Name: ID: Discussion Section:

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Name: ID: Discussion Section:"

Transcription

1 Math 28 Midterm 3 Spring 2009 Name: ID: Discussion Section: This exam consists of 6 questions: 4 multiple choice questions worth 5 points each 2 hand-graded questions worth a total of 30 points. INSTRUCTIONS: Read each problem carefully and answer the question as written. You may use a non-graphing calculator and a standard sized (no larger than 4 6 ) index card worth of notes for the exam, but you may use no other aids. Record your answer to the multiple choice questions on the accompanying answer card. Show your work on the written problems and write clearly.. Suppose we wish to use the method of Lagrange multipliers to minimize the function f x, y = 2 x2 3 x y y 2 2 subject to the constraint y=x. We can find that this minimum occurs at the point (5, 4) What is the absolute value of the Lagrange Multiplier, λ corresponding to this minimum point? (a) (b) 2 (c) 3 (d) 4 (e) 5 (f) 6 (g) 7 (h) 8 (i) 9 (j) 0 Solution: (g). We construct F x, y, = f x, y g x, y = 2 x2 3 x y y 2 2 y x. Taking partial derivatives, we get: F x = x 3 y, F y = 3 x 2 y, F = y x. Setting F x =F y =0 and solving for λ we get = x 3 y= 3 x 2 y. Plugging our point (5, 4) into these equations, we get = 7. Page of 3

2 Math 28 Midterm 3 Spring Find the equation of the straight line that minimizes the least-squares error for the points (,), (2,3), (3,6). (a) y=2 x 3 2 (b) y= 5 2 x 5 3 (c) y= 3 2 x 3 2 (d) y=3 x (e) y= 5 2 x (f) y=3 x 2 (g) y= 3 2 x 4 3 (h) y= 7 3 x 3 (i) y=5 x 4 Solution: (b). We calculate: N =3, x=6, y=0, x 2 =4, x y=25. So, A= = = Then, B= = Page 2 of 3

3 Math 28 Midterm 3 Spring Let f x = x. Let p 2 x be the second Taylor polynomial of f(x) at x =. Find p 2 3. (Choose the nearest answer.) (a). (b).2 (c).3 (d).4 (e).5 (f).6 (g).7 (h).8 (i).9 (j) 2.0 Solution: (e). We calculate: f ' x =, f ' ' x = /2 2 x 4 x. 3 /2 So, f =, f ' = 2, f ' ' = 4. So, p x = / 4 2 x 2 2! x 2. So, p 2 3 = 2 =.5. Page 3 of 3

4 Math 28 Midterm 3 Spring Which of the following is the third Taylor polynomial, p 3 x for f x =sin 3 x at x=0? (a) 4.5 x 2 (b) 3 x 27 x 3 (c) 3 x 4.5 x 3 (d) 9 x 2 (e) 3 x 9 x 2 27 x 3 (f) 3 x 9 x 2 27 x 3 (g) 3 x 4.5x x 3 (h) 3 3x 4.5 x x 3 (i) 3 x 4.5 x 3 (j) x x 2 x 3 Solution: (i). Notice that f ' x =3 cos3 x, f ' ' x = 9sin 3 x, f ' ' ' x = 27 cos 3 x. So, f 0 =0, f ' 0 =3, f ' ' 0 =0, f ' ' ' 0 = 27. So, p 3 x =3 x 27 3! x 3 =3 x 4.5 x 3. Page 4 of 3

5 Math 28 Midterm 3 Spring 2009 k 5. Consider the infinite series k = answer. k 2. Find s 2, the second partial sum of the series. Select the closest (a) 0.97 (b).05 (c).4 (d).22 (e).33 (f).4 (g).67 (h).8 (i).98 (j) 2.22 Solution: (b) Recall that s 2 is just the sum of the first two terms of the series. In this case, it will be Page 5 of 3

6 Math 28 Midterm 3 Spring 2009 Answers for Problems 6 and 7: (a) none of them (b) I only (c) II only (d) III only (e) I and II only (f) I and III only (g) II and III only (h) all of them 6. Which of the following series converge? 3 I. k = k II. III. k = k n= 2 k 3/2 Solution: (d). All three are p-series. III is the only one where p >. 7. Which of the following series converge? I. 3 k k = 5 k II. III. k = n= k 3 k 2 3 k 5 k Solution: (h). I. is a geometric series with r= 3 5. II. converges by comparison to 3 k. III. converges by comparison to 2 3 k. Page 6 of 3

7 Math 28 Midterm 3 Spring 2009 Answers for Problems 8 and 9: (Choose the closest answer in each case.) (a) 2.0 (b) 2.4 (c) 2.8 (d) 3.2 (e) 3.6 (f) 4.0 (g) 4.4 (h) 4.8 (i) 5.2 (j) The series diverges. 8. Find the sum of the geometric series: Solution: (c). This is a geometric series with a = 5 and r = Thus the sum is Find the sum of the geometric series: Solution: (j). This is a geometric series with a = 5 and r = -.2. Since r the series must diverge. Page 7 of 3

8 Math 28 Midterm 3 Spring Suppose that a $20 billion income tax is proposed. Assuming that every member of the population has the same 92% marginal propensity to consume, how much additional aggregate spending will be generated by the tax cut? (Hint: Express the additional consumption in the form of a geometric series.) (a) $8.6 billion (b) $23 billion (c) $25 billion (d) $96 billion (e) $2 billion (f) $20 billion (g) $60 billion (h) $230 billion (i) $250 billion Solution: (h). Recall that the additional spending will be given by: = =230 Page 8 of 3

