# Objectives. Materials

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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 handheld to approximate the sum of a series Materials TI-8 Plus / TI-83 Plus Introduction One ancient mathematical puzzle is sometimes called the Racecourse Paradox. A person is running a -mile racecourse. Of course, the runner must complete half the course (/ mile) before completing the entire course. But when the first half of the course is completed, the runner must complete half of what remains (/ mile) before finishing, and then complete half of what remains after that (/8 mile) before finishing, and so on. Because there are infinitely many distances that the runner must complete before finishing the race, how can the runner ever hope to complete the entire course? Our everyday experience convinces you that the runner will certainly complete the race, yet adding up infinitely many positive distances would reasonably seem to sum up to an infinite total, or at least would take forever to actually compute! Exploration In symbols, the question behind the Racecourse Paradox is whether it makes sense to add the infinitely many terms and obtain a finite sum, namely. Such an infinite summation is called an infinite series. Another notation that could be used for the infinite series in the Racecourse Paradox is n = where the capital Greek letter sigma denotes a sum and the infinitely many terms are represented by substituting n =,, 3, into the expression -.

2 Calculus Activities The mathematical resolution of the Racecourse Paradox does not involve actually adding up infinitely many terms, but rather by considering the infinite series as representing an infinite sequence of partial sums, each of which has only a finite number of terms. The sequence of partial sums corresponding to n = is n = lim S = n n = lim n = and the series converges. An infinite series whose sequence of partial sums has no limit is a series that diverges. The infinite series encountered in the Racecourse Paradox is an example of a geometric series. A geometric series is one in which the ratio of any term to the one preceding it is always the same. In other words, there is a factor r (the constant ratio) such that each term in the series is r times the term immediately before it. You can see that the ratio is r = for the infinite series n = = S = S = + = S = = S 3 = + + = 8 S n = =. The sum of the infinite series is defined as the limit of the corresponding sequence of partial sums. In this case,

3 Activity 5: Exploring Infinite Series Two other examples of geometric series are and L L3 The ratio for the first example is r =, 5 and the ratio for the second example is r =. 3 In general, any geometric series will have the form a + ar + ar + ar ar n + where a is the first term in the series and r is the constant ratio. In summation notation, a geometric series is written as Note that here the index n starts at, so the first term is ar 0 = a = a. It is not only easy to determine whether geometric series converge or diverge, but their exact sums can also be computed when they converge. The partial sums of a geometric series ar n ar n are S 0 =.. a S = a + ar S = a+ ar+ ar S 3 = a+ ar+ ar + ar 3 S n = a+ ar + ar + ar ar n

4 Calculus Activities To determine whether the series converges, decide whether lim n S n exists. If the limit exists, then that value is the sum of the series. S n can be expressed conveniently, as shown below. First, multiply S n by r to get rs n = ar + ar + ar ar n + ar n + Now, subtract rs n from S n : S n rs n = ( a+ ar + ar + ar ar n ) ( ar + ar + ar ar n + ar n + ) Rearranging the terms on the right side of the equation, S n rs n = a ar+ ar ar + ar ar 3 + ar 3 ar n + ar n ar n + = a ar n + If r, then S n can be solved for by factoring the left side of the equation and dividing both sides of the equation by ( r). S a ar n + n = = r a( r n + ) r This form is used to find a limit for S n. If L < r <, then lim n S n a( r n + ) = lim = n r a r because r n+ approaches 0 as n increases. If either r > or r < L, then the absolute value of r n+ will grow without bound as n increases, and the limit will not exist. Two special values of r still need to be addressed. If r =, then S n = a+ a + a + + a = ( n + )a will also grow without bound as n increases, and the limit will not exist. If r L, then S n = a a+ a a+ +( L) n a is equal to 0 if n is odd and is equal to a if n is even, so the partial sums oscillate in value, and the limit still does not exist.

5 Activity 5: Exploring Infinite Series In summary, the geometric series a converges and has sum if L < r <, and diverges if r or r. r Returning to the prior two examples converges to the sum ( L( 3 ) ) because the first term is a = 5 and the ratio r = L3 is between L and. 5 6 ar n = = = 0 7 As for the second example, L diverges because the ratio r = is greater than. 3 Keep in mind that an infinite series represents a sequence of partial sums. On the home screen, the seq command generates a list based on a formula, an index variable, a starting value, and an ending value. The sum command sums the terms in a list. Used together, you can calculate partial sums.

