Limits of Functions (a, L)
|
|
- Alannah Hutchinson
- 5 years ago
- Views:
Transcription
1 Limits of Functions f(x) (a, L) L f(x) x a x x 20
2 Informal Definition: If the values of can be made as close to as we like by taking values of sufficiently close to [but not equal to ] then we write or as 21
3 Observe: " " means can approach from either side On a sketch, the graph of approaches the 2-D plane location [destination] called,, but the graph itself may have no point, occupying that location! may not be The language to describe how the outputs behave as the inputs approaches number 22
4 Example: We examine the graph? 1 1 Domain of 1 1 0, 0 1 0, 1 1, 0 23
5 4 3,5 3 2,5 2 1,5 1 0, Conjecture:
6 General definition: Let be a function and a real number (that may be or may be not in the domain of ). We say that the it as approaches of is L, written if can be made arbitrarily close to by choosing sufficiently close to (but not equal to). If no such number exists, then we say that does not exist. Warning: Not all its exist! 25
7 Example: from the left, 1 0 from the right, 1 So has no meaning! 26
8 Two-Sided and One-Sided Limits Notation approaches from the left [minus in a superscript position] or [comes up to ] or approaches from the right [plus in a superscript position] or [comes down to ] or 27
9 Relationship between Two-Sided and One-Side Limits: Theorem [if and only if]: exists exists and both equal 28
10 Example: 1 does not exist 1 29
11 The Algebra of Limits as Basic Limits as,, used in polynomial functions and rational functions. The constant function The identity function: The reciprocal ( flip over ) function:
12 Limits of Sums, Differences, Products, Quotients and Roots The Rules of Algebra for Limits Let be any real number and then 31
13 Provided 0 L Provided when = even then 0 32
14 Limits of Polynomial Function Polynomial Expressions A monomial (one-term polynomial) has the form A real number constant called a coefficient Subscript is a label Two monomial with the same degree and same variable are called like terms : ; - like terms A polynomial in one variable has the standard form: [higher powers lower powers] 0 leading coefficient 33 - a variable n=0,1,2, 3, not negative called the degree of the monomial
15 By the Rules of Algebra for Limits we can break down polynomials into simpler parts Example:
16 For any polynomial function For polynomial, this it is the same as substitution of for 35
17 Limits of Rational Functions and the appearance of There are 3 cases to consider Case 1: 0 Limit Example:
18 Case 2: 0 0 Limit does not exist (division by 0!) Classic Examples:
19 a) 1 b) 1 38
20 Case 3: 0 0 Limit determine whether the it exists or not, without more work! Example: an indeterminate form: We cannot This is only one particularly technique! Does not work always! 39
21 The Algebra of Limits as : End Behavior Basic Limits: The constant function and The identity function: 40
22 The reciprocal ( flip over ) function:
23 Limits of Sums, Differences, Products, Quotient and Roots The Rules of Algebra for Limits applied to or We only state for case As before, suppose: then 42
24 L Provided 0 Provided when = even then 0 43
25 A polynomial function Where 0 Limits of Polynomial Functions: Two End Behaviors The two end behaviors are that as (the rightward end) or (the leftward end) Then Observe: The two possibilities! 44
26 So, the end behavior of matches the end behavior of Theorem: Example:
27 Limits of Rational Functions: Three Types of End Behavior The Degree of a polynomial is the exponent of the highest power of in the polynomial 46
28 Type 1. Deg(top)=Deg(bottom) Example:
29 Type 2. Deg(top)<Deg(bottom) Example: Always zero the (-axis) is a horizontal asymptote 48
30 Type 3. Deg(top)>Deg(bottom) If 0 If 0 Always one of these Always one of these Example:
31 Limits of and note, that makes no sense
32 Limits of Trigonometric Functions
33 Some more techniques for computing its a) Rational functions: divide the top and the bottom to cancel (reduce): factor and cancel
34 b) If we have some roots: expand the top and the bottom with a factor: The standard thing to do with a square root in a sum or difference is rationalize
35 c) One-sided its:,, Substitution,, replace and by : : 3 3 0
36 Finding a Limit by Squeezing Old problem: How do we calculate Answer: we squeeze Theorem [ squeezing Theorem] 0 If for all in some open interval containing and too. Then 55