9 Math 28 Midterm 3 Spring Which of the following statements describes a correct application of the comparison test to the infinite series ln k k? 3 / 2 (a) (b) (c) (d) (e) (f) (g) (h) ln k k 3/ 2 ln k ln k k 3/ 2 ln k ln k k 3/ 2 ln k ln k k 3/ 2 ln k ln k k 3/ 2 ln k k 3/ 2 ln k k 3/ 2 ln k k 3/ 2 for all k and so for all k and so for all k and so for all k and so k 3/ 2 for all k and so k 3/ 2 for all k and so k 3/ 2 for all k and so k 3/ 2 for all k and so converges. diverges. converges. diverges. converges. diverges. converges. diverges. Solution: (e). We know that it is a p-series with p>. ln k k 3/ 2 k 3/ 2 and we also know that k 3 / 2 is convergent since Page 9 of 3

10 Math 28 Midterm 3 Spring The Taylor series for sin x at x = 0 is given by: sin x= x 3! x3 5! x5 7! x7 Which of the following is the Taylor series at x = 0 for f x =x sin x? (a) 2 x2 4 x4 42 x6 (b) x 2 x2 24 x4 720 x6 (c) x 2 6 x4 20 x x8. (d) 6 x2 20 x4 720 x6 (e) 2 x2 20 x x6 (f) x 6 x3 20 x x7 (g) x 2 2 x4 20 x x8 (h) 6 x2 4 x4 42 x6 (i) 2 x2 24 x x6 (j) x 2 6 x4 20 x x8 Solution: (c). The Taylor Series expansion for sin x is x 6 x3 20 x x7. So, the Taylor series for x sin x is x 2 6 x4 20 x x8. So, the Taylor series for x sin x is x 2 6 x4 20 x x8. Page 0 of 3

11 Math 28 Midterm 3 Spring 2009 Answers for Problems 3 and 4: (a) (b) (c) (d) (e) (f) x 2 x 4 x 6 x 2 x 4 x 6 x x 2 x 3 x x 2 x 3 x 4 2 x 3 x 2 4 x 3 2 x 3 x2 4 x3 (g) x 2 x2 3 x3 4 x4 (h) x 3 5 x5 7 x7 9 x9 (i) 2 x 3 x2 4 x3 3. Which of these series the Taylor series for at x = 0? x 2 Solution: (a). The Taylor series at x = 0 for is given by: x x = x x2 x 3 Substituting x 2 for x gives us: x = x2 x 4 x 6 4. Which of these is the Taylor series for ln x at x = 0? Solution: (g). Notice that ln x C= x dx= x x 2 x 3 dx So, ln x C=x 2 x2 3 x3 4 x4. Plugging in x=0, we find that ln C =0 and so C=0. Page of 3

12 Math 28 Midterm 3 Spring 2009 Name: ID: Discussion Section: Written Problem Instructions: Answer below. Show your work. Write clearly. You may receive partial credit for partially worked-out answers. 5. (5 Points) A certain manufacturer's production is governed by the Cobb-Douglas function f x, y =240 x 0.25 y 0.75 where x and y represent input units of labor and capital respectively. In addition, suppose that each unit of labor costs $50 and each unit of capital costs $00. (a) (5 Points) Find the cost of inputs and amount of production when x=00 and y=200. Solution: Production is given by f 00, 200 = ,363. The cost of inputs is given by C x, y = =25,000. (b) (0 Points) Use the method of Lagrange multipliers to find the levels of labor and capital input which maximize production and whose cost of inputs totals In addition, find this maximum production level. Solution: Our objective function is f(x,y) and our constraint is C x, y =25,000. We formulate this as a Lagrange multiplier problem by considering the function F x, y, where: F x, y, =240 x 0.25 y , x 00 y. We then calculate: F x =60 x 0.75 y and F y =80 x 0.25 y So, we find that =.2 x 0.75 y 0.75 =.8 x 0.25 y We multiply through by x 0.75 y 0.25 to get.2 y=.8 x or y=.5 x. Plugging back to the constraint, we then find that 25,000=50 x 00.5 x =200 x and so x=25. From here, we find that y=87.5. So, production is optimized at 25 units of labor and 87.5 units of capital. Plugging these into out production formula, we calculate our optimal production to be: f 25,87.5 = ,662. Page 2 of 3

13 Math 28 Midterm 3 Spring 2009 Name: ID: Discussion Section: Written Problem Instructions: Answer below. Show your work. Write clearly. You may receive partial credit for partially worked-out answers. 6. (a) (5 Points) Find the Taylor series for e x2 at x=0. (Hint: You should know the Taylor Series for e x at x=0. ) You do not need to use sigma-notation, but give at least the first four non-zero terms. Solution: We know that e x = x x2 2! x3 x. So, e 2= x 2 x 4 3! 2! x6 3!. (b) (5 Points) Find a series that converges to 0 e x 2 dx x Solution: 0 e 2 dx= 0 x2 x4 2! x6 3! dx= x x3 3 x5 5 2! x7 7 3! = (c) (5 Points) Sum up the first four terms of the series from part (b) to get an estimate for the integral 0 e x 2 dx. Solution: Summing up the first four terms of the series, we get the estimate: = Page 3 of 3

Practice Final Math 122 Spring 12 Instructor: Jeff Lang

Practice Final Math 122 Spring 12 Instructor: Jeff Lang Practice Final Math Spring Instructor: Jeff Lang. Find the limit of the sequence a n = ln (n 5) ln (3n + 8). A) ln ( ) 3 B) ln C) ln ( ) 3 D) does not exist. Find the limit of the sequence a n = (ln n)6

