# Mathematical Procedures

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 CHAPTER 6 Mathematical Procedures

2 168 CHAPTER 6 Mathematical Procedures The multidisciplinary approach to medicine has incorporated a wide variety of mathematical procedures from the fields of physics, chemistry, and engineering. The information presented in this chapter is designed as a self-teaching refresher course to be used as a review of basic mathematical procedures. Some of the more advanced mathematical concepts, including the section on descriptive statistics, should also help the practitioner to interpret data presented in medical journals and scientific articles. Fundamental Axioms Commutative Axiom a + b 5 b + a ab 5 ba When two or more numbers are added or multiplied together, their order does not affect the result. Associative Axiom (a + b) + c 5 a + (b + c) (ab)c 5 a(bc) When three or more numbers are added together, the way they are grouped or associated makes no difference in the result. The same holds true for multiplication. Distributive Axiom a(b + c) 5 ab + ac A coefficient (multiplier) of a sum may be distributed as a multiplier of each term. Order of Precedence A convention has been established for the order in which numerical operations are performed. This is to prevent confusion when evaluating expressions such as , which could be either 18 or 36. The following rules apply:

3 Fractions If the numerical expression does not contain fences (such as parentheses), then operations are carried out in the following order: a. Raising numbers to powers or extracting roots of numbers. b. Multiplication or division. c. Addition or subtraction If the numerical expression does contain fences, then follow the procedure in Rule 1, starting with the innermost set of parentheses. The sequence is round fences (parentheses), square fences [brackets], double fences {braces}. Once the fences have been eliminated, the expression can be evaluated following Rule {3 3 2 [ ( ) 20] + 12} {3 3 2 [ (6 4) 20] + 12} {3 3 2 [ ] + 12} {3 3 2 [ ] + 12} { } { } Fractions When a number is expressed as a fraction (e.g., 3 5), the number above the line (3) is called the numerator and the number below the line (5) the denominator. Multiplication Property of Fractions a 3 c b 3 c 5 a b 1c 2 02 The numerator and denominator of a fraction may be multiplied or divided by the same nonzero number to produce a fraction of equal value.

4 170 CHAPTER 6 Mathematical Procedures Simplify (reduce) the fraction 9 12 Solution 1. Find the largest integer that will evenly divide both the numerator and denominator. 2. Divide both the numerator and denominator by that number. The largest whole number is a9 3 b 4 a12 3 b Multiplication of Fractions a b 3 c d 5 ac bd ? Solution 1. Multiply the numerators Multiply the denominators Simplify the resulting fraction if possible Division of Fractions a b 4 c d 5 a b 3 d c 5 ad bc

5 Fractions 171 Find the quotient: Solution 1. Invert the divisor. Change 2 3 to Multiply the dividend by the inverted divisor Simplify if possible. Addition and Subtraction of Fractions with the Same Denominator a b 1 c 1a 1 c2 5 b b a b 2 c 1a 2 c2 5 b b Solution ? 1. Combine numerators Write the resultant fraction with the new numerator and the same denominator. 3. Simplify if possible

6 172 CHAPTER 6 Mathematical Procedures Addition and Subtraction of Fractions with Different Denominators To add or subtract fractions that do not have the same denominator, it is first necessary to express them as fractions having the same denominators. To find a common denominator, find an integer that is evenly divisible by each denominator. The smallest or least common denominator (LCD) is the most convenient. Solution 1. First find the LCD as follows: ? a. Express each denominator as the product of primes (integers greater than 1 that are evenly divisible by only themselves and 1). b. Note the greatest power to which an integer occurs in any denominator. c. The product of the integers noted in part b is the LCD. 2. Write fractions as equivalent fractions with denominators equal to the LCD. 3. Combine the numerators and use the LCD as the denominator. 4. Simplify if possible. a b. 2 2 is the greatest power of 2 in either denominator, 3 2 is the greatest power of 3 in either denominator. c

7 Ratios, Proportions, and Unit Conversion 173 Ratios, Proportions, and Unit Conversion The ratio of two numbers may be written as follows: a/b 5 a:b Two equivalent ratios form a proportion. a/b 5 c/d a:b 5 c:d a:b :: c:d Regardless of how the above proportions are expressed, they are read a is to b as c is to d. Ratios provide a convenient method for converting units. To change the units of a quantity, multiply by ratios whose values are equal to 1 (which does not change the value of the quantity). Select the dimensions of the ratios such that the unit to be changed occurs as a factor of the numerator or as a factor of the denominator. Thus, when the quantity is multiplied by the ratio, the unit is canceled and replaced by an equivalent unit and quantity. Convert 2 kilometers/hour to feet/second. Solution 1. Write an equation expressing the problem. 2 km/hr 5 x ft/s 2. Multiply the known quantity by ratios whose value is equal to 1, such that the desired unit remains after canceling pairs of equal dimensions that appear in both the numerator and the denominator. 1 km 5 1,000 m 1 m ft 1 hr 5 60 min 1 min 5 60 s 2 km hr 3 1,000 m 1 km 3.3 ft 3 1 m 3 1 hr 60 min 3 1 min 6,600 ft 1.8 ft s 3,600 s s