37 Idea: h(c) g(c) c 56
38 ? 57 1
39 Area
40 Since The squeeze Theorem implies, that
41 Exercise
42 Exercise
43 Continuous Function Continuous Function at a single point What properties of a function cause breach or holes in the graph? (so, it does not continue ) Definition: A function is continuous at provided all three conditions are satisfied. 1. is defined [ exists at ] 2. exists [equal a real number] 3. If not, the is discontinuous at 62
44 63
45 Intuitively, is continuous at if the Graph of does not break at If is not continuous at (i.e. if the Graph of does break at ), then is a discontinuity of Note: If is an endpoint for the Domain of, then in the definition is replaced by the appropriate one-sided it, e.g. Is defined on 0, and is continuous at 0 because 0 0 and 0 64
46 Function Continuous on an Interval is continuous on, or, if is continuous at each in the interval. [two-sided its are possible here at each ] What about defined on, [No two-sided its at or ] Definition: at is continuous from the left, if from the right if if 65
47 Definition: is continuous on,, if 1. is continuous on, 2. is continuous from the right at 3. is continuous from the left at 66
48 Properties and combinations of continuous functions Recall: If is a polynomial function, then So, every polynomial function is continuous everywhere. Suppose,, are continuous at Theorem: ; ; are all also continuous at is continuous at provided 0 [otherwise, is discontinuous at ] So, every rational function is continuous at every point where the bottom is not zero. 67
49 Lastly: the composition of continuous functions is also continuous Theorem: If is continuous at and is continuous at then is continuous at 68
50 The Intermediate Value Theorem and Approximating Roots: Intermediate Value Theorem If is continuous on, and is between and, or equal to one of them, then there is at least one value of in, such that 69
51 Theorem: If is continuous on, and, are non zero with opposite signs, then there is at least one solution of 0 in, 70
6.1 Polynomial Functions
6.1 Polynomial Functions Definition. A polynomial function is any function p(x) of the form p(x) = p n x n + p n 1 x n 1 + + p 2 x 2 + p 1 x + p 0 where all of the exponents are non-negative integers and
More informationLimits and Continuity
Limits and Continuity MATH 151 Calculus for Management J. Robert Buchanan Department of Mathematics Fall 2018 Objectives After this lesson we will be able to: Determine the left-hand and right-hand limits
More information( ) = 1 x. g( x) = x3 +2
Rational Functions are ratios (quotients) of polynomials, written in the form f x N ( x ) and D x ( ) are polynomials, and D x ( ) does not equal zero. The parent function for rational functions is f x
More informationRational Functions 4.5
Math 4 Pre-Calculus Name Date Rational Function Rational Functions 4.5 g ( ) A function is a rational function if f ( ), where g ( ) and ( ) h ( ) h are polynomials. Vertical asymptotes occur at -values
More information2.2 The Limit of a Function
2.2 The Limit of a Function Introductory Example: Consider the function f(x) = x is near 0. x f(x) x f(x) 1 3.7320508 1 4.236068 0.5 3.8708287 0.5 4.1213203 0.1 3.9748418 0.1 4.0248457 0.05 3.9874607 0.05
More informationMaking Connections with Rational Functions and Equations
Section 3.5 Making Connections with Rational Functions and Equations When solving a problem, it's important to read carefully to determine whether a function is being analyzed (Finding key features) or
More informationMATH 151 Engineering Mathematics I
MATH 151 Engineering Mathematics I Fall 2018, WEEK 3 JoungDong Kim Week 3 Section 2.3, 2.5, 2.6, Calculating Limits Using the Limit Laws, Continuity, Limits at Infinity; Horizontal Asymptotes. Section
More information1.5 Inverse Trigonometric Functions
1.5 Inverse Trigonometric Functions Remember that only one-to-one functions have inverses. So, in order to find the inverse functions for sine, cosine, and tangent, we must restrict their domains to intervals
More informationLIMITS AT INFINITY MR. VELAZQUEZ AP CALCULUS
LIMITS AT INFINITY MR. VELAZQUEZ AP CALCULUS RECALL: VERTICAL ASYMPTOTES Remember that for a rational function, vertical asymptotes occur at values of x = a which have infinite its (either positive or
More informationContinuity and One-Sided Limits
