +b 1 x b +b = x c 1x 1 c therefore is onotone trnsfortion. Hence there is no dded exibility in preferences by llowing + b 6= 1 (nd this sipli es clcultions). Thus, I will typiclly use utility function u (x 1 ; x ) = x 1x 1 Moreover, those of you who re cofortble with logrits y note tht ln x 1x 1 = ln x1 + (1 ) ln x nd you re free to use tht trnsfortions whenever you like. The FOC then ieditely becoes so this sves soe work. + 1 x 1 x 1 p1 = 0; 3..3 The Possibility of Corner Solutions y f(b) 6 f(x) - b x Figure 3 Derivtive My be Non-Zero if Solution t the Boundry 44

9 Recll tht if the highest vlue of the function is chieved t boundry point, then the rst order condition need not necessrily hold t the xiu. I ll skip the detils, but it is strightforwrd to work out (nd intuitive) tht for our utility xiiztion proble 1. If solution is t the lower end of the boundry (with x 1 being the choice vrible in the reduced proble you get fter plugging in the budget constrint), then tht ens tht the slope of the indi erence curve t point (x 1 ; x ) = 0; p is tter thn the budget line.. If solution is t the upper end of the boundry, then tht ens tht the slope of the indi erence curve t point (x 1 ; x ) = ; 0 is steeper thn the budget line. x 6 c cc c cc c cc c c - x 1 Figure 4 A Corner Solution To see tht this is rel possibility in siple exple, let u (x 1 ; x ) = p x 1 + x Following the se steps s in the previous exple this results in utility xiiztion proble tht y be written s p x x1 + x 1 0x 1 If there is n interior solution, we thus hve tht the rst order condition (if f (x) = p x then df(x) dx = 1 p x ) 1 p x 1 = 0 45

10 ust hold for soe 0 x 1 Solving this condition for x 1 we get x 1 = p For siplicity, set = nd = 1; in which cse x 1 = 1 Then, we cn recover the consuption of x through the budget constrint s x 1 + p x = 1 + x = ) x = 1 The point with this is tht we notice tht if < 1; then our cndidte solution for x is negtive, which doesn t ke ny sense t ll. We thus conclude tht the solution ust be t corner nd (it y be useful for you to plot the function p x x 1 to see this) the solution is insted to use ll incoe on x 1 ; i.e. (x 1; x ) = (; 0) solves the proble. If you hve hrd tie to see this you should 1) plot p x x 1 ; nd/or ) plug in (x 1 ; x ) = (; 0) nd note tht u (; 0) = p ; 3) plug in (x 1 ; x ) = 0; nd note tht u 0; = < p 46

### Math 135 Circles and Completing the Square Examples

Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for

### Integration by Substitution

Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is

### Graphs on Logarithmic and Semilogarithmic Paper

0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl

### Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding

1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde

### 2.016 Hydrodynamics Prof. A.H. Techet

.01 Hydrodynics Reding #.01 Hydrodynics Prof. A.H. Techet Added Mss For the cse of unstedy otion of bodies underwter or unstedy flow round objects, we ust consider the dditionl effect (force) resulting

### Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.

2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this

### Econ 4721 Money and Banking Problem Set 2 Answer Key

Econ 472 Money nd Bnking Problem Set 2 Answer Key Problem (35 points) Consider n overlpping genertions model in which consumers live for two periods. The number of people born in ech genertion grows in

### PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY

MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive

### Binary Representation of Numbers Autar Kaw

Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse- rel number to its binry representtion,. convert binry number to n equivlent bse- number. In everydy

### 5.6 POSITIVE INTEGRAL EXPONENTS

54 (5 ) Chpter 5 Polynoils nd Eponents 5.6 POSITIVE INTEGRAL EXPONENTS In this section The product rule for positive integrl eponents ws presented in Section 5., nd the quotient rule ws presented in Section

### Algebra Review. How well do you remember your algebra?

Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then

### Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )

Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +

### MATH 150 HOMEWORK 4 SOLUTIONS

MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive

### Integration. 148 Chapter 7 Integration

48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but

### Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.

Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd

### Lecture 5. Inner Product

Lecture 5 Inner Product Let us strt with the following problem. Given point P R nd line L R, how cn we find the point on the line closest to P? Answer: Drw line segment from P meeting the line in right

### Linear Equations in Two Variables

Liner Equtions in Two Vribles In this chpter, we ll use the geometry of lines to help us solve equtions. Liner equtions in two vribles If, b, ndr re rel numbers (nd if nd b re not both equl to 0) then

### LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of

### 9 CONTINUOUS DISTRIBUTIONS

9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete

### and thus, they are similar. If k = 3 then the Jordan form of both matrices is

Homework ssignment 11 Section 7. pp. 249-25 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If

### Physics 43 Homework Set 9 Chapter 40 Key

Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nm-wide region t x

### Problem Set 2: Solutions ECON 301: Intermediate Microeconomics Prof. Marek Weretka. Problem 1 (Marginal Rate of Substitution)

Proble Set 2: Solutions ECON 30: Interediate Microeconoics Prof. Marek Weretka Proble (Marginal Rate of Substitution) (a) For the third colun, recall that by definition MRS(x, x 2 ) = ( ) U x ( U ). x

### 5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.

5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous rel-vlued

### 4.11 Inner Product Spaces

314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces

### Section 7-4 Translation of Axes

62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 7-4 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the

### All pay auctions with certain and uncertain prizes a comment

CENTER FOR RESEARC IN ECONOMICS AND MANAGEMENT CREAM Publiction No. 1-2015 All py uctions with certin nd uncertin prizes comment Christin Riis All py uctions with certin nd uncertin prizes comment Christin

### COMPLEX FRACTIONS. section. Simplifying Complex Fractions

58 (6-6) Chpter 6 Rtionl Epressions undles tht they cn ttch while working together for 0 hours. 00 600 6 FIGURE FOR EXERCISE 9 95. Selling. George sells one gzine suscription every 0 inutes, wheres Theres

### 1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator

AP Clculus Finl Review Sheet When you see the words. This is wht you think of doing. Find the zeros Find roots. Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor.

### Bayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Byesin Updting with Continuous Priors Clss 3, 8.05, Spring 04 Jeremy Orloff nd Jonthn Bloom Lerning Gols. Understnd prmeterized fmily of distriutions s representing continuous rnge of hypotheses for the

### SPECIAL PRODUCTS AND FACTORIZATION

MODULE - Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come

### Operations with Polynomials

38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply

### Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100

hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by

### Basic Analysis of Autarky and Free Trade Models

Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently

### EQUATIONS OF LINES AND PLANES

EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in point-direction nd twopoint

### MODULE 3. 0, y = 0 for all y

Topics: Inner products MOULE 3 The inner product of two vectors: The inner product of two vectors x, y V, denoted by x, y is (in generl) complex vlued function which hs the following four properties: i)

### Experiment 6: Friction

Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht

### Module Summary Sheets. C3, Methods for Advanced Mathematics (Version B reference to new book) Topic 2: Natural Logarithms and Exponentials

MEI Mthemtics in Ection nd Instry Topic : Proof MEI Structured Mthemtics Mole Summry Sheets C, Methods for Anced Mthemtics (Version B reference to new book) Topic : Nturl Logrithms nd Eponentils Topic

### Exponential and Logarithmic Functions

Nme Chpter Eponentil nd Logrithmic Functions Section. Eponentil Functions nd Their Grphs Objective: In this lesson ou lerned how to recognize, evlute, nd grph eponentil functions. Importnt Vocbulr Define

### LECTURE #05. Learning Objective. To describe the geometry in and around a unit cell in terms of directions and planes.

LECTURE #05 Chpter 3: Lttice Positions, Directions nd Plnes Lerning Objective To describe the geometr in nd round unit cell in terms of directions nd plnes. 1 Relevnt Reding for this Lecture... Pges 64-83.

### Reasoning to Solve Equations and Inequalities

Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing

### Lecture 3 Gaussian Probability Distribution

Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike

### Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:

Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you

### Factoring Polynomials

Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles

### Homework 3 Solutions

CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.

