Partial derivatives BUSINESS MATHEMATICS
|
|
- Elfrieda Watson
- 5 years ago
- Views:
Transcription
1 Partial derivatives BUSINESS MATHEMATICS 1
2 CONTENTS Derivatives for functions of two variables Higher-order partial derivatives Derivatives for functions of many variables Old exam question Further study 2
3 DERIVATIVES FOR FUNCTIONS OF TWO VARIABLES Can we find the extreme values of a function g x, y of two variables x and y? Try g x, y = fails! g x, y = x 3 y + x 2 y 2 + x + y 2 3
4 DERIVATIVES FOR FUNCTIONS OF TWO VARIABLES Can we find the extreme values of a function g x, y of two variables x and y? g x, y = x 3 y + x 2 y 2 + x + y 2 Try g x, y = fails! Recall definition of derivative of a function f x of one variable: f x = df x dx = lim f x + h f x h 0 h Generalization to partial derivative of g x, y of 2 variables: g x, y = lim h 0 g x + h, y g x, y h 4
5 DERIVATIVES FOR FUNCTIONS OF TWO VARIABLES Differentiate g x, y = x 3 y + x 2 y 2 + x + y 2 with respect to x (and keeping y fixed) g = 3x2 y + 2xy
6 DERIVATIVES FOR FUNCTIONS OF TWO VARIABLES Differentiate g x, y = x 3 y + x 2 y 2 + x + y 2 with respect to x (and keeping y fixed) g = 3x2 y + 2xy and with respect to y (and keeping x fixed) g y = x3 + 2x 2 y + 2y 6
7 DERIVATIVES FOR FUNCTIONS OF TWO VARIABLES Differentiate g x, y = x 3 y + x 2 y 2 + x + y 2 with respect to x (and keeping y fixed) g = 3x2 y + 2xy and with respect to y (and keeping x fixed) g y = x3 + 2x 2 y + 2y Clearly, in this case g g y. Therefore, never write f for a function of two variables! 7
8 DERIVATIVES FOR FUNCTIONS OF TWO VARIABLES So we define the partial derivative of f with respect to x f x, y = lim h 0 f x + h, y f x, y h and similar with respect to y f x, y y = lim h 0 f x, y + h f x, y h 8
9 DERIVATIVES FOR FUNCTIONS OF TWO VARIABLES f x, y f x, y y 9
10 DERIVATIVES FOR FUNCTIONS OF TWO VARIABLES Alternative notations f, f x,y, f x, f 1, f x, f 1, x f, Not important to remember, but important to recognize so, basically a lot of choice, but never write df or f dx 10
11 DERIVATIVES FOR FUNCTIONS OF TWO VARIABLES The partial derivative in a point is a number: f x,y x,y = 2, 5 = 3 The partial derivative over a range of points is a function of x and y: f x,y = 2x + 3y 6 11
12 DERIVATIVES FOR FUNCTIONS OF TWO VARIABLES Example: Cobb-Douglas production function describing how a firm s output q depends on capital input (K) and labour input (L): q K, L = A K α L β where A, α, and β are positive constants. 12
13 DERIVATIVES FOR FUNCTIONS OF TWO VARIABLES Example: Cobb-Douglas production function describing how a firm s output q depends on capital input (K) and labour input (L): q K, L = A K α L β where A, α, and β are positive constants. Marginal productivity of capital q K = A α Kα 1 L β Note that when 0 < α < 1, q K diminishing marginal returns. is a decreasing function of K leading to 13
14 EXERCISE 1 Given is f x, y = x y. Find f and f y. 14
15 HIGHER-ORDER PARTIAL DERIVATIVES Recall the second derivative d dx df x dx = d2 f x dx 2 = f x Four possibilities for function g x, y : y y g g y g g y = 2 g 2 = 2 g y 2 = 2 g y = 2 g y 16