Continuity and One-Sided Limits 1. Welcome to continuity and one-sided limits. My name is Tuesday Johnson and I m a lecturer at the University of Texas El Paso. 2. With each lecture I present, I will start
More informationChapter 2: Functions, Limits and Continuity
Chapter 2: Functions, Limits and Continuity Functions Limits Continuity Chapter 2: Functions, Limits and Continuity 1 Functions Functions are the major tools for describing the real world in mathematical
More information2.1 Limits, Rates of Change and Slopes of Tangent Lines
2.1 Limits, Rates of Change and Slopes of Tangent Lines (1) Average rate of change of y f x over an interval x 0,x 1 : f x 1 f x 0 x 1 x 0 Instantaneous rate of change of f x at x x 0 : f x lim 1 f x 0
More informationIntroduction. A rational function is a quotient of polynomial functions. It can be written in the form
RATIONAL FUNCTIONS Introduction A rational function is a quotient of polynomial functions. It can be written in the form where N(x) and D(x) are polynomials and D(x) is not the zero polynomial. 2 In general,
More information1.1 Introduction to Limits
Chapter 1 LIMITS 1.1 Introduction to Limits Why Limit? Suppose that an object steadily moves forward, with s(t) denotes the position at time t. The average speed over the interval [1,2] is The average
More informationTo get horizontal and slant asymptotes algebraically we need to know about end behaviour for rational functions.
Concepts: Horizontal Asymptotes, Vertical Asymptotes, Slant (Oblique) Asymptotes, Transforming Reciprocal Function, Sketching Rational Functions, Solving Inequalities using Sign Charts. Rational Function
More informationLimits at Infinity. Horizontal Asymptotes. Definition (Limits at Infinity) Horizontal Asymptotes
Limits at Infinity If a function f has a domain that is unbounded, that is, one of the endpoints of its domain is ±, we can determine the long term behavior of the function using a it at infinity. Definition
More informationCalculus First Semester Review Name: Section: Evaluate the function: (g o f )( 2) f (x + h) f (x) h. m(x + h) m(x)
Evaluate the function: c. (g o f )(x + 2) d. ( f ( f (x)) 1. f x = 4x! 2 a. f( 2) b. f(x 1) c. f (x + h) f (x) h 4. g x = 3x! + 1 Find g!! (x) 5. p x = 4x! + 2 Find p!! (x) 2. m x = 3x! + 2x 1 m(x + h)
More informationLecture 3 (Limits and Derivatives)
Lecture 3 (Limits and Derivatives) Continuity In the previous lecture we saw that very often the limit of a function as is just. When this is the case we say that is continuous at a. Definition: A function
More informationThe function graphed below is continuous everywhere. The function graphed below is NOT continuous everywhere, it is discontinuous at x 2 and
Section 1.4 Continuity A function is a continuous at a point if its graph has no gaps, holes, breaks or jumps at that point. If a function is not continuous at a point, then we say it is discontinuous
More information2. 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 informationReteach Multiplying and Dividing Rational Expressions
8-2 Multiplying and Dividing Rational Expressions Examples of rational expressions: 3 x, x 1, and x 3 x 2 2 x 2 Undefined at x 0 Undefined at x 0 Undefined at x 2 When simplifying a rational expression:
More informationContinuity and One-Sided Limits. By Tuesday J. Johnson
Continuity and One-Sided Limits By Tuesday J. Johnson Suggested Review Topics Algebra skills reviews suggested: Evaluating functions Rationalizing numerators and/or denominators Trigonometric skills reviews
More informationInfinite Limits. Infinite Limits. Infinite Limits. Previously, we discussed the limits of rational functions with the indeterminate form 0/0.
Infinite Limits Return to Table of Contents Infinite Limits Infinite Limits Previously, we discussed the limits of rational functions with the indeterminate form 0/0. Now we will consider rational functions
More informationNAME DATE PERIOD. Power and Radical Functions. New Vocabulary Fill in the blank with the correct term. positive integer.
2-1 Power and Radical Functions What You ll Learn Scan Lesson 2-1. Predict two things that you expect to learn based on the headings and Key Concept box. 1. 2. Lesson 2-1 Active Vocabulary extraneous solution
More informationH-Pre-Calculus Targets Chapter I can write quadratic functions in standard form and use the results to sketch graphs of the function.