### Review guide for the final exam in Math 233

Review guide for the finl exm in Mth 33 1 Bsic mteril. This review includes the reminder of the mteril for mth 33. The finl exm will be cumultive exm with mny of the problems coming from the mteril covered

### Econ 100A: Intermediate Microeconomics Notes on Consumer Theory

Econ 100A: Interediate Microeconoics Notes on Consuer Theory Linh Bun Winter 2012 (UCSC 1. Consuer Theory Utility Functions 1.1. Types of Utility Functions The following are soe of the type of the utility

Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology

Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology

Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology

### Redistributing the Gains from Trade through Non-linear. Lump-sum Transfers

Redistributing the Gins from Trde through Non-liner Lump-sum Trnsfers Ysukzu Ichino Fculty of Economics, Konn University April 21, 214 Abstrct I exmine lump-sum trnsfer rules to redistribute the gins from

### www.mathsbox.org.uk e.g. f(x) = x domain x 0 (cannot find the square root of negative values)

www.mthsbo.org.uk CORE SUMMARY NOTES Functions A function is rule which genertes ectl ONE OUTPUT for EVERY INPUT. To be defined full the function hs RULE tells ou how to clculte the output from the input

### v T R x m Version PREVIEW Practice 7 carroll (11108) 1

Version PEVIEW Prctice 7 crroll (08) his print-out should he 5 questions. Multiple-choice questions y continue on the next colun or pge find ll choices before nswering. Atwood Mchine 05 00 0.0 points A

### Unit 6: Exponents and Radicals

Eponents nd Rdicls -: The Rel Numer Sstem Unit : Eponents nd Rdicls Pure Mth 0 Notes Nturl Numers (N): - counting numers. {,,,,, } Whole Numers (W): - counting numers with 0. {0,,,,,, } Integers (I): -

### 15.6. The mean value and the root-mean-square value of a function. Introduction. Prerequisites. Learning Outcomes. Learning Style

The men vlue nd the root-men-squre vlue of function 5.6 Introduction Currents nd voltges often vry with time nd engineers my wish to know the verge vlue of such current or voltge over some prticulr time

### 2 DIODE CLIPPING and CLAMPING CIRCUITS

2 DIODE CLIPPING nd CLAMPING CIRCUITS 2.1 Ojectives Understnding the operting principle of diode clipping circuit Understnding the operting principle of clmping circuit Understnding the wveform chnge of

### Demand. Lecture 3. August 2015. Reading: Perlo Chapter 4 1 / 58

Demand Lecture 3 Reading: Perlo Chapter 4 August 2015 1 / 58 Introduction We saw the demand curve in chapter 2. We learned about consumer decision making in chapter 3. Now we bridge the gap between the

### Economic 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

### Warm-up for Differential Calculus

Summer Assignment Wrm-up for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte:

### Inequalities for the internal angle-bisectors of a triangle

Mtheticl Counictions 2(997), 4-45 4 Inequlities for the internl ngle-bisectors of tringle Wlther Jnous nd Šefket Arslngić Abstrct. Severl ne inequlities of type α ± for ngle-bisectors re proved. Certin

### RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS

RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS Known for over 500 yers is the fct tht the sum of the squres of the legs of right tringle equls the squre of the hypotenuse. Tht is +b c. A simple proof is

### CHAPTER 11 Numerical Differentiation and Integration

CHAPTER 11 Numericl Differentition nd Integrtion Differentition nd integrtion re bsic mthemticl opertions with wide rnge of pplictions in mny res of science. It is therefore importnt to hve good methods

### 9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes

The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is so-clled becuse when the sclr product of two vectors

Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology

### CS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001

CS99S Lortory 2 Preprtion Copyright W. J. Dlly 2 Octoer, 2 Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes to oserve logic

### Techniques for Requirements Gathering and Definition. Kristian Persson Principal Product Specialist

Techniques for Requirements Gthering nd Definition Kristin Persson Principl Product Specilist Requirements Lifecycle Mngement Elicit nd define business/user requirements Vlidte requirements Anlyze requirements

### Distributions. (corresponding to the cumulative distribution function for the discrete case).

Distributions Recll tht n integrble function f : R [,] such tht R f()d = is clled probbility density function (pdf). The distribution function for the pdf is given by F() = (corresponding to the cumultive

