1.2 Supplement: Mathematical Models: A Catalog of Essential Functions

Size: px
Start display at page:

Download "1.2 Supplement: Mathematical Models: A Catalog of Essential Functions"

Transcription

1 Math 131 -copyright Angela Allen, Fall Supplement: Mathematical Models: A Catalog of Essential Functions Note: Some of these examples and figures come from your textbook Single Variable Calculus: Concepts & Contexts, 4th ed., by Stewart. Steps of Mathematical Modeling 1. Given a real-world problem, identify (and name) the independent and dependent variables. 2. Use knowledge (or data) of the physical situation and mathematical skills to form equations relating the variables. 3. Apply mathematics (calculus) to the mathematical model in order to derive mathematical conclusions. 4. Interpret the mathematical conclusions (and possibly use the information to make predictions). 5. Test the predictions by checking against new real data. If the predictions don t compare well with reality, refine the model or formulate a new model. Note: A mathematical model is an of a physical situation. Types of Mathematical Models Linear Models: A linear function is of the form f (x) = mx + b, where m is the slope and b is the y intercept. The slope is the the corresponding change in y. of y with respect to x. For every unit increase in x, the slope gives Linear functions have a constant rate of change (i.e. slope). Example: Consider the function f (x) = 4x 1. Find and interpret the slope. Question: What types of slopes correspond to increasing, decreasing, horizontal and vertical lines? Example: As dry air moves upward, it expands and cools. If the ground temperature is 25 C and the temperature at a height of 1 km is 10 C, a) express the temperature T (in C) as a function of the height H (in kilometers), assuming that a linear model is appropriate, and interpret the slope, b) determine the temperature at a height of 3 km.

2 Math 131 -copyright Angela Allen, Fall Linear Regression: We can construct an empirical model based on collected data if there is no physical law or principle to help us create a model. *We will find a curve (called the least squares line) that fits the data points best. Note: The line does NOT have to go through any of the points. Example: (p. 27) The table below lists the average carbon dioxide level in the atmosphere (in parts per million) at Mauna Loa Observatory from 1980 to Year CO 2 level (ppm) a) Make a scatter plot of the data and find a linear model for the carbon dioxide level. Scatterplot: Note: Our text does not align the data when the inputs are in years. We will align the data. In order for the answers to correspond with the text, the input variable must be the number of years since the beginning of the first input year. 1) Press STAT, EDIT, and enter the inputs in L 1 and outputs in L 2. Note: If you have data in your lists and you want to clear it, scroll up to the list name and hit CLEAR. DO NOT HIT DELETE on the list name; the entire list will be deleted. 2) Press 2nd, Y =, and ENTER. 3) Turn the Stat Plot on by highlighting on and pressing ENTER. 4) Highlight the first graph, and be sure you have X List: L 1 and Y List: L 2. (To call L 1, press 2nd, 1.) 5) Press ZOOM, 9 to graph the data points. Note: On the Y = screen, Plot 1 should be on (highlighted). If you are not using your stat plots, make sure they are turned off (unhighlighted) on this screen. Question: Does a linear model seem appropriate? Linear Regression: (after data has been entered using the instructions above) 1) From the home screen ( 2nd, MODE ), press STAT, CALC, and 4. 2) On the home screen, you should see LinReg(ax + b). Call L 1, L 2, Y 1 and press ENTER to run. The comma is above the 7, and to get Y 1, press VARS, Y-VARS, FUNCTION, and 1. (This will store your model in Y 1.) 3) Press ZOOM, 9 to see the data and the model graphed. Note: When you press STAT, CALC, you will see several regression models besides linear regression. They include Quadratic, Cubic, Quartic, Exponential, Logarithmic, Logistic, and Sine models. Answer:

3 Math 131 -copyright Angela Allen, Fall b) Use your unrounded model (which should still be stored in your calculator) to estimate the average CO 2 level for 1987 and to predict the level for Note: When we use a model to estimate values between observed values, we are using use a model to predict values outside the region of observations, we are using.. When we Finding Function Values: 1) Enter the function in slope-intercept form into Y 1 by pressing Y =. 2) On the home screen ( 2nd, MODE ), call Y 1 by pressing VARS, Y-VARS, FUNCTION,and 1. Then, open parenthesis, type the input value, and close parenthesis. You should have something like Y 1 (2). Press ENTER. Answer: c) Use your unrounded model to determine during what year the CO 2 level will exceed 400 ppm. Intersecting Functions (to solve equations): 1) Press Y = and enter the left side of the equation into Y 1 and the right side into Y 2. (Make sure your Stat Plots 1, 2, and 3 are off unhighlighted.) 2) Try ZOOM, 6 ; ZOOM, 0 ; or WINDOW (adjust) and then GRAPH. (Remember, you must be able to see the intersection point in your window.) 3) Press 2nd, TRACE, and 5. 4) Move your cursor close to the intersection point, and press ENTER for the first curve, ENTER for the second curve, and then ENTER again for Guess?. Answer: Polynomials: A function P is a polynomial if P(x) = a n x n + a n 1 x n a 2 x 2 + a 1 x + a 0, where n is a nonnegative integer and the numbers a 0,a 1,a 2,...,a n are constants called the coefficients. What is the domain of a polynomial? If the leading coefficient a n 0, then the degree of the polynomial is n. A polynomial of degree 2 is of the form P(x) = ax 2 + bx + c and is called a and is either concave up or concave down: function. Its graph is a