More information

MATH 2300 review problems for Exam 3 ANSWERS

MATH 2300 review problems for Exam 3 ANSWERS MATH 300 review problems for Exam 3 ANSWERS. Check whether the following series converge or diverge. In each case, justify your answer by either computing the sum or by by showing which convergence test

More information

HOMEWORK 4 SOLUTIONS. All questions are from Vector Calculus, by Marsden and Tromba

HOMEWORK 4 SOLUTIONS. All questions are from Vector Calculus, by Marsden and Tromba HOMEWORK SOLUTIONS All questions are from Vector Calculus, by Marsden and Tromba Question :..6 Let w = f(x, y) be a function of two variables, and let x = u + v, y = u v. Show that Solution. By the chain

More information

f (x) = x 2 (x) = (1 + x 2 ) 2 (x) = (1 + x 2 ) 2 + 8x 2 (1 + x 2 ) 3 8x (1 + x 2 ) x (1 + x 2 ) 3 48x3 f(0) = tan 1 (0) = 0 f (0) = 2

f (x) = x 2 (x) = (1 + x 2 ) 2 (x) = (1 + x 2 ) 2 + 8x 2 (1 + x 2 ) 3 8x (1 + x 2 ) x (1 + x 2 ) 3 48x3 f(0) = tan 1 (0) = 0 f (0) = 2 Math 5 Exam # Practice Problem Solutions. Find the Maclaurin series for tan (x (feel free just to write out the first few terms. Answer: Let f(x = tan (x. Then the first few derivatives of f are: f (x

More information

Taylor and Maclaurin Series

Taylor and Maclaurin Series Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions

More information

SOLUTIONS. f x = 6x 2 6xy 24x, f y = 3x 2 6y. To find the critical points, we solve

SOLUTIONS. f x = 6x 2 6xy 24x, f y = 3x 2 6y. To find the critical points, we solve SOLUTIONS Problem. Find the critical points of the function f(x, y = 2x 3 3x 2 y 2x 2 3y 2 and determine their type i.e. local min/local max/saddle point. Are there any global min/max? Partial derivatives

More information

Sample Midterm Solutions

Sample Midterm Solutions Sample Midterm Solutions Instructions: Please answer both questions. You should show your working and calculations for each applicable problem. Correct answers without working will get you relatively few

More information

Math 53 Worksheet Solutions- Minmax and Lagrange

Math 53 Worksheet Solutions- Minmax and Lagrange Math 5 Worksheet Solutions- Minmax and Lagrange. Find the local maximum and minimum values as well as the saddle point(s) of the function f(x, y) = e y (y x ). Solution. First we calculate the partial

More information

A power series about x = a is the series of the form

A power series about x = a is the series of the form POWER SERIES AND THE USES OF POWER SERIES Elizabeth Wood Now we are finally going to start working with a topic that uses all of the information from the previous topics. The topic that we are going to

More information

EXAM. Practice for Third Exam. Math , Fall Dec 1, 2003 ANSWERS

EXAM. Practice for Third Exam. Math , Fall Dec 1, 2003 ANSWERS EXAM Practice for Third Exam Math 35-006, Fall 003 Dec, 003 ANSWERS i Problem series.) A.. In each part determine if the series is convergent or divergent. If it is convergent find the sum. (These are

More information

Constrained optimization.

Constrained optimization. ams/econ 11b supplementary notes ucsc Constrained optimization. c 2010, Yonatan Katznelson 1. Constraints In many of the optimization problems that arise in economics, there are restrictions on the values

More information

Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

More information

Introduction to Series and Sequences Math 121 Calculus II D Joyce, Spring 2013

Introduction to Series and Sequences Math 121 Calculus II D Joyce, Spring 2013 Introduction to Series and Sequences Math Calculus II D Joyce, Spring 03 The goal. The main purpose of our study of series and sequences is to understand power series. A power series is like a polynomial

More information

Lecture 24: Series finale

Lecture 24: Series finale Lecture 24: Series finale Nathan Pflueger 2 November 20 Introduction This lecture will discuss one final convergence test and error bound for series, which concerns alternating series. A general summary

More information

MATH 132: CALCULUS II SYLLABUS

MATH 132: CALCULUS II SYLLABUS MATH 32: CALCULUS II SYLLABUS Prerequisites: Successful completion of Math 3 (or its equivalent elsewhere). Math 27 is normally not a sufficient prerequisite for Math 32. Required Text: Calculus: Early

More information

Math 201 Lecture 23: Power Series Method for Equations with Polynomial

Math 201 Lecture 23: Power Series Method for Equations with Polynomial Math 201 Lecture 23: Power Series Method for Equations with Polynomial Coefficients Mar. 07, 2012 Many examples here are taken from the textbook. The first number in () refers to the problem number in

More information

The Method of Partial Fractions Math 121 Calculus II Spring 2015

The Method of Partial Fractions Math 121 Calculus II Spring 2015 Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method

More information

Money and Banking Prof. Yamin Ahmad ECON 354 Spring 2007

Money and Banking Prof. Yamin Ahmad ECON 354 Spring 2007 Money and Banking Prof. Yamin Ahmad ECON 354 Spring 2007 Midterm Exam # 2 Name Id # Instructions: There are two parts to this midterm. Part A consists of multiple choice questions. Please mark the answers

More information

6.8 Taylor and Maclaurin s Series

6.8 Taylor and Maclaurin s Series 6.8. TAYLOR AND MACLAURIN S SERIES 357 6.8 Taylor and Maclaurin s Series 6.8.1 Introduction The previous section showed us how to find the series representation of some functions by using the series representation