8 174 CHAPTER 6 Mathematical Procedures Exponents When a product is the result of multiplying a factor by itself several times, it is convenient to use a shorthand notation that shows the number used as a factor (a) and the number of factors (n) in the product. In general, such numbers are expressed in the form a n, where a is the base and n the exponent. The rules for these numbers are shown in Table 6 7. Table 6 7 Rules for Exponents Rule a n? a m 5 a n1m x 2? x 3 5 x 5 a n a m 5 a n2m z 7 z 5 5 z 2 (ab) n 5 a n b n (2wy) 2 5 4w 2 y 2 a 0 5 1, (a? 0) 9x a 1 5 a 3x 1 5 3x a 2n 5 1/a n x /x 2 a 1/n 5 2 n a y 1/ y a m/n 5 2 n a m z 2/ z 2 (a m ) n 5 a mn (w 2 ) 3 5 w 6 Scientific Notation A number expressed as a multiple of a power of 10, such as , is said to be written in scientific notation. Numbers written in this way have two parts: a number between 1 and 10 called the coefficient, multiplied by a power of 10 called the exponent. This notation has three distinct advantages: a. It simplifies the expression of very large or very small numbers that would otherwise require many zeros. For example, 681,000, and b. Scientific notation clarifies the number of significant figures in a large number. For example, if the radius of the earth is written as 6,378,000 m, it is not clear whether any of the zeros after the 8 is significant. However, when the same number is written as m, it is understood that only the first four digits are significant. c. Calculations that involve very large or very small numbers are greatly simplified using scientific notation.

9 Scientific Notation 175 Addition and Subtraction Solution Convert all numbers to the same power of 10 as the number with the highest exponent. 2. Add or subtract the coefficients and retain the same exponent in the answer. Multiplication and Division Solution ( )( ) Multiply (or divide) the coefficients. ( )( ) 5 8( ) 2. Combine the powers of 10 using the rules for exponents. Powers and Roots Solution ( ) 2 8( ) Raise the coefficient to the indicated power. ( ) 2 5 (4 2 )(10 5 ) (10 5 ) 2 2. Multiply the exponent by the indicated power. 16(10 5 ) ( )

10 176 CHAPTER 6 Mathematical Procedures Significant Figures By convention, the number of digits used to express a measured number roughly indicates the error. For example, if a measurement is reported as 35.2 cm, one would assume that the true length was between and cm (i.e., the error is about 0.05 cm). The last digit (2) in the reported measurement is uncertain, although one can reliably state that it is either 1 or 2. The digit to the right of 2, however, can be any number (5, 6, 7, 8, 9, 0, 1, 2, 3, 4). If the measurement is reported as cm, it would indicate that the error is even less (0.005 cm). The number of reliably known digits in a measurement is the number of significant figures. Thus, the number 35.2 cm has three significant figures, and the number cm has four. The number of significant figures is independent of the decimal point. The numbers 35.2 cm and m are the same quantities, both having three significant figures and expressing the same degree of accuracy. The use of significant figures to indicate the accuracy of a result is not as precise as giving the actual error, but is sufficient for most purposes. Zeros as Significant Figures Final zeros to the right of the decimal point that are used to indicate accuracy are significant: significant figures significant figures significant figures For numbers less than one, zeros between the decimal point and the first digit are not significant: significant figure significant figures Zeros between digits are significant: significant figures significant figures significant figures

11 Significant Figures 177 If a number is written with no decimal point, the final zeros may or may not be significant. For instance, the distance between Earth and the sun might be written as 92,900,000 miles, although the accuracy may be only ±5000 miles. This would make only the first zero after the 9 significant. On the other hand, a value of 50 ml measured with a graduated cylinder would be expected to have two significant figures owing to the greater accuracy of the measurement. To avoid ambiguity, numbers are often written as powers of 10 (scientific notation), making all digits significant. Using this convention, 92,900,000 would be written , indicating that there are four significant figures. Calculations Using Significant Figures The least precise measurement used in a calculation determines the number of significant figures in the answer. Thus, rather than , since the least precise number (73.5) is accurate to only one decimal place. Similarly, , with only one significant figure. For multiplication or division, the rule of thumb is: The product or quotient has the same number of significant figures as the term with the fewest significant figures. As an example, in / , the term with the fewest significant figures is 4.6. Since this number has at most two significant figures, the result should be rounded off to 1.6. Rounding Off The results of mathematical computations are often rounded off to specific numbers of significant figures. This is done so that one does not infer an accuracy in the result that was not present in the measurements. The following rules are universally accepted and will ensure bias-free reporting of results (the number of significant figures desired should be determined first). 1. If the final digits of the number are 1, 2, 3, or 4, they are rounded down (dropped) and the preceding figure is retained unaltered. 2. If the final digits are 6, 7, 8, or 9, they are rounded up (i.e., they are dropped and the preceding figure is increased by one). 3. If the digit to be dropped is a 5, it is rounded down if the preceding figure is even and rounded up if the preceding figure is odd. Thus, 2.45 and 6.15 are rounded off to 2.4 and 6.2, respectively.