4 Math 131 -copyright Angela Allen, Fall A polynomial of degree 3 is of the form P(x) = ax 3 + bx 2 + cx + d and is called a function has an inflection point. The following are example of cubic functions: function. A cubic Example: (p. 30) A ball is dropped from the upper observation deck of the CN Tower, 450 m above the ground, and its height h above the ground is recorded at 1-second intervals in the following table: Time (sec) Height (m) Find an appropriate model for the data and use the unrounded model to predict the time at which the ball hits the ground. Note: Even though a cubic function will fit the data, we should NOT consider a cubic function because there is no inflection point in the scatter plot, and we want the least complicated function possible. Power Functions: A function of the form f (x) = x a, where a is a constant, is called a power function. If a = n, where n is a positive integer, the power function is a polynomial with only one term. For example, f (x) = x, f (x) = x 2, etc. the shape of f (x) = x n depends on whether n is even or odd. Also, as n increases, the graph of y = x n becomes flatter near 0 and steeper when x 1. For example:

5 Math 131 -copyright Angela Allen, Fall If a = 1 n, where n is a positive integer, then f (x) = x 1 n = n x is called a root function. For example: If a = 1, f (x) = x 1 = 1 x is called the reciprocal function. its graph is a hyperbola with the coordinate axes as its asymptotes: Rational Functions: A rational function f is a ratio of two polynomials: f (x) = P(x), where P and Q are polynomials. Q(x) In general, what is the domain of a rational function? Example: Find the domain of the function f (x) = x2 3x 4 x 2 2x 8. Algebraic Functions: A function f is called an algebraic function if it can be constructed using algebraic operations (such as addition, subtraction, multiplication, division, and taking roots) starting with polynomials. Any rational function is automatically an algebraic function. Is g(x) = x4 16x x+ x + (x 2) 3 x + 1 an algebraic function? The graphs of algebraic functions have a variety of shapes.

6 Math 131 -copyright Angela Allen, Fall Trigonometric Functions: Trigonometry and trigonometric functions are reviewed on Reference Page 2 and also in Appendix C. We always use radian measure. The graphs of the sine and cosine functions are shown below: Figure 18, pg. 33 What is the domain of both functions? What is the range? The zeros of the sine function occur at the integer multiples of π. In other words, sinx = 0 when x = nπ, where n is an integer. Sine and cosine are periodic and have a period of The periodic nature of these functions makes them suitable for modeling repetitive phenomena. The tangent function is given by the equation tanx = cosx sinx, and its graph is shown below:. Thus, sin(x + 2π) = sinx and cos(x + 2π) = cosx. Figure 19, pg. 34 When is tanx undefined? What is the range of tanx? The tangent function has a period of. So, tan(x + π) = tanx for all x. The cosecant, secant, and cotangent functions are reciprocals of the sine, cosine, and tangent functions (respectively). Their graphs are shown in Appendix C.

7 Math 131 -copyright Angela Allen, Fall Exponential Functions: Exponential functions are of the form f (x) = a x, where the base a is a positive constant. If a > 1, then the exponential function is increasing (exponential growth). If 0 < a < 1, then it is decreasing (exponential decay): What is the domain of an exponential function? What is the range of an exponential function? Logarithmic Functions: Logarithmic (Log) functions are of the form f (x) = log a x, where the base a is a positive constant. Log functions are inverse functions of exponential functions. The following is the graph of a log function: What is the domain of a log function? What is the range of a log function? Transcendental Functions - These include the trig, inverse trig, exponential, and log functions, as well as a vast number of other functions that have never been named. Example: Classify the function h(x) = x 12.

8 Math 131 -copyright Angela Allen, Fall Regression Examples (Courtesy of Sherry Scarborough and Heather Ramsey) 1. The table below shows the temperature inside a vehicle based upon the number of minutes after the ignition is turned off when it is 80 F outside. (Source: The Eagle, July 29, 2007) a) Find an appropriate model for the data. Minutes Temperature ( F) b) Use your unrounded model to estimate the temperature inside a vehicle 70 minutes after the ignition is turned off when the outside temperature is 80 F. Round to the nearest degree (i.e. integer). 2. The per capita consumption of chocolate milk in Dairyland has increased in the past 10 years as shown in the table. a) Find an appropriate model to fit the data. Year Liters per person per year b) Use your unrounded model to estimate the per capita consumption in c) According to the unrounded model, when will per capita chocolate milk consumption exceed 5 liters per person per year?

FUNCTIONS AND MODELS

FUNCTIONS AND MODELS 1 FUNCTIONS AND MODELS FUNCTIONS AND MODELS 1.2 MATHEMATICAL MODELS: A CATALOG OF ESSENTIAL FUNCTIONS In this section, we will learn about: The purpose of mathematical models. MATHEMATICAL MODELS A mathematical

More information

1.2 Mathematical Models: A Catalog of Essential Functions

1.2 Mathematical Models: A Catalog of Essential Functions 1.2 Mathematical Models: A Catalog of Essential Functions A is a mathematical description of a real-world phenomenon. e.g., the size of a population, the demand for a product, the speed of a falling object,

More information

AP CALCULUS SUMMER WORKSHEET

AP CALCULUS SUMMER WORKSHEET AP CALCULUS SUMMER WORKSHEET DUE: First Day of School, 2011 Complete this assignment at your leisure during the summer. I strongly recommend you complete a little each week. It is designed to help you

More information

AP CALCULUS SUMMER WORKSHEET

AP CALCULUS SUMMER WORKSHEET AP CALCULUS SUMMER WORKSHEET DUE: First Day of School Aug. 19, 2010 Complete this assignment at your leisure during the summer. It is designed to help you become more comfortable with your graphing calculator,

More information

MATH 1040 Objectives List

MATH 1040 Objectives List MATH 1040 Objectives List Textbook: Calculus, Early Transcendentals, 7th edition, James Stewart Students should expect test questions that require synthesis of these objectives. Unit 1 WebAssign problems

