V =! Things to remember: E(p) = - pf'(p)
|
|
- Moris Burke
- 6 years ago
- Views:
Transcription
1 dx (B) From (2), d!(4x + 5) (5x + 100) Setting x 150 in (1), we get 45, ,000 or !15 ± 55 2 "15 ± and 2 Since 0, 20. Now, for x 150, 20 and dx d -[4(150) + 5(20)]("6) 5(150) + 100(20) , 20-6, we have Thus, the rice is increasing at the rate of $1.53 er month. 31. Volume V πr 2 h, where h thickness of the circular oil slick. Since h 0.1 1, we have: 10 V! 10 R2 Differentiating with resect to t: # d % " dv $ Given: dr dv! 5 & 10 R2 ( '! dr 2R 10! 5 R dr 0.32 when R 500. Therefore, (500)(0.32) 100π(0.32) cubic feet er minute. EXERCISE 4-7 Things to remember: 1. RELATIVE AND PERCENTAGE RATES OF CHANGE The RELATIVE RATE OF CHANGE of a function f(x) is f'(x) f(x). The PERCENTAGE RATE OF CHANGE is 100 f'(x) f(x). 2. ELASTICITY OF DEMAND If rice and demand are related by x f(), then the ELASTICITY OF DEMAND is given by E() - f'() f() 198 CHAPTER 4 ADDITIONAL DERIVATIVE TOPICS
2 3. INTERPRETATION OF ELASTICITY OF DEMAND E() Demand Interretation 0 < E() < 1 Inelastic Demand is not sensitive to changes in rice. A change in rice roduces a smaller change in demand. E() > 1 Elastic Demand is sensitive to changes in rice. A change in rice roduces a larger change in demand. E() 1 Unit A change in rice roduces the same change in demand. 4. REVENUE AND ELASTICITY OF DEMAND If R() f() is the revenue function, then R'() and [1 - E()] always have the same sign. Demand is inelastic [E() < 1, R'() > 0]: A rice increase will increase revenue. A rice decrease will decrease revenue. Demand is elastic [E() > 1, R'() < 0]: A rice increase will decrease revenue. A rice decrease will increase revenue. 1. f(x) 25 f (x) 0 Relative rate of change of f: f "(x) f(x) f(x) 30x f (x) 30 Relative rate of change of f: f "(x) f(x) 30 30x 1 x 5. f(x) 10x f'(x) 10 Relative rate of change of f: f'(x) f(x) 10 10x x f(x) 100x - 0.5x 2 f'(x) x Relative rate of change of f: f'(x) f(x) 100! x 100x! 0.5x 2 9. f(x) 4 + 2e -2x f'(x) -4e -2x Relative rate of change of f: f'(x) f(x)!4e!2x 4 + 2e!2x - 2e!2x 2 + e!2x " e 2x e 2x e 2x EXERCISE
3 11. f(x) 25x + 3x ln x f'(x) ln x ln x Relative rate of change of f: f'(x) f(x) 13. x f() 12, f'() -20 Elasticity of demand: E()!f'() f() ln x 25x + 3x ln x ,000! (A) At 10: E(10) 12,000! , Demand is inelastic (B) At 20: E(20) 12,000! ; unit elasticity ,000 (C) At 30: E(30) 12,000! 9,000 18,000 3,000 6 Demand is elastic. 15. x f() f'() Elasticity of demand: E()!f'() f() ! 2! (A) At 30: E(30) 950! 60! Demand is inelastic (B) At 50: E(50) 950! 100! ; unit elasticity (C) At 70: E(70) 950! 140! Demand is elastic x 30 (A) x 30! , 0 30 (B) f() f'() -200 Elasticity of demand: E()!f'() f() ! ! 10 (C) At 10: E(10) 30! If the rice increases by 10%, the demand will decrease by aroximately 0.5(10%) 5%. 200 CHAPTER 4 ADDITIONAL DERIVATIVE TOPICS
4 25 (D) At 25: E(25) 30! 25 5 If the rice increases by 10%, the demand will decrease by aroximately 5(10%) 50%. 15 (E) At 15: E(15) 30! 15 1 If the rice increases by 10%, the demand will decrease by aroximately 10% x + 60 (A) x 60! , 0 60 (B) R() ( ) (C) f() f'() -50 Elasticity of demand: E()!f'() f() ! 50 (D) Elastic: E() Inelastic: E() 60! > 1 > 60 - > 30, 30 < < 60 60! < 1 < 60 - < 30, 0 < < 30 60! (E) R'() f() [1 - E()] (equation (9)) R'() > 0 if E() < 1; R'() < 0 if E() > 1 Therefore, revenue is increasing for 0 < < 30 and decreasing for 30 < < 60. (F) If $10 and the rice is decreased, revenue will also decrease. (G) If $40 and the rice is decreased, revenue will increase. 21. x f() 10( - 30) 2, 0 30 f'() 20( - 30)![20(! 30)] Elasticity of demand: E() 10(! 30) 2 Elastic: E() - Inelastic: E() -!2! 30 2! 30 > 1-2 < - 30 ( - 30 < 0 reverses inequality) -3 < -30 > 10; 10 < < 30 2! 30 < 1-2 > - 30 ( - 30 < 0 reverses inequality) -3 > -30 < 10; 0 < < 10 EXERCISE
5 23. x f() 144 " 2, 0 72 f'() 1 2 (144-2)-1/2 (-2) Elasticity of demand: E() Elastic: E() Inelastic: E() 144! 2 "1 144 " 2 144! 2 > 1 > > 144 > 48, 48 < < ! 2 < 1 < < 144 < 48, 0 < < x f() 2,500 " f'() 1 2 (2, ) -1/2 (-4)!2 (2, 500! 2 2 ) Elasticity of demand: E() 2,500! ,250! 2 2 Elastic: E() 1,250! 2 > 1 Inelastic: E() 2 > 1, > 1,250 2 > 625 > 25, 25 < < ,250! 2 < 1 2 < 1, < 1,250 2 < 625 < 25, 0 < < CHAPTER 4 ADDITIONAL DERIVATIVE TOPICS
6 27. x f() 20(10 - ) 0 10 R() f() 20(10 - ) R'() Critical value: R'() ; 5 Sign chart for R'(): Test Numbers R'() R() Increasing Decreasing Demand: Inelastic Elastic x R'() 0 200(+) 10!200(!) 29. x f() 40( - 15) R() f() 40( - 15) 2 R'() 40( - 15) (2)( - 15) 40( - 15)[ ] 40( - 15)(3-15) 120( - 15)( - 5) Critical values [in (0, 15)]: 5 Sign chart for R'(): Test Numbers R'() R() Increasing Decreasing Demand: Inelastic Elastic x R'() 0 (+) 10 (!) 31. x f() R() f() R'() / Critical values: R'() Sign chart for R'(): R'() R() Increasing Decreasing Demand: Inelastic Elastic x 2; 4 Test Numbers R'() 0 30(+) 5 (!) EXERCISE
7 33. g(x) x g'(x) -0.1 E(x) - g(x) xg'(x) - 50! 0.1x!0.1x E(200) g(x) 50-2 x 500 x - 1 g'(x) - 1 x E(x) - g(x) xg'(x) - 50 " 2 x # x % & " ( E(400) $ 1 x ' 50 x x f() A -k, A, k ositive constants f'() -Ak -k-1 E()!f'() f() Ak!k A!k k 39. The comany's daily cost is increasing by 1.25(20) $25 er day. 41. x ,000 x f() 2, f'() -400 Elasticity of demand: E() E(2) 2 3 < ,000! 400 5! The demand is inelastic; a rice increase will increase revenue. 43. x + 1, x f() 800-1,000 f'() -1,000 Elasticity of demand: E() 1, ! 1, ! 5 E(0.30) 1.5 4! < 1 The demand is inelastic; a rice decrease will decrease revenue. 45. From Problem 41, R() f() 2, R'() 2, Critical values: R'() 2, R"() -800 Since 2.50 is the only critical value and R"(2.50) -800 < 0, the maximum revenue occurs when the rice $ CHAPTER 4 ADDITIONAL DERIVATIVE TOPICS
8 y 47. f(t) 0.34t , 0 t 50 f'(t) Percentage rate of change: 100 f'(t) f(t) t r(t) ln t r'(t) t Relative rate of change of f(t): f'(t)!3.6 f(t) t 11.3! 3.6 ln t! t! 3.6 t ln t C(t) Relative rate of change in 2002: C(12)! (12)! 3.6(12)ln(12) The relative rate of change for robberies annually er 1,000 oulation age 12 and over is aroximately t CHAPTER 4 REVIEW 1. A(t) 2000e 0.09t A(5) 2000e 0.09(5) 2000e or $ A(10) 2000e 0.09(10) 2000e or $ A(20) 2000e 0.09(20) 2000e , or $12, (4-1) 2. d dx (2 ln x + 3ex ) 2 d dx ln x + 3 d dx ex 2 x + 3ex (4-2) 3. d dx e2x-3 e 2x-3 d (2x - 3) (by the chain rule) dx 2e 2x-3 (4-4) 4. y ln(2x + 7) 1 y' 2x + 7 (2) 2 2x + 7 (by the chain rule) (4-4) 5. y ln u, where u 3 + e x. (A) y ln[3 + e x ] (B) dy dx dy du du dx 1 u (ex ) e x (ex ) e x 3 + e x (4-4) CHAPTER 4 REVIEW 205
: Chain Rule, Rules for Exponential and Logarithmic Functions, and Elasticity
4.3-4.5: Chain Rule, Rules for Exponential and Logarithmic Functions, and Elasticity The Chain Rule: Given y = f(g(x)). If the derivatives g (x) and f (g(x)) both exist, then y exists and (f(g(x))) = f
More informationMATH 142 Business Mathematics II
MATH 142 Business Mathematics II Summer, 2016, WEEK 2 JoungDong Kim Week 2: 4.1, 4.2, 4.3, 4.4, 4.5 Chapter 4 Rules for the Derivative Section 4.1 Derivatives of Powers, Exponents, and Sums Differentiation
More informationf(u) can take on many forms. Several of these forms are presented in the following examples. dx, x is a variable.