12 178 CHAPTER 6 Mathematical Procedures Functions A function is a particular type of relation between groups of numbers. The uniqueness of a function is that each member of one group is associated with exactly one member of another group. In general, let the variable x stand for the values of one group of numbers and the variable y stand for the values of another group. If each value of x is associated with a unique value of y, then this relation is a function. Specifically, y is said to be a function of x and is denoted y 5 f(x). With this notation, x is called the independent variable and y the dependent variable. A function may be represented graphically by using a two-dimensional coordinate (Cartesian) plane formed by two perpendicular axes intersecting each other at a point with coordinates designated as x 5 0, y 5 0 (Fig. 6 1). The vertical axis denotes values of y and the horizontal axis values of x. The function is plotted as a series of points whose coordinates are the values of x with their corresponding values of y as determined by the function. 10 Y axis y = f(x) X axis Figure 6 1 Graphic representation of the function y = ƒ(x), where ƒ(x)= 0.3x + 1. Linear Functions One of the simplest functions is expressed by the formula y 5 ax

13 Functions 179 where y and x are variables a is a constant The constant a is sometimes referred to as the constant of proportionality and y is said to be directly proportional to x (if y is expressed as y 5 a/x, y is said to be inversely proportional to x and the function is no longer linear). The graph of the equation y 5 ax is a straight line. The constant a is the slope of the line. General Linear Equation where y 5 ax + b y 5 dependent variable a 5 slope x 5 independent variable b 5 y-intercept (the value of y at which the graph of the equation crosses the y-axis) Solving Linear Equations To solve a linear equation, 1. Combine similar terms. 2. Use inverse operations to undo remaining additions and subtractions (i.e., add or subtract the same quantities to both sides of the equation). Get all terms with the unknown variable on one side of the equation. 3. If the equation involves fractions, multiply both sides by the least common denominator. 4. If there are multiplications or divisions indicated in the variable term, use inverse operations to find the value of the variable. 5. Check the result by substituting the value into the original equation.

14 180 CHAPTER 6 Mathematical Procedures s x x x 1 2x 3 10x x 1 2x x x 1 2x x 2 5x 2 2x x 5 5x 1 2x x 2 2x a5x 2 2x 3 b 5 3(39) 4. 13x x x 13 15x 2 2x x (9) (9) (9) Quadratic Equations A function of the form y 5 ax 2 is called a quadratic function. It is sometimes expressed in the more general form where a, b, and c are constants 3 y 5 ax 2 + bx + c

15 Quadratic Equations 181 The graph of this equation is a parabola. Frequently, it is of interest to know where the parabola intersects the x-axis. The value of y at any point on the x-axis is zero. Therefore, to find the values of x where the graph intersects the x-axis, the quadratic equation is expressed in standard form: ax 2 + bx + c 5 0 with a? 0. The solution of any quadratic equation expressed in standard form may be found using the quadratic formula: x 5 2b ; 2b2 2 4ac 2a where a, b, and c are the coefficients in the quadratic equation Solution 1. Write the equation in standard form. Solve 3x x 3x x Note the coefficients a, b, and c. a 5 3, b 5 210, c Substitute these values in the quadratic formula: 4. Simplify. x 5 2b ; 2b2 2 4ac 2a x ; x 5 10 ; ; or 1

16 182 CHAPTER 6 Mathematical Procedures Logarithms The logarithm of a number (N) is the exponent (x) to which the base (a) must be raised to produce N. Thus, if a x 5 N then log a N 5 x for a. 0 and a? 0. For example, log (read: the log to the base 2 equals 3) because Logarithms are written as numbers with two parts: an integer, called the characteristic, and a decimal, called the mantissa (e.g., log ). Common Logarithms Common logarithms are those that have the base 10. In this book, the base number will be omitted with the assumption that log means log 10. Table 6 1 shows the general rules of common logarithms. x log x log log log Table 6 1 Rules of Common Logarithms Rule 1. log ab 5 log a + log b x 5 (746)(384) (a. 0, b. 0) log x 5 log log 384 log log log x x 5 286, log 1/a 5 2log a x 5 1/273 (a. 0) log x 5 2log x

17 Logarithms 183 Table 6 1 Rules of Common Logarithms (continued) Rule 3. log a/b 5 log a 2 log b x 5 478/21 (a. 0, b. 0) log x 5 log log 21 log log log x x log a n 5 n log a x (a. 0, n is a real number) 5 (374) 1/3 log x 5 1/3 log 374 log log x 5 1/3(2.5729) x The Characteristic The integer or characteristic of the logarithm of a number is determined by the position of the decimal point in the number. The characteristic of a number can easily be found by expressing the number in scientific notation. Once in this form, the exponent is used as the characteristic. s Number Characteristic Note: The characteristics of logarithms of numbers less than 1 can be written in several ways. Thus, log log (not ) or Written as a negative number (as with handheld calculators) log