More information

Dear Future CALCULUS Student,

Dear Future CALCULUS Student, Dear Future CALCULUS Student, I am looking forward to teaching the AP Calculus AB class this coming year and hope that you are looking forward to the class as well. Here a few things you need to know prior

More information

Dear Future CALCULUS Student,

Dear Future CALCULUS Student, Dear Future CALCULUS Student, I am looking forward to teaching the AP Calculus AB class this coming year and hope that you are looking forward to the class as well. Here a few things you need to know prior

More information

NAME: DATE: CLASS: AP CALCULUS AB SUMMER MATH 2018

NAME: DATE: CLASS: AP CALCULUS AB SUMMER MATH 2018 NAME: DATE: CLASS: AP CALCULUS AB SUMMER MATH 2018 A] Refer to your pre-calculus notebook, the internet, or the sheets/links provided for assistance. B] Do not wait until the last minute to complete this

More information

Section 0.2 & 0.3 Worksheet. Types of Functions

Section 0.2 & 0.3 Worksheet. Types of Functions MATH 1142 NAME Section 0.2 & 0.3 Worksheet Types of Functions Now that we have discussed what functions are and some of their characteristics, we will explore different types of functions. Section 0.2

More information

Review of Section 1.1. Mathematical Models. Review of Section 1.1. Review of Section 1.1. Functions. Domain and range. Piecewise functions

Review of Section 1.1. Mathematical Models. Review of Section 1.1. Review of Section 1.1. Functions. Domain and range. Piecewise functions Review of Section 1.1 Functions Mathematical Models Domain and range Piecewise functions January 19, 2017 Even and odd functions Increasing and decreasing functions Mathematical Models January 19, 2017

More information

ALGEBRA & TRIGONOMETRY FOR CALCULUS MATH 1340

ALGEBRA & TRIGONOMETRY FOR CALCULUS MATH 1340 ALGEBRA & TRIGONOMETRY FOR CALCULUS Course Description: MATH 1340 A combined algebra and trigonometry course for science and engineering students planning to enroll in Calculus I, MATH 1950. Topics include:

More information

Grade 11 or 12 Pre-Calculus

Grade 11 or 12 Pre-Calculus Grade 11 or 12 Pre-Calculus Strands 1. Polynomial, Rational, and Radical Relationships 2. Trigonometric Functions 3. Modeling with Functions Strand 1: Polynomial, Rational, and Radical Relationships Standard

More information

Chapter 1 Functions and Models

Chapter 1 Functions and Models Chapter 1 Functions and Models 1.2 Mathematical Models: A catalog of Essential Functions A mathematical model is a mathematical description of a real world situations such as the size of a population,

More information

Calculus: Early Transcendental Functions Lecture Notes for Calculus 101. Feras Awad Mahmoud

Calculus: Early Transcendental Functions Lecture Notes for Calculus 101. Feras Awad Mahmoud Calculus: Early Transcendental Functions Lecture Notes for Calculus 101 Feras Awad Mahmoud Last Updated: August 2, 2012 1 2 Feras Awad Mahmoud Department of Basic Sciences Philadelphia University JORDAN

More information

A Library of Functions

A Library of Functions LibraryofFunctions.nb 1 A Library of Functions Any study of calculus must start with the study of functions. Functions are fundamental to mathematics. In its everyday use the word function conveys to us

More information

function independent dependent domain range graph of the function The Vertical Line Test

function independent dependent domain range graph of the function The Vertical Line Test Functions A quantity y is a function of another quantity x if there is some rule (an algebraic equation, a graph, a table, or as an English description) by which a unique value is assigned to y by a corresponding

More information

Core 3 (A2) Practice Examination Questions

Core 3 (A2) Practice Examination Questions Core 3 (A) Practice Examination Questions Trigonometry Mr A Slack Trigonometric Identities and Equations I know what secant; cosecant and cotangent graphs look like and can identify appropriate restricted

More information

x and y, called the coordinates of the point.

x and y, called the coordinates of the point. P.1 The Cartesian Plane The Cartesian Plane The Cartesian Plane (also called the rectangular coordinate system) is the plane that allows you to represent ordered pairs of real numbers by points. It is

More information

Math Academy I Fall Study Guide. CHAPTER ONE: FUNDAMENTALS Due Thursday, December 8

Math Academy I Fall Study Guide. CHAPTER ONE: FUNDAMENTALS Due Thursday, December 8 Name: Math Academy I Fall Study Guide CHAPTER ONE: FUNDAMENTALS Due Thursday, December 8 1-A Terminology natural integer rational real complex irrational imaginary term expression argument monomial degree

More information

Algebra I Calculator Activities

Algebra I Calculator Activities First Nine Weeks SOL Objectives Calculating Measures of Central Tendency SOL A.17 Organize a set of data Calculate the mean, median, mode, and range of a set of data Describe the relationships between

More information

Objectives List. Important Students should expect test questions that require a synthesis of these objectives.

Objectives List. Important Students should expect test questions that require a synthesis of these objectives. MATH 1040 - of One Variable, Part I Textbook 1: : Algebra and Trigonometry for ET. 4 th edition by Brent, Muller Textbook 2:. Early Transcendentals, 3 rd edition by Briggs, Cochran, Gillett, Schulz s List

More information

Math 121: Calculus 1 - Fall 2013/2014 Review of Precalculus Concepts

Math 121: Calculus 1 - Fall 2013/2014 Review of Precalculus Concepts Introduction Math 121: Calculus 1 - Fall 201/2014 Review of Precalculus Concepts Welcome to Math 121 - Calculus 1, Fall 201/2014! This problems in this packet are designed to help you review the topics

More information

Next, we ll use all of the tools we ve covered in our study of trigonometry to solve some equations.