H-Pre-Calculus Targets Chapter Section. Sketch and analyze graphs of quadratic functions.. I can write quadratic functions in standard form and use the results to sketch graphs of the function. Identify
More informationUMUC MATH-107 Final Exam Information
UMUC MATH-07 Final Exam Information What should you know for the final exam? Here are some highlights of textbook material you should study in preparation for the final exam. Review this material from
More informationReview Guideline for Final
Review Guideline for Final Here is the outline of the required skills for the final exam. Please read it carefully and find some corresponding homework problems in the corresponding sections to practice.
More informationAnalyzing Rational Functions
Analyzing Rational Functions These notes are intended as a summary of section 2.3 (p. 105 112) in your workbook. You should also read the section for more complete explanations and additional examples.
More informationter. on Can we get a still better result? Yes, by making the rectangles still smaller. As we make the rectangles smaller and smaller, the
Area and Tangent Problem Calculus is motivated by two main problems. The first is the area problem. It is a well known result that the area of a rectangle with length l and width w is given by A = wl.
More informationNotes: 1. Regard as the maximal output error and as the corresponding maximal input error
Limits and Continuity One of the major tasks in analysis is to classify a function by how nice it is Of course, nice often depends upon what you wish to do or model with the function Below is a list of
More informationContinuity. To handle complicated functions, particularly those for which we have a reasonable formula or formulas, we need a more precise definition.
Continuity Intuitively, a function is continuous if its graph can be traced on paper in one motion without lifting the pencil from the paper. Thus the graph has no tears or holes. To handle complicated
More informationFive-Minute Check (over Lesson 8 3) CCSS Then/Now New Vocabulary Key Concept: Vertical and Horizontal Asymptotes Example 1: Graph with No Horizontal
Five-Minute Check (over Lesson 8 3) CCSS Then/Now New Vocabulary Key Concept: Vertical and Horizontal Asymptotes Example 1: Graph with No Horizontal Asymptote Example 2: Real-World Example: Use Graphs
More informationOctober 27, 2018 MAT186 Week 3 Justin Ko. We use the following notation to describe the limiting behavior of functions.
October 27, 208 MAT86 Week 3 Justin Ko Limits. Intuitive Definitions of Limits We use the following notation to describe the iting behavior of functions.. (Limit of a Function A it is written as f( = L
More informationRational Functions. A rational function is a function that is a ratio of 2 polynomials (in reduced form), e.g.
Rational Functions A rational function is a function that is a ratio of polynomials (in reduced form), e.g. f() = p( ) q( ) where p() and q() are polynomials The function is defined when the denominator
More informationAP 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 informationFinal Exam Review Packet
1 Exam 1 Material Sections A.1, A.2 and A.6 were review material. There will not be specific questions focused on this material but you should know how to: Simplify functions with exponents. Factor quadratics
More informationFinal Exam Review Packet
1 Exam 1 Material Sections A.1, A.2 and A.6 were review material. There will not be specific questions focused on this material but you should know how to: Simplify functions with exponents. Factor quadratics
More informationThe most factored form is usually accomplished by common factoring the expression. But, any type of factoring may come into play.
MOST FACTORED FORM The most factored form is the most factored version of a rational expression. Being able to find the most factored form is an essential skill when simplifying the derivatives found using
More informationCalculus I Exam 1 Review Fall 2016
Problem 1: Decide whether the following statements are true or false: (a) If f, g are differentiable, then d d x (f g) = f g. (b) If a function is continuous, then it is differentiable. (c) If a function
More informationMath 115 Spring 11 Written Homework 10 Solutions
Math 5 Spring Written Homework 0 Solutions. For following its, state what indeterminate form the its are in and evaluate the its. (a) 3x 4x 4 x x 8 Solution: This is in indeterminate form 0. Algebraically,
More information2.3 Differentiation Formulas. Copyright Cengage Learning. All rights reserved.