More information

Student Name: Instructor:

Student Name: Instructor: Math 30 - Final Exam Version Student Name: Instructor: INSTRUCTIONS READ THIS NOW Print your name in CAPITAL letters. It is your responsibility that your test paper is submitted to your professor. Please

More information

Representation of functions as power series

Representation of functions as power series Representation of functions as power series Dr. Philippe B. Laval Kennesaw State University November 9, 008 Abstract This document is a summary of the theory and techniques used to represent functions

More information

(a) We have x = 3 + 2t, y = 2 t, z = 6 so solving for t we get the symmetric equations. x 3 2. = 2 y, z = 6. t 2 2t + 1 = 0,

(a) We have x = 3 + 2t, y = 2 t, z = 6 so solving for t we get the symmetric equations. x 3 2. = 2 y, z = 6. t 2 2t + 1 = 0, Name: Solutions to Practice Final. Consider the line r(t) = 3 + t, t, 6. (a) Find symmetric equations for this line. (b) Find the point where the first line r(t) intersects the surface z = x + y. (a) We

More information

Lecture 4: Equality Constrained Optimization. Tianxi Wang

Lecture 4: Equality Constrained Optimization. Tianxi Wang Lecture 4: Equality Constrained Optimization Tianxi Wang wangt@essex.ac.uk 2.1 Lagrange Multiplier Technique (a) Classical Programming max f(x 1, x 2,..., x n ) objective function where x 1, x 2,..., x

More information

EXERCISES FOR CHAPTER 6: Taylor and Maclaurin Series

EXERCISES FOR CHAPTER 6: Taylor and Maclaurin Series EXERCISES FOR CHAPTER 6: Taylor and Maclaurin Series. Find the first 4 terms of the Taylor series for the following functions: (a) ln centered at a=, (b) centered at a=, (c) sin centered at a = 4. (a)

More information

Lectures 5-6: Taylor Series

Lectures 5-6: Taylor Series Math 1d Instructor: Padraic Bartlett Lectures 5-: Taylor Series Weeks 5- Caltech 213 1 Taylor Polynomials and Series As we saw in week 4, power series are remarkably nice objects to work with. In particular,

More information

Problem Set #3-Key. (1) What is the largest possible number of pizza slices you could consume in a month given your budget?

Problem Set #3-Key. (1) What is the largest possible number of pizza slices you could consume in a month given your budget? Problem Set #3-Key Sonoma State University Economics 305-Intermediate Microeconomic Theory Dr. Cuellar Suppose that you consume only two goods pizza (X) and beer (Y). Your budget on pizza and beer is $120

More information

To give it a definition, an implicit function of x and y is simply any relationship that takes the form:

To give it a definition, an implicit function of x and y is simply any relationship that takes the form: 2 Implicit function theorems and applications 21 Implicit functions The implicit function theorem is one of the most useful single tools you ll meet this year After a while, it will be second nature to

More information

Taylor Polynomials and Taylor Series Math 126

Taylor Polynomials and Taylor Series Math 126 Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will

More information

Geometric Series and Annuities

Geometric Series and Annuities Geometric Series and Annuities Our goal here is to calculate annuities. For example, how much money do you need to have saved for retirement so that you can withdraw a fixed amount of money each year for

More information

Power Series. We already know how to express one function as a series. Take a look at following equation: 1 1 r = r n. n=0

Power Series. We already know how to express one function as a series. Take a look at following equation: 1 1 r = r n. n=0 Math -0 Calc Power, Taylor, and Maclaurin Series Survival Guide One of the harder concepts that we have to become comfortable with during this semester is that of sequences and series We are working with

More information

x 2 x 2 cos 1 x x2, lim 1. If x > 0, multiply all three parts by x > 0, we get: x x cos 1 x x, lim lim x cos 1 lim = 5 lim sin 5x

x 2 x 2 cos 1 x x2, lim 1. If x > 0, multiply all three parts by x > 0, we get: x x cos 1 x x, lim lim x cos 1 lim = 5 lim sin 5x Homework 4 3.4,. Show that x x cos x x holds for x 0. Solution: Since cos x, multiply all three parts by x > 0, we get: x x cos x x, and since x 0 x x 0 ( x ) = 0, then by Sandwich theorem, we get: x 0

More information

1 3 4 = 8i + 20j 13k. x + w. y + w

1 3 4 = 8i + 20j 13k. x + w. y + w ) Find the point of intersection of the lines x = t +, y = 3t + 4, z = 4t + 5, and x = 6s + 3, y = 5s +, z = 4s + 9, and then find the plane containing these two lines. Solution. Solve the system of equations

More information

The Geometric Series

The Geometric Series The Geometric Series Professor Jeff Stuart Pacific Lutheran University c 8 The Geometric Series Finite Number of Summands The geometric series is a sum in which each summand is obtained as a common multiple

More information

Infinite series, improper integrals, and Taylor series

Infinite series, improper integrals, and Taylor series Chapter Infinite series, improper integrals, and Taylor series. Introduction This chapter has several important and challenging goals. The first of these is to understand how concepts that were discussed

More information

Course Notes for Math 162: Mathematical Statistics Approximation Methods in Statistics

Course Notes for Math 162: Mathematical Statistics Approximation Methods in Statistics Course Notes for Math 16: Mathematical Statistics Approximation Methods in Statistics Adam Merberg and Steven J. Miller August 18, 6 Abstract We introduce some of the approximation methods commonly used