6 Calculus Activities An example follows. On the home screen, enter seq(/^k,k,,0). Note: The seq command can be found in either the CATALOG or in the LIST Menu under OPS. To convert the entries in the list to fraction form, select :Frac from the MATH Menu. Use the sum command to find the sum of the ten terms in the list. This results in the value of 0 n = the tenth partial sum of the series n = The sequence mode on the graphing handheld is a good environment for investigating infinite series with tables and graphs. After you select the Seq MODE, input the values shown in the screenshot in the Y= editor:

7 Activity 5: Exploring Infinite Series In sequence mode, the independent variable key produces the index variable n. Again, the sum command sums up the terms in a list, and the seq command generates a list based on a formula, an index variable, a starting value, and an ending value. Used together in this way, u(n) represents the nth partial sum of our series. 0 n = View the TABLE to see a list of results for these partial sums for different values of n. To see these results displayed graphically (with values of n on the horizontal axis), first check the FORMAT Menu to make sure that Time is selected. Then set up the viewing window parameters as shown. Select Dot mode, and press s to see the partial sums plotted.

8 Calculus Activities. Write the geometric series in summation notation. Use the graphing handheld to generate a list of the first eleven terms ( to ) in both decimal and fraction form, and calculate the partial sum for. Then enter the sequence of partial sums in Y= editor, and plot each of the partial sums for to for the series.. Write the geometric series in summation notation. Use the graphing handheld to generate a list of the first eleven terms ( to ) in both decimal and fraction form, and calculate the partial sum for. Then enter the sequence of partial sums in the Y= editor, and plot each of the partial sums for to for the series. 3. Explain why the series is a geometric series. Does it converge? If so, find its sum. Use the graphing handheld to generate a list of the first eleven terms ( to ) in both decimal and fraction form, and calculate the partial sum for. Then enter the sequence of partial sums in the Y= editor, and plot each of the partial sums for to for the series.. Explain why the series L is a geometric series. Does it converge? If so, find its sum. Use the graphing handheld to generate a list of the first eleven terms ( to ) in both decimal and fraction form, and calculate the partial sum for. Then enter the sequence of partial sums in the Y= editor, and plot each of the partial sums for to n =0 for the series in GRAPH.

9 Activity 5: Exploring Infinite Series 5. Explain why the series n + is not a geometric series. Do you think it converges? Use the graphing handheld to find the partial sum for, 0, and 50 in y 0. Then plot each of the partial sums for to 0 for the series in GRAPH. The infinite series in Question 5 is an important example called the harmonic series. In summation notation, the harmonic series is written as n = n and it diverges, although very slowly. The graph of its partial sums should remind you of the graph of a logarithmic function. Logarithmic functions also grow without bound, and also very slowly. Infinite series like the harmonic series show why relying on computation of partial sums to decide whether an infinite series converges is not practical. In the case of the harmonic series, the individual terms certainly approach 0. When n becomes large enough, becomes smaller than the precision limits that n the graphing handheld can detect. However, the contributions these small terms make to the total are enough to make the partial sums keep growing without bound. With a geometric series, there is not only a test telling to identify whether it converges or diverges but also a formula that computes the exact sum of the series (in other words, the limit of the partial sums) when it does converge. There are many important series that are not geometric, but it is often very difficult to find the exact sum of a convergent series. The next best thing is to have a test that at least identifies whether a given series converges or diverges. Some tests also give a bound on the error that would occur if a partial sum as an approximation to the sum of the whole series were used if the series were convergent. Series that have terms that alternate in sign (positive and negative) provide us with a simple test that provides both an answer to the convergence question and an error bound.