2.3 Differentiation Formulas Copyright Cengage Learning. All rights reserved. Differentiation Formulas Let s start with the simplest of all functions, the constant function f (x) = c. The graph of this
More information2. If the values for f(x) can be made as close as we like to L by choosing arbitrarily large. lim
Limits at Infinity and Horizontal Asymptotes As we prepare to practice graphing functions, we should consider one last piece of information about a function that will be helpful in drawing its graph the
More informationLearning Objectives. Zeroes. The Real Zeros of a Polynomial Function
The Real Zeros of a Polynomial Function 1 Learning Objectives 1. Use the Remainder and Factor Theorems 2. Use the Rational Zeros Theorem to list the potential rational zeros of a polynomial function 3.
More informationPolynomial Expressions and Functions
Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 1 - of 36 Topic 32: Polynomial Expressions and Functions Recall the definitions of polynomials and terms. Definition: A polynomial
More informationRational Functions. p x q x. f x = where p(x) and q(x) are polynomials, and q x 0. Here are some examples: x 1 x 3.
Rational Functions In mathematics, rational means in a ratio. A rational function is a ratio of two polynomials. Rational functions have the general form p x q x, where p(x) and q(x) are polynomials, and
More informationof multiplicity two. The sign of the polynomial is shown in the table below
161 Precalculus 1 Review 5 Problem 1 Graph the polynomial function P( ) ( ) ( 1). Solution The polynomial is of degree 4 and therefore it is positive to the left of its smallest real root and to the right
More informationAnalysis of Functions
Lecture for Week 11 (Secs. 5.1 3) Analysis of Functions (We used to call this topic curve sketching, before students could sketch curves by typing formulas into their calculators. It is still important
More informationWellston City Schools Calculus Curriculum Calendar
Wellston City Schools Calculus 2006-2007 Curriculum Calendar Grading Period 1:Week 1: Review 11 th grade standards Learn to represent functions using: *Words *Tables of values *Graphs *Formulas Present
More informationUNIT 3. Rational Functions Limits at Infinity (Horizontal and Slant Asymptotes) Infinite Limits (Vertical Asymptotes) Graphing Rational Functions
UNIT 3 Rational Functions Limits at Infinity (Horizontal and Slant Asymptotes) Infinite Limits (Vertical Asymptotes) Graphing Rational Functions Recall From Unit Rational Functions f() is a rational function
More information1.4 CONTINUITY AND ITS CONSEQUENCES
Continuity: Informal Idea We say that a function is continuous on an interval if its graph on that t interval can be drawn without t interruption, ti that is, without lifting the pencil from the paper.
More informationRational Functions. Elementary Functions. Algebra with mixed fractions. Algebra with mixed fractions
Rational Functions A rational function f (x) is a function which is the ratio of two polynomials, that is, Part 2, Polynomials Lecture 26a, Rational Functions f (x) = where and are polynomials Dr Ken W
More informationQUIZ ON CHAPTERS 1 AND 2 - SOLUTIONS REVIEW / LIMITS AND CONTINUITY; MATH 150 SPRING 2017 KUNIYUKI 105 POINTS TOTAL, BUT 100 POINTS = 100%
QUIZ ON CHAPTERS AND 2 - SOLUTIONS REVIEW / LIMITS AND CONTINUITY; MATH 50 SPRING 207 KUNIYUKI 05 POINTS TOTAL, BUT 00 POINTS = 00% ) For a), b), and c) below, bo in the correct answer. (6 points total;
More informationMATH 151 Engineering Mathematics I
MATH 151 Engineering Mathematics I Spring 2018, WEEK 3 JoungDong Kim Week 3 Section 2.5, 2.6, 2.7, Continuity, Limits at Infinity; Horizontal Asymptotes, Derivatives and Rates of Change. Section 2.5 Continuity
More informationNAME DATE PERIOD. Operations with Polynomials. Review Vocabulary Evaluate each expression. (Lesson 1-1) 3a 2 b 4, given a = 3, b = 2
5-1 Operations with Polynomials What You ll Learn Skim the lesson. Predict two things that you expect to learn based on the headings and the Key Concept box. 1. Active Vocabulary 2. Review Vocabulary Evaluate
More informationChapter 2. Limits and Continuity 2.6 Limits Involving Infinity; Asymptotes of Graphs
2.6 Limits Involving Infinity; Asymptotes of Graphs Chapter 2. Limits and Continuity 2.6 Limits Involving Infinity; Asymptotes of Graphs Definition. Formal Definition of Limits at Infinity.. We say that
More informationAlgebra 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 information5.3. Polynomials and Polynomial Functions
5.3 Polynomials and Polynomial Functions Polynomial Vocabulary Term a number or a product of a number and variables raised to powers Coefficient numerical factor of a term Constant term which is only a
More informationAlgebra 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 informationCHAPTER 8A- RATIONAL FUNCTIONS AND RADICAL FUNCTIONS Section Multiplying and Dividing Rational Expressions
Name Objectives: Period CHAPTER 8A- RATIONAL FUNCTIONS AND RADICAL FUNCTIONS Section 8.3 - Multiplying and Dividing Rational Expressions Multiply and divide rational expressions. Simplify rational expressions,
More informationSection 5.1 Determine if a function is a polynomial function. State the degree of a polynomial function.