More information

constraint. Let us penalize ourselves for making the constraint too big. We end up with a

constraint. Let us penalize ourselves for making the constraint too big. We end up with a Chapter 4 Constrained Optimization 4.1 Equality Constraints (Lagrangians) Suppose we have a problem: Maximize 5, (x 1, 2) 2, 2(x 2, 1) 2 subject to x 1 +4x 2 =3 If we ignore the constraint, we get the

More information

MATH 31B: MIDTERM 1 REVIEW. 1. Inverses. yx 3y = 1. x = 1 + 3y y 4( 1) + 32 = 1

MATH 31B: MIDTERM 1 REVIEW. 1. Inverses. yx 3y = 1. x = 1 + 3y y 4( 1) + 32 = 1 MATH 3B: MIDTERM REVIEW JOE HUGHES. Inverses. Let f() = 3. Find the inverse g() for f. Solution: Setting y = ( 3) and solving for gives and g() = +3. y 3y = = + 3y y. Let f() = 4 + 3. Find a domain on

More information

Algebra 2 Final Exam Review Ch 7-14

Algebra 2 Final Exam Review Ch 7-14 Algebra 2 Final Exam Review Ch 7-14 Short Answer 1. Simplify the radical expression. Use absolute value symbols if needed. 2. Simplify. Assume that all variables are positive. 3. Multiply and simplify.

More information

Fourier Series. Chapter Some Properties of Functions Goal Preliminary Remarks

Fourier Series. Chapter Some Properties of Functions Goal Preliminary Remarks Chapter 3 Fourier Series 3.1 Some Properties of Functions 3.1.1 Goal We review some results about functions which play an important role in the development of the theory of Fourier series. These results

More information

12.3 Geometric Sequences. Copyright Cengage Learning. All rights reserved.

12.3 Geometric Sequences. Copyright Cengage Learning. All rights reserved. 12.3 Geometric Sequences Copyright Cengage Learning. All rights reserved. 1 Objectives Geometric Sequences Partial Sums of Geometric Sequences What Is an Infinite Series? Infinite Geometric Series 2 Geometric

More information

Average rate of change of y = f(x) with respect to x as x changes from a to a + h:

Average rate of change of y = f(x) with respect to x as x changes from a to a + h: L15-1 Lecture 15: Section 3.4 Definition of the Derivative Recall the following from Lecture 14: For function y = f(x), the average rate of change of y with respect to x as x changes from a to b (on [a,

More information

Graduate Introductory Macroeconomics: Exam 1

Graduate Introductory Macroeconomics: Exam 1 Graduate Introductory Macroeconomics: Exam Name: Student ID: You have 60 minutes. Answer in English or Japanese.. Consumption tax in a one-period model 0 points) The household determines the consumption

More information

ANALYTICAL MATHEMATICS FOR APPLICATIONS 2016 LECTURE NOTES Series

ANALYTICAL MATHEMATICS FOR APPLICATIONS 2016 LECTURE NOTES Series ANALYTICAL MATHEMATICS FOR APPLICATIONS 206 LECTURE NOTES 8 ISSUED 24 APRIL 206 A series is a formal sum. Series a + a 2 + a 3 + + + where { } is a sequence of real numbers. Here formal means that we don

More information

Project: The Harmonic Series, the Integral Test

Project: The Harmonic Series, the Integral Test . Let s start with the sequence a n = n. Does this sequence converge or diverge? Explain. To determine if a sequence converges, we just take a limit: lim n n sequence converges to 0. = 0. The. Now consider

More information

Math 150, Fall 2009 Solutions to Practice Final Exam [1] The equation of the tangent line to the curve. cosh y = x + sin y + cos y

Math 150, Fall 2009 Solutions to Practice Final Exam [1] The equation of the tangent line to the curve. cosh y = x + sin y + cos y Math 150, Fall 2009 Solutions to Practice Final Exam [1] The equation of the tangent line to the curve at the point (0, 0) is cosh y = x + sin y + cos y Answer : y = x Justification: The equation of the

More information

Math 120 Final Exam Practice Problems, Form: A

Math 120 Final Exam Practice Problems, Form: A Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,

More information

Lecture VI. Review of even and odd functions Definition 1 A function f(x) is called an even function if. f( x) = f(x)

Lecture VI. Review of even and odd functions Definition 1 A function f(x) is called an even function if. f( x) = f(x) ecture VI Abstract Before learning to solve partial differential equations, it is necessary to know how to approximate arbitrary functions by infinite series, using special families of functions This process

More information

1.(6pts) Find symmetric equations of the line L passing through the point (2, 5, 1) and perpendicular to the plane x + 3y z = 9.

1.(6pts) Find symmetric equations of the line L passing through the point (2, 5, 1) and perpendicular to the plane x + 3y z = 9. .(6pts Find symmetric equations of the line L passing through the point (, 5, and perpendicular to the plane x + 3y z = 9. (a x = y + 5 3 = z (b x (c (x = ( 5(y 3 = z + (d x (e (x + 3(y 3 (z = 9 = y 3

More information

Deriving Demand Functions - Examples 1

Deriving Demand Functions - Examples 1 Deriving Demand Functions - Examples 1 What follows are some examples of different preference relations and their respective demand functions. In all the following examples, assume we have two goods x

More information

10.2 Series and Convergence

10.2 Series and Convergence 10.2 Series and Convergence Write sums using sigma notation Find the partial sums of series and determine convergence or divergence of infinite series Find the N th partial sums of geometric series and