16 HIGHER-ORDER PARTIAL DERIVATIVES Recall the second derivative d dx df x dx = d2 f x dx 2 = f x Four possibilities for function g x, y : y y g g y g g y = 2 g 2 = 2 g y 2 = 2 g y = 2 g y so, never d2 g dx2 or g Alternative notations: 2 g g x,y, y y, g yx, g 21, g yx, g 21, xy g,. 17
17 HIGHER-ORDER PARTIAL DERIVATIVES Example: g x, y = x 3 y + x 2 y 2 + x + y 2 2 g 2 = 6xy + 2y2 2 g = y 2 2x g = y 3x2 + 4xy 2 g = y 3x2 + 4xy 18
18 HIGHER-ORDER PARTIAL DERIVATIVES Example: g x, y = x 3 y + x 2 y 2 + x + y 2 2 g 2 = 6xy + 2y2 2 g = y 2 2x g = y 3x2 + 4xy 2 g = y 3x2 + 4xy For almost all functions 2 g y = 2 g y and certainly for all functions we encounter in business and economics. 19
19 EXERCISE 2 Given is f x, y = 4x 3 y 2 3y 4 e 2x. Find 2 f y in x, y = 1,0. 20
20 HIGHER-ORDER PARTIAL DERIVATIVES Likewise, we can define third-order derivatives f x,y = 3 f 3 y y f x,y = 3 f y 2 How many are there? How many are different? 22
21 HIGHER-ORDER PARTIAL DERIVATIVES Likewise, we can define third-order derivatives f x,y = 3 f 3 y y f x,y = 3 f y 2 How many are there? How many are different? And even higher-order partial derivatives n f, n n f n 1 y,. 23
22 DERIVATIVES FOR FUNCTIONS OF MANY VARIABLES For functions f x 1, x 2, x 3,, x n partial derivatives we can form n first-order f, f,, 1 2 f n and many many second-order partial derivatives 24
23 EXERCISE 3 Given is g x = 1 σ n i=1 n x i. Find g 4. 25
24 OLD EXAM QUESTION 27 March 2015, Q1b 27
25 OLD EXAM QUESTION 22 October 2014, Q1h 28
26 FURTHER STUDY Sydsæter et al. 5/E Tutorial exercises week 2 partial derivatives higher-order partial derivatives partial derivatives graphically 29
Constrained optimization BUSINESS MATHEMATICS
Constrained optimization BUSINESS MATHEMATICS 1 CONTENTS Unconstrained optimization Constrained optimization Lagrange method Old exam question Further study 2 UNCONSTRAINED OPTIMIZATION Recall extreme
More informationMatrices BUSINESS MATHEMATICS
Matrices BUSINESS MATHEMATICS 1 CONTENTS Matrices Special matrices Operations with matrices Matrix multipication More operations with matrices Matrix transposition Symmetric matrices Old exam question
More informationMATH 19520/51 Class 5
MATH 19520/51 Class 5 Minh-Tam Trinh University of Chicago 2017-10-04 1 Definition of partial derivatives. 2 Geometry of partial derivatives. 3 Higher derivatives. 4 Definition of a partial differential
More informationf( x) f( y). Functions which are not one-to-one are often called many-to-one. Take the domain and the range to both be all the real numbers:
I. UNIVARIATE CALCULUS Given two sets X and Y, a function is a rule that associates each member of X with exactly one member of Y. That is, some x goes in, and some y comes out. These notations are used
More informationParis. Optimization. Philippe Bich (Paris 1 Panthéon-Sorbonne and PSE) Paris, Philippe Bich
Paris. Optimization. (Paris 1 Panthéon-Sorbonne and PSE) Paris, 2017. Lecture 1: About Optimization A For QEM-MMEF, the course (3H each week) and tutorial (4 hours each week) from now to october 22. Exam
More informationg(t) = f(x 1 (t),..., x n (t)).
Reading: [Simon] p. 313-333, 833-836. 0.1 The Chain Rule Partial derivatives describe how a function changes in directions parallel to the coordinate axes. Now we shall demonstrate how the partial derivatives
More informationz = f (x; y) = x 3 3x 2 y x 2 3
BEE Mathematics for Economists Week, ecture Thursday..7 Functions in two variables Dieter Balkenborg Department of Economics University of Exeter Objective This lecture has the purpose to make you familiar
More informationTopic 7. Part I Partial Differentiation Part II Marginal Functions Part II Partial Elasticity Part III Total Differentiation Part IV Returns to scale
Topic 7 Part I Partial Differentiation Part II Marginal Functions Part II Partial Elasticity Part III Total Differentiation Part IV Returns to scale Jacques (4th Edition): 5.1-5.3 1 Functions of Several
More informationConstrained optimization.
ams/econ 11b supplementary notes ucsc Constrained optimization. c 2016, Yonatan Katznelson 1. Constraints In many of the optimization problems that arise in economics, there are restrictions on the values
More informationMath Review ECON 300: Spring 2014 Benjamin A. Jones MATH/CALCULUS REVIEW
MATH/CALCULUS REVIEW SLOPE, INTERCEPT, and GRAPHS REVIEW (adapted from Paul s Online Math Notes) Let s start with some basic review material to make sure everybody is on the same page. The slope of a line
More informationIntermediate Macroeconomics, EC2201. L1: Economic growth I
Intermediate Macroeconomics, EC2201 L1: Economic growth I Anna Seim Department of Economics, Stockholm University Spring 2017 1 / 44 Contents and literature Growth facts. Production. Literature: Jones
More informationAdvanced Microeconomics
Advanced Microeconomics Ivan Etzo University of Cagliari ietzo@unica.it Dottorato in Scienze Economiche e Aziendali, XXXIII ciclo Ivan Etzo (UNICA) Lecture 1: Technolgy 1 / 61 Overview 1 Firms behavior
More informationIntroduction to systems of equations
Introduction to systems of equations A system of equations is a collection of two or more equations that contains the same variables. This is a system of two equations with two variables: In solving a
More informationLECTURE NOTES ON MICROECONOMICS
LECTURE NOTES ON MICROECONOMICS ANALYZING MARKETS WITH BASIC CALCULUS William M. Boal Part : Mathematical tools Chapter : Introduction to multivariate calculus But those skilled in mathematical analysis
More informationIOP2601. Some notes on basic mathematical calculations
IOP601 Some notes on basic mathematical calculations The order of calculations In order to perform the calculations required in this module, there are a few steps that you need to complete. Step 1: Choose
More informationFunctions of Several Variables
Functions of Several Variables Partial Derivatives Philippe B Laval KSU March 21, 2012 Philippe B Laval (KSU) Functions of Several Variables March 21, 2012 1 / 19 Introduction In this section we extend
More informationSAMPLING, THE CLT, AND THE STANDARD ERROR. Business Statistics
SAMPLING, THE CLT, AND THE STANDARD ERROR Business Statistics CONTENTS Sampling The central limit theorem Point and interval estimates for μ Confidence intervals for μ Old exam question Further study SAMPLING
More informationModelling Production
Modelling Production David N. DeJong University of Pittsburgh Econ. 1540, Spring 2010 DND () Production Econ. 1540, Spring 2010 1 / 23 Introduction The production function is the foundation upon which
More informationPartial Differentiation
CHAPTER 7 Partial Differentiation From the previous two chapters we know how to differentiate functions of one variable But many functions in economics depend on several variables: output depends on both
More informationDifferentiation. 1. What is a Derivative? CHAPTER 5
CHAPTER 5 Differentiation Differentiation is a technique that enables us to find out how a function changes when its argument changes It is an essential tool in economics If you have done A-level maths,
More informationIn economics, the amount of a good x demanded is a function of the price of that good. In other words,
I. UNIVARIATE CALCULUS Given two sets X and Y, a function is a rule that associates each member of X with eactly one member of Y. That is, some goes in, and some y comes out. These notations are used to
More information5 Systems of Equations
Systems of Equations Concepts: Solutions to Systems of Equations-Graphically and Algebraically Solving Systems - Substitution Method Solving Systems - Elimination Method Using -Dimensional Graphs to Approximate
More informationMath 1314 Test 4 Review Lesson 16 Lesson Use Riemann sums with midpoints and 6 subdivisions to approximate the area between
Math 1314 Test 4 Review Lesson 16 Lesson 24 1. Use Riemann sums with midpoints and 6 subdivisions to approximate the area between and the x-axis on the interval [1, 9]. Recall: RectangleSum[,
More informationMATH 19520/51 Class 4
MATH 19520/51 Class 4 Minh-Tam Trinh University of Chicago 2017-10-02 1 Functions and independent ( nonbasic ) vs. dependent ( basic ) variables. 2 Cobb Douglas production function and its interpretation.
More informationMultiple Regression: Example
Multiple Regression: Example Cobb-Douglas Production Function The Cobb-Douglas production function for observed economic data i = 1,..., n may be expressed as where O i is output l i is labour input c
More informationDifferentiation - Quick Review From Calculus
Differentiation - Quick Review From Calculus Philippe B. Laval KSU Current Semester Philippe B. Laval (KSU) Differentiation - Quick Review From Calculus Current Semester 1 / 13 Introduction In this section,
More informationSection 2.4: Add and Subtract Rational Expressions
CHAPTER Section.: Add and Subtract Rational Expressions Section.: Add and Subtract Rational Expressions Objective: Add and subtract rational expressions with like and different denominators. You will recall
More information(x x 0 ) 2 + (y y 0 ) 2 = ε 2, (2.11)
2.2 Limits and continuity In order to introduce the concepts of limit and continuity for functions of more than one variable we need first to generalise the concept of neighbourhood of a point from R to
More informationMultivariate calculus
Multivariate calculus Lecture note 5 Outline 1. Multivariate functions in Euclidean space 2. Continuity 3. Multivariate differentiation 4. Differentiability 5. Higher order derivatives 6. Implicit functions
More informationTutorial 3: Optimisation
Tutorial : Optimisation ECO411F 011 1. Find and classify the extrema of the cubic cost function C = C (Q) = Q 5Q +.. Find and classify the extreme values of the following functions (a) y = x 1 + x x 1x
More informationEconomics 203: Intermediate Microeconomics. Calculus Review. A function f, is a rule assigning a value y for each value x.
Economics 203: Intermediate Microeconomics Calculus Review Functions, Graphs and Coordinates Econ 203 Calculus Review p. 1 Functions: A function f, is a rule assigning a value y for each value x. The following
More informationComparative Statics. Autumn 2018
Comparative Statics Autumn 2018 What is comparative statics? Contents 1 What is comparative statics? 2 One variable functions Multiple variable functions Vector valued functions Differential and total
More information14.3 Partial Derivatives
14 14.3 Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. On a hot day, extreme humidity makes us think the temperature is higher than it really is, whereas
More informationMathematical Economics (ECON 471) Lecture 3 Calculus of Several Variables & Implicit Functions
Mathematical Economics (ECON 471) Lecture 3 Calculus of Several Variables & Implicit Functions Teng Wah Leo 1 Calculus of Several Variables 11 Functions Mapping between Euclidean Spaces Where as in univariate
More informationSometimes the domains X and Z will be the same, so this might be written:
II. MULTIVARIATE CALCULUS The first lecture covered functions where a single input goes in, and a single output comes out. Most economic applications aren t so simple. In most cases, a number of variables
More informationMATH 2554 (Calculus I)
MATH 2554 (Calculus I) Dr. Ashley K. University of Arkansas February 21, 2015 Table of Contents Week 6 1 Week 6: 16-20 February 3.5 Derivatives as Rates of Change 3.6 The Chain Rule 3.7 Implicit Differentiation
More informationMathematical Economics: Lecture 9
Mathematical Economics: Lecture 9 Yu Ren WISE, Xiamen University October 17, 2011 Outline 1 Chapter 14: Calculus of Several Variables New Section Chapter 14: Calculus of Several Variables Partial Derivatives
More informationBEE1024 Mathematics for Economists
BEE1024 Mathematics for Economists Dieter and Jack Rogers and Juliette Stephenson Department of Economics, University of Exeter February 1st 2007 1 Objective 2 Isoquants 3 Objective. The lecture should
More informationCHAPTER 4: HIGHER ORDER DERIVATIVES. Likewise, we may define the higher order derivatives. f(x, y, z) = xy 2 + e zx. y = 2xy.
April 15, 2009 CHAPTER 4: HIGHER ORDER DERIVATIVES In this chapter D denotes an open subset of R n. 1. Introduction Definition 1.1. Given a function f : D R we define the second partial derivatives as
More information14.30 Introduction to Statistical Methods in Economics Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 4.30 Introduction to Statistical Methods in Economics Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
More informationMain topics for the First Midterm Exam
Main topics for the First Midterm Exam The final will cover Sections.-.0, 2.-2.5, and 4.. This is roughly the material from first three homeworks and three quizzes, in addition to the lecture on Monday,
More informationLecture 1: Introduction to IO Tom Holden
Lecture 1: Introduction to IO Tom Holden http://io.tholden.org/ Email: t.holden@surrey.ac.uk Standard office hours: Thursday, 12-1PM + 3-4PM, 29AD00 However, during term: CLASSES will be run in the first
More informationMULTIPLE REGRESSION ANALYSIS AND OTHER ISSUES. Business Statistics
MULTIPLE REGRESSION ANALYSIS AND OTHER ISSUES Business Statistics CONTENTS Multiple regression Dummy regressors Assumptions of regression analysis Predicting with regression analysis Old exam question
More informationMicroeconomic Theory -1- Introduction
Microeconomic Theory -- Introduction. Introduction. Profit maximizing firm with monopoly power 6 3. General results on maximizing with two variables 8 4. Model of a private ownership economy 5. Consumer
More informationThe Real Business Cycle Model
The Real Business Cycle Model Macroeconomics II 2 The real business cycle model. Introduction This model explains the comovements in the fluctuations of aggregate economic variables around their trend.
More informationTutorial Code and TA (circle one): T1 Charles Tsang T2 Stephen Tang
Department of Computer & Mathematical Sciences University of Toronto at Scarborough MATA33H3Y: Calculus for Management II Final Examination August, 213 Examiner: A. Chow Surname (print): Given Name(s)
More informationMath 1302 Notes 2. How many solutions? What type of solution in the real number system? What kind of equation is it?
Math 1302 Notes 2 We know that x 2 + 4 = 0 has How many solutions? What type of solution in the real number system? What kind of equation is it? What happens if we enlarge our current system? Remember
More informationMath 126 Enhanced 10.3 Series with positive terms The University of Kansas 1 / 12
Section 10.3 Convergence of series with positive terms 1. Integral test 2. Error estimates for the integral test 3. Comparison test 4. Limit comparison test (LCT) Math 126 Enhanced 10.3 Series with positive
More informationFinal Exam Advanced Mathematics for Economics and Finance
Final Exam Advanced Mathematics for Economics and Finance Dr. Stefanie Flotho Winter Term /5 March 5 General Remarks: ˆ There are four questions in total. ˆ All problems are equally weighed. ˆ This is
More informationMath Exam 2, October 14, 2008
Math 96 - Exam 2, October 4, 28 Name: Problem (5 points Find all solutions to the following system of linear equations, check your work: x + x 2 x 3 2x 2 2x 3 2 x x 2 + x 3 2 Solution Let s perform Gaussian
More informationIE 5531 Midterm #2 Solutions
IE 5531 Midterm #2 s Prof. John Gunnar Carlsson November 9, 2011 Before you begin: This exam has 9 pages and a total of 5 problems. Make sure that all pages are present. To obtain credit for a problem,
More information7.1 Functions of Two or More Variables
7.1 Functions of Two or More Variables Hartfield MATH 2040 Unit 5 Page 1 Definition: A function f of two variables is a rule such that each ordered pair (x, y) in the domain of f corresponds to exactly
More informationAssumption 5. The technology is represented by a production function, F : R 3 + R +, F (K t, N t, A t )
6. Economic growth Let us recall the main facts on growth examined in the first chapter and add some additional ones. (1) Real output (per-worker) roughly grows at a constant rate (i.e. labor productivity
More informationAN INTRODUCTION TO CURVILINEAR ORTHOGONAL COORDINATES
AN INTRODUCTION TO CURVILINEAR ORTHOGONAL COORDINATES Overview Throughout the first few weeks of the semester, we have studied vector calculus using almost exclusively the familiar Cartesian x,y,z coordinate
More informationContent by Week Week of October 14 27
Content by Week Week of October 14 27 Learning objectives By the end of this week, you should be able to: Understand the purpose and interpretation of confidence intervals for the mean, Calculate confidence
More informationMathematical Economics: Lecture 16
Mathematical Economics: Lecture 16 Yu Ren WISE, Xiamen University November 26, 2012 Outline 1 Chapter 21: Concave and Quasiconcave Functions New Section Chapter 21: Concave and Quasiconcave Functions Concave
More informationBusiness Mathematics. Lecture Note #13 Chapter 7-(1)
1 Business Mathematics Lecture Note #13 Chapter 7-(1) Applications of Partial Differentiation 1. Differentials and Incremental Changes 2. Production functions: Cobb-Douglas production function, MP L, MP
More informationDefinition: Absolute Value The absolute value of a number is the distance that the number is from zero. The absolute value of x is written x.
R. Absolute Values We begin this section by recalling the following definition. Definition: Absolute Value The absolute value of a number is the distance that the number is from zero. The absolute value
More information3/1/2016. Intermediate Microeconomics W3211. Lecture 3: Preferences and Choice. Today s Aims. The Story So Far. A Short Diversion: Proofs
1 Intermediate Microeconomics W3211 Lecture 3: Preferences and Choice Introduction Columbia University, Spring 2016 Mark Dean: mark.dean@columbia.edu 2 The Story So Far. 3 Today s Aims 4 So far, we have
More information1. (4 % each, total 20 %) Answer each of the following. (No need to show your work for this problem). 3 n. n!? n=1
NAME: EXAM 4 - Math 56 SOlutions Instruction: Circle your answers and show all your work CLEARLY Partial credit will be given only when you present what belongs to part of a correct solution (4 % each,
More informationPractice Problems #1 Practice Problems #2
Practice Problems #1 Interpret the following equations where C is the cost, and Q is quantity produced by the firm a) C(Q) = 10 + Q Costs depend on quantity. If the firm produces nothing, costs are 10,
More informationEcon 110: Introduction to Economic Theory. 8th Class 2/7/11
Econ 110: Introduction to Economic Theory 8th Class 2/7/11 go over problem answers from last time; no new problems today given you have your problem set to work on; we'll do some problems for these concepts
More informationMath 131. The Derivative and the Tangent Line Problem Larson Section 2.1
Math 131. The Derivative and the Tangent Line Problem Larson Section.1 From precalculus, the secant line through the two points (c, f(c)) and (c +, f(c + )) is given by m sec = rise f(c + ) f(c) f(c +
More informationMAT1300 Final Review. Pieter Hofstra. December 4, 2009
December 4, 2009 Sections from the book to study (8th Edition) Chapter 0: 0.1: Real line and Order 0.2: Absolute Value and Distance 0.3: Exponents and Radicals 0.4: Factoring Polynomials (you may omit
More informationChain Rule. MATH 311, Calculus III. J. Robert Buchanan. Spring Department of Mathematics
3.33pt Chain Rule MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Spring 2019 Single Variable Chain Rule Suppose y = g(x) and z = f (y) then dz dx = d (f (g(x))) dx = f (g(x))g (x)
More informationSection 13.3 Concavity and Curve Sketching. Dr. Abdulla Eid. College of Science. MATHS 104: Mathematics for Business II
Section 13.3 Concavity and Curve Sketching College of Science MATHS 104: Mathematics for Business II (University of Bahrain) Concavity 1 / 18 Concavity Increasing Function has three cases (University of
More informationA. Incorrect! Replacing is not a method for solving systems of equations.
ACT Math and Science - Problem Drill 20: Systems of Equations No. 1 of 10 1. What methods were presented to solve systems of equations? (A) Graphing, replacing, and substitution. (B) Solving, replacing,
More informationTangent Lines and Derivatives
The Derivative and the Slope of a Graph Tangent Lines and Derivatives Recall that the slope of a line is sometimes referred to as a rate of change. In particular, we are referencing the rate at which the
More informationSection 5.3: Linear Inequalities
336 Section 5.3: Linear Inequalities In the first section, we looked at a company that produces a basic and premium version of its product, and we determined how many of each item they should produce fully
More informationMath 211 Business Calculus TEST 3. Question 1. Section 2.2. Second Derivative Test.