Test Instructions Objectives Section 5.1 Section 5.1 Determine if a function is a polynomial function. State the degree of a polynomial function. Form a polynomial whose zeros and degree are given. Graph
More informationAlgebra 1 Summer Assignment 2018
Algebra 1 Summer Assignment 2018 The following packet contains topics and definitions that you will be required to know in order to succeed in Algebra 1 this coming school year. You are advised to be familiar
More informationWed. Sept 28th: 1.3 New Functions from Old Functions: o vertical and horizontal shifts o vertical and horizontal stretching and reflecting o
Homework: Appendix A: 1, 2, 3, 5, 6, 7, 8, 11, 13-33(odd), 34, 37, 38, 44, 45, 49, 51, 56. Appendix B: 3, 6, 7, 9, 11, 14, 16-21, 24, 29, 33, 36, 37, 42. Appendix D: 1, 2, 4, 9, 11-20, 23, 26, 28, 29,
More informationDay 6: 6.4 Solving Polynomial Equations Warm Up: Factor. 1. x 2-2x x 2-9x x 2 + 6x + 5
Day 6: 6.4 Solving Polynomial Equations Warm Up: Factor. 1. x 2-2x - 15 2. x 2-9x + 14 3. x 2 + 6x + 5 Solving Equations by Factoring Recall the factoring pattern: Difference of Squares:...... Note: There
More information6.1 Using Properties of Exponents 1. Use properties of exponents to evaluate and simplify expressions involving powers. Product of Powers Property
6.1 Using Properties of Exponents Objectives 1. Use properties of exponents to evaluate and simplify expressions involving powers. 2. Use exponents and scientific notation to solve real life problems.
More information2.4 The Precise Definition of a Limit
2.4 The Precise Definition of a Limit Reminders/Remarks: x 4 < 3 means that the distance between x and 4 is less than 3. In other words, x lies strictly between 1 and 7. So, x a < δ means that the distance
More informationChapter. Part 1: Consider the function
Chapter 9 9.2 Analysing rational Functions Pages 446 456 Part 1: Consider the function a) What value of x is important to consider when analysing this function? b) Now look at the graph of this function
More informationThe final is cumulative, but with more emphasis on chapters 3 and 4. There will be two parts.
Math 141 Review for Final The final is cumulative, but with more emphasis on chapters 3 and 4. There will be two parts. Part 1 (no calculator) graphing (polynomial, rational, linear, exponential, and logarithmic
More informationPre-Calculus: Functions and Their Properties (Solving equations algebraically and graphically, matching graphs, tables, and equations, and
Pre-Calculus: 1.1 1.2 Functions and Their Properties (Solving equations algebraically and graphically, matching graphs, tables, and equations, and finding the domain, range, VA, HA, etc.). Name: Date:
More informationFinding Limits Analytically
Finding Limits Analytically Most of this material is take from APEX Calculus under terms of a Creative Commons License In this handout, we explore analytic techniques to compute its. Suppose that f(x)
More informationTopic 3 Outline. What is a Limit? Calculating Limits Infinite Limits Limits at Infinity Continuity. 1 Limits and Continuity
Topic 3 Outline 1 Limits and Continuity What is a Limit? Calculating Limits Infinite Limits Limits at Infinity Continuity D. Kalajdzievska (University of Manitoba) Math 1520 Fall 2015 1 / 27 Topic 3 Learning
More informationReview for Chapter 2 Test
Review for Chapter 2 Test This test will cover Chapter (sections 2.1-2.7) Know how to do the following: Use a graph of a function to find the limit (as well as left and right hand limits) Use a calculator
More informationGUIDED NOTES 5.6 RATIONAL FUNCTIONS
GUIDED NOTES 5.6 RATIONAL FUNCTIONS LEARNING OBJECTIVES In this section, you will: Use arrow notation. Solve applied problems involving rational functions. Find the domains of rational functions. Identify
More informationCalculus I Practice Test Problems for Chapter 2 Page 1 of 7
Calculus I Practice Test Problems for Chapter Page of 7 This is a set of practice test problems for Chapter This is in no way an inclusive set of problems there can be other types of problems on the actual
More informationMATH 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 informationInverse Variation. y varies inversely as x. REMEMBER: Direct variation y = kx where k is not equal to 0.