More information

1 Mathematical Induction

1 Mathematical Induction Extra Credit Homework Problems Note: these problems are of varying difficulty, so you might want to assign different point values for the different problems. I have suggested the point values each problem

More information

PROBLEM SET. Practice Problems for Exam #2. Math 2350, Fall Nov. 7, 2004 Corrected Nov. 10 ANSWERS

PROBLEM SET. Practice Problems for Exam #2. Math 2350, Fall Nov. 7, 2004 Corrected Nov. 10 ANSWERS PROBLEM SET Practice Problems for Exam #2 Math 2350, Fall 2004 Nov. 7, 2004 Corrected Nov. 10 ANSWERS i Problem 1. Consider the function f(x, y) = xy 2 sin(x 2 y). Find the partial derivatives f x, f y,

More information

Series Convergence Tests Math 122 Calculus III D Joyce, Fall 2012

Series Convergence Tests Math 122 Calculus III D Joyce, Fall 2012 Some series converge, some diverge. Series Convergence Tests Math 22 Calculus III D Joyce, Fall 202 Geometric series. We ve already looked at these. We know when a geometric series converges and what it

More information

Power Series Lecture Notes

Power Series Lecture Notes Power Series Lecture Notes A power series is a polynomial with infinitely many terms. Here is an example: $ 0ab œ â Like a polynomial, a power series is a function of. That is, we can substitute in different

More information

Solutions for Review Problems

Solutions for Review Problems olutions for Review Problems 1. Let be the triangle with vertices A (,, ), B (4,, 1) and C (,, 1). (a) Find the cosine of the angle BAC at vertex A. (b) Find the area of the triangle ABC. (c) Find a vector

More information

1 Lecture: Integration of rational functions by decomposition

1 Lecture: Integration of rational functions by decomposition Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.

More information

Sequences and Series

Sequences and Series Sequences and Series Consider the following sum: 2 + 4 + 8 + 6 + + 2 i + The dots at the end indicate that the sum goes on forever. Does this make sense? Can we assign a numerical value to an infinite

More information

AFM Ch.12 - Practice Test

AFM Ch.12 - Practice Test AFM Ch.2 - Practice Test Multiple Choice Identify the choice that best completes the statement or answers the question.. Form a sequence that has two arithmetic means between 3 and 89. a. 3, 33, 43, 89

More information

REVIEW OF MICROECONOMICS

REVIEW OF MICROECONOMICS ECO 352 Spring 2010 Precepts Weeks 1, 2 Feb. 1, 8 REVIEW OF MICROECONOMICS Concepts to be reviewed Budget constraint: graphical and algebraic representation Preferences, indifference curves. Utility function

More information

CHAPTER 4 Consumer Choice

CHAPTER 4 Consumer Choice CHAPTER 4 Consumer Choice CHAPTER OUTLINE 4.1 Preferences Properties of Consumer Preferences Preference Maps 4.2 Utility Utility Function Ordinal Preference Utility and Indifference Curves Utility and

More information

Definition. Sequences. Sequence As A Function. Finding the n th Term. Converging Sequences. Divergent Sequences 12/13/2010. A of numbers. Lesson 9.

Definition. Sequences. Sequence As A Function. Finding the n th Term. Converging Sequences. Divergent Sequences 12/13/2010. A of numbers. Lesson 9. Definition Sequences Lesson 9.1 A of numbers Listed according to a given Typically written as a 1, a 2, a n Often shortened to { a n } Example 1, 3, 5, 7, 9, A sequence of numbers Finding the n th Term

More information

Objectives. Materials

Objectives. Materials Activity 5 Exploring Infinite Series Objectives Identify a geometric series Determine convergence and sum of geometric series Identify a series that satisfies the alternating series test Use a graphing

More information

Approximating functions by Taylor Polynomials.

Approximating functions by Taylor Polynomials. Chapter 4 Approximating functions by Taylor Polynomials. 4.1 Linear Approximations We have already seen how to approximate a function using its tangent line. This was the key idea in Euler s method. If

More information

Economics 326: Marshallian Demand and Comparative Statics. Ethan Kaplan

Economics 326: Marshallian Demand and Comparative Statics. Ethan Kaplan Economics 326: Marshallian Demand and Comparative Statics Ethan Kaplan September 17, 2012 Outline 1. Utility Maximization: General Formulation 2. Marshallian Demand 3. Homogeneity of Degree Zero of Marshallian

More information

MATH 105: PRACTICE PROBLEMS FOR SERIES: SPRING Problems

MATH 105: PRACTICE PROBLEMS FOR SERIES: SPRING Problems MATH 05: PRACTICE PROBLEMS FOR SERIES: SPRING 20 INSTRUCTOR: STEVEN MILLER (SJM@WILLIAMS.EDU).. Sequences and Series.. Problems Question : Let a n =. Does the series +n+n 2 n= a n converge or diverge?

