Topic 2 Limits and Continuity c and d) Continuity Handout Notes Assigned Problems: Intro book pg 73, 1-3 and 6-8
|
|
- Jade Carpenter
- 5 years ago
- Views:
Transcription
1 c&d. Continuity Handout. Page 1 of 5 Topic Limits and Continuity c and d) Continuity Handout Notes Assigned Problems: Intro book pg 73, 1-3 and 6-8 Recall Limits and Function Values: We have already studied all the concepts necessary to understand continuity To recap, let s look at the following graphs and answer three questions Does f(a) exist? Does lim f ( x ) exist? Does lim f ( x ) = f ( a )? x a x a 1) ) 3) 4) 5) For a function to be continuous, it has to pass all three tests above, if it fails even one it is NOT continuous. Which of the above functions are continuous at a? Informal Definition of Continuity: A function f is continuous at x = a if and only if 1. f(a) is defined. lim f ( x ) exists x a 3. lim f ( x ) = f ( a ) x a If a function f is not continuous at x = a, it is said to be discontinuous at x = a Types of Discontinuities: If the limit of the function exists (and is finite) at x = a, then continuity fails because either f(a) is not defined, or lim f ( x ) f ( a ), the discontinuity is classified as removable. This is because by simply x a redefining f(a) we can make f continuous at a. Which of the above pictures has a removable discontinuity? How would you fix this? If the limit of the function is infinite, then it is called an infinite discontinuity. Which of the above pictures has an infinite discontinuity? If the limit from the right and left of a are not the same, then the function is said to have a jump discontinuity
2 c&d. Continuity Handout. Page of 5 Continuous on an Interval: A function can be continuous from the right and left at a number (basically you only look at a right or left hand limit) A function can be continuous on an interval if it is continuous for all the points in that interval A function is said to be continuous if it is continuous on its entire domain While the same word (i.e. continuous) is used to describe all of these situations, it is important to keep in mind what you are referring to. Here they are from least strict to most strict continuous at a from the left (or right) continuous at a continuous on an interval continuous The following functions are continuous on their domains: polynomials, rational functions, root functions, trig functions, inverse trig functions, exponential functions, log functions. But wait, I thought log functions were not defined for negative values? How could, say log(x) satisfy the above statement if it isn t even defined for x < 0? Formal Definition of Continuity: A function f : D Re is continuous at a point c if and only if ε > 0 δ > 0 s.t. for x D and x c < δ then f( x) f( c) <ε For a function to be continuous, it must be continuous for all points in its domain Proving Continuity at a Point: Example 1: Show f(x) = x + 1 is continuous at x =. Let x < δ for δ > 0 Then f( x) f() = x+ 1 (4+ 1) = x 4 = x < δ So f( x) f() < ε for ε δ < ε i.e. ε > 0 δ > 0 δ < s.t. for x D and x < δ then f( x) f() < ε
3 Example : Show f ( x) = x f : Re Re is continuous at x = Let x < δ for δ > 0 Then f x f x x ( ) () = = 4 Let s assume this is true (for now) so we can find delta ε < x 4 < ε < x < + ε 4 ε 4 4 ε < x < 4+ ε ε < 4 4 ε < x < 4+ ε So take δ = min( 4 ε, 4 +ε ) In fact, further investigation shows that δ = 4+ ε. ( ) c&d. Continuity Handout. Page 3 of 5 ε > 0 ( wlog.... ε < 4) δ > 0 δ = 4 + ε s.t. for x D and x < δ then f( x) f() <ε. Actually, let s plug in some values to see how this works e d = sqrt(4+e)- - d/ f( - d/) - f() < e? + d/ f( + d/) - f() < e? Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Just so you can see there is more than one way to prove Let x < δ for δ > 0 Then since a+ b a + b, we have x+ x + f ( x) = x f : Re Re is cts at x = And since a b a b, we have that x x < δ or δ < x < + δ If we let δ =, then x+ x + < 4+ =6 So, = = + <. So let δ = min(, ε / 6) f( x) f() x x x 6δ This requires more thought but is actually much cleaner