Inverse Variation y varies inversely as x. REMEMBER: Direct variation y = kx where k is not equal to 0. Inverse variation xy = k or y = k where k is not equal to 0. x Identify whether the following functions
More informationEQ: What are limits, and how do we find them? Finite limits as x ± Horizontal Asymptote. Example Horizontal Asymptote
Finite limits as x ± The symbol for infinity ( ) does not represent a real number. We use to describe the behavior of a function when the values in its domain or range outgrow all finite bounds. For example,
More informationSEE and DISCUSS the pictures on pages in your text. Key picture:
Math 6 Notes 1.1 A PREVIEW OF CALCULUS There are main problems in calculus: 1. Finding a tangent line to a curve though a point on the curve.. Finding the area under a curve on some interval. SEE and DISCUSS
More informationMTH4100 Calculus I. Lecture notes for Week 4. Thomas Calculus, Sections 2.4 to 2.6. Rainer Klages
MTH4100 Calculus I Lecture notes for Week 4 Thomas Calculus, Sections 2.4 to 2.6 Rainer Klages School of Mathematical Sciences Queen Mary University of London Autumn 2009 One-sided its and its at infinity
More informationChapter 2. Exercise 5. Evaluate the limit
Chapter Exercise 5 Evaluate the limit f x_ : x 9 x x As x - this produces a 0 in both the numerator and denominator, which means we should factor and simplify before we evaluate. num Factor Numerator f
More informationSection Properties of Rational Expressions
88 Section. - Properties of Rational Expressions Recall that a rational number is any number that can be written as the ratio of two integers where the integer in the denominator cannot be. Rational Numbers:
More informationNOTES: EXPONENT RULES
NOTES: EXPONENT RULES DAY 2 Topic Definition/Rule Example(s) Multiplication (add exponents) x a x b = x a+b x 4 x 8 x 5 y 2 x 2 y Power to a Power (multiply exponents) x a ( ) b = x ab ( x ) 7 ( x ) 2
More informationChapter 2. Limits and Continuity. 2.1 Rates of change and Tangents to Curves. The average Rate of change of y = f(x) with respect to x over the
Chapter 2 Limits and Continuity 2.1 Rates of change and Tangents to Curves Definition 2.1.1 : interval [x 1, x 2 ] is The average Rate of change of y = f(x) with respect to x over the y x = f(x 2) f(x
More informationPre-calculus 12 Curriculum Outcomes Framework (110 hours)
Curriculum Outcomes Framework (110 hours) Trigonometry (T) (35 40 hours) General Curriculum Outcome: Students will be expected to develop trigonometric reasoning. T01 Students will be expected to T01.01
More informationINDEX UNIT 3 TSFX REFERENCE MATERIALS 2014 ALGEBRA AND ARITHMETIC
INDEX UNIT 3 TSFX REFERENCE MATERIALS 2014 ALGEBRA AND ARITHMETIC Surds Page 1 Algebra of Polynomial Functions Page 2 Polynomial Expressions Page 2 Expanding Expressions Page 3 Factorising Expressions
More informationRational and Radical Functions. College Algebra
Rational and Radical Functions College Algebra Rational Function A rational function is a function that can be written as the quotient of two polynomial functions P(x) and Q(x) f x = P(x) Q(x) = a )x )
More informationCalculus. Weijiu Liu. Department of Mathematics University of Central Arkansas 201 Donaghey Avenue, Conway, AR 72035, USA
Calculus Weijiu Liu Department of Mathematics University of Central Arkansas 201 Donaghey Avenue, Conway, AR 72035, USA 1 Opening Welcome to your Calculus I class! My name is Weijiu Liu. I will guide you
More information8. Limit Laws. lim(f g)(x) = lim f(x) lim g(x), (x) = lim x a f(x) g lim x a g(x)
8. Limit Laws 8.1. Basic Limit Laws. If f and g are two functions and we know the it of each of them at a given point a, then we can easily compute the it at a of their sum, difference, product, constant
More informationReview: Limits of Functions - 10/7/16
Review: Limits of Functions - 10/7/16 1 Right and Left Hand Limits Definition 1.0.1 We write lim a f() = L to mean that the function f() approaches L as approaches a from the left. We call this the left
More information1.3 Limits and Continuity
.3 Limits and Continuity.3. Limits Problem 8. What will happen to the functional values of as x gets closer and closer to 2? f(x) = Solution. We can evaluate f(x) using x values nearer and nearer to 2
More informationUnit 1 Vocabulary. A function that contains 1 or more or terms. The variables may be to any non-negative power.