### 6.2 Volumes of Revolution: The Disk Method

mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine so-clled volumes of

### Treatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.

The nlysis of vrince (ANOVA) Although the t-test is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the t-test cn be used to compre the mens of only

### The Definite Integral

Chpter 4 The Definite Integrl 4. Determining distnce trveled from velocity Motivting Questions In this section, we strive to understnd the ides generted by the following importnt questions: If we know

### n Using the formula we get a confidence interval of 80±1.64

9.52 The professor of sttistics oticed tht the rks i his course re orlly distributed. He hs lso oticed tht his orig clss verge is 73% with stdrd devitio of 12% o their fil exs. His fteroo clsses verge

### Regular Sets and Expressions

Regulr Sets nd Expressions Finite utomt re importnt in science, mthemtics, nd engineering. Engineers like them ecuse they re super models for circuits (And, since the dvent of VLSI systems sometimes finite

### AREA OF A SURFACE OF REVOLUTION

AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.

### SUBSTITUTION I.. f(ax + b)

Integrtion SUBSTITUTION I.. f(x + b) Grhm S McDonld nd Silvi C Dll A Tutoril Module for prctising the integrtion of expressions of the form f(x + b) Tble of contents Begin Tutoril c 004 g.s.mcdonld@slford.c.uk

### Firm Objectives. The Theory of the Firm II. Cost Minimization Mathematical Approach. First order conditions. Cost Minimization Graphical Approach

Pro. Jy Bhttchry Spring 200 The Theory o the Firm II st lecture we covered: production unctions Tody: Cost minimiztion Firm s supply under cost minimiztion Short vs. long run cost curves Firm Ojectives

### Quick Reference Guide: One-time Account Update

Quick Reference Guide: One-time Account Updte How to complete The Quick Reference Guide shows wht existing SingPss users need to do when logging in to the enhnced SingPss service for the first time. 1)

### The Velocity Factor of an Insulated Two-Wire Transmission Line

The Velocity Fctor of n Insulted Two-Wire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the

### TIME VALUE OF MONEY PROBLEMS CHAPTERS THREE TO TEN

TIME VLUE OF MONEY PROBLEMS CHPTERS THREE TO TEN Probles In how any years \$ will becoe \$265 if = %? 265 ln n 933844 9 34 years ln( 2 In how any years will an aount double if = 76%? ln 2 n 9 46 years ln76

### Version 001 Summer Review #03 tubman (IBII20142015) 1

Version 001 Summer Reiew #03 tubmn (IBII20142015) 1 This print-out should he 35 questions. Multiple-choice questions my continue on the next column or pge find ll choices before nswering. Concept 20 P03

### Babylonian Method of Computing the Square Root: Justifications Based on Fuzzy Techniques and on Computational Complexity

Bbylonin Method of Computing the Squre Root: Justifictions Bsed on Fuzzy Techniques nd on Computtionl Complexity Olg Koshelev Deprtment of Mthemtics Eduction University of Texs t El Pso 500 W. University

### c 2008 Je rey A. Miron We have described the constraints that a consumer faces, i.e., discussed the budget constraint.

Lecture 2b: Utility c 2008 Je rey A. Miron Outline: 1. Introduction 2. Utility: A De nition 3. Monotonic Transformations 4. Cardinal Utility 5. Constructing a Utility Function 6. Examples of Utility Functions

### PROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 1

PROBLEMS - APPLICATIONS OF DERIVATIVES Pge ( ) Wter seeps out of conicl filter t the constnt rte of 5 cc / sec. When the height of wter level in the cone is 5 cm, find the rte t which the height decreses.