10 Calculus Activities If is an alternating series such that decrease in absolute value) and then this alternating series converges. ( L) n a n a n + a n lim a n In this situation, the Nth partial sum N ( L ) n a = a n 0 a + a a 3 + ± a N for all n (the size of the terms approximates the sum of with an error no larger than remainder of the series. ( L) n a n a N +, the absolute value of the first term in the Note that this test does not give a formula for the exact sum of the alternating series, but it prescribes a way to approximate that sum with a predetermined level of accuracy of our choosing. For example, the alternating harmonic series is given by ( L) n + - = n n = ( L) n n Although the harmonic series diverges, the alternating harmonic series satisfies the conditions of the alternating series test and must converge. Suppose that you wanted to calculate its sum within 0.0 of the actual value. The absolute value of the 00th term is 00 = 0.0. Calculate the 99th partial sum 99 ( L) n + - = n n = ( L)

11 Activity 5: Exploring Infinite Series The error will be less than 0.0. Use the y 0 feature on the graphing handheld to make this calculation by defining the appropriate sequence of partial sums in the Y= editor, as shown below. You can also compute it on the Home screen. Note: It is known that the alternating harmonic series converges exactly to ln(), the natural logarithm of. Compare this to three decimal places, ln 0.693, to verify that the approximation is actually closer than 0.0 to the exact sum of the series. Keep in mind that the alternating series test establishes a worst-case error bound guarantee and approximations using partial sums could be much closer than this bound.

12 Calculus Activities For each of the following alternating series, determine whether each meets the conditions of the Alternating Series Test. If it does, find the smallest value n such that the nth partial sum would approximate the sum of the series to within 0.0, and find the value of that partial sum n = n = ( L) n + - n ( L) n + - n +! = ( L) n n ( L) n + = +! 3!! 5! - + ( n + )! 8. ( L) n - n + ( n + )! = 3 + 3! 5 + ( L) n + - 5! n + + ( n + )! 9. ( L) n ( L) n - 3 n = ( n)! 3 +! ! 3 n + ( n)! 0. n = ( L) n + n n + 3 ( L) n + n = ( n + ). ( L) n - n + 3 ( L) n = ( - + n + ). n = ( L) n + n = 3 ( L) n + n

### 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

### Section 1.3 P 1 = 1 2. = 1 4 2 8. P n = 1 P 3 = Continuing in this fashion, it should seem reasonable that, for any n = 1, 2, 3,..., = 1 2 4.

Difference Equations to Differential Equations Section. The Sum of a Sequence This section considers the problem of adding together the terms of a sequence. Of course, this is a problem only if more than

### I remember that when I

8. Airthmetic and Geometric Sequences 45 8. ARITHMETIC AND GEOMETRIC SEQUENCES Whenever you tell me that mathematics is just a human invention like the game of chess I would like to believe you. But I

### GEOMETRIC SEQUENCES AND SERIES

4.4 Geometric Sequences and Series (4 7) 757 of a novel and every day thereafter increase their daily reading by two pages. If his students follow this suggestion, then how many pages will they read during

### TI-83 Plus Graphing Calculator Keystroke Guide

TI-83 Plus Graphing Calculator Keystroke Guide In your textbook you will notice that on some pages a key-shaped icon appears next to a brief description of a feature on your graphing calculator. In this

### Some sequences have a fixed length and have a last term, while others go on forever.

Sequences and series Sequences A sequence is a list of numbers (actually, they don t have to be numbers). Here is a sequence: 1, 4, 9, 16 The order makes a difference, so 16, 9, 4, 1 is a different sequence.

### Lies My Calculator and Computer Told Me

Lies My Calculator and Computer Told Me 2 LIES MY CALCULATOR AND COMPUTER TOLD ME Lies My Calculator and Computer Told Me See Section.4 for a discussion of graphing calculators and computers with graphing

### 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

### NUMBER SYSTEMS. William Stallings

NUMBER SYSTEMS William Stallings The Decimal System... The Binary System...3 Converting between Binary and Decimal...3 Integers...4 Fractions...5 Hexadecimal Notation...6 This document available at WilliamStallings.com/StudentSupport.html

### 9.2 Summation Notation

9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a

### 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

### To discuss this topic fully, let us define some terms used in this and the following sets of supplemental notes.

INFINITE SERIES SERIES AND PARTIAL SUMS What if we wanted to sum up the terms of this sequence, how many terms would I have to use? 1, 2, 3,... 10,...? Well, we could start creating sums of a finite number

### 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

### TI-86 Graphing Calculator Keystroke Guide

TI-86 Graphing Calculator Keystroke Guide In your textbook you will notice that on some pages a key-shaped icon appears next to a brief description of a feature on your graphing calculator. In this guide

### Review of Fundamental Mathematics

Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools

### How do you compare numbers? On a number line, larger numbers are to the right and smaller numbers are to the left.

The verbal answers to all of the following questions should be memorized before completion of pre-algebra. Answers that are not memorized will hinder your ability to succeed in algebra 1. Number Basics