18 184 CHAPTER 6 Mathematical Procedures The Mantissa The mantissa is the decimal part of the logarithm of a number. The mantissa of a series of digits is the same regardless of the position of the decimal point. Thus, the logarithms of 1.7, 17, and 170 all have the same mantissa, which is Antilogarithms The number having a given logarithm is called the antilogarithm (antilog). The logarithm of 125 is approximately Therefore, the antilog of is , which is approximately 125. Antilogs of Negative Logarithms Using a calculator, negative logarithms can be solved simply by using the 10 x key. For example, the antilog of 2 is However, log and antilog tables in reference books are used with positive mantissas only. Therefore, a negative logarithm must be changed to a log with a positive mantissa to find its antilog. This change of form is accomplished by first adding and then subtracting 1, which does not alter the original value of the logarithm. Solution 1. Write the log as a negative characteristic minus the mantissa. 2. Subtract 1 from the characteristic and add 1 to the mantissa. Find antilog Express the result as a log having a negative characteristic and a positive mantissa Use the tables to find the antilog. antilog antilog

19 Logarithms 185 Natural Logarithms When a logarithmic function must be differentiated or integrated, it is convenient to rewrite the function with the number e as a base. The number e is approximately equal to Logarithms which have the base e are called natural logarithms and are denoted by ln (read: ell-en ). If e x 5 N, then log e N 5 ln N 5 x Any number of the form a x may be rewritten with e as the base: a x 5 e x ln a Note: e x is sometimes written as exp(x). The same rules apply to natural logarithms that apply to common logarithms. See Table 6 2. Table 6 2 Rules of Natural Logarithms 1. ln ab 5 ln a + ln b (a. 0, b. 0) 2. ln 1/a 5 2 ln a (a. 0) 3. ln a/b 5 ln a 2 ln b (a. 0, b. 0) 4. ln a x 5 x ln a (a. 0, x is a real number) 5. ln e ln e x 5 x 5 e ln x 7. a x 5 e x ln x (a. 0) 8. ln x 5 (ln 10)(log x) (log x) (x. 0) Change of Base Logarithms to one base can easily be changed to logarithms of another base using the following equation. log a x 5 log b x log b a

21 Trigonometry 187 Trigonometric Functions In trigonometry, an angle is considered positive if it is generated by a counterclockwise rotation from standard position, and negative if it is generated by a clockwise rotation (Fig. 6 2). The trigonometric functions of a positive acute angle u can be defined as ratios of the sides of a right triangle: sine of u 5 sin u 5 y/r cosine of u 5 cos u 5 x/r tangent of u 5 tan u 5 y/x cotangent of u 5 cot u 5 x/y secant of u 5 sec u 5 r/x cosecant of u 5 csc u 5 r/y These functions can also be expressed in terms of sine and cosine alone: tan u 5 sin u/cos u cot u 5 cos u/sin u sec u 5 1/cos u csc u 5 1/sin u Y X θ r x y Positive Negative Figure 6 2 An angle is generated by rotating a ray (or half-line) about the origin of a circle. The angle is positive if it is generated by a counterclockwise rotation from the x-axis and negative for a clockwise rotation.

22 188 CHAPTER 6 Mathematical Procedures Basic Trigonometric Identities sin 2 u + cos 2 u 5 1 sec 2 u tan 2 u Probability The probability of an event A is denoted p(a). It is defined as follows: If an event can occur in p number of ways and can fail to occur in q number of ways, then the probability of the event occurring is p/(p + q). The odds in favor of an event occurring are p to q. Addition Rule If A and B are any events, then p(a or B) 5 p(a) + p(b) p(a and B) The probability of drawing either a king or a black card from a deck of 52 playing cards is p(king or black card) 5 p(king) 1 p(black card) 2 p(king also black) 5 4/ /52 2 2/52 5 7/13 Note: If events A and B cannot occur at the same time, they are said to be mutually exclusive, and the addition rule can be simplified to p(a or B) 5 p(a) + p(b) Multiplication Rule If A and B are any events, then p(a and B) 5 p(a B) 3 p(b) where p(a B) 5 the probability of event A given that event B has occurred

### Algebra and Geometry Review (61 topics, no due date)

Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties

### Mathematics Placement

Mathematics Placement The ACT COMPASS math test is a self-adaptive test, which potentially tests students within four different levels of math including pre-algebra, algebra, college algebra, and trigonometry.

### Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}

### how to use dual base log log slide rules

how to use dual base log log slide rules by Professor Maurice L. Hartung The University of Chicago Pickett The World s Most Accurate Slide Rules Pickett, Inc. Pickett Square Santa Barbara, California 93102

### Understanding Basic Calculus

Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other

### 1 TRIGONOMETRY. 1.0 Introduction. 1.1 Sum and product formulae. Objectives

TRIGONOMETRY Chapter Trigonometry Objectives After studying this chapter you should be able to handle with confidence a wide range of trigonometric identities; be able to express linear combinations of

### MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab

MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab MATH 0110 is established to accommodate students desiring non-course based remediation in developmental mathematics. This structure will