More information

Problem 1 (10 pts) Find the radius of convergence and interval of convergence of the series

Problem 1 (10 pts) Find the radius of convergence and interval of convergence of the series 1 Problem 1 (10 pts) Find the radius of convergence and interval of convergence of the series a n n=1 n(x + 2) n 5 n 1. n(x + 2)n Solution: Do the ratio test for the absolute convergence. Let a n =. Then,

More information

SOLUTIONS TO HOMEWORK ASSIGNMENT #5, Math 253

SOLUTIONS TO HOMEWORK ASSIGNMENT #5, Math 253 SOLUTIONS TO HOMEWORK ASSIGNMENT #5, Math 53. For what values of the constant k does the function f(x, y) =kx 3 + x +y 4x 4y have (a) no critical points; (b) exactly one critical point; (c) exactly two

More information

1 Functions, Graphs and Limits

1 Functions, Graphs and Limits 1 Functions, Graphs and Limits 1.1 The Cartesian Plane In this course we will be dealing a lot with the Cartesian plane (also called the xy-plane), so this section should serve as a review of it and its

More information

ECON Mathematical Economics - ANSWERS FINAL EXAM. 1. (a)- The consumer's utility maximization problem is written as:

ECON Mathematical Economics - ANSWERS FINAL EXAM. 1. (a)- The consumer's utility maximization problem is written as: ECON 331 - Mathematical Economics - ANSWERS FINAL EXAM 1. a- The consumer's utility maximization problem is written as: max x,y [xy + y2 + 2x + 2y] s.t. 6x + 10y = m, x 0, y 0 - The associated Lagrangian

More information

The variable λ is a dummy variable called a Lagrange multiplier ; we only really care about the values of x, y, and z.

The variable λ is a dummy variable called a Lagrange multiplier ; we only really care about the values of x, y, and z. Math a Lagrange Multipliers Spring, 009 The method of Lagrange multipliers allows us to maximize or minimize functions with the constraint that we only consider points on a certain surface To find critical

More information

1 Introduction to Differential Equations

1 Introduction to Differential Equations 1 Introduction to Differential Equations A differential equation is an equation that involves the derivative of some unknown function. For example, consider the equation f (x) = 4x 3. (1) This equation

More information

Math 018 Review Sheet v.3

Math 018 Review Sheet v.3 Math 018 Review Sheet v.3 Tyrone Crisp Spring 007 1.1 - Slopes and Equations of Lines Slopes: Find slopes of lines using the slope formula m y y 1 x x 1. Positive slope the line slopes up to the right.

More information

Chapter 4 Online Appendix: The Mathematics of Utility Functions

Chapter 4 Online Appendix: The Mathematics of Utility Functions Chapter 4 Online Appendix: The Mathematics of Utility Functions We saw in the text that utility functions and indifference curves are different ways to represent a consumer s preferences. Calculus can

More information

Economics 101 Homework #1 Fall 2014 Due 09/18/2014 in lecture

Economics 101 Homework #1 Fall 2014 Due 09/18/2014 in lecture Economics 101 Homework #1 Fall 2014 Due 09/18/2014 in lecture Directions: The homework will be collected in a box before the lecture. Please place your name, TA name and section number on top of the homework

More information

AP Calculus BC. All students enrolling in AP Calculus BC should have successfully completed AP Calculus AB.

AP Calculus BC. All students enrolling in AP Calculus BC should have successfully completed AP Calculus AB. AP Calculus BC Course Description: Advanced Placement Calculus BC is primarily concerned with developing the students understanding of the concepts of calculus and providing experiences with its methods

More information

Nonhomogeneous Linear Equations

Nonhomogeneous Linear Equations Nonhomogeneous Linear Equations In this section we learn how to solve second-order nonhomogeneous linear differential equations with constant coefficients, that is, equations of the form ay by cy G x where

More information

Constrained Optimization: The Method of Lagrange Multipliers:

Constrained Optimization: The Method of Lagrange Multipliers: Constrained Optimization: The Method of Lagrange Multipliers: Suppose the equation p(x,) x 60x 7 00 models profit when x represents the number of handmade chairs and is the number of handmade rockers produced

More information

Linear and quadratic Taylor polynomials for functions of several variables.

Linear and quadratic Taylor polynomials for functions of several variables. ams/econ 11b supplementary notes ucsc Linear quadratic Taylor polynomials for functions of several variables. c 010, Yonatan Katznelson Finding the extreme (minimum or maximum) values of a function, is

More information

1 Calculus of Several Variables

1 Calculus of Several Variables 1 Calculus of Several Variables Reading: [Simon], Chapter 14, p. 300-31. 1.1 Partial Derivatives Let f : R n R. Then for each x i at each point x 0 = (x 0 1,..., x 0 n) the ith partial derivative is defined

More information

Differential Equations

Differential Equations Differential Equations A differential equation is an equation that contains an unknown function and one or more of its derivatives. Here are some examples: y = 1, y = x, y = xy y + 2y + y = 0 d 3 y dx

More information

Math 432 HW 2.5 Solutions

Math 432 HW 2.5 Solutions Math 432 HW 2.5 Solutions Assigned: 1-10, 12, 13, and 14. Selected for Grading: 1 (for five points), 6 (also for five), 9, 12 Solutions: 1. (2y 3 + 2y 2 ) dx + (3y 2 x + 2xy) dy = 0. M/ y = 6y 2 + 4y N/

More information

An important theme in this book is to give constructive definitions of mathematical objects. Thus, for instance, if you needed to evaluate.

An important theme in this book is to give constructive definitions of mathematical objects. Thus, for instance, if you needed to evaluate. Chapter 10 Series and Approximations An important theme in this book is to give constructive definitions of mathematical objects. Thus, for instance, if you needed to evaluate 1 0 e x2 dx, you could set

More information

Solutions to old Exam 1 problems

Solutions to old Exam 1 problems Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections

More information

4.3 Lagrange Approximation

4.3 Lagrange Approximation 206 CHAP. 4 INTERPOLATION AND POLYNOMIAL APPROXIMATION Lagrange Polynomial Approximation 4.3 Lagrange Approximation Interpolation means to estimate a missing function value by taking a weighted average

More information

Increasing for all. Convex for all. ( ) Increasing for all (remember that the log function is only defined for ). ( ) Concave for all.