### Overview. Essential Questions. Precalculus, Quarter 4, Unit 4.5 Build Arithmetic and Geometric Sequences and Series

Sequences and Series Overview Number of instruction days: 4 6 (1 day = 53 minutes) Content to Be Learned Write arithmetic and geometric sequences both recursively and with an explicit formula, use them

### 2. THE x-y PLANE 7 C7

2. THE x-y PLANE 2.1. The Real Line When we plot quantities on a graph we can plot not only integer values like 1, 2 and 3 but also fractions, like 3½ or 4¾. In fact we can, in principle, plot any real

### Fractions and Decimals

Fractions and Decimals Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles December 1, 2005 1 Introduction If you divide 1 by 81, you will find that 1/81 =.012345679012345679... The first

### 6.1. The Exponential Function. Introduction. Prerequisites. Learning Outcomes. Learning Style

The Exponential Function 6.1 Introduction In this block we revisit the use of exponents. We consider how the expression a x is defined when a is a positive number and x is irrational. Previously we have

### Maths Workshop for Parents 2. Fractions and Algebra

Maths Workshop for Parents 2 Fractions and Algebra What is a fraction? A fraction is a part of a whole. There are two numbers to every fraction: 2 7 Numerator Denominator 2 7 This is a proper (or common)

### Calculator Notes for the TI-Nspire and TI-Nspire CAS

INTRODUCTION Calculator Notes for the Getting Started: Navigating Screens and Menus Your handheld is like a small computer. You will always work in a document with one or more problems and one or more

### 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

### Grade 6 Mathematics Performance Level Descriptors

Limited Grade 6 Mathematics Performance Level Descriptors A student performing at the Limited Level demonstrates a minimal command of Ohio s Learning Standards for Grade 6 Mathematics. A student at this

### Basic Use of the TI-84 Plus

Basic Use of the TI-84 Plus Topics: Key Board Sections Key Functions Screen Contrast Numerical Calculations Order of Operations Built-In Templates MATH menu Scientific Notation The key VS the (-) Key Navigation

### 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

### GEOMETRIC SEQUENCES AND SERIES

GEOMETRIC SEQUENCES AND SERIES Summary 1. Geometric sequences... 2 2. Exercise... 6 3. Geometric sequence applications to financial mathematics... 6 4. Vocabulary... 7 5. Exercises :... 8 6. Geometric

### 1 Error in Euler s Method

1 Error in Euler s Method Experience with Euler s 1 method raises some interesting questions about numerical approximations for the solutions of differential equations. 1. What determines the amount of

### Math Workshop October 2010 Fractions and Repeating Decimals

Math Workshop October 2010 Fractions and Repeating Decimals This evening we will investigate the patterns that arise when converting fractions to decimals. As an example of what we will be looking at,

### Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum

Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum UNIT I: The Hyperbolic Functions basic calculus concepts, including techniques for curve sketching, exponential and logarithmic

### Limit processes are the basis of calculus. For example, the derivative. f f (x + h) f (x)

SEC. 4.1 TAYLOR SERIES AND CALCULATION OF FUNCTIONS 187 Taylor Series 4.1 Taylor Series and Calculation of Functions Limit processes are the basis of calculus. For example, the derivative f f (x + h) f

### 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

### Section 6-3 Arithmetic and Geometric Sequences

466 6 SEQUENCES, SERIES, AND PROBABILITY Section 6- Arithmetic and Geometric Sequences Arithmetic and Geometric Sequences nth-term Formulas Sum Formulas for Finite Arithmetic Series Sum Formulas for Finite

### LIES MY CALCULATOR AND COMPUTER TOLD ME

LIES MY CALCULATOR AND COMPUTER TOLD ME See Section Appendix.4 G for a discussion of graphing calculators and computers with graphing software. A wide variety of pocket-size calculating devices are currently

### Prentice Hall: Middle School Math, Course 1 2002 Correlated to: New York Mathematics Learning Standards (Intermediate)

New York Mathematics Learning Standards (Intermediate) Mathematical Reasoning Key Idea: Students use MATHEMATICAL REASONING to analyze mathematical situations, make conjectures, gather evidence, and construct

### Temperature Scales. The metric system that we are now using includes a unit that is specific for the representation of measured temperatures.