### A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions

A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25

### 13. Write the decimal approximation of 9,000,001 9,000,000, rounded to three significant

æ If 3 + 4 = x, then x = 2 gold bar is a rectangular solid measuring 2 3 4 It is melted down, and three equal cubes are constructed from this gold What is the length of a side of each cube? 3 What is the

### HIBBING COMMUNITY COLLEGE COURSE OUTLINE

HIBBING COMMUNITY COLLEGE COURSE OUTLINE COURSE NUMBER & TITLE: - Beginning Algebra CREDITS: 4 (Lec 4 / Lab 0) PREREQUISITES: MATH 0920: Fundamental Mathematics with a grade of C or better, Placement Exam,

### Extra Credit Assignment Lesson plan. The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam.

Extra Credit Assignment Lesson plan The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam. The extra credit assignment is to create a typed up lesson

### Exam 1 Sample Question SOLUTIONS. y = 2x

Exam Sample Question SOLUTIONS. Eliminate the parameter to find a Cartesian equation for the curve: x e t, y e t. SOLUTION: You might look at the coordinates and notice that If you don t see it, we can

### PURSUITS IN MATHEMATICS often produce elementary functions as solutions that need to be

Fast Approximation of the Tangent, Hyperbolic Tangent, Exponential and Logarithmic Functions 2007 Ron Doerfler http://www.myreckonings.com June 27, 2007 Abstract There are some of us who enjoy using our

### Mathematics Review for MS Finance Students

Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,

### Copyrighted Material. Chapter 1 DEGREE OF A CURVE

Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two

### 2010 Solutions. a + b. a + b 1. (a + b)2 + (b a) 2. (b2 + a 2 ) 2 (a 2 b 2 ) 2

00 Problem If a and b are nonzero real numbers such that a b, compute the value of the expression ( ) ( b a + a a + b b b a + b a ) ( + ) a b b a + b a +. b a a b Answer: 8. Solution: Let s simplify the

### The Australian Curriculum Mathematics

The Australian Curriculum Mathematics Mathematics ACARA The Australian Curriculum Number Algebra Number place value Fractions decimals Real numbers Foundation Year Year 1 Year 2 Year 3 Year 4 Year 5 Year

### Connections Across Strands Provides a sampling of connections that can be made across strands, using the theme (integers) as an organizer.

Overview Context Connections Positions integers in a larger context and shows connections to everyday situations, careers, and tasks. Identifies relevant manipulatives, technology, and web-based resources

### Section 1.1. Introduction to R n

The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to

### SPECIAL PRODUCTS AND FACTORS

CHAPTER 442 11 CHAPTER TABLE OF CONTENTS 11-1 Factors and Factoring 11-2 Common Monomial Factors 11-3 The Square of a Monomial 11-4 Multiplying the Sum and the Difference of Two Terms 11-5 Factoring the

### AP Physics 1 and 2 Lab Investigations

AP Physics 1 and 2 Lab Investigations Student Guide to Data Analysis New York, NY. College Board, Advanced Placement, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks

### Algebra 1 Course Information

Course Information Course Description: Students will study patterns, relations, and functions, and focus on the use of mathematical models to understand and analyze quantitative relationships. Through

### Polynomial Operations and Factoring

Algebra 1, Quarter 4, Unit 4.1 Polynomial Operations and Factoring Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned Identify terms, coefficients, and degree of polynomials.

### DRAFT. New York State Testing Program Grade 8 Common Core Mathematics Test. Released Questions with Annotations

DRAFT New York State Testing Program Grade 8 Common Core Mathematics Test Released Questions with Annotations August 2014 Developed and published under contract with the New York State Education Department

### Florida Math for College Readiness

Core Florida Math for College Readiness Florida Math for College Readiness provides a fourth-year math curriculum focused on developing the mastery of skills identified as critical to postsecondary readiness

### 2009 Chicago Area All-Star Math Team Tryouts Solutions

1. 2009 Chicago Area All-Star Math Team Tryouts Solutions If a car sells for q 1000 and the salesman earns q% = q/100, he earns 10q 2. He earns an additional 100 per car, and he sells p cars, so his total

### SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS

(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic

### Polynomials. Dr. philippe B. laval Kennesaw State University. April 3, 2005

Polynomials Dr. philippe B. laval Kennesaw State University April 3, 2005 Abstract Handout on polynomials. The following topics are covered: Polynomial Functions End behavior Extrema Polynomial Division

### Chapter 7 Outline Math 236 Spring 2001

Chapter 7 Outline Math 236 Spring 2001 Note 1: Be sure to read the Disclaimer on Chapter Outlines! I cannot be responsible for misfortunes that may happen to you if you do not. Note 2: Section 7.9 will

### 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,

### 04 Mathematics CO-SG-FLD004-03. Program for Licensing Assessments for Colorado Educators

04 Mathematics CO-SG-FLD004-03 Program for Licensing Assessments for Colorado Educators Readers should be advised that this study guide, including many of the excerpts used herein, is protected by federal

### L 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has

The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:

### Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year.