MATH 56: INTEGRATION USING u-du SUBSTITUTION: u-substitution and the Indefinite Integral: An antiderivative of a function f is a function F such that F (x) = f (x). Any two antiderivatives of f differ
More informationFinal Examination Re - Calculus I 21 December 2015
. (5 points) Given the graph of f below, determine each of the following. Use, or does not exist where appropriate. y (a) (b) x 3 x 2 + (c) x 2 (d) x 2 (e) f(2) = (f) x (g) x (h) f (3) = 3 2 6 5 4 3 2
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Assn.1-.3 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) How long will it take for the value of an account to be $890 if $350 is deposited
More informationCalculus for Business Economics Life Sciences and Social Sciences 13th Edition Barnett SOLUTIONS MANUAL Full download at:
Calculus for Business Economics Life Sciences and Social Sciences 1th Edition Barnett TEST BANK Full download at: https://testbankreal.com/download/calculus-for-business-economics-life-sciencesand-social-sciences-1th-edition-barnett-test-bank/
More informationInstructor: Elhoussine Ghardi Course: calcmanagementspring2018
Student: Date: Instructor: Elhoussine Ghardi Course: calcmanagementspring018 Assignment: HW3spring018 1. Differentiate the following function. f (x) = f(x) = 7 4x + 9 e x. f(x) = 6 ln x + 5x 7 3. Differentiate
More information4.1 Exponential Functions. For Formula 1, the value of n is based on the frequency of compounding. Common frequencies include:
4.1 Exponential Functions Hartfield MATH 2040 Unit 4 Page 1 Recall from algebra the formulas for Compound Interest: Formula 1 For Discretely Compounded Interest A t P 1 r n nt Formula 2 Continuously Compounded
More informationMATH 105 CHAPTER 2 page 1
MATH 105 CHAPTER 2 page 1 RATE OF CHANGE EXAMPLE: A company determines that the cost in dollars to manufacture x cases ofcdʼs Imitations of the Rich and Famous by Kevin Connors is given by C(x) =100 +15x
More informationExam 2 Review (Sections Covered: and )
Exam 2 Review (Sections Covered: 4.1-4.5 and 5.1-5.6) 1. Find the derivative of the following. (a) f(x) = 1 2 x6 3x 4 + 6e x (b) A(s) = s 1/2 ln s ln(13) (c) f(x) = 5e x 8 ln x 2. Given below is the price-demand
More informationName: Math 10250, Final Exam - Version A May 8, 2007
Math 050, Final Exam - Version A May 8, 007 Be sure that you have all 6 pages of the test. Calculators are allowed for this examination. The exam lasts for two hours. The Honor Code is in effect for this
More informationFinal Exam Sample Problems
MATH 00 Sec. Final Exam Sample Problems Please READ this! We will have the final exam on Monday, May rd from 0:0 a.m. to 2:0 p.m.. Here are sample problems for the new materials and the problems from the
More informationEXAM #2 Review. Spring Name: MATH 142, Drost Section # Seat #
Spring 2010 1 EXAM #2 Review Name: MATH 142, Drost Section # Seat # 1. Katy s Kitchen has a total cost function of C(x) = x + 25 to make x jars of jam, and C(x) is measured in dollars. The revenue in dollars,
More information25 Increasing and Decreasing Functions
- 25 Increasing and Decreasing Functions It is useful in mathematics to define whether a function is increasing or decreasing. In this section we will use the differential of a function to determine this
More informationDSC1520 ASSIGNMENT 3 POSSIBLE SOLUTIONS
DSC1520 ASSIGNMENT 3 POSSIBLE SOLUTIONS Question 1 Find the derivative of the function: ( ) Replace with, expand the brackets and simplify before differentiating Apply the Power Rule of differentiation.
More informationTHE USE OF A CALCULATOR, CELL PHONE, OR ANY OTHER ELECTRONIC DEVICE IS NOT PERMITTED DURING THIS EXAMINATION.
MATH 110 FINAL EXAM **Test** December 14, 2009 TEST VERSION A NAME STUDENT NUMBER INSTRUCTOR SECTION NUMBER This examination will be machine processed by the University Testing Service. Use only a number
More informationANSWERS. Part 1 2. i) 1000 ii) iii) iv) 501 v) x x a) i) 4 ii) 3,4 b) p=10,9
ANSWERS. Part. i) 000 ii) 000 iii) 50 iv) 50 v) x +0x+0.. a) i) ii), b) p=0,9. a) i) 0 ii) 9,09 iii) 00 b) The INCREASE in cost incurred when you clean the lake above 50%, i.e. the marginal cost of further
More information2. Find the marginal profit if a profit function is (2x 2 4x + 4)e 4x and simplify.