Temperature Scales INTRODUCTION The metric system that we are now using includes a unit that is specific for the representation of measured temperatures. The unit of temperature in the metric system is

### Properties of sequences Since a sequence is a special kind of function it has analogous properties to functions:

Sequences and Series A sequence is a special kind of function whose domain is N - the set of natural numbers. The range of a sequence is the collection of terms that make up the sequence. Just as the word

### Basic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704.

Basic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704. The purpose of this Basic Math Refresher is to review basic math concepts so that students enrolled in PUBP704:

### STATISTICS FOR PSYCH MATH REVIEW GUIDE

STATISTICS FOR PSYCH MATH REVIEW GUIDE ORDER OF OPERATIONS Although remembering the order of operations as BEDMAS may seem simple, it is definitely worth reviewing in a new context such as statistics formulae.

### Squares Inside Squares

The High School Math Project Focus on Algebra Squares Inside Squares (Sequences) Objective Students use the TI-92 to generate a sequence by determining the areas of squares inscribed in squares. They then

### 1, 1 2, 1 3, 1 4,... 2 nd term. 1 st term

1 Sequences 11 Overview A (numerical) sequence is a list of real numbers in which each entry is a function of its position in the list The entries in the list are called terms For example, 1, 1, 1 3, 1

### LESSON PLANS FOR PERCENTAGES, FRACTIONS, DECIMALS, AND ORDERING Lesson Purpose: The students will be able to:

LESSON PLANS FOR PERCENTAGES, FRACTIONS, DECIMALS, AND ORDERING Lesson Purpose: The students will be able to: 1. Change fractions to decimals. 2. Change decimals to fractions. 3. Change percents to decimals.

### How to Type Math Symbols in MS Word

Word has some great tools for helping you input math symbols and math equations into your assignment or Word document! On the next few pages, you will see screens and instructions for how to input various

### EXPONENTS. To the applicant: KEY WORDS AND CONVERTING WORDS TO EQUATIONS

To the applicant: The following information will help you review math that is included in the Paraprofessional written examination for the Conejo Valley Unified School District. The Education Code requires

### x if x 0, x if x < 0.

Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the

### Estimated Pre Calculus Pacing Timeline

Estimated Pre Calculus Pacing Timeline 2010-2011 School Year The timeframes listed on this calendar are estimates based on a fifty-minute class period. You may need to adjust some of them from time to

### > 2. Error and Computer Arithmetic

> 2. Error and Computer Arithmetic Numerical analysis is concerned with how to solve a problem numerically, i.e., how to develop a sequence of numerical calculations to get a satisfactory answer. Part

### IB Maths SL Sequence and Series Practice Problems Mr. W Name

IB Maths SL Sequence and Series Practice Problems Mr. W Name Remember to show all necessary reasoning! Separate paper is probably best. 3b 3d is optional! 1. In an arithmetic sequence, u 1 = and u 3 =

### Stanford Math Circle: Sunday, May 9, 2010 Square-Triangular Numbers, Pell s Equation, and Continued Fractions

Stanford Math Circle: Sunday, May 9, 00 Square-Triangular Numbers, Pell s Equation, and Continued Fractions Recall that triangular numbers are numbers of the form T m = numbers that can be arranged in

### TYPES OF NUMBERS. Example 2. Example 1. Problems. Answers

TYPES OF NUMBERS When two or more integers are multiplied together, each number is a factor of the product. Nonnegative integers that have exactly two factors, namely, one and itself, are called prime

### No Solution Equations Let s look at the following equation: 2 +3=2 +7

5.4 Solving Equations with Infinite or No Solutions So far we have looked at equations where there is exactly one solution. It is possible to have more than solution in other types of equations that are

Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function

### Texas Instruments TI-83, TI-83 Plus Graphics Calculator I.1 Systems of Linear Equations

Part I: Texas Instruments TI-83, TI-83 Plus Graphics Calculator I.1 Systems of Linear Equations I.1.1 Basics: Press the ON key to begin using your TI-83 calculator. If you need to adjust the display contrast,

### 4/1/2017. PS. Sequences and Series FROM 9.2 AND 9.3 IN THE BOOK AS WELL AS FROM OTHER SOURCES. TODAY IS NATIONAL MANATEE APPRECIATION DAY