This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra

### ACCUPLACER. Testing & Study Guide. Prepared by the Admissions Office Staff and General Education Faculty Draft: January 2011

ACCUPLACER Testing & Study Guide Prepared by the Admissions Office Staff and General Education Faculty Draft: January 2011 Thank you to Johnston Community College staff for giving permission to revise

### Objectives After completing this section, you should be able to:

Chapter 5 Section 1 Lesson Angle Measure Objectives After completing this section, you should be able to: Use the most common conventions to position and measure angles on the plane. Demonstrate an understanding

### Algebra II and Trigonometry

Algebra II and Trigonometry Textbooks: Algebra 2: California Publisher: McDougal Li@ell/Houghton Mifflin (2006 EdiHon) ISBN- 13: 978-0618811816 Course descriphon: Algebra II complements and expands the

### Find the length of the arc on a circle of radius r intercepted by a central angle θ. Round to two decimal places.

SECTION.1 Simplify. 1. 7π π. 5π 6 + π Find the measure of the angle in degrees between the hour hand and the minute hand of a clock at the time shown. Measure the angle in the clockwise direction.. 1:0.

### (1.) The air speed of an airplane is 380 km/hr at a bearing of. Find the ground speed of the airplane as well as its

(1.) The air speed of an airplane is 380 km/hr at a bearing of 78 o. The speed of the wind is 20 km/hr heading due south. Find the ground speed of the airplane as well as its direction. Here is the diagram:

### SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS

SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS Assume f ( x) is a nonconstant polynomial with real coefficients written in standard form. PART A: TECHNIQUES WE HAVE ALREADY SEEN Refer to: Notes 1.31

### Chapter 111. Texas Essential Knowledge and Skills for Mathematics. Subchapter B. Middle School

Middle School 111.B. Chapter 111. Texas Essential Knowledge and Skills for Mathematics Subchapter B. Middle School Statutory Authority: The provisions of this Subchapter B issued under the Texas Education

### DEFINITION 5.1.1 A complex number is a matrix of the form. x y. , y x

Chapter 5 COMPLEX NUMBERS 5.1 Constructing the complex numbers One way of introducing the field C of complex numbers is via the arithmetic of matrices. DEFINITION 5.1.1 A complex number is a matrix of

### A Quick Algebra Review

1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals

### Display Format To change the exponential display format, press the [MODE] key 3 times.

Tools FX 300 MS Calculator Overhead OH 300 MS Handouts Other materials Applicable activities Activities for the Classroom FX-300 Scientific Calculator Quick Reference Guide (inside the calculator cover)

### Examples of Tasks from CCSS Edition Course 3, Unit 5

Examples of Tasks from CCSS Edition Course 3, Unit 5 Getting Started The tasks below are selected with the intent of presenting key ideas and skills. Not every answer is complete, so that teachers can

### Content. Chapter 4 Functions 61 4.1 Basic concepts on real functions 62. Credits 11

Content Credits 11 Chapter 1 Arithmetic Refresher 13 1.1 Algebra 14 Real Numbers 14 Real Polynomials 19 1.2 Equations in one variable 21 Linear Equations 21 Quadratic Equations 22 1.3 Exercises 28 Chapter

### 2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

### Transition To College Mathematics

Transition To College Mathematics In Support of Kentucky s College and Career Readiness Program Northern Kentucky University Kentucky Online Testing (KYOTE) Group Steve Newman Mike Waters Janis Broering

### FSCJ PERT. Florida State College at Jacksonville. assessment. and Certification Centers

FSCJ Florida State College at Jacksonville Assessment and Certification Centers PERT Postsecondary Education Readiness Test Study Guide for Mathematics Note: Pages through are a basic review. Pages forward

### LAKE ELSINORE UNIFIED SCHOOL DISTRICT

LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1-Semester 2 Grade Level: 10-12 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:

### FACTORING POLYNOMIALS

296 (5-40) Chapter 5 Exponents and Polynomials where a 2 is the area of the square base, b 2 is the area of the square top, and H is the distance from the base to the top. Find the volume of a truncated

### In this section, you will develop a method to change a quadratic equation written as a sum into its product form (also called its factored form).

CHAPTER 8 In Chapter 4, you used a web to organize the connections you found between each of the different representations of lines. These connections enabled you to use any representation (such as a graph,

### Polynomial Degree and Finite Differences

CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial

### Secondary Mathematics Syllabuses

Secondary Mathematics Syllabuses Copyright 006 Curriculum Planning and Development Division. This publication is not for sale. All rights reserved. No part of this publication may be reproduced without

### Functional Math II. Information CourseTitle. Types of Instruction

Functional Math II Course Outcome Summary Riverdale School District Information CourseTitle Functional Math II Credits 0 Contact Hours 135 Instructional Area Middle School Instructional Level 8th Grade

### MATD 0390 - Intermediate Algebra Review for Pretest

MATD 090 - Intermediate Algebra Review for Pretest. Evaluate: a) - b) - c) (-) d) 0. Evaluate: [ - ( - )]. Evaluate: - -(-7) + (-8). Evaluate: - - + [6 - ( - 9)]. Simplify: [x - (x - )] 6. Solve: -(x +