Next, we ll use all of the tools we ve covered in our study of trigonometry to solve some equations. Section 6.3 - Solving Trigonometric Equations Next, we ll use all of the tools we ve covered in our study of trigonometry to solve some equations. These are equations from algebra: Linear Equation: Solve:

More information

APPLICATIONS OF DIFFERENTIATION

APPLICATIONS OF DIFFERENTIATION 4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION The method we used to sketch curves in Section 4.5 was a culmination of much of our study of differential calculus. The graph was the final

More information

West Windsor-Plainsboro Regional School District Algebra and Trigonometry Grades 11-12

West Windsor-Plainsboro Regional School District Algebra and Trigonometry Grades 11-12 West Windsor-Plainsboro Regional School District Algebra and Trigonometry Grades 11-12 Unit 1: Linear Relationships & Functions Content Area: Mathematics Course & Grade Level: Algebra & Trigonometry, 11

More information

Welcome to AP Calculus!!!

Welcome to AP Calculus!!! Welcome to AP Calculus!!! In preparation for next year, you need to complete this summer packet. This packet reviews & expands upon the concepts you studied in Algebra II and Pre-calculus. Make sure you

More information

WAYNESBORO AREA SCHOOL DISTRICT CURRICULUM ALGEBRA II

WAYNESBORO AREA SCHOOL DISTRICT CURRICULUM ALGEBRA II UNIT: Review of Basic Algebra Skills as Needed SR1 and any Supplemental Materials UNIT : What skills from Algebra I are used in Algebra II? Review Algebra I Skills as Needed SR1 and any additional resources

More information

AP CALCULUS AB SUMMER ASSIGNMNET NAME: READ THE FOLLOWING DIRECTIONS CAREFULLY

AP CALCULUS AB SUMMER ASSIGNMNET NAME: READ THE FOLLOWING DIRECTIONS CAREFULLY AP CALCULUS AB SUMMER ASSIGNMNET NAME: READ THE FOLLOWING DIRECTIONS CAREFULLY 1. This packet is to be handed in on the first day of school. 2. All work must be shown in the space provided in the packet.

More information

Region 16 Board of Education. Precalculus Curriculum

Region 16 Board of Education. Precalculus Curriculum Region 16 Board of Education Precalculus Curriculum 2008 1 Course Description This course offers students an opportunity to explore a variety of concepts designed to prepare them to go on to study calculus.

More information

Chapter 1. Functions 1.1. Functions and Their Graphs

Chapter 1. Functions 1.1. Functions and Their Graphs 1.1 Functions and Their Graphs 1 Chapter 1. Functions 1.1. Functions and Their Graphs Note. We start by assuming that you are familiar with the idea of a set and the set theoretic symbol ( an element of

More information

Pre-Calculus Chapter 0. Solving Equations and Inequalities 0.1 Solving Equations with Absolute Value 0.2 Solving Quadratic Equations

Pre-Calculus Chapter 0. Solving Equations and Inequalities 0.1 Solving Equations with Absolute Value 0.2 Solving Quadratic Equations Pre-Calculus Chapter 0. Solving Equations and Inequalities 0.1 Solving Equations with Absolute Value 0.1.1 Solve Simple Equations Involving Absolute Value 0.2 Solving Quadratic Equations 0.2.1 Use the

More information

AP Calculus AB Summer Preparation

AP Calculus AB Summer Preparation AP Calculus AB Summer Preparation Name Topic #1: GRAPHING CALCULATOR SKILLS All students are required to have a graphing calculator (GC) for use with our course. Our course is taught with the Texas Instrument

More information

HUDSONVILLE HIGH SCHOOL COURSE FRAMEWORK

HUDSONVILLE HIGH SCHOOL COURSE FRAMEWORK HUDSONVILLE HIGH SCHOOL COURSE FRAMEWORK COURSE / SUBJECT P r e c a l c u l u s ( A ) KEY COURSE OBJECTIVES/ENDURING UNDERSTANDINGS OVERARCHING/ESSENTIAL SKILLS OR QUESTIONS and Graphs Polynomial, Power,

More information

Math 4C Fall 2008 Final Exam Study Guide Format 12 questions, some multi-part. Questions will be similar to sample problems in this study guide,

Math 4C Fall 2008 Final Exam Study Guide Format 12 questions, some multi-part. Questions will be similar to sample problems in this study guide, Math 4C Fall 2008 Final Exam Study Guide Format 12 questions, some multi-part. Questions will be similar to sample problems in this study guide, homework problems, lecture examples or examples from the

More information

DuVal High School Summer Review Packet AP Calculus

DuVal High School Summer Review Packet AP Calculus DuVal High School Summer Review Packet AP Calculus Welcome to AP Calculus AB. This packet contains background skills you need to know for your AP Calculus. My suggestion is, you read the information and

More information

Math 121: Calculus 1 - Winter 2012/2013 Review of Precalculus Concepts

Math 121: Calculus 1 - Winter 2012/2013 Review of Precalculus Concepts Introduction Math 11: Calculus 1 - Winter 01/01 Review of Precalculus Concepts Welcome to Math 11 - Calculus 1, Winter 01/01! This problems in this packet are designed to help you review the topics from

More information

Seymour Public Schools Curriculum

Seymour Public Schools Curriculum The Mathematics Department believes its students must learn the importance of mathematics, the integration of different branches of mathematics, the application of math to real-life problems, and the connections

More information

Curriculum Mapper - Complete Curriculum Maps CONTENT. 1.2 Evaluate expressions (p.18 Activity 1.2).