Math 211 Business Calculus TEST 3 Question 1. Section 2.2. Second Derivative Test. p. 1/?? Math 211 Business Calculus TEST 3 Question 1. Section 2.2. Second Derivative Test. Question 2. Section 2.3. Graph
More informationz = f (x; y) f (x ; y ) f (x; y) f (x; y )
BEEM0 Optimization Techiniques for Economists Lecture Week 4 Dieter Balkenborg Departments of Economics University of Exeter Since the fabric of the universe is most perfect, and is the work of a most
More informationMath 116 Practice for Exam 2
Math 6 Practice for Exam Generated October 6, 5 Name: Instructor: Section Number:. This exam has 5 questions. Note that the problems are not of equal difficulty, so you may want to skip over and return
More informationMath 155 Prerequisite Review Handout
Math 155 Prerequisite Review Handout August 23, 2010 Contents 1 Basic Mathematical Operations 2 1.1 Examples...................................... 2 1.2 Exercises.......................................
More informationCES functions and Dixit-Stiglitz Formulation
CES functions and Dixit-Stiglitz Formulation Weijie Chen Department of Political and Economic Studies University of Helsinki September, 9 4 8 3 7 Labour 6 5 4 5 Labour 5 Capital 3 4 6 8 Capital Any suggestion
More informationCSC236 Week 11. Larry Zhang
CSC236 Week 11 Larry Zhang 1 Announcements Next week s lecture: Final exam review This week s tutorial: Exercises with DFAs PS9 will be out later this week s. 2 Recap Last week we learned about Deterministic
More informationDifferentiation of x n.
Differentiation of x n. In this unit we derive the result, from first principles, that The result is illustrated with several examples. if y = x n dx = nx n 1. In order to master the techniques explained
More informationMath 128 Midterm 2 Spring 2009
Name: ID: Discussion Section: This exam consists of 16 questions: 14 multiple choice questions worth 5 points each 2 hand-graded questions worth a total of 30 points. INSTRUCTIONS: Read each problem carefully
More informationSEVERAL μs AND MEDIANS: MORE ISSUES. Business Statistics
SEVERAL μs AND MEDIANS: MORE ISSUES Business Statistics CONTENTS Post-hoc analysis ANOVA for 2 groups The equal variances assumption The Kruskal-Wallis test Old exam question Further study POST-HOC ANALYSIS
More informationLecture 5: The neoclassical growth model
THE UNIVERSITY OF SOUTHAMPTON Paul Klein Office: Murray Building, 3005 Email: p.klein@soton.ac.uk URL: http://paulklein.se Economics 3010 Topics in Macroeconomics 3 Autumn 2010 Lecture 5: The neoclassical
More informationEconS Cost Structures
EconS 425 - Cost Structures Eric Dunaway Washington State University eric.dunaway@wsu.edu Industrial Organization Eric Dunaway (WSU) EconS 425 Industrial Organization 1 / 34 Introduction Today, we ll review
More informationECON 186 Class Notes: Derivatives and Differentials
ECON 186 Class Notes: Derivatives and Differentials Jijian Fan Jijian Fan ECON 186 1 / 27 Partial Differentiation Consider a function y = f (x 1,x 2,...,x n ) where the x i s are all independent, so each
More informationECON2285: Mathematical Economics
ECON2285: Mathematical Economics Yulei Luo Economics, HKU September 17, 2018 Luo, Y. (Economics, HKU) ME September 17, 2018 1 / 46 Static Optimization and Extreme Values In this topic, we will study goal
More informationWeek 10: Theory of the Firm (Jehle and Reny, Chapter 3)
Week 10: Theory of the Firm (Jehle and Reny, Chapter 3) Tsun-Feng Chiang* *School of Economics, Henan University, Kaifeng, China November 22, 2015 First Last (shortinst) Short title November 22, 2015 1
More informationIntegrals. D. DeTurck. January 1, University of Pennsylvania. D. DeTurck Math A: Integrals 1 / 61
Integrals D. DeTurck University of Pennsylvania January 1, 2018 D. DeTurck Math 104 002 2018A: Integrals 1 / 61 Integrals Start with dx this means a little bit of x or a little change in x If we add up
More informationThe Consumer, the Firm, and an Economy