Additional Review Exam 2 MATH 2053 The only formula that will be provided is for economic lot size (section 12.3) as announced in class, no WebWork questions were given on this. km q = 2a Please note not
More information1.12 Exercises EXERCISES Use integration by parts to compute. ln(x) dx. 2. Compute 1 x ln(x) dx. Hint: Use the substitution u = ln(x).
2 EXERCISES 27 2 Exercises Use integration by parts to compute lnx) dx 2 Compute x lnx) dx Hint: Use the substitution u = lnx) 3 Show that tan x) =/cos x) 2 and conclude that dx = arctanx) + C +x2 Note:
More informationMacroeconomics Module 3: Cobb-Douglas production function practice problems. (The attached PDF file has better formatting.)
Macroeconomics Module 3: Cobb-Douglas production function practice problems (The attached PDF file has better formatting.) The final exam has three types of problems on economic growth! Problems on convergence
More informationWhere It s Used. R. 1 Find f x. using the chain rule. d dx. Page 1 of 7
Elasticity of Demand Learning Objectives: Find relative rates of change Find percentage rates of change Determine whether demand is elastic, inelastic, or has unit elasticity Find elasticity of demand
More informationAdditional Review Exam 1 MATH 2053 Please note not all questions will be taken off of this. Study homework and in class notes as well!
Additional Review Exam 1 MATH 2053 Please note not all questions will be taken off of this. Study homework and in class notes as well! x 2 1 1. Calculate lim x 1 x + 1. (a) 2 (b) 1 (c) (d) 2 (e) the limit
More informationFinal Exam Review - Business Calculus - Spring x x
Final Exam Review - Business Calculus - Spring 2016 Name: 1. (a) Find limit lim x 1 x 1 x 1 (b) Find limit lim x 0 x + 2 4 x 1 2. Use the definition of derivative: dy dx = lim f(x + h) f(x) h 0 h Given
More informationFinal Exam Review. b) lim. 3. Find the limit, if it exists. If the limit is infinite, indicate whether it is + or. [Sec. 2.
Final Exam Review Math 42G 2x, x >. Graph f(x) = { 8 x, x Find the following limits. a) lim x f(x). Label at least four points. [Sec. 2.4, 2.] b) lim f(x) x + c) lim f(x) = Exist/DNE (Circle one) x 2,
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. C) y = - 39x - 80 D) y = x + 8 5
Assn 3.4-3.7 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the equation of the tangent line to the curve when x has the given value. 1)
More informationLogarithmic and Exponential Functions
Asymptotes and Intercepts Logarithmic and exponential functions have asymptotes and intercepts. Consider the functions f(x) = log ax and f(x) = lnx. Both have an x-intercept at (1, 0) and a vertical asymptote
More informationQuantitative Techniques (Finance) 203. Derivatives for Functions with Multiple Variables
Quantitative Techniques (Finance) 203 Derivatives for Functions with Multiple Variables Felix Chan October 2006 1 Introduction In the previous lecture, we discussed the concept of derivative as approximation
More informationTest # 4 Review Math MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Test # 4 Review Math 25 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the integral. ) 4(2x + 5) A) 4 (2x + 5) 4 + C B) 4 (2x + 5) 4 +
More informationMidterm 3. Math Summer Last Name: First Name: Student Number: Section (circle one): 921 (Warren Code) or 922 (Marc Carnovale)
Math 184 - Summer 2011 Midterm 3 Last Name: First Name: Student Number: Section (circle one): 921 (Warren Code) or 922 (Marc Carnovale) Read all of the following information before starting the exam: Calculators
More informationImportance Sampling for Option Pricing. Steven R. Dunbar. Put Options. Monte Carlo Method. Importance. Sampling. Examples.
for for January 25, 2016 1 / 26 Outline for 1 2 3 4 2 / 26 Put Option for A put option is the right to sell an asset at an established price at a certain time. The established price is the strike price,
More informationMath 122 Calculus for Business Admin. and Social Sciences
Math 122 Calculus for Business Admin. and Social Sciences Instructor: Ann Clifton Name: Exam #1 A July 3, 2018 Do not turn this page until told to do so. You will have a total of 1 hour 40 minutes to complete
More informationLecture : The Definite Integral & Fundamental Theorem of Calculus MTH 124. We begin with a theorem which is of fundamental importance.
We begin with a theorem which is of fundamental importance. The Fundamental Theorem of Calculus (FTC) If F (t) is continuous for a t b, then b a F (t) dt = F (b) F (a). Moreover the antiderivative F is
More information3.1 Exponential Functions and Their Graphs Date: Exponential Function
3.1 Exponential Functions and Their Graphs Date: Exponential Function Exponential Function: A function of the form f(x) = b x, where the b is a positive constant other than, and the exponent, x, is a variable.