Curriculum Mapper - Complete Curriculum Maps CONTENT. 1.2 Evaluate expressions (p.18 Activity 1.2). Page 1 of 9 Close Window Print Page Layout Show Standards View Paragraph Format View Course Description MATH 3 (MASTER MAP) School: Binghamton High School Teacher: Master Map Email: Course #: 203 Grade

More information

Polynomial Functions and Models

Polynomial Functions and Models 1 CA-Fall 2011-Jordan College Algebra, 4 th edition, Beecher/Penna/Bittinger, Pearson/Addison Wesley, 2012 Chapter 4: Polynomial Functions and Rational Functions Section 4.1 Polynomial Functions and Models

More information

Math 121: Calculus 1 - Fall 2012/2013 Review of Precalculus Concepts

Math 121: Calculus 1 - Fall 2012/2013 Review of Precalculus Concepts Introduction Math 11: Calculus 1 - Fall 01/01 Review of Precalculus Concepts Welcome to Math 11 - Calculus 1, Fall 01/01! This problems in this packet are designed to help you review the topics from Algebra

More information

List of PreCalculus Algebra Mathematical Concept Practice Sheets (Updated Spring 2015)

List of PreCalculus Algebra Mathematical Concept Practice Sheets (Updated Spring 2015) List of PreCalculus Algebra Mathematical Concept Practice Sheets (Updated Spring 2015) MAT 155P MAT 155 1 Absolute Value Equations P 7 P 3 2 Absolute Value Inequalities P 9 P 4 3 Algebraic Expressions:

More information

Class Syllabus. Accessible Topic - Topics accessible to visually impaired students using a screen reader.

Class Syllabus. Accessible Topic - Topics accessible to visually impaired students using a screen reader. Class Syllabus Class: MATH 1112 Spring 2019 Pilot - MWF 9:05 to 9:55 am Subject : College Algebra with Trigonometry Class Code: WQJQT-3GQW4 Inst ruct or: Ritter Class Dat es: 01/04/2019-05/10/2019 Class

More information

8.7 MacLaurin Polynomials

8.7 MacLaurin Polynomials 8.7 maclaurin polynomials 67 8.7 MacLaurin Polynomials In this chapter you have learned to find antiderivatives of a wide variety of elementary functions, but many more such functions fail to have an antiderivative

More information

2015 SUMMER MATH PACKET

2015 SUMMER MATH PACKET Name: Date: 05 SUMMER MATH PACKET College Algebra Trig. - I understand that the purpose of the summer packet is for my child to review the topics they have already mastered in previous math classes and

More information

Lectures. Section Theoretical (Definitions & Theorem) Examples Exercises HW

Lectures. Section Theoretical (Definitions & Theorem) Examples Exercises HW King Abdul-Aziz University Academic year 1437-1438 Department of Mathematics 2016-2017 Math 110 (S & E) Syllabus / Term (1) Book: Calculus Early Transcendentals by James Stewart 7 th edition Lectures Chapter

More information

Chapter 4E - Combinations of Functions

Chapter 4E - Combinations of Functions Fry Texas A&M University!! Math 150!! Chapter 4E!! Fall 2015! 121 Chapter 4E - Combinations of Functions 1. Let f (x) = 3 x and g(x) = 3+ x a) What is the domain of f (x)? b) What is the domain of g(x)?

More information

CP Algebra 2. Unit 3B: Polynomials. Name: Period:

CP Algebra 2. Unit 3B: Polynomials. Name: Period: CP Algebra 2 Unit 3B: Polynomials Name: Period: Learning Targets 10. I can use the fundamental theorem of algebra to find the expected number of roots. Solving Polynomials 11. I can solve polynomials by

More information

AP Calculus Summer Prep

AP Calculus Summer Prep AP Calculus Summer Prep Topics from Algebra and Pre-Calculus (Solutions are on the Answer Key on the Last Pages) The purpose of this packet is to give you a review of basic skills. You are asked to have

More information

2. Algebraic functions, power functions, exponential functions, trig functions

2. Algebraic functions, power functions, exponential functions, trig functions Math, Prep: Familiar Functions (.,.,.5, Appendix D) Name: Names of collaborators: Main Points to Review:. Functions, models, graphs, tables, domain and range. Algebraic functions, power functions, exponential

More information

High School Mathematics Honors PreCalculus

High School Mathematics Honors PreCalculus High School Mathematics Honors PreCalculus This is an accelerated course designed for the motivated math students with an above average interest in mathematics. It will cover all topics presented in Precalculus.

More information

Indiana Academic Standards for Precalculus

Indiana Academic Standards for Precalculus PRECALCULUS correlated to the Indiana Academic Standards for Precalculus CC2 6/2003 2004 Introduction to Precalculus 2004 by Roland E. Larson and Robert P. Hostetler Precalculus thoroughly explores topics

More information

NORTH ALLEGHENY SCHOOL DISTRICT MATHEMATICS DEPARTMENT HONORS PRE-CALCULUS SYLLABUS COURSE NUMBER: 3421

NORTH ALLEGHENY SCHOOL DISTRICT MATHEMATICS DEPARTMENT HONORS PRE-CALCULUS SYLLABUS COURSE NUMBER: 3421 NORTH ALLEGHENY SCHOOL DISTRICT MATHEMATICS DEPARTMENT HONORS PRE-CALCULUS SYLLABUS COURSE NUMBER: 3421 Units of Credit: 1.0 credits, honors weight Course Length: 184 days (full year) Course Overview This

More information

CK- 12 Algebra II with Trigonometry Concepts 1

CK- 12 Algebra II with Trigonometry Concepts 1 14.1 Graphing Sine and Cosine 1. A.,1 B. (, 1) C. 3,0 D. 11 1, 6 E. (, 1) F. G. H. 11, 4 7, 1 11, 3. 3. 5 9,,,,,,, 4 4 4 4 3 5 3, and, 3 3 CK- 1 Algebra II with Trigonometry Concepts 1 4.ans-1401-01 5.

More information

NYS Algebra II and Trigonometry Suggested Sequence of Units (P.I's within each unit are NOT in any suggested order)

NYS Algebra II and Trigonometry Suggested Sequence of Units (P.I's within each unit are NOT in any suggested order) 1 of 6 UNIT P.I. 1 - INTEGERS 1 A2.A.1 Solve absolute value equations and inequalities involving linear expressions in one variable 1 A2.A.4 * Solve quadratic inequalities in one and two variables, algebraically

More information

Curriculum Guide College Algebra

Curriculum Guide College Algebra Unit 1: Review of Basic Algebra Polynomial literally means many names. What is the significance of the many names for God in the Bible? 15 Lessons CA#1 1. Add/subtract/multiply/divide polynomials. 2. Factor

More information

Algebra II Notes Quadratic Functions Unit Applying Quadratic Functions. Math Background

Algebra II Notes Quadratic Functions Unit Applying Quadratic Functions. Math Background Applying Quadratic Functions Math Background Previously, you Graphed and solved quadratic functions. Solved literal equations for a given variable. Found the inverse for a linear function. Verified by

More information

correlated to the Indiana Academic Standards for Precalculus CC2

correlated to the Indiana Academic Standards for Precalculus CC2 correlated to the Indiana Academic Standards for Precalculus CC2 6/2003 2003 Introduction to Advanced Mathematics 2003 by Richard G. Brown Advanced Mathematics offers comprehensive coverage of precalculus

More information

Summer Assignment MAT 414: Calculus

Summer Assignment MAT 414: Calculus Summer Assignment MAT 414: Calculus Calculus - Math 414 Summer Assignment Due first day of school in September Name: 1. If f ( x) = x + 1, g( x) = 3x 5 and h( x) A. f ( a+ ) x+ 1, x 1 = then find: x+ 7,

More information

You should be comfortable with everything below (and if you aren t you d better brush up).

You should be comfortable with everything below (and if you aren t you d better brush up). Review You should be comfortable with everything below (and if you aren t you d better brush up).. Arithmetic You should know how to add, subtract, multiply, divide, and work with the integers Z = {...,,,

More information

Pre-Calculus Mathematics Curriculum

Pre-Calculus Mathematics Curriculum Pre-Calculus Mathematics Curriculum First day introductions, materials, policies, procedures and Summer Exam (2 days) Unit 1 Estimated time frame for unit 1 Big Ideas Essential Question Competencies Lesson

More information

Topic Outline for Algebra 2 & and Trigonometry One Year Program

Topic Outline for Algebra 2 & and Trigonometry One Year Program Topic Outline for Algebra 2 & and Trigonometry One Year Program Algebra 2 & and Trigonometry - N - Semester 1 1. Rational Expressions 17 Days A. Factoring A2.A.7 B. Rationals A2.N.3 A2.A.17 A2.A.16 A2.A.23

More information

Session 4 2:40 3:30. If neither the first nor second differences repeat, we need to try another

Session 4 2:40 3:30. If neither the first nor second differences repeat, we need to try another Linear Quadratics & Exponentials using Tables We can classify a table of values as belonging to a particular family of functions based on the math operations found on any calculator. First differences

More information

Section 6.2 Notes Page Trigonometric Functions; Unit Circle Approach

Section 6.2 Notes Page Trigonometric Functions; Unit Circle Approach Section Notes Page Trigonometric Functions; Unit Circle Approach A unit circle is a circle centered at the origin with a radius of Its equation is x y = as shown in the drawing below Here the letter t

More information

Unit Calendar. Date Sect. Topic Homework HW On-Time Apr , 2, 3 Quadratic Equations & Page 638: 3-11 Page 647: 3-29, odd

Unit Calendar. Date Sect. Topic Homework HW On-Time Apr , 2, 3 Quadratic Equations & Page 638: 3-11 Page 647: 3-29, odd Name/Period: Unit Calendar Date Sect. Topic Homework HW On-Time Apr. 4 10.1, 2, 3 Quadratic Equations & Page 638: 3-11 Graphs Page 647: 3-29, odd Apr. 6 9.4 10.4 Solving Quadratic Equations by Factoring

More information

Pre-Calculus 40 Final Outline/Review:

Pre-Calculus 40 Final Outline/Review: 2016-2017 Pre-Calculus 40 Final Outline/Review: Non-Calculator Section: 16 multiple choice (32 pts) and 6 open ended (24 pts). Calculator Section: 8 multiple choice (16 pts) and 11 open ended (36 pts).

More information

b n x n + b n 1 x n b 1 x + b 0

b n x n + b n 1 x n b 1 x + b 0 Math Partial Fractions Stewart 7.4 Integrating basic rational functions. For a function f(x), we have examined several algebraic methods for finding its indefinite integral (antiderivative) F (x) = f(x)

More information

Topic Outline for Integrated Algebra 2 and Trigonometry-R One Year Program with Regents in June

Topic Outline for Integrated Algebra 2 and Trigonometry-R One Year Program with Regents in June Topic Outline for Integrated Algebra 2 and Trigonometry-R One Year Program with Regents in June Integrated Algebra 2 & Trigonometry - R Semester 1 1. Rational Expressions 7 Days A. Factoring A2.A.7 Factor

More information

REQUIRED MATHEMATICAL SKILLS FOR ENTERING CADETS

REQUIRED MATHEMATICAL SKILLS FOR ENTERING CADETS REQUIRED MATHEMATICAL SKILLS FOR ENTERING CADETS The Department of Applied Mathematics administers a Math Placement test to assess fundamental skills in mathematics that are necessary to begin the study

More information

Precalculus Review. Functions to KNOW! 1. Polynomial Functions. Types: General form Generic Graph and unique properties. Constants. Linear.