Andrew McLennan October 28, 2014 Economics 7250 Advanced Mathematical Techniques for Economics Second Semester 2014 Lecture 15 The Consumer, the Firm, and an Economy I. Introduction A. The material discussed
More information10/22/16. 1 Math HL - Santowski SKILLS REVIEW. Lesson 15 Graphs of Rational Functions. Lesson Objectives. (A) Rational Functions
Lesson 15 Graphs of Rational Functions SKILLS REVIEW! Use function composition to prove that the following two funtions are inverses of each other. 2x 3 f(x) = g(x) = 5 2 x 1 1 2 Lesson Objectives! The
More informationADVANCED MACRO TECHNIQUES Midterm Solutions
36-406 ADVANCED MACRO TECHNIQUES Midterm Solutions Chris Edmond hcpedmond@unimelb.edu.aui This exam lasts 90 minutes and has three questions, each of equal marks. Within each question there are a number
More informationSeptember Math Course: First Order Derivative
September Math Course: First Order Derivative Arina Nikandrova Functions Function y = f (x), where x is either be a scalar or a vector of several variables (x,..., x n ), can be thought of as a rule which
More informationECON 186 Class Notes: Optimization Part 2
ECON 186 Class Notes: Optimization Part 2 Jijian Fan Jijian Fan ECON 186 1 / 26 Hessians The Hessian matrix is a matrix of all partial derivatives of a function. Given the function f (x 1,x 2,...,x n ),
More informationa factors The exponential 0 is a special case. If b is any nonzero real number, then
0.1 Exponents The expression x a is an exponential expression with base x and exponent a. If the exponent a is a positive integer, then the expression is simply notation that counts how many times the
More informationTheory of the Firm. Production Technology
Theory of the Firm Production Technology The Firm What is a firm? In reality, the concept firm and the reasons for the existence of firms are complex. Here we adopt a simple viewpoint: a firm is an economic
More information14.05: Section Handout #1 Solow Model
14.05: Section Handout #1 Solow Model TA: Jose Tessada September 16, 2005 Today we will review the basic elements of the Solow model. Be prepared to ask any questions you may have about the derivation
More informationECON Advanced Economic Theory-Microeconomics. Prof. W. M. Semasinghe
ECON 53015 Advanced Economic Theory-Microeconomics Prof. W. M. Semasinghe Theory of Production Theory of production deals with the question of How to produce? It discusses the supply side of the pricing
More informationProblem Set 2. E. Charlie Nusbaum Econ 204A October 12, 2015
E. Charlie Nusbaum Econ 204A October 12, 2015 Problem Set 2 Romer Problem 1.3 Describe how, if at all, each of the following developments affects the breaeven and actual investment lines in our basic diagram
More informationDRAFT - Math 101 Lecture Note - Dr. Said Algarni
2 Limits 2.1 The Tangent Problems The word tangent is derived from the Latin word tangens, which means touching. A tangent line to a curve is a line that touches the curve and a secant line is a line that
More informationUnderstanding Exponents Eric Rasmusen September 18, 2018
Understanding Exponents Eric Rasmusen September 18, 2018 These notes are rather long, but mathematics often has the perverse feature that if someone writes a long explanation, the reader can read it much
More informationFirms and returns to scale -1- Firms and returns to scale
Firms and returns to scale -1- Firms and returns to scale. Increasing returns to scale and monopoly pricing 2. Constant returns to scale 19 C. The CRS economy 25 D. pplication to trade 47 E. Decreasing
More informationMath 163: Lecture notes
Math 63: Lecture notes Professor Monika Nitsche March 2, 2 Special functions that are inverses of known functions. Inverse functions (Day ) Go over: early exam, hw, quizzes, grading scheme, attendance
More informationSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Study Questions for OPMT 5701 Most Questions come from Chapter 17. The Answer Key has a section code for each. Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers
More information