### How To Network A Smll Business

Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology

### On the Efficiency of Public and Private Health Care Systems: An Application to Alternative Health Policies in the United Kingdom

On the Efficiency of ublic nd rivte Helth Cre Systes: An Appliction to Alterntive Helth olicies in the United Kingdo In W.H. rry Jnury 2001 Discussion per 01-07 Resources for the Future 1616 Street, W

### Derivatives and Rates of Change

Section 2.1 Derivtives nd Rtes of Cnge 2010 Kiryl Tsiscnk Derivtives nd Rtes of Cnge Te Tngent Problem EXAMPLE: Grp te prbol y = x 2 nd te tngent line t te point P(1,1). Solution: We ve: DEFINITION: Te

### 2005-06 Second Term MAT2060B 1. Supplementary Notes 3 Interchange of Differentiation and Integration

Source: http://www.mth.cuhk.edu.hk/~mt26/mt26b/notes/notes3.pdf 25-6 Second Term MAT26B 1 Supplementry Notes 3 Interchnge of Differentition nd Integrtion The theme of this course is bout vrious limiting

### Review Problems for the Final of Math 121, Fall 2014

Review Problems for the Finl of Mth, Fll The following is collection of vrious types of smple problems covering sections.,.5, nd.7 6.6 of the text which constitute only prt of the common Mth Finl. Since

### . At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2

7 CHAPTER THREE. Cross Product Given two vectors = (,, nd = (,, in R, the cross product of nd written! is defined to e: " = (!,!,! Note! clled cross is VECTOR (unlike which is sclr. Exmple (,, " (4,5,6

### Lectures 8 and 9 1 Rectangular waveguides

1 Lectures 8 nd 9 1 Rectngulr wveguides y b x z Consider rectngulr wveguide with 0 < x b. There re two types of wves in hollow wveguide with only one conductor; Trnsverse electric wves

### How fast can we sort? Sorting. Decision-tree model. Decision-tree for insertion sort Sort a 1, a 2, a 3. CS 3343 -- Spring 2009

CS 4 -- Spring 2009 Sorting Crol Wenk Slides courtesy of Chrles Leiserson with smll chnges by Crol Wenk CS 4 Anlysis of Algorithms 1 How fst cn we sort? All the sorting lgorithms we hve seen so fr re comprison

### AP STATISTICS SUMMER MATH PACKET

AP STATISTICS SUMMER MATH PACKET This pcket is review of Algebr I, Algebr II, nd bsic probbility/counting. The problems re designed to help you review topics tht re importnt to your success in the clss.

### Week 7 - Perfect Competition and Monopoly

Week 7 - Perfect Competition nd Monopoly Our im here is to compre the industry-wide response to chnges in demnd nd costs by monopolized industry nd by perfectly competitive one. We distinguish between

### Helicopter Theme and Variations

Helicopter Theme nd Vritions Or, Some Experimentl Designs Employing Pper Helicopters Some possible explntory vribles re: Who drops the helicopter The length of the rotor bldes The height from which the

### 19. The Fermat-Euler Prime Number Theorem

19. The Fermt-Euler Prime Number Theorem Every prime number of the form 4n 1 cn be written s sum of two squres in only one wy (side from the order of the summnds). This fmous theorem ws discovered bout

### FUNCTIONS AND EQUATIONS. xεs. The simplest way to represent a set is by listing its members. We use the notation

FUNCTIONS AND EQUATIONS. SETS AND SUBSETS.. Definition of set. A set is ny collection of objects which re clled its elements. If x is n element of the set S, we sy tht x belongs to S nd write If y does

### Section 5-4 Trigonometric Functions

5- Trigonometric Functions Section 5- Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form

### Introduction to Integration Part 2: The Definite Integral

Mthemtics Lerning Centre Introduction to Integrtion Prt : The Definite Integrl Mr Brnes c 999 Universit of Sdne Contents Introduction. Objectives...... Finding Ares 3 Ares Under Curves 4 3. Wht is the

### I calculate the unemployment rate as (In Labor Force Employed)/In Labor Force

Introduction to the Prctice of Sttistics Fifth Edition Moore, McCbe Section 4.5 Homework Answers to 98, 99, 100,102, 103,105, 107, 109,110, 111, 112, 113 Working. In the lnguge of government sttistics,

### CURVES ANDRÉ NEVES. that is, the curve α has finite length. v = p q p q. a i.e., the curve of smallest length connecting p to q is a straight line.

CURVES ANDRÉ NEVES 1. Problems (1) (Ex 1 of 1.3 of Do Crmo) Show tht the tngent line to the curve α(t) (3t, 3t 2, 2t 3 ) mkes constnt ngle with the line z x, y. (2) (Ex 6 of 1.3 of Do Crmo) Let α(t) (e