More information19/01/2017. Profit maximization and competitive supply
Perfectly Cometitive Markets Profit Maximization Marginal Revenue, Marginal Cost, and Profit Maximization Choosing Outut in the Short Run The Cometitive Firm s Short-Run Suly Curve The Short-Run Market
More informationx f(x) D.N.E
Limits Consider the function f(x) x2 x. This function is not defined for x, but if we examine the value of f for numbers close to, we can observe something interesting: x 0 0.5 0.9 0.999.00..5 2 f(x).5.9.999
More informationLaboratory I.9 Applications of the Derivative
Laboratory I.9 Applications of the Derivative Goals The student will determine intervals where a function is increasing or decreasing using the first derivative. The student will find local minima and
More informationPercentage Change and Elasticity
ucsc supplementary notes math 105a Percentage Change and Elasticity 1. Relative and percentage rates of change The derivative of a differentiable function y = fx) describes how the function changes. The
More informationTN 2 - Basic Calculus with Financial Applications
G.S. Questa, 016 TN Basic Calculus with Finance [016-09-03] Page 1 of 16 TN - Basic Calculus with Financial Applications 1 Functions and Limits Derivatives 3 Taylor Series 4 Maxima and Minima 5 The Logarithmic
More information( ) 4 ( )! x f) h(x) = 2cos x + 1
Chapter Prerequisite Skills BLM -.. Identifying Types of Functions. Identify the type of function (polynomial, rational, logarithmic, etc.) represented by each of the following. Justify your response.
More informationExam Review. Exam Review
Chain Rule Chain Rule d dx g(f (x)) = g (f (x))f (x) Chain Rule d dx g(f (x)) = g (f (x))f (x) Write all roots as powers Chain Rule d dx g(f (x)) = g (f (x))f (x) Write all roots as powers ( d dx ) 1 2
More information0 Review: Lines, Fractions, Exponents Lines Fractions Rules of exponents... 5
Contents 0 Review: Lines, Fractions, Exponents 3 0.1 Lines................................... 3 0.2 Fractions................................ 4 0.3 Rules of exponents........................... 5 1 Functions
More informationNotation for the Derivative:
Notation for the Derivative: MA 15910 Lesson 13 Notes Section 4.1 (calculus part of textbook, page 196) Techniques for Finding Derivatives The derivative of a function y f ( x) may be written in any of
More informationNote: I gave a few examples of nearly each of these. eg. #17 and #18 are the same type of problem.
Study Guide for Exam 3 Sections covered: 3.6, Ch 5 and Ch 7 Exam highlights 1 implicit differentiation 3 plain derivatives 3 plain antiderivatives (1 with substitution) 1 Find and interpret Partial Derivatives
More informationChapter 2 Rocket Launch: AREA BETWEEN CURVES
ANSWERS Mathematics (Mathematical Analysis) page 1 Chapter Rocket Launch: AREA BETWEEN CURVES RL-. a) 1,.,.; $8, $1, $18, $0, $, $6, $ b) x; 6(x ) + 0 RL-. a), 16, 9,, 1, 0; 1,,, 7, 9, 11 c) D = (-, );
More informationCentral Limit Theorem, Joint Distributions Spring 2018
Central Limit Theorem, Joint Distributions 18.5 Spring 218.5.4.3.2.1-4 -3-2 -1 1 2 3 4 Exam next Wednesday Exam 1 on Wednesday March 7, regular room and time. Designed for 1 hour. You will have the full
More informationSolving the Black-Scholes Equation
Solving the Black-Scholes Equation An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Initial Value Problem for the European Call rf = F t + rsf S + 1 2 σ2 S 2 F SS for (S,
More informationLecture 11 - Business and Economics Optimization Problems and Asymptotes
Lecture 11 - Business and Economics Optimization Problems and Asymptotes 11.1 More Economics Applications Price Elasticity of Demand One way economists measure the responsiveness of consumers to a change
More informationMarket Demand Demand Elasticity Elasticity & Revenue Marginal Revenue. Market Demand Chapter 15
Market Demand Chapter 15 Outline Deriving market demand from individual demands How responsive is q d to a change in price? (elasticity) What is the relationship between revenue and demand elasticity?
More informationRisk Neutral Modelling Exercises
Risk Neutral Modelling Exercises Geneviève Gauthier Exercise.. Assume that the rice evolution of a given asset satis es dx t = t X t dt + X t dw t where t = ( + sin (t)) and W = fw t : t g is a (; F; P)
More informationAuction theory. Filip An. U.U.D.M. Project Report 2018:35. Department of Mathematics Uppsala University
U.U.D.M. Project Report 28:35 Auction theory Filip An Examensarbete i matematik, 5 hp Handledare: Erik Ekström Examinator: Veronica Crispin Quinonez Augusti 28 Department of Mathematics Uppsala University
More informationNear-Expiry Asymptotics of the Implied Volatility in Local and Stochastic Volatility Models
Mathematical Finance Colloquium, USC September 27, 2013 Near-Expiry Asymptotics of the Implied Volatility in Local and Stochastic Volatility Models Elton P. Hsu Northwestern University (Based on a joint
More informationSTOCHASTIC CALCULUS AND BLACK-SCHOLES MODEL
STOCHASTIC CALCULUS AND BLACK-SCHOLES MODEL YOUNGGEUN YOO Abstract. Ito s lemma is often used in Ito calculus to find the differentials of a stochastic process that depends on time. This paper will introduce
More informationIntroduction to Computational Finance and Financial Econometrics Return Calculations
You can t see this text! Introduction to Computational Finance and Financial Econometrics Return Calculations Eric Zivot Spring 2015 Eric Zivot (Copyright 2015) Return Calculations 1 / 56 Outline 1 The
More informationAuctions 1: Common auctions & Revenue equivalence & Optimal mechanisms. 1 Notable features of auctions. use. A lot of varieties.