### Course Outlines. 1. Name of the Course: Algebra I (Standard, College Prep, Honors) Course Description: ALGEBRA I STANDARD (1 Credit)

Course Outlines 1. Name of the Course: Algebra I (Standard, College Prep, Honors) Course Description: ALGEBRA I STANDARD (1 Credit) This course will cover Algebra I concepts such as algebra as a language,

### Factoring and Applications

Factoring and Applications What is a factor? The Greatest Common Factor (GCF) To factor a number means to write it as a product (multiplication). Therefore, in the problem 48 3, 4 and 8 are called the

### The Greatest Common Factor; Factoring by Grouping

296 CHAPTER 5 Factoring and Applications 5.1 The Greatest Common Factor; Factoring by Grouping OBJECTIVES 1 Find the greatest common factor of a list of terms. 2 Factor out the greatest common factor.

### 9 MATRICES AND TRANSFORMATIONS

9 MATRICES AND TRANSFORMATIONS Chapter 9 Matrices and Transformations Objectives After studying this chapter you should be able to handle matrix (and vector) algebra with confidence, and understand the

### averages simple arithmetic average (arithmetic mean) 28 29 weighted average (weighted arithmetic mean) 32 33

537 A accumulated value 298 future value of a constant-growth annuity future value of a deferred annuity 409 future value of a general annuity due 371 future value of an ordinary general annuity 360 future

### SECTION P.5 Factoring Polynomials

BLITMCPB.QXP.0599_48-74 /0/0 0:4 AM Page 48 48 Chapter P Prerequisites: Fundamental Concepts of Algebra Technology Eercises Critical Thinking Eercises 98. The common cold is caused by a rhinovirus. The

### Licensed to: Printed in the United States of America 1 2 3 4 5 6 7 16 15 14 13 12

Licensed to: CengageBrain User This is an electronic version of the print textbook. Due to electronic rights restrictions, some third party content may be suppressed. Editorial review has deemed that any

### David Bressoud Macalester College, St. Paul, MN. NCTM Annual Mee,ng Washington, DC April 23, 2009

David Bressoud Macalester College, St. Paul, MN These slides are available at www.macalester.edu/~bressoud/talks NCTM Annual Mee,ng Washington, DC April 23, 2009 The task of the educator is to make the

### Blue Pelican Alg II First Semester

Blue Pelican Alg II First Semester Teacher Version 1.01 Copyright 2009 by Charles E. Cook; Refugio, Tx (All rights reserved) Alg II Syllabus (First Semester) Unit 1: Solving linear equations and inequalities

### DOE FUNDAMENTALS HANDBOOK MATHEMATICS Volume 2 of 2

DOE-HDBK-1014/2-92 JUNE 1992 DOE FUNDAMENTALS HANDBOOK MATHEMATICS Volume 2 of 2 U.S. Department of Energy Washington, D.C. 20585 FSC-6910 Distribution Statement A. Approved for public release; distribution

### MATH 90 CHAPTER 1 Name:.

MATH 90 CHAPTER 1 Name:. 1.1 Introduction to Algebra Need To Know What are Algebraic Expressions? Translating Expressions Equations What is Algebra? They say the only thing that stays the same is change.

### correct-choice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were:

Topic 1 2.1 mode MultipleSelection text How can we approximate the slope of the tangent line to f(x) at a point x = a? This is a Multiple selection question, so you need to check all of the answers that

### YOU CAN COUNT ON NUMBER LINES

Key Idea 2 Number and Numeration: Students use number sense and numeration to develop an understanding of multiple uses of numbers in the real world, the use of numbers to communicate mathematically, and

### Algebra 1 If you are okay with that placement then you have no further action to take Algebra 1 Portion of the Math Placement Test

Dear Parents, Based on the results of the High School Placement Test (HSPT), your child should forecast to take Algebra 1 this fall. If you are okay with that placement then you have no further action

### 6.1 The Greatest Common Factor; Factoring by Grouping

386 CHAPTER 6 Factoring and Applications 6.1 The Greatest Common Factor; Factoring by Grouping OBJECTIVES 1 Find the greatest common factor of a list of terms. 2 Factor out the greatest common factor.

### Determinants can be used to solve a linear system of equations using Cramer s Rule.

2.6.2 Cramer s Rule Determinants can be used to solve a linear system of equations using Cramer s Rule. Cramer s Rule for Two Equations in Two Variables Given the system This system has the unique solution

### SECTION 1-6 Quadratic Equations and Applications

58 Equations and Inequalities Supply the reasons in the proofs for the theorems stated in Problems 65 and 66. 65. Theorem: The complex numbers are commutative under addition. Proof: Let a bi and c di be

### Calculus. Contents. Paul Sutcliffe. Office: CM212a.

Calculus Paul Sutcliffe Office: CM212a. www.maths.dur.ac.uk/~dma0pms/calc/calc.html Books One and several variables calculus, Salas, Hille & Etgen. Calculus, Spivak. Mathematical methods in the physical

### BX in ( u, v) basis in two ways. On the one hand, AN = u+

1. Let f(x) = 1 x +1. Find f (6) () (the value of the sixth derivative of the function f(x) at zero). Answer: 7. We expand the given function into a Taylor series at the point x = : f(x) = 1 x + x 4 x

### 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

### POLYNOMIALS and FACTORING

POLYNOMIALS and FACTORING Exponents ( days); 1. Evaluate exponential expressions. Use the product rule for exponents, 1. How do you remember the rules for exponents?. How do you decide which rule to use

### The following table lists metric prefixes that come up frequently in physics. Learning these prefixes will help you in the various exercises.