Increasing for all. Convex for all. ( ) Increasing for all (remember that the log function is only defined for ). ( ) Concave for all. 1. Differentiation The first derivative of a function measures by how much changes in reaction to an infinitesimal shift in its argument. The largest the derivative (in absolute value), the faster is evolving.

More information

Determine whether the sequence is arithmetic, geometric, or neither. 18, 11, 4,

Determine whether the sequence is arithmetic, geometric, or neither. 18, 11, 4, Over Lesson 10 1 Determine whether the sequence is arithmetic, geometric, or neither. 18, 11, 4, A. arithmetic B. geometric C. neither Over Lesson 10 1 Determine whether the sequence is arithmetic, geometric,

More information

Integrals of Rational Functions

Integrals of Rational Functions Integrals of Rational Functions Scott R. Fulton Overview A rational function has the form where p and q are polynomials. For example, r(x) = p(x) q(x) f(x) = x2 3 x 4 + 3, g(t) = t6 + 4t 2 3, 7t 5 + 3t

More information

Introduction to Fourier Series

Introduction to Fourier Series Introduction to Fourier Series MA 16021 October 15, 2014 Even and odd functions Definition A function f(x) is said to be even if f( x) = f(x). The function f(x) is said to be odd if f( x) = f(x). Graphically,

More information

Math 2280 Section 002 [SPRING 2013] 1

Math 2280 Section 002 [SPRING 2013] 1 Math 2280 Section 002 [SPRING 2013] 1 Today well learn about a method for solving systems of differential equations, the method of elimination, that is very similar to the elimination methods we learned

More information

In this section, we will consider techniques for solving problems of this type.

In this section, we will consider techniques for solving problems of this type. Constrained optimisation roblems in economics typically involve maximising some quantity, such as utility or profit, subject to a constraint for example income. We shall therefore need techniques for solving

More information

1 Algebra - Partial Fractions

1 Algebra - Partial Fractions 1 Algebra - Partial Fractions Definition 1.1 A polynomial P (x) in x is a function that may be written in the form P (x) a n x n + a n 1 x n 1 +... + a 1 x + a 0. a n 0 and n is the degree of the polynomial,

More information

Partial Fractions. (x 1)(x 2 + 1)

Partial Fractions. (x 1)(x 2 + 1) Partial Fractions Adding rational functions involves finding a common denominator, rewriting each fraction so that it has that denominator, then adding. For example, 3x x 1 3x(x 1) (x + 1)(x 1) + 1(x +

More information

Solutions for 2015 Algebra II with Trigonometry Exam Written by Ashley Johnson and Miranda Bowie, University of North Alabama

Solutions for 2015 Algebra II with Trigonometry Exam Written by Ashley Johnson and Miranda Bowie, University of North Alabama Solutions for 0 Algebra II with Trigonometry Exam Written by Ashley Johnson and Miranda Bowie, University of North Alabama. For f(x = 6x 4(x +, find f(. Solution: The notation f( tells us to evaluate the

More information

Long-Run Average Cost. Econ 410: Micro Theory. Long-Run Average Cost. Long-Run Average Cost. Economies of Scale & Scope Minimizing Cost Mathematically

Long-Run Average Cost. Econ 410: Micro Theory. Long-Run Average Cost. Long-Run Average Cost. Economies of Scale & Scope Minimizing Cost Mathematically Slide 1 Slide 3 Econ 410: Micro Theory & Scope Minimizing Cost Mathematically Friday, November 9 th, 2007 Cost But, at some point, average costs for a firm will tend to increase. Why? Factory space and

More information

13.4 THE CROSS PRODUCT

13.4 THE CROSS PRODUCT 710 Chapter Thirteen A FUNDAMENTAL TOOL: VECTORS 62. Use the following steps and the results of Problems 59 60 to show (without trigonometry) that the geometric and algebraic definitions of the dot product

More information

Math 241, Exam 1 Information.

Math 241, Exam 1 Information. Math 241, Exam 1 Information. 9/24/12, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.1-14.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)

More information

Economics 2020a / HBS 4010 / HKS API-111 FALL 2010 Solutions to Practice Problems for Lectures 1 to 4

Economics 2020a / HBS 4010 / HKS API-111 FALL 2010 Solutions to Practice Problems for Lectures 1 to 4 Economics 00a / HBS 4010 / HKS API-111 FALL 010 Solutions to Practice Problems for Lectures 1 to 4 1.1. Quantity Discounts and the Budget Constraint (a) The only distinction between the budget line with

More information

CALCULUS Lab set Practice 5

CALCULUS Lab set Practice 5 CALCULUS Lab set Practice 5 New Maple Commands plot([seq( [n,a(n)],..n)], style=point) Plots the sequence a(n) sum(a(n),..n )Adds the N first terms of the sequence a(n) sum(a(n),..infinity) Finds the sum

More information

6. Define log(z) so that π < I log(z) π. Discuss the identities e log(z) = z and log(e w ) = w.

6. Define log(z) so that π < I log(z) π. Discuss the identities e log(z) = z and log(e w ) = w. hapter omplex integration. omplex number quiz. Simplify 3+4i. 2. Simplify 3+4i. 3. Find the cube roots of. 4. Here are some identities for complex conjugate. Which ones need correction? z + w = z + w,

More information

The Mathematics Diagnostic Test

The Mathematics Diagnostic Test The Mathematics iagnostic Test Mock Test and Further Information 010 In welcome week, students will be asked to sit a short test in order to determine the appropriate lecture course, tutorial group, whether

More information