1 Notable features of auctions Ancient market mechanisms. use. A lot of varieties. Widespread in Auctions 1: Common auctions & Revenue equivalence & Optimal mechanisms Simple and transparent games (mechanisms).
More informationLecture 7: Computation of Greeks
Lecture 7: Computation of Greeks Ahmed Kebaier kebaier@math.univ-paris13.fr HEC, Paris Outline 1 The log-likelihood approach Motivation The pathwise method requires some restrictive regularity assumptions
More informationFinance II. May 27, F (t, x)+αx f t x σ2 x 2 2 F F (T,x) = ln(x).
Finance II May 27, 25 1.-15. All notation should be clearly defined. Arguments should be complete and careful. 1. (a) Solve the boundary value problem F (t, x)+αx f t x + 1 2 σ2 x 2 2 F (t, x) x2 =, F
More informationComputing Derivatives With Formulas (pages 12-13), Solutions
Computing Derivatives With Formulas (pages 12-13), Solutions This worksheet focuses on computing derivatives using the shortcut formulas, including the power rule, product rule, and quotient rule. We will
More informationPricing Barrier Options under Local Volatility
Abstract Pricing Barrier Options under Local Volatility Artur Sepp Mail: artursepp@hotmail.com, Web: www.hot.ee/seppar 16 November 2002 We study pricing under the local volatility. Our research is mainly
More informationSolving the Black-Scholes Equation
Solving the Black-Scholes Equation An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2014 Initial Value Problem for the European Call The main objective of this lesson is solving
More information1ACE Exercise 3. Name Date Class
1ACE Exercise 3 Investigation 1 3. A rectangular pool is L feet long and W feet wide. A tiler creates a border by placing 1-foot square tiles along the edges of the pool and triangular tiles on the corners,
More informationName: Practice B Exam 2. October 8, 2014
Department of Mathematics University of Notre Dame Math 10250 Elem. of Calc. I Name: Instructor: Practice B Exam 2 October 8, 2014 This exam is in 2 parts on 10 pages and contains 14 problems worth a total
More informationExam M Fall 2005 PRELIMINARY ANSWER KEY
Exam M Fall 005 PRELIMINARY ANSWER KEY Question # Answer Question # Answer 1 C 1 E C B 3 C 3 E 4 D 4 E 5 C 5 C 6 B 6 E 7 A 7 E 8 D 8 D 9 B 9 A 10 A 30 D 11 A 31 A 1 A 3 A 13 D 33 B 14 C 34 C 15 A 35 A
More information2) Endpoints of a diameter (-1, 6), (9, -2) A) (x - 2)2 + (y - 4)2 = 41 B) (x - 4)2 + (y - 2)2 = 41 C) (x - 4)2 + y2 = 16 D) x2 + (y - 2)2 = 25
Math 101 Final Exam Review Revised FA17 (through section 5.6) The following problems are provided for additional practice in preparation for the Final Exam. You should not, however, rely solely upon these
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
MGF 1107 Practice Final Dr. Schnackenberg MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Graph the equation. Select integers for x, -3 x 3. 1) y
More informationAlgebra 1 Semester 1 Final Exam Review (Chapters 2, 3, 4, & 5)
Name: Date Block: Algebra 1 Semester 1 Final Exam Review (Chapters 2, 3, 4, & 5) Chapter 2: Solving Equations 1. Solve each equation. Show your work! 5 1 3n y a) f + 9.8 = 8 b) h + = c) 2p = 54 d) - 15
More informationRho and Delta. Paul Hollingsworth January 29, Introduction 1. 2 Zero coupon bond 1. 3 FX forward 2. 5 Rho (ρ) 4. 7 Time bucketing 6
Rho and Delta Paul Hollingsworth January 29, 2012 Contents 1 Introduction 1 2 Zero coupon bond 1 3 FX forward 2 4 European Call under Black Scholes 3 5 Rho (ρ) 4 6 Relationship between Rho and Delta 5
More informationPRINTABLE VERSION. Practice Final Exam
Page 1 of 25 PRINTABLE VERSION Practice Final Exam Question 1 The following table of values gives a company's annual profits in millions of dollars. Rescale the data so that the year 2003 corresponds to
More informationNet Benefits Test SPP EIS Market May 2012
Net Benefits Test SPP EIS Market May 2012 Topics Net Benefits Test Threshold Steps for Determining Net Benefits Test Threshold Results for May 2011 through May 2012 3 Net Benefits Threshold Price The Net
More informationAlgebra I April 2017 EOC Study Guide Practice Test 1
Name: Algebra I April 2017 EOC Study Guide Practice Test 1 Score: Top 3 Items to Study: 1. 2. 3. 1) The distance a car travels can be found using the formula d = rt, where d is the distance, r is the rate
More informationAMH4 - ADVANCED OPTION PRICING. Contents
AMH4 - ADVANCED OPTION PRICING ANDREW TULLOCH Contents 1. Theory of Option Pricing 2 2. Black-Scholes PDE Method 4 3. Martingale method 4 4. Monte Carlo methods 5 4.1. Method of antithetic variances 5
More informationMidterm Answers 1. a. We can solve for K as a function of L and take the derivative holding Q constant: 1/a. = - b a. K L dl. dk + dl = - b Ê.