Chapter 0 Solutions Circles Learning Goal: To calculate the circumference or area of a circle Every day, we see circles in compact disks, coins, and wheels, just to name a few examples Circles are also

### Polynomials and Factoring

Lesson 2 Polynomials and Factoring A polynomial function is a power function or the sum of two or more power functions, each of which has a nonnegative integer power. Because polynomial functions are built

### Interpretation of Test Scores for the ACCUPLACER Tests

Interpretation of Test Scores for the ACCUPLACER Tests ACCUPLACER is a trademark owned by the College Entrance Examination Board. Visit The College Board on the Web at: www.collegeboard.com/accuplacer

### Such As Statements, Kindergarten Grade 8

Such As Statements, Kindergarten Grade 8 This document contains the such as statements that were included in the review committees final recommendations for revisions to the mathematics Texas Essential

### 8th Grade Texas Mathematics: Unpacked Content

8th Grade Texas Mathematics: Unpacked Content What is the purpose of this document? To increase student achievement by ensuring educators understand specifically what the new standards mean a student must

### Algebra Isn t Hard or Demystifying Math: A Gentle Approach to Algebra. by Peter J. Hutnick

Algebra Isn t Hard or Demystifying Math: A Gentle Approach to Algebra by Peter J. Hutnick 17th July 2004 2 Contents I Fundamentals of Algebra 17 1 Arithmetic 19 1.1 Vocabulary...............................

### Mathematics: Content Knowledge

The Praxis Study Companion Mathematics: Content Knowledge 5161 www.ets.org/praxis Welcome to the Praxis Study Companion Welcome to the Praxis Study Companion Prepare to Show What You Know You have been

### Inversion. Chapter 7. 7.1 Constructing The Inverse of a Point: If P is inside the circle of inversion: (See Figure 7.1)

Chapter 7 Inversion Goal: In this chapter we define inversion, give constructions for inverses of points both inside and outside the circle of inversion, and show how inversion could be done using Geometer

### December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

### College Algebra. Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1

College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1 Course Description This course provides

### Microsoft Mathematics for Educators:

Microsoft Mathematics for Educators: Familiarize yourself with the interface When you first open Microsoft Mathematics, you ll see the following elements displayed: 1. The Calculator Pad which includes

### Definitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder).

Math 50, Chapter 8 (Page 1 of 20) 8.1 Common Factors Definitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder). Find all the factors of a. 44 b. 32

### For example, estimate the population of the United States as 3 times 10⁸ and the

CCSS: Mathematics The Number System CCSS: Grade 8 8.NS.A. Know that there are numbers that are not rational, and approximate them by rational numbers. 8.NS.A.1. Understand informally that every number

Polynomials and Quadratics Want to be an environmental scientist? Better be ready to get your hands dirty!.1 Controlling the Population Adding and Subtracting Polynomials............703.2 They re Multiplying

### To define function and introduce operations on the set of functions. To investigate which of the field properties hold in the set of functions

Chapter 7 Functions This unit defines and investigates functions as algebraic objects. First, we define functions and discuss various means of representing them. Then we introduce operations on functions

### 2After completing this chapter you should be able to

After completing this chapter you should be able to solve problems involving motion in a straight line with constant acceleration model an object moving vertically under gravity understand distance time

### Factoring a Difference of Two Squares. Factoring a Difference of Two Squares

284 (6 8) Chapter 6 Factoring 87. Tomato soup. The amount of metal S (in square inches) that it takes to make a can for tomato soup is a function of the radius r and height h: S 2 r 2 2 rh a) Rewrite this

### Common Core Standards Practice Week 8

Common Core Standards Practice Week 8 Selected Response 1. Describe the end behavior of the polynomial f(x) 5 x 8 8x 1 6x. A down and down B down and up C up and down D up and up Constructed Response.

### CENTRAL TEXAS COLLEGE SYLLABUS FOR DSMA 0306 INTRODUCTORY ALGEBRA. Semester Hours Credit: 3

CENTRAL TEXAS COLLEGE SYLLABUS FOR DSMA 0306 INTRODUCTORY ALGEBRA Semester Hours Credit: 3 (This course is equivalent to DSMA 0301. The difference being that this course is offered only on those campuses

### Indiana University Purdue University Indianapolis. Marvin L. Bittinger. Indiana University Purdue University Indianapolis. Judith A.

STUDENT S SOLUTIONS MANUAL JUDITH A. PENNA Indiana Universit Purdue Universit Indianapolis COLLEGE ALGEBRA: GRAPHS AND MODELS FIFTH EDITION Marvin L. Bittinger Indiana Universit Purdue Universit Indianapolis