MODULE 1 1 Polynomial A function that contains 1 or more or terms. The variables may be to any non-negative power. 1 Modeling Mathematical modeling is the process of using, and to represent real world
More informationRelations and Functions (for Math 026 review)
Section 3.1 Relations and Functions (for Math 026 review) Objective 1: Understanding the s of Relations and Functions Relation A relation is a correspondence between two sets A and B such that each element
More informationInduction, sequences, limits and continuity
Induction, sequences, limits and continuity Material covered: eclass notes on induction, Chapter 11, Section 1 and Chapter 2, Sections 2.2-2.5 Induction Principle of mathematical induction: Let P(n) be
More informationUNIT 4: RATIONAL AND RADICAL EXPRESSIONS. 4.1 Product Rule. Objective. Vocabulary. o Scientific Notation. o Base
UNIT 4: RATIONAL AND RADICAL EXPRESSIONS M1 5.8, M2 10.1-4, M3 5.4-5, 6.5,8 4.1 Product Rule Objective I will be able to multiply powers when they have the same base, including simplifying algebraic expressions
More information2.1 Quadratic Functions
Date:.1 Quadratic Functions Precalculus Notes: Unit Polynomial Functions Objective: The student will sketch the graph of a quadratic equation. The student will write the equation of a quadratic function.
More information1) If f x symmetric about what? (Box in one:) (2 points) the x-axis the y-axis the origin none of these
QUIZ ON CHAPTERS AND - SOLUTIONS REVIEW / LIMITS AND CONTINUITY; MATH 50 FALL 06 KUNIYUKI 05 POINTS TOTAL, BUT 00 POINTS = 00% = x /, then the graph of y = f ( x) in the usual (Cartesian) xy-plane is )
More informationMATH 1314 College Algebra Scott Travis Fall 2014 Review for Exam #2
MATH 1314 College Algebra Scott Travis Fall 2014 Review for Exam #2 There are eight sections from Chapters 4 and 5 included in the exam: 4.1, 4.3, 5.1 to 5.6. This review should help you prepare. For each
More informationSection 3.1 Quadratic Functions
Chapter 3 Lecture Notes Page 1 of 72 Section 3.1 Quadratic Functions Objectives: Compare two different forms of writing a quadratic function Find the equation of a quadratic function (given points) Application
More informationCalculus I. George Voutsadakis 1. LSSU Math 151. Lake Superior State University. 1 Mathematics and Computer Science
Calculus I George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 151 George Voutsadakis (LSSU) Calculus I November 2014 1 / 67 Outline 1 Limits Limits, Rates
More informationMath 110 Midterm 1 Study Guide October 14, 2013
Name: For more practice exercises, do the study set problems in sections: 3.4 3.7, 4.1, and 4.2. 1. Find the domain of f, and express the solution in interval notation. (a) f(x) = x 6 D = (, ) or D = R
More informationChapter 2 Polynomial and Rational Functions
Chapter 2 Polynomial and Rational Functions Overview: 2.2 Polynomial Functions of Higher Degree 2.3 Real Zeros of Polynomial Functions 2.4 Complex Numbers 2.5 The Fundamental Theorem of Algebra 2.6 Rational
More information