Midterm Answers. a. e can solve for K as a function of and take the derivative holding Q constant: K Q d - b Q Á --b/a a < 0 Econ 58 Gary Smith Fall 004 Alternatively, we can take the total derivative:
More informationPlease make sure you bubble in your answers carefully on the bubble sheet and circle your answers on your test booklet.
Math 128 Exam #1 Fall 2017 SPECIAL CODE: 101701 Name Signature: ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Academic Honesty Statement: By signing my name above, I acknowledge
More informationSTOCHASTIC INTEGRALS
Stat 391/FinMath 346 Lecture 8 STOCHASTIC INTEGRALS X t = CONTINUOUS PROCESS θ t = PORTFOLIO: #X t HELD AT t { St : STOCK PRICE M t : MG W t : BROWNIAN MOTION DISCRETE TIME: = t < t 1
More informationCOPYRIGHTED MATERIAL. I.1 Basic Calculus for Finance
I.1 Basic Calculus for Finance I.1.1 INTRODUCTION This chapter introduces the functions that are commonly used in finance and discusses their properties and applications. For instance, the exponential
More informationWITH SKETCH ANSWERS. Postgraduate Certificate in Finance Postgraduate Certificate in Economics and Finance
WITH SKETCH ANSWERS BIRKBECK COLLEGE (University of London) BIRKBECK COLLEGE (University of London) Postgraduate Certificate in Finance Postgraduate Certificate in Economics and Finance SCHOOL OF ECONOMICS,
More informationYou are responsible for upholding the University of Maryland Honor Code while taking this exam.
Econ 300 Spring 013 First Midterm Exam version W Answers This exam consists of 5 multiple choice questions. The maximum duration of the exam is 50 minutes. 1. In the spaces provided on the scantron, write
More informationInformation and uncertainty in a queueing system
Information and uncertainty in a queueing system Refael Hassin December 7, 7 Abstract This aer deals with the effect of information and uncertainty on rofits in an unobservable single server queueing system.
More informationQueens College, CUNY, Department of Computer Science Computational Finance CSCI 365 / 765 Fall 2017 Instructor: Dr. Sateesh Mane.
Queens College, CUNY, Department of Computer Science Computational Finance CSCI 365 / 765 Fall 217 Instructor: Dr. Sateesh Mane c Sateesh R. Mane 217 13 Lecture 13 November 15, 217 Derivation of the Black-Scholes-Merton
More informationChapter 2: Random Variables (Cont d)
Chapter : Random Variables (Cont d) Section.4: The Variance of a Random Variable Problem (1): Suppose that the random variable X takes the values, 1, 4, and 6 with probability values 1/, 1/6, 1/, and 1/6,
More information1 Geometric Brownian motion
Copyright c 05 by Karl Sigman Geometric Brownian motion Note that since BM can take on negative values, using it directly for modeling stock prices is questionable. There are other reasons too why BM is
More informationMacroeconomic Analysis ECON 6022A Fall 2011 Problem Set 4
Macroeconomic Analysis ECON 6022A Fall 2011 Problem Set 4 November 2, 2011 1 The price level and money demand Suppose the price level in the economy is P. Real money demand L(Y, i) is the same as we ve
More informationAlgebra 2 Unit 11 Practice Test Name:
Algebra 2 Unit 11 Practice Test Name: 1. A study of the annual population of the red-winged blackbird in Ft. Mill, South Carolina, shows the population,, can be represented by the function, where the t
More informationDepartment of Mathematics
Department of Mathematics TIME: 3 Hours Setter: AM DATE: 27 July 2015 GRADE 12 PRELIM EXAMINATION MATHEMATICS: PAPER I Total marks: 150 Moderator: JH Name of student: PLEASE READ THE FOLLOWING INSTRUCTIONS
More informationGrowth 2. Chapter 6 (continued)
Growth 2 Chapter 6 (continued) 1. Solow growth model continued 2. Use the model to understand growth 3. Endogenous growth 4. Labor and goods markets with growth 1 Solow Model with Exogenous Labor-Augmenting
More informationAlgebra with Calculus for Business: Review (Summer of 07)
Algebra with Calculus for Business: Review (Summer of 07) 1. Simplify (5 1 m 2 ) 3 (5m 2 ) 4. 2. Simplify (cd) 3 2 (c 3 ) 1 4 (d 1 4 ) 3. 3. Simplify (x 1 2 + y 1 2 )(x 1 2 y 1 2 ) 4. Solve the equation
More informationChapter 5 Integration
Chapter 5 Integration Integration Anti differentiation: The Indefinite Integral Integration by Substitution The Definite Integral The Fundamental Theorem of Calculus 5.1 Anti differentiation: The Indefinite
More informationEcon 214Q Second Midterm August 4, 2005
Econ 214Q Second Midterm August 4, 2005 Name: Answer the questions fully to your best ability. Use the space provided. If you run out of room, use the backsides. No partial credit will be given if you
More informationMath 1314 Week 6 Session Notes
Math 1314 Week 6 Session Notes A few remaining examples from Lesson 7: 0.15 Example 17: The model Nt ( ) = 34.4(1 +.315 t) gives the number of people in the US who are between the ages of 45 and 55. Note,
More informationIntegrating rational functions (Sect. 8.4)
Integrating rational functions (Sect. 8.4) Integrating rational functions, p m(x) q n (x). Polynomial division: p m(x) The method of partial fractions. p (x) (x r )(x r 2 ) p (n )(x). (Repeated roots).