PS. Sequences and Series FROM 9.2 AND 9.3 IN THE BOOK AS WELL AS FROM OTHER SOURCES. TODAY IS NATIONAL MANATEE APPRECIATION DAY 1 Oh the things you should learn How to recognize and write arithmetic sequences

### CHAPTER 3 Numbers and Numeral Systems

CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,

### Mathematical Procedures

CHAPTER 6 Mathematical Procedures 168 CHAPTER 6 Mathematical Procedures The multidisciplinary approach to medicine has incorporated a wide variety of mathematical procedures from the fields of physics,

### 6.4 Normal Distribution

Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under

### MATH BOOK OF PROBLEMS SERIES. New from Pearson Custom Publishing!

MATH BOOK OF PROBLEMS SERIES New from Pearson Custom Publishing! The Math Book of Problems Series is a database of math problems for the following courses: Pre-algebra Algebra Pre-calculus Calculus Statistics

### Student Performance Q&A:

Student Performance Q&A: AP Calculus AB and Calculus BC Free-Response Questions The following comments on the free-response questions for AP Calculus AB and Calculus BC were written by the Chief Reader,

### Unit 1 Number Sense. In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions.

Unit 1 Number Sense In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions. BLM Three Types of Percent Problems (p L-34) is a summary BLM for the material

### 3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.

### Multiplying Fractions

. Multiplying Fractions. OBJECTIVES 1. Multiply two fractions. Multiply two mixed numbers. Simplify before multiplying fractions 4. Estimate products by rounding Multiplication is the easiest of the four

### A.2. Exponents and Radicals. Integer Exponents. What you should learn. Exponential Notation. Why you should learn it. Properties of Exponents

Appendix A. Exponents and Radicals A11 A. Exponents and Radicals What you should learn Use properties of exponents. Use scientific notation to represent real numbers. Use properties of radicals. Simplify

### Mathematics 31 Pre-calculus and Limits

Mathematics 31 Pre-calculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals

### Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below.

Infinite Algebra 1 Kuta Software LLC Common Core Alignment Software version 2.05 Last revised July 2015 Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below. High School

### Matrix Operations on a TI-83 Graphing Calculator

Matrix Operations on a TI-83 Graphing Calculator Christopher Carl Heckman Department of Mathematics and Statistics, Arizona State University checkman@math.asu.edu (This paper is based on a talk given in

### 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

### 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

### Maths Targets Year 1 Addition and Subtraction Measures. N / A in year 1.

Number and place value Maths Targets Year 1 Addition and Subtraction Count to and across 100, forwards and backwards beginning with 0 or 1 or from any given number. Count, read and write numbers to 100

### Charlesworth School Year Group Maths Targets

Charlesworth School Year Group Maths Targets Year One Maths Target Sheet Key Statement KS1 Maths Targets (Expected) These skills must be secure to move beyond expected. I can compare, describe and solve

### In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data.

MATHEMATICS: THE LEVEL DESCRIPTIONS In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data. Attainment target

### Microeconomics Sept. 16, 2010 NOTES ON CALCULUS AND UTILITY FUNCTIONS

DUSP 11.203 Frank Levy Microeconomics Sept. 16, 2010 NOTES ON CALCULUS AND UTILITY FUNCTIONS These notes have three purposes: 1) To explain why some simple calculus formulae are useful in understanding

### Georgia Standards of Excellence Curriculum Map. Mathematics. GSE 8 th Grade

Georgia Standards of Excellence Curriculum Map Mathematics GSE 8 th Grade These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement. GSE Eighth Grade

### 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

### MITES Physics III Summer Introduction 1. 3 Π = Product 2. 4 Proofs by Induction 3. 5 Problems 5

MITES Physics III Summer 010 Sums Products and Proofs Contents 1 Introduction 1 Sum 1 3 Π Product 4 Proofs by Induction 3 5 Problems 5 1 Introduction These notes will introduce two topics: A notation which

### MATH-0910 Review Concepts (Haugen)

Unit 1 Whole Numbers and Fractions MATH-0910 Review Concepts (Haugen) Exam 1 Sections 1.5, 1.6, 1.7, 1.8, 2.1, 2.2, 2.3, 2.4, and 2.5 Dividing Whole Numbers Equivalent ways of expressing division: a b,

### EQUATIONS and INEQUALITIES

EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line

### Absolute Value of Reasoning

About Illustrations: Illustrations of the Standards for Mathematical Practice (SMP) consist of several pieces, including a mathematics task, student dialogue, mathematical overview, teacher reflection