Precalculus Review. Functions to KNOW! 1. Polynomial Functions. Types: General form Generic Graph and unique properties. Constants. Linear. Precalculus Review Functions to KNOW! 1. Polynomial Functions Types: General form Generic Graph and unique properties Constants Linear Quadratic Cubic Generalizations for Polynomial Functions - The domain

More information

Contents. CHAPTER P Prerequisites 1. CHAPTER 1 Functions and Graphs 69. P.1 Real Numbers 1. P.2 Cartesian Coordinate System 14

Contents. CHAPTER P Prerequisites 1. CHAPTER 1 Functions and Graphs 69. P.1 Real Numbers 1. P.2 Cartesian Coordinate System 14 CHAPTER P Prerequisites 1 P.1 Real Numbers 1 Representing Real Numbers ~ Order and Interval Notation ~ Basic Properties of Algebra ~ Integer Exponents ~ Scientific Notation P.2 Cartesian Coordinate System

More information

CURRICULUM GUIDE. Honors Algebra II / Trigonometry

CURRICULUM GUIDE. Honors Algebra II / Trigonometry CURRICULUM GUIDE Honors Algebra II / Trigonometry The Honors course is fast-paced, incorporating the topics of Algebra II/ Trigonometry plus some topics of the pre-calculus course. More emphasis is placed

More information

STANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA II. 2 nd Nine Weeks,

STANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA II. 2 nd Nine Weeks, STANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA II 2 nd Nine Weeks, 2016-2017 1 OVERVIEW Algebra II Content Review Notes are designed by the High School Mathematics Steering Committee as a resource

More information

Advanced Mathematics Unit 2 Limits and Continuity

Advanced Mathematics Unit 2 Limits and Continuity Advanced Mathematics 3208 Unit 2 Limits and Continuity NEED TO KNOW Expanding Expanding Expand the following: A) (a + b) 2 B) (a + b) 3 C) (a + b)4 Pascals Triangle: D) (x + 2) 4 E) (2x -3) 5 Random Factoring

More information

Advanced Mathematics Unit 2 Limits and Continuity

Advanced Mathematics Unit 2 Limits and Continuity Advanced Mathematics 3208 Unit 2 Limits and Continuity NEED TO KNOW Expanding Expanding Expand the following: A) (a + b) 2 B) (a + b) 3 C) (a + b)4 Pascals Triangle: D) (x + 2) 4 E) (2x -3) 5 Random Factoring

More information

North Toronto Christian School MATH DEPARTMENT

North Toronto Christian School MATH DEPARTMENT North Toronto Christian School MATH DEPARTMENT 2017-2018 Course: Advanced Functions, Grade 12 MHF4U Teacher: D. Brooks (dbrooks@ntcs.on.ca) Course description: This course extends students experience with

More information

Math 12 Final Exam Review 1

Math 12 Final Exam Review 1 Math 12 Final Exam Review 1 Part One Calculators are NOT PERMITTED for this part of the exam. 1. a) The sine of angle θ is 1 What are the 2 possible values of θ in the domain 0 θ 2π? 2 b) Draw these angles

More information

Algebra II Learning Targets

Algebra II Learning Targets Chapter 0 Preparing for Advanced Algebra LT 0.1 Representing Functions Identify the domain and range of functions LT 0.2 FOIL Use the FOIL method to multiply binomials LT 0.3 Factoring Polynomials Use

More information

Using a graphic display calculator

Using a graphic display calculator 12 Using a graphic display calculator CHAPTER OBJECTIVES: This chapter shows you how to use your graphic display calculator (GDC) to solve the different types of problems that you will meet in your course.

More information

Quantile Textbook Report

Quantile Textbook Report Quantile Textbook Report Algebra 2 Author Charles, Randall I., et al StateEdition West Virginia Grade Algebra 2 1 Expressions, Equations, and Inequalities 1.1 Patterns and Expressions 930Q 1.2 Properties

More information

AP Calculus Summer Assignment Summer 2017 Expectations for Summer Assignment on the first day of the school year.

AP Calculus Summer Assignment Summer 2017 Expectations for Summer Assignment on the first day of the school year. Summer 07 Expectations for Summer Assignment This packet is to be submitted to your Calculus BC teacher on the first day of the school year. All work must be shown in the packet OR on separate paper attached

More information

Pre Calculus Gary Community School Corporation Unit Planning Map

Pre Calculus Gary Community School Corporation Unit Planning Map UNIT/TIME FRAME STANDARDS Functions and Graphs (6 weeks) PC.F.1: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities,

More information

PRECALCULUS BISHOP KELLY HIGH SCHOOL BOISE, IDAHO. Prepared by Kristina L. Gazdik. March 2005

PRECALCULUS BISHOP KELLY HIGH SCHOOL BOISE, IDAHO. Prepared by Kristina L. Gazdik. March 2005 PRECALCULUS BISHOP KELLY HIGH SCHOOL BOISE, IDAHO Prepared by Kristina L. Gazdik March 2005 1 TABLE OF CONTENTS Course Description.3 Scope and Sequence 4 Content Outlines UNIT I: FUNCTIONS AND THEIR GRAPHS

More information

George Washington Carver Engineering and Science High School 2018 Summer Enrichment

George Washington Carver Engineering and Science High School 2018 Summer Enrichment George Washington Carver Engineering and Science High School 2018 Summer Enrichment CALCULUS ASSIGNMENT #1: SUMMER PROJECT (Due Tuesday September 11 th ) (Worth 100 points for the packet and 100 points

More information

5.4 - Quadratic Functions

5.4 - Quadratic Functions Fry TAMU Spring 2017 Math 150 Notes Section 5.4 Page! 92 5.4 - Quadratic Functions Definition: A function is one that can be written in the form f (x) = where a, b, and c are real numbers and a 0. (What

More information

Summer Packet for Students Taking Introduction to Calculus in the Fall

Summer Packet for Students Taking Introduction to Calculus in the Fall Summer Packet for Students Taking Introduction to Calculus in the Fall Algebra 2 Topics Needed for Introduction to Calculus Need to know: à Solve Equations Linear Quadratic Absolute Value Polynomial Rational

More information

Topics from Algebra and Pre-Calculus. (Key contains solved problems)

Topics from Algebra and Pre-Calculus. (Key contains solved problems) Topics from Algebra and Pre-Calculus (Key contains solved problems) Note: The purpose of this packet is to give you a review of basic skills. You are asked not to use the calculator, except on p. (8) and

More information

MyMathLab for School Precalculus Graphical, Numerical, Algebraic Common Core Edition 2016

MyMathLab for School Precalculus Graphical, Numerical, Algebraic Common Core Edition 2016 A Correlation of MyMathLab for School Precalculus Common Core Edition 2016 to the Tennessee Mathematics Standards Approved July 30, 2010 Bid Category 13-090-10 , Standard 1 Mathematical Processes Course

More information

Algebra and Trigonometry

Algebra and Trigonometry Algebra and Trigonometry 978-1-63545-098-9 To learn more about all our offerings Visit Knewtonalta.com Source Author(s) (Text or Video) Title(s) Link (where applicable) OpenStax Jay Abramson, Arizona State

More information

CARLISLE AREA SCHOOL DISTRICT Carlisle, PA PRE-CALCULUS. GRADES 11 and 12

CARLISLE AREA SCHOOL DISTRICT Carlisle, PA PRE-CALCULUS. GRADES 11 and 12 CARLISLE AREA SCHOOL DISTRICT Carlisle, PA 17013 PRE-CALCULUS GRADES 11 and 12 Date of Board Approval: April 17, 2014 CARLISLE AREA SCHOOL DISTRICT PLANNED INSTRUCTION COVER PAGE TITLE OF SUBJECT: Math

More information

AP CALCULUS AB Study Guide for Midterm Exam 2017

AP CALCULUS AB Study Guide for Midterm Exam 2017 AP CALCULUS AB Study Guide for Midterm Exam 2017 CHAPTER 1: PRECALCULUS REVIEW 1.1 Real Numbers, Functions and Graphs - Write absolute value as a piece-wise function - Write and interpret open and closed

More information

x is also called the abscissa y is also called the ordinate "If you can create a t-table, you can graph anything!"

x is also called the abscissa y is also called the ordinate If you can create a t-table, you can graph anything! Senior Math Section 6-1 Notes Rectangular Coordinates and Lines Label the following 1. quadrant 1 2. quadrant 2 3. quadrant 3 4. quadrant 4 5. origin 6. x-axis 7. y-axis 8. Ordered Pair (x, y) at (2, 1)

More information

a factors The exponential 0 is a special case. If b is any nonzero real number, then

a factors The exponential 0 is a special case. If b is any nonzero real number, then 0.1 Exponents The expression x a is an exponential expression with base x and exponent a. If the exponent a is a positive integer, then the expression is simply notation that counts how many times the

More information

As we know, the three basic trigonometric functions are as follows: Figure 1

As we know, the three basic trigonometric functions are as follows: Figure 1 Trigonometry Basic Functions As we know, the three basic trigonometric functions are as follows: sin θ = cos θ = opposite hypotenuse adjacent hypotenuse tan θ = opposite adjacent Where θ represents an

More information

1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.

1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents. Math120 - Precalculus. Final Review. Fall, 2011 Prepared by Dr. P. Babaali 1 Algebra 1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.

More information

Algebra 2 Khan Academy Video Correlations By SpringBoard Activity

Algebra 2 Khan Academy Video Correlations By SpringBoard Activity SB Activity Activity 1 Creating Equations 1-1 Learning Targets: Create an equation in one variable from a real-world context. Solve an equation in one variable. 1-2 Learning Targets: Create equations in

More information

AP CALCULUS AB. Summer Assignment. Page 1

AP CALCULUS AB. Summer Assignment. Page 1 AP CALCULUS AB Summer Assignment Page 1 Welcome to AP Calculus AB. This will be the toughest class yet in your mathematical careers but the benefit you will receive by having this experience in high school

More information

Algebra 2 Khan Academy Video Correlations By SpringBoard Activity

Algebra 2 Khan Academy Video Correlations By SpringBoard Activity SB Activity Activity 1 Creating Equations 1-1 Learning Targets: Create an equation in one variable from a real-world context. Solve an equation in one variable. 1-2 Learning Targets: Create equations in

More information

Algebra Review. Finding Zeros (Roots) of Quadratics, Cubics, and Quartics. Kasten, Algebra 2. Algebra Review

Algebra Review. Finding Zeros (Roots) of Quadratics, Cubics, and Quartics. Kasten, Algebra 2. Algebra Review Kasten, Algebra 2 Finding Zeros (Roots) of Quadratics, Cubics, and Quartics A zero of a polynomial equation is the value of the independent variable (typically x) that, when plugged-in to the equation,

More information

Algebra 3-4 Honors PUHSD Curriculum. PARCC MODEL CONTENT FRAMEWORK FOR ALGEBRA 3-4 Honors

Algebra 3-4 Honors PUHSD Curriculum. PARCC MODEL CONTENT FRAMEWORK FOR ALGEBRA 3-4 Honors PARCC MODEL CONTENT FRAMEWORK FOR ALGEBRA 3-4 Honors Building on their work in Algebra I with linear and quadratic functions, students in Algebra II expand their repertoire by working with rational and

More information