More informationKØBENHAVNS UNIVERSITET (Blok 2, 2011/2012) Naturvidenskabelig kandidateksamen Continuous time finance (FinKont) TIME ALLOWED : 3 hours
This question paper consists of 3 printed pages FinKont KØBENHAVNS UNIVERSITET (Blok 2, 211/212) Naturvidenskabelig kandidateksamen Continuous time finance (FinKont) TIME ALLOWED : 3 hours This exam paper
More informationMath 1130 Exam 2 Review SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Math 1130 Exam 2 Review Provide an appropriate response. 1) Write the following in terms of ln x, ln(x - 3), and ln(x + 1): ln x 3 (x - 3)(x + 1) 2 1) 2) Write the following in terms of ln x, ln(x - 3),
More informationLecture 2. Main Topics: (Part II) Chapter 2 (2-7), Chapter 3. Bayes Theorem: Let A, B be two events, then. The probabilities P ( B), probability of B.
STT315, Section 701, Summer 006 Lecture (Part II) Main Toics: Chater (-7), Chater 3. Bayes Theorem: Let A, B be two events, then B A) = A B) B) A B) B) + A B) B) The robabilities P ( B), B) are called
More informationMicroeconomic theory focuses on a small number of concepts. The most fundamental concept is the notion of opportunity cost.
Microeconomic theory focuses on a small number of concepts. The most fundamental concept is the notion of opportunity cost. Opportunity Cost (or "Wow, I coulda had a V8!") The underlying idea is derived
More informationF A S C I C U L I M A T H E M A T I C I
F A S C I C U L I M A T H E M A T I C I Nr 38 27 Piotr P luciennik A MODIFIED CORRADO-MILLER IMPLIED VOLATILITY ESTIMATOR Abstract. The implied volatility, i.e. volatility calculated on the basis of option
More informationExploring Volatility Derivatives: New Advances in Modelling. Bruno Dupire Bloomberg L.P. NY
Exploring Volatility Derivatives: New Advances in Modelling Bruno Dupire Bloomberg L.P. NY bdupire@bloomberg.net Global Derivatives 2005, Paris May 25, 2005 1. Volatility Products Historical Volatility
More informationFakultät III Wirtschaftswissenschaften Univ.-Prof. Dr. Jan Franke-Viebach
Exam WS 03-4: International Trade ( nd Exam Period) Universität Siegen Fakultät III Wirtschatswissenschaten Univ.-Pro. Dr. Jan Franke-Viebach Exam International Trade Winter Semester 03-4 ( nd Exam Period)
More informationA discretionary stopping problem with applications to the optimal timing of investment decisions.
A discretionary stopping problem with applications to the optimal timing of investment decisions. Timothy Johnson Department of Mathematics King s College London The Strand London WC2R 2LS, UK Tuesday,
More informationLecture 4. Finite difference and finite element methods
Finite difference and finite element methods Lecture 4 Outline Black-Scholes equation From expectation to PDE Goal: compute the value of European option with payoff g which is the conditional expectation
More informationSection 7.1: Continuous Random Variables
Section 71: Continuous Random Variables Discrete-Event Simulation: A First Course c 2006 Pearson Ed, Inc 0-13-142917-5 Discrete-Event Simulation: A First Course Section 71: Continuous Random Variables
More information508-B (Statistics Camp, Wash U, Summer 2016) Asymptotics. Author: Andrés Hincapié and Linyi Cao. This Version: August 9, 2016
Asymtotics Author: Anrés Hincaié an Linyi Cao This Version: August 9, 2016 Asymtotics 3 In arametric moels, we usually assume that the oulation follows some istribution F (x θ) with unknown θ. Knowing
More information