### Square Roots. Learning Objectives. Pre-Activity

Section 1. Pre-Activity Preparation Square Roots Our number system has two important sets of numbers: rational and irrational. The most common irrational numbers result from taking the square root of non-perfect

### Week 13 Trigonometric Form of Complex Numbers

Week Trigonometric Form of Complex Numbers Overview In this week of the course, which is the last week if you are not going to take calculus, we will look at how Trigonometry can sometimes help in working

### Solving Inequalities Examples

Solving Inequalities Examples 1. Joe and Katie are dancers. Suppose you compare their weights. You can make only one of the following statements. Joe s weight is less than Kate s weight. Joe s weight is

### Angles and Quadrants. Angle Relationships and Degree Measurement. Chapter 7: Trigonometry

Chapter 7: Trigonometry Trigonometry is the study of angles and how they can be used as a means of indirect measurement, that is, the measurement of a distance where it is not practical or even possible

### Figure 1. A typical Laboratory Thermometer graduated in C.

SIGNIFICANT FIGURES, EXPONENTS, AND SCIENTIFIC NOTATION 2004, 1990 by David A. Katz. All rights reserved. Permission for classroom use as long as the original copyright is included. 1. SIGNIFICANT FIGURES

### Paramedic Program Pre-Admission Mathematics Test Study Guide

Paramedic Program Pre-Admission Mathematics Test Study Guide 05/13 1 Table of Contents Page 1 Page 2 Page 3 Page 4 Page 5 Page 6 Page 7 Page 8 Page 9 Page 10 Page 11 Page 12 Page 13 Page 14 Page 15 Page

### Algebra I Credit Recovery

Algebra I Credit Recovery COURSE DESCRIPTION: The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics,

### 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

### 5.1 Radical Notation and Rational Exponents

Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots

### Appendix F: Mathematical Induction

Appendix F: Mathematical Induction Introduction In this appendix, you will study a form of mathematical proof called mathematical induction. To see the logical need for mathematical induction, take another

### Arithmetic Progression

Worksheet 3.6 Arithmetic and Geometric Progressions Section 1 Arithmetic Progression An arithmetic progression is a list of numbers where the difference between successive numbers is constant. The terms

### Mathematics, Basic Math and Algebra

NONRESIDENT TRAINING COURSE Mathematics, Basic Math and Algebra NAVEDTRA 14139 DISTRIBUTION STATEMENT A: Approved for public release; distribution is unlimited. PREFACE About this course: This is a self-study

### Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

### Parallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular.

CONDENSED L E S S O N. Parallel and Perpendicular In this lesson you will learn the meaning of parallel and perpendicular discover how the slopes of parallel and perpendicular lines are related use slopes

### Decimals and Percentages

Decimals and Percentages Specimen Worksheets for Selected Aspects Paul Harling b recognise the number relationship between coordinates in the first quadrant of related points Key Stage 2 (AT2) on a line

### Utah Core Curriculum for Mathematics

Core Curriculum for Mathematics correlated to correlated to 2005 Chapter 1 (pp. 2 57) Variables, Expressions, and Integers Lesson 1.1 (pp. 5 9) Expressions and Variables 2.2.1 Evaluate algebraic expressions

### PLOTTING DATA AND INTERPRETING GRAPHS

PLOTTING DATA AND INTERPRETING GRAPHS Fundamentals of Graphing One of the most important sets of skills in science and mathematics is the ability to construct graphs and to interpret the information they

### Unit Essential Question: When do we need standard symbols, operations, and rules in mathematics? (CAIU)

Page 1 Whole Numbers Unit Essential : When do we need standard symbols, operations, and rules in mathematics? (CAIU) M6.A.3.2.1 Whole Number Operations Dividing with one digit (showing three forms of answers)

### Grade 6 Mathematics Assessment. Eligible Texas Essential Knowledge and Skills

Grade 6 Mathematics Assessment Eligible Texas Essential Knowledge and Skills STAAR Grade 6 Mathematics Assessment Mathematical Process Standards These student expectations will not be listed under a separate

### Developmental Math Course Outcomes and Objectives

Developmental Math Course Outcomes and Objectives I. Math 0910 Basic Arithmetic/Pre-Algebra Upon satisfactory completion of this course, the student should be able to perform the following outcomes and