4 Example 3: Show f( x) = x f :[0, ) Re is cts at x = 1 Let x 1 < δ for δ > 0 x + 1 x 1 f( x) f(1) = x 1 = ( x 1) = x + 1 x Now, x 0 x 0 x x + 1 x 1 So, f( x) f(1) < δ 1 ε > 0 δ > 0 δ = ε s.t. for x D and x 1 < δ then f( x) f(1) <ε. Therefore, ( ) Again, values to see e d = e 1 - d/ f(1 - d/) - f(1) < e? 1 + d/ f(1 + d/) - f(1) < e? Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y 0 1 Y Y c&d. Continuity Handout. Page 4 of 5
5 Showing Continuous on a Domain: Example 4: Show f( x) = x f :Re Re is continuous Let x c < δ for δ > 0 and c Re Then since a+ b a + b, we have x + c x + c And since a b a b, we have that x c x c < δ or c δ < x < c + δ If we let δ = c, then x + c x + c < c + c = 3c. So, f ( x) f( c) x c x c x c δ 3c = = + <. So let δ = min( c, ε / c). But this means c = 0 is a special case. If c = 0, then x Therefore, < δ, and ( ) c&d. Continuity Handout. Page 5 of 5 f( x) f(0) = x = x < δ. ε > 0 δ > 0 δ = min( c, ε / c) when c 0, δ = ε when c= 0 s.t. for x, c Re and 0 < x c < δ then f( x) f( c) < ε.
The 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 information1.10 Continuity Brian E. Veitch
1.10 Continuity Definition 1.5. A function is continuous at x = a if 1. f(a) exists 2. lim x a f(x) exists 3. lim x a f(x) = f(a) If any of these conditions fail, f is discontinuous. Note: From algebra
More informationMAT01A1: Precise Definition of a Limit and Continuity
MAT01A1: Precise Definition of a Limit and Continuity Dr Craig 7 March 2018 Semester Test 1 D1 LAB 110 Be seated by 08h15. Everything up to and including Ch 2.3. Bring student cards. No bags. No calculators.
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 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 informationContinuity, Intermediate Value Theorem (2.4)
Continuity, Intermediate Value Theorem (2.4) Xiannan Li Kansas State University January 29th, 2017 Intuitive Definition: A function f(x) is continuous at a if you can draw the graph of y = f(x) without
More information2.1 The Tangent and Velocity Problems
2.1 The Tangent and Velocity Problems Ex: When you jump off a swing, where do you go? Ex: Can you approximate this line with another nearby? How would you get a better approximation? Ex: A cardiac monitor
More informationSolution to Review Problems for Midterm #1
Solution to Review Problems for Midterm # Midterm I: Wednesday, September in class Topics:.,.3 and.-.6 (ecept.3) Office hours before the eam: Monday - and 4-6 p.m., Tuesday - pm and 4-6 pm at UH 080B)
More informationContinuity. MATH 161 Calculus I. J. Robert Buchanan. Fall Department of Mathematics
Continuity MATH 161 Calculus I J. Robert Buchanan Department of Mathematics Fall 2017 Intuitive Idea A process or an item can be described as continuous if it exists without interruption. The mathematical
More informationTHE LIMIT PROCESS (AN INTUITIVE INTRODUCTION)
The Limit Process THE LIMIT PROCESS (AN INTUITIVE INTRODUCTION) We could begin by saying that limits are important in calculus, but that would be a major understatement. Without limits, calculus would
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 informationTHS Step By Step Calculus Chapter 1
Name: Class Period: Throughout this packet there will be blanks you are epected to fill in prior to coming to class. This packet follows your Larson Tetbook. Do NOT throw away! Keep in 3 ring binder until
More informationMath Section Bekki George: 08/28/18. University of Houston. Bekki George (UH) Math /28/18 1 / 37
Math 1431 Section 14616 Bekki George: bekki@math.uh.edu University of Houston 08/28/18 Bekki George (UH) Math 1431 08/28/18 1 / 37 Office Hours: Tuesdays and Thursdays 12:30-2pm (also available by appointment)
More informationCH 2: Limits and Derivatives
2 The tangent and velocity problems CH 2: Limits and Derivatives the tangent line to a curve at a point P, is the line that has the same slope as the curve at that point P, ie the slope of the tangