2.2 Limit of a Function and Limit Laws
|
|
- Mitchell Henderson
- 6 years ago
- Views:
Transcription
1 Limit of a Function and Limit Laws Section Notes Page Let s look at the graph y What is y()? That s right, its undefined, but what if we wanted to find the y value the graph is approaching as we get close to an value of? This y-value that it is approaching is called a it Here s some notation for the problem we were just describing What this means is that we want to find what y-value the graph is approaching as gets close to Since we need to get really close to an -value of, let s make a table of values We want to pick values for that are very close to We can pick a couple above and below To find these, you need to put each value into the formula y You can use your table feature on your graphing calculator to do this one Because there are many types of graphing calculators, see me after class if you need to know how to use the table function The table of values should look like: y() By looking at this table, it appears the y-value is approaching 05, or / So you would write your answer as: Now let s use some algebra (this will be eplained more in a later chapter) We can factor the denominator so now the problem becomes To solve this, just put a in for ( )( ) and we get: This is the same as our estimate Limits must approach the same number For eample, let s look at the following graph of f(): Let s look at f ( ) Notice as we approach from the left and from the right we approach two different numbers When this happens, we say the it does not eist
2 What about f ( ) if the graph below is of f()? Section Notes Page Here as approaches from each side, the y values approach two different values, so again the it does not eist What about f ( ) if the graph below is of f()? This one is called an unbounded it since the y-values keep increasing to Since infinity is not a real number, the it does not eist Let s look at the graph below: First, does f() eist? Yes, f() = What is f ( )? Does not eist Approaches two different y-values What is f ( )? This eists As approaches negative one we want to see what y-value the graph approaches This would be zero More on net page
3 Limit Laws Section Notes Page ) f ( ) g( ) f ( ) g( ) c c ) k f ( ) k f ( ) c ) c c c f ( ) g( ) f ( ) g( ) c ) f ( ) c g( ) f ( ) c g( ) c n 5) [ f ( )] f ( ) n c c ) n f ( ) n f ( ) c c 7 5 by using it properties to break it down The properties tell us we can break up this it like this: (5) What are we really doing here? We are simply plugging in the c value into our epression You are not required to show the break down of each it unless the questions specifically ask you to Here, we are just going to replace with since this is our c value () () () 9 The problem with this one is that if I put in a - for I will be dividing by zero which is undefined However if we factor the denominator you will be able to cancel out the part that make the bottom zero Once this part is einated then we can plug in the - for : 9 ( )( )
4 5 8 Section Notes Page This is another one where you will divide by zero if you put in a for Again you would want to factor both the numerator and denominator and then cancel Finally you can then plug in for 5 ( )( ) 8 ( )( ) Since plugging in a won t give us a zero in the denominator, we can just plug in and get the answer: For this one, just plug in - for Remember you are allowed to take the odd root of a negative number 7 0 The problem with this one is that if we put in a zero for we will be dividing by zero so we must do something to this to get rid of the Almost always the operation you will do is to multiply the top and bottom by the conjugate A conjugate (of the numerator in this case) is the same thing but with the opposite sign So we will multiply top and bottom by Then we will cancel: 0 0 Now when we multiply across the top you can use the difference of squares formula, which is ( a b)( a b) a b So if we have then this will equal: and when we simplify we get So let s continue: Now cancel the s to get: Now plug in 0 for : 0 0
5 Section Notes Page 5 This is another one we need to multiply by the conjugate, but this time we will multiply by the conjugate of the denominator, which would be We then will follow the same steps as shown above You can still use the difference of squares formula when you do ( ) Limits with Trigonometric Functions With these its we can still plug in the c value into the epression to get the it You will just need to make sure you have your unit circle or trig tables ready tan Just put in the pi for You will get: tan tan 0 cos cos cos cos 0 sin sin sin Sometimes you may need to use trig identities to simplify before you plug in tan sec If we plug in pi right now we will be dividing by zero We will write these in terms of sine and cosine tan sec sin cos cos sin sin
Limits. f(x) and lim. g(x) g(x)
Limits Limit Laws Suppose c is constant, n is a positive integer, and f() and g() both eist. Then,. [f() + g()] = f() + g() 2. [f() g()] = f() g() [ ] 3. [c f()] = c f() [ ] [ ] 4. [f() g()] = f() g()
More informationThe method of rationalizing
Roberto s Notes on Differential Calculus Chapter : Resolving indeterminate forms Section The method of rationalizing What you need to know already: The concept of it and the factor-and-cancel method of
More informationWelcome. Please Sign-In
Welcome Please Sign-In Day 1 Session 1 Self-Evaluation Topics to be covered: Equations Systems of Equations Solving Inequalities Absolute Value Equations Equations Equations An equation says two things
More informationThe method of rationalizing
Roberto s Notes on Differential Calculus Chapter : Resolving indeterminate forms Section The method of rationalizing What you need to know already: The concept of it and the factor-and-cancel method of
More informationTrigonometry Review Day 1
Name Trigonometry Review Day 1 Algebra II Rotations and Angle Terminology II Terminal y I Positive angles rotate in a counterclockwise direction. Reference Ray Negative angles rotate in a clockwise direction.
More informationSUM AND DIFFERENCES. Section 5.3 Precalculus PreAP/Dual, Revised 2017
SUM AND DIFFERENCES Section 5. Precalculus PreAP/Dual, Revised 2017 Viet.dang@humbleisd.net 8/1/2018 12:41 AM 5.4: Sum and Differences of Trig Functions 1 IDENTITY Question 1: What is Cosine 45? Question
More informationUnit 6 Introduction to Trigonometry The Unit Circle (Unit 6.3)
Unit Introduction to Trigonometr The Unit Circle Unit.) William Bill) Finch Mathematics Department Denton High School Introduction Trig Functions Circle Quadrental Angles Other Angles Unit Circle Periodic
More informationMA 180 Lecture Chapter 7 College Algebra and Calculus by Larson/Hodgkins Limits and Derivatives
MA 180 Lecture Chapter 7 College Algebra and Calculus by Larson/Hodgkins Limits and Derivatives 7.1) Limits An important concept in the study of mathematics is that of a it. It is often one of the harder
More informationFinal Exam: Precalculus
Final Exam: Precalculus Apr. 17, 2018 ANSWERS Without Notes or Calculators Version A 1. Consider the unit circle: a. Angle in degrees: What is the angle in radians? What are the coordinates? b. Coordinates:
More informationSection 6.2 Graphs of the Other Trig Functions
Section 62 Graphs of the Other Trig Functions 369 Section 62 Graphs of the Other Trig Functions In this section, we will explore the graphs of the other four trigonometric functions We ll begin with the
More informationCalculus I (part 1): Limits and Continuity (by Evan Dummit, 2016, v. 2.01)
Calculus I (part ): Limits and Continuity (by Evan Dummit, 206, v. 2.0) Contents Limits and Continuity. Limits (Informally)...............................................2 Limits and the Limit Laws..........................................
More informationAP Calculus Summer Review Packet
AP Calculus Summer Review Packet Name: Date began: Completed: **A Formula Sheet has been stapled to the back for your convenience!** Email anytime with questions: danna.seigle@henry.k1.ga.us Complex Fractions
More informationChapter 4 Using Fundamental Identities Section USING FUNDAMENTAL IDENTITIES. Fundamental Trigonometric Identities. Reciprocal Identities
Chapter 4 Using Fundamental Identities Section 4.1 4.1 USING FUNDAMENTAL IDENTITIES Fundamental Trigonometric Identities Reciprocal Identities csc x sec x cot x Quotient Identities tan x cot x Pythagorean
More informationLimits and Derivatives (Review of Math 249 or 251)
Chapter 3 Limits and Derivatives (Review of Math 249 or 251) 3.1 Overview This is the first of two chapters reviewing material from calculus; its and derivatives are discussed in this chapter, and integrals
More informationSummer Assignment for students entering: Algebra 2 Trigonometry Honors
Summer Assignment for students entering: Algebra Trigonometry Honors Please have the following worksheets completed and ready to be handed in on the first day of class in the fall. Make sure you show your
More informationAlbertson AP Calculus AB AP CALCULUS AB SUMMER PACKET DUE DATE: The beginning of class on the last class day of the first week of school.
Albertson AP Calculus AB Name AP CALCULUS AB SUMMER PACKET 2017 DUE DATE: The beginning of class on the last class day of the first week of school. This assignment is to be done at you leisure during the
More informationTable of Laplace Transforms
Table of Laplace Transforms 1 1 2 3 4, p > -1 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 Heaviside Function 27 28. Dirac Delta Function 29 30. 31 32. 1 33 34. 35 36. 37 Laplace Transforms
More information1. The Pythagorean Theorem
. The Pythagorean Theorem The Pythagorean theorem states that in any right triangle, the sum of the squares of the side lengths is the square of the hypotenuse length. c 2 = a 2 b 2 This theorem can be
More informationGraphing Trig Functions - Sine & Cosine
Graphing Trig Functions - Sine & Cosine Up to this point, we have learned how the trigonometric ratios have been defined in right triangles using SOHCAHTOA as a memory aid. We then used that information
More informationSection 7.6 Graphs of the Sine and Cosine Functions
Section 7.6 Graphs of the Sine and Cosine Functions We are going to learn how to graph the sine and cosine functions on the xy-plane. Just like with any other function, it is easy to do by plotting points.
More informationWalt Whitman High School SUMMER REVIEW PACKET. For students entering AP CALCULUS BC
Walt Whitman High School SUMMER REVIEW PACKET For students entering AP CALCULUS BC Name: 1. This packet is to be handed in to your Calculus teacher on the first day of the school year.. All work must be
More informationRadical Expressions and Functions What is a square root of 25? How many square roots does 25 have? Do the following square roots exist?
Hartfield Intermediate Algebra (Version 2014-2D) Unit 4 Page 1 Topic 4 1 Radical Epressions and Functions What is a square root of 25? How many square roots does 25 have? Do the following square roots
More informationPreCalculus 300. Algebra 2 Review
PreCalculus 00 Algebra Review Algebra Review The following topics are a review of some of what you learned last year in Algebra. I will spend some time reviewing them in class. You are responsible for
More informationCalculus Chapter 1 Limits. Section 1.2 Limits
Calculus Chapter 1 Limits Section 1.2 Limits Limit Facts part 1 1. The answer to a limit is a y-value. 2. The limit tells you to look at a certain x value. 3. If the x value is defined (in the domain),
More informationMastery. PRECALCULUS Student Learning Targets
PRECALCULUS Student Learning Targets Big Idea: Sequences and Series 1. I can describe a sequence as a function where the domain is the set of natural numbers. Connections (Pictures, Vocabulary, Definitions,
More informationAn Introduction to Maple This lab is adapted from a lab created by Bob Milnikel.
Some quick tips for getting started with Maple: An Introduction to Maple This lab is adapted from a lab created by Bob Milnikel. [Even before we start, take note of the distinction between Tet mode and
More informationTABLE OF CONTENTS CHAPTER 1 LIMIT AND CONTINUITY... 26
TABLE OF CONTENTS CHAPTER LIMIT AND CONTINUITY... LECTURE 0- BASIC ALGEBRAIC EXPRESSIONS AND SOLVING EQUATIONS... LECTURE 0- INTRODUCTION TO FUNCTIONS... 9 LECTURE 0- EXPONENTIAL AND LOGARITHMIC FUNCTIONS...
More informationIn section 8.1, we began by introducing the sine function using a circle in the coordinate plane:
Chapter 8.: Degrees and Radians, Reference Angles In section 8.1, we began by introducing the sine function using a circle in the coordinate plane: y (3,3) θ x We now return to the coordinate plane, but
More informationSecondary Math 3- Honors. 7-4 Inverse Trigonometric Functions
Secondary Math 3- Honors 7-4 Inverse Trigonometric Functions Warm Up Fill in the Unit What You Will Learn How to restrict the domain of trigonometric functions so that the inverse can be constructed. How
More informationUse the indicated strategy from your notes to simplify each expression. Each section may use the indicated strategy AND those strategies before.
Pre Calculus Worksheet: Fundamental Identities Day 1 Use the indicated strategy from your notes to simplify each expression. Each section may use the indicated strategy AND those strategies before. Strategy
More informationRadical Functions Review
Radical Functions Review Specific Outcome 3 Graph and analyze radical functions (limited to functions involving one radical) Acceptable Standard sketch and analyze (domain, range, invariant points, - and
More informationThe Straight Line. m is undefined. Use. Show that mab
The Straight Line What is the gradient of a horizontal line? What is the equation of a horizontal line? So the equation of the x-axis is? What is the gradient of a vertical line? What is the equation of
More informationPre Calculus Worksheet: Fundamental Identities Day 1
Pre Calculus Worksheet: Fundamental Identities Day 1 Use the indicated strategy from your notes to simplify each expression. Each section may use the indicated strategy AND those strategies before. Strategy
More informationObjectives: After completing this section, you should be able to do the following: Calculate the lengths of sides and angles of a right triangle using
Ch 13 - RIGHT TRIANGLE TRIGONOMETRY Objectives: After completing this section, you should be able to do the following: Calculate the lengths of sides and angles of a right triangle using trigonometric
More informationMath 3 Coordinate Geometry Part 2 Graphing Solutions
Math 3 Coordinate Geometry Part 2 Graphing Solutions 1 SOLVING SYSTEMS OF EQUATIONS GRAPHICALLY The solution of two linear equations is the point where the two lines intersect. For example, in the graph
More informationFunctional Analysis Functions in Review
Functional Analysis Functions in Review You have spent a great deal of time over the past three years studying all of the algebraic types of functions linear, absolute value, quadratic, cubic, quartic,
More informationMATH EXAM 1 - SPRING 2018 SOLUTION
MATH 140 - EXAM 1 - SPRING 018 SOLUTION 8 February 018 Instructor: Tom Cuchta Instructions: Show all work, clearly and in order, if you want to get full credit. If you claim something is true you must
More informationChapter 7: Analytic Trigonometry
Chapter 7: Analytic Trigonometry 7. Trigonometric Identities Below are the basic trig identities discussed in previous chapters. Reciprocal csc(x) sec(x) cot(x) sin(x) cos(x) tan(x) Quotient sin(x) cos(x)
More informationTrigonometric Integrals
Most trigonometric integrals can be solved by using trigonometric identities or by following a strategy based on the form of the integrand. There are some that are not so easy! Basic Trig Identities and
More information2.1 Basics of Functions and Their Graphs
.1 Basics of Functions and Their Graphs Section.1 Notes Page 1 Domain: (input) all the x-values that make the equation defined Defined: There is no division by zero or square roots of negative numbers
More informationSession 3. Rational and Radical Equations. Math 30-1 R 3. (Revisit, Review and Revive)
Session 3 Rational and Radical Equations Math 30-1 R 3 (Revisit, Review and Revive) Rational Functions Review Specific Outcome 14 Graph and analyze rational functions (limited to numerators and denominators
More information14.1 Similar Triangles and the Tangent Ratio Per Date Trigonometric Ratios Investigate the relationship of the tangent ratio.
14.1 Similar Triangles and the Tangent Ratio Per Date Trigonometric Ratios Investigate the relationship of the tangent ratio. Using the space below, draw at least right triangles, each of which has one
More informationSlide 1 / 180. Radicals and Rational Exponents
Slide 1 / 180 Radicals and Rational Exponents Slide 2 / 180 Roots and Radicals Table of Contents: Square Roots Intro to Cube Roots n th Roots Irrational Roots Rational Exponents Operations with Radicals
More information7.2 Trigonometric Integrals
7. Trigonometric Integrals The three identities sin x + cos x, cos x (cos x + ) and sin x ( cos x) can be used to integrate expressions involving powers of Sine and Cosine. The basic idea is to use an
More informationA Quick Review of Trigonometry
A Quick Review of Trigonometry As a starting point, we consider a ray with vertex located at the origin whose head is pointing in the direction of the positive real numbers. By rotating the given ray (initial
More informationMath 144 Activity #2 Right Triangle Trig and the Unit Circle
1 p 1 Right Triangle Trigonometry Math 1 Activity #2 Right Triangle Trig and the Unit Circle We use right triangles to study trigonometry. In right triangles, we have found many relationships between the
More informationUnit 7 Solving Right triangles math 2.notebook April 17, 2018
Warm Up Calculate the value of. Unit 7 Learning Intention: Given a right triangle, students will be able to write and use trigonometric ratios to solve right triangles. Success Criteria: 1. I will be able
More informationSection 1.1: Four Ways to Represent a Function
Section.: Four Ways to Represent a Function. The Definition of a Function Functions are one of the most basic tools in mathematics, so we start by considering the definition of a function and all related
More informationUnit 4 Trigonometry. Study Notes 1 Right Triangle Trigonometry (Section 8.1)
Unit 4 Trigonometr Stud Notes 1 Right Triangle Trigonometr (Section 8.1) Objective: Evaluate trigonometric functions of acute angles. Use a calculator to evaluate trigonometric functions. Use trigonometric
More informationNow, we need to refresh our memory of some axioms from Geometry. 3 sides known
9.3 The Law of Sines First we need the definition for an oblique triangle. This is nothing but a triangle that is not a right triangle. In other words, all angles in the triangle are not of a measure of
More information18.01 Single Variable Calculus Fall 2006
MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Lecture 6: Trigonometric
More information4.1: Angles & Angle Measure
4.1: Angles & Angle Measure In Trigonometry, we use degrees to measure angles in triangles. However, degree is not user friendly in many situations (just as % is not user friendly unless we change it into
More informationYou can take the arccos of both sides to get θ by itself.
.7 SOLVING TRIG EQUATIONS Example on p. 8 How do you solve cos ½ for? You can tae the arccos of both sides to get by itself. cos - (cos ) cos - ( ½) / However, arccos only gives us an answer between 0
More informationReview of Trigonometry
Worksheet 8 Properties of Trigonometric Functions Section Review of Trigonometry This section reviews some of the material covered in Worksheets 8, and The reader should be familiar with the trig ratios,
More informationf(x) lim does not exist.
Indeterminate Forms and L Hopital s Rule When we computed its of quotients, i.e. its of the form f() a g(), we came across several different things that could happen: f(). a g() = f(a) g(a) when g(). a
More informationRelated Angles WS # 6.1. Co-Related Angles WS # 6.2. Solving Linear Trigonometric Equations Linear
UNIT 6 TRIGONOMETRIC IDENTITIES AND EQUATIONS Date Lesson Text TOPIC Homework 13 6.1 (60) 7.1 Related Angles WS # 6.1 15 6. (61) 7.1 Co-Related Angles WS # 6. 16 6.3 (63) 7. Compound Angle Formulas I Sine
More informationGraphs of Other Trig Functions
Graph y = tan. y 0 0 6 3 3 3 5 6 3 3 1 Graphs of Other Trig Functions.58 3 1.7 undefined 3 3 3 1.7-1 0.58 3 CHAT Pre-Calculus 3 The Domain is all real numbers ecept multiples of. (We say the domain is
More informationRelation: Pairs of items that are related in a predictable way.
We begin this unit on a Friday, after a quiz. We may or may not go through these ideas in class. Note that there are links to Kahn Academy lessons on my website. Objective 1. Recognize a relation vs. a
More informationSection 1.4 Limits involving infinity
Section. Limits involving infinit (/3/08) Overview: In later chapters we will need notation and terminolog to describe the behavior of functions in cases where the variable or the value of the function
More informationSection 10.1 Polar Coordinates
Section 10.1 Polar Coordinates Up until now, we have always graphed using the rectangular coordinate system (also called the Cartesian coordinate system). In this section we will learn about another system,
More informationSection 4.4 Rational Functions and Their Graphs. 1, the line x = 0 (y-axis) is its vertical asymptote.
Section 4.4 Rational Functions and Their Graphs p( ) A rational function can be epressed as where p() and q() are q( ) 3 polynomial functions and q() is not equal to 0. For eample, 16 is a rational function.
More informationby Kevin M. Chevalier
Precalculus Review Handout.4 Trigonometric Functions: Identities, Graphs, and Equations, Part I by Kevin M. Chevalier Angles, Degree and Radian Measures An angle is composed of: an initial ray (side) -
More informationSquare roots: We say that the square root of 16 is 4. We write this as 16 4.
Intermediate algebra Class notes Radicals and Radical Functions (section 10.1) These are square roots, cube roots, etc. Square roots: We say that the square root of 16 is 4. We write this as 16 4. Root
More informationto and go find the only place where the tangent of that
Study Guide for PART II of the Spring 14 MAT187 Final Exam. NO CALCULATORS are permitted on this part of the Final Exam. This part of the Final exam will consist of 5 multiple choice questions. You will
More informationUnit Circle. Project Response Sheet
NAME: PROJECT ACTIVITY: Trigonometry TOPIC Unit Circle GOALS MATERIALS Explore Degree and Radian Measure Explore x- and y- coordinates on the Unit Circle Investigate Odd and Even functions Investigate
More informationSection 4.4 Rational Functions and Their Graphs
Section 4.4 Rational Functions and Their Graphs p( ) A rational function can be epressed as where p() and q() are q( ) 3 polynomial functions and q() is not equal to 0. For eample, is a 16 rational function.
More informationAlgebra II. Chapter 13 Notes Sections 13.1 & 13.2
Algebra II Chapter 13 Notes Sections 13.1 & 13.2 Name Algebra II 13.1 Right Triangle Trigonometry Day One Today I am using SOHCAHTOA and special right triangle to solve trig problems. I am successful
More informationBasics of Computational Geometry
Basics of Computational Geometry Nadeem Mohsin October 12, 2013 1 Contents This handout covers the basic concepts of computational geometry. Rather than exhaustively covering all the algorithms, it deals
More information0 COORDINATE GEOMETRY
0 COORDINATE GEOMETRY Coordinate Geometr 0-1 Equations of Lines 0- Parallel and Perpendicular Lines 0- Intersecting Lines 0- Midpoints, Distance Formula, Segment Lengths 0- Equations of Circles 0-6 Problem
More informationUsing Fundamental Identities. Fundamental Trigonometric Identities. Reciprocal Identities. sin u 1 csc u. sec u. sin u Quotient Identities
3330_050.qxd /5/05 9:5 AM Page 374 374 Chapter 5 Analytic Trigonometry 5. Using Fundamental Identities What you should learn Recognize and write the fundamental trigonometric identities. Use the fundamental
More informationCheck In before class starts:
Name: Date: Lesson 5-3: Graphing Trigonometric Functions Learning Goal: How do I use the critical values of the Sine and Cosine curve to graph vertical shift and vertical stretch? Check In before class
More informationAlgebra II. Slide 1 / 162. Slide 2 / 162. Slide 3 / 162. Trigonometric Functions. Trig Functions
Slide 1 / 162 Algebra II Slide 2 / 162 Trigonometric Functions 2015-12-17 www.njctl.org Trig Functions click on the topic to go to that section Slide 3 / 162 Radians & Degrees & Co-terminal angles Arc
More informationCalculus I Review Handout 1.3 Introduction to Calculus - Limits. by Kevin M. Chevalier
Calculus I Review Handout 1.3 Introduction to Calculus - Limits by Kevin M. Chevalier We are now going to dive into Calculus I as we take a look at the it process. While precalculus covered more static
More informationAlgebra II Radical Equations
1 Algebra II Radical Equations 2016-04-21 www.njctl.org 2 Table of Contents: Graphing Square Root Functions Working with Square Roots Irrational Roots Adding and Subtracting Radicals Multiplying Radicals
More informationCW High School. Advanced Math A. 1.1 I can make connections between the algebraic equation or description for a function, its name, and its graph.
1. Functions and Math Models (10.00%) 1.1 I can make connections between the algebraic equation or description for a function, its name, and its graph. 4 Pro cient I can make connections between the algebraic
More informationSummer Packet Geometry PAP
Summer Packet Geometry PAP IMPORTANT INSTRUCTIONS FOR STUDENTS!!! We understand that students come to Geometry with different strengths and needs. For this reason, students have options for completing
More informationMATH STUDENT BOOK. 12th Grade Unit 4
MATH STUDENT BOOK th Grade Unit Unit GRAPHING AND INVERSE FUNCTIONS MATH 0 GRAPHING AND INVERSE FUNCTIONS INTRODUCTION. GRAPHING 5 GRAPHING AND AMPLITUDE 5 PERIOD AND FREQUENCY VERTICAL AND HORIZONTAL
More informationVerifying Trigonometric Identities
Verifying Trigonometric Identities What you should learn Verify trigonometric identities. Why you should learn it You can use trigonometric identities to rewrite trigonometric equations that model real-life
More informationChoose the correct answer below. 2. Convert the angle to a decimal in degrees.
1. Choose the figure that shows an angle of in standard position. Choose the correct answer below. 2. Convert the angle to a decimal in degrees. (Do not round until the final answer. Then round to two
More informationChapter 1. Limits and Continuity. 1.1 Limits
Chapter Limits and Continuit. Limits The its is the fundamental notion of calculus. This underling concept is the thread that binds together virtuall all of the calculus ou are about to stud. In this section,
More informationMATH056 IMMERSION Dr Jason Samuels simplify the expression tutoring M-Th M1203
MATH056 IMMERSION Dr Jason Samuels simplify the expression tutoring M-Th 1230-530 M1203 # identify every calculation that order of operations tells you to do in the next step # check your signs solve for
More informationAW Math 10 UNIT 7 RIGHT ANGLE TRIANGLES
AW Math 10 UNIT 7 RIGHT ANGLE TRIANGLES Assignment Title Work to complete Complete 1 Triangles Labelling Triangles 2 Pythagorean Theorem 3 More Pythagorean Theorem Eploring Pythagorean Theorem Using Pythagorean
More informationSection 2.5: Continuity
Section 2.5: Continuity 1. The Definition of Continuity We start with a naive definition of continuity. Definition 1.1. We say a function f() is continuous if we can draw its graph without lifting out
More informationLesson Title 2: Problem TK Solving with Trigonometric Ratios
Part UNIT RIGHT solving TRIANGLE equations TRIGONOMETRY and inequalities Lesson Title : Problem TK Solving with Trigonometric Ratios Georgia Performance Standards MMG: Students will define and apply sine,
More informationTrigonometry and the Unit Circle. Chapter 4
Trigonometry and the Unit Circle Chapter 4 Topics Demonstrate an understanding of angles in standard position, expressed in degrees and radians. Develop and apply the equation of the unit circle. Solve
More informationAP Calculus AB. Table of Contents. Slide 1 / 180. Slide 2 / 180. Slide 3 / 180. Review Unit
Slide 1 / 180 Slide 2 / 180 P alculus Review Unit 2015-10-20 www.njctl.org Table of ontents lick on the topic to go to that section Slide 3 / 180 Slopes Equations of Lines Functions Graphing Functions
More information: Find the values of the six trigonometric functions for θ. Special Right Triangles:
ALGEBRA 2 CHAPTER 13 NOTES Section 13-1 Right Triangle Trig Understand and use trigonometric relationships of acute angles in triangles. 12.F.TF.3 CC.9- Determine side lengths of right triangles by using
More informationGraphing Trigonometric Functions
LESSON Graphing Trigonometric Functions Graphing Sine and Cosine UNDERSTAND The table at the right shows - and f ()-values for the function f () 5 sin, where is an angle measure in radians. Look at the
More informationAP Calculus AB. Table of Contents. Slide 1 / 180. Slide 2 / 180. Slide 3 / 180. Review Unit
Slide 1 / 180 Slide 2 / 180 P alculus Review Unit 2015-10-20 www.njctl.org Table of ontents lick on the topic to go to that section Slide 3 / 180 Slopes Equations of Lines Functions Graphing Functions
More informationSolving for the Unknown: Basic Operations & Trigonometry ID1050 Quantitative & Qualitative Reasoning
Solving for the Unknown: Basic Operations & Trigonometry ID1050 Quantitative & Qualitative Reasoning What is Algebra? An expression is a combination of numbers and operations that leads to a numerical
More informationA lg e b ra II. Trig o n o m e tric F u n c tio
1 A lg e b ra II Trig o n o m e tric F u n c tio 2015-12-17 www.njctl.org 2 Trig Functions click on the topic to go to that section Radians & Degrees & Co-terminal angles Arc Length & Area of a Sector
More informationPRECALCULUS MR. MILLER
PRECALCULUS MR. MILLER I. COURSE DESCRIPTION This course requires students to use symbolic reasoning and analytical methods to represent mathematical situations, to express generalizations, and to study
More informationTrigonometric ratios provide relationships between the sides and angles of a right angle triangle. The three most commonly used ratios are:
TRIGONOMETRY TRIGONOMETRIC RATIOS If one of the angles of a triangle is 90º (a right angle), the triangle is called a right angled triangle. We indicate the 90º (right) angle by placing a box in its corner.)
More informationSM 2. Date: Section: Objective: The Pythagorean Theorem: In a triangle, or
SM 2 Date: Section: Objective: The Pythagorean Theorem: In a triangle, or. It doesn t matter which leg is a and which leg is b. The hypotenuse is the side across from the right angle. To find the length
More informationMath 111 Lecture Notes Section 3.3: Graphing Rational Functions
Math 111 Lecture Notes Section 3.3: Graphing Rational Functions A rational function is of the form R() = p() q() where p and q are polnomial functions. The zeros of a rational function occur where p()
More information( 3) ( 4 ) 1. Exponents and Radicals ( ) ( xy) 1. MATH 102 College Algebra. still holds when m = n, we are led to the result
Exponents and Radicals ZERO & NEGATIVE EXPONENTS If we assume that the relation still holds when m = n, we are led to the result m m a m n 0 a = a = a. Consequently, = 1, a 0 n n a a a 0 = 1, a 0. Then
More informationExample 1: Given below is the graph of the quadratic function f. Use the function and its graph to find the following: Outputs
Quadratic Functions: - functions defined by quadratic epressions (a 2 + b + c) o the degree of a quadratic function is ALWAYS 2 - the most common way to write a quadratic function (and the way we have
More information5.2 Verifying Trigonometric Identities
360 Chapter 5 Analytic Trigonometry 5. Verifying Trigonometric Identities Introduction In this section, you will study techniques for verifying trigonometric identities. In the next section, you will study
More informationToday we will focus on solving for the sides and angles of non-right triangles when given two angles and a side.
5.5 The Law of Sines Pre-Calculus. Use the Law of Sines to solve non-right triangles. Today we will focus on solving for the sides and angles of non-right triangles when given two angles and a side. Derivation:
More information8B.2: Graphs of Cosecant and Secant
Opp. Name: Date: Period: 8B.: Graphs of Cosecant and Secant Or final two trigonometric functions to graph are cosecant and secant. Remember that So, we predict that there is a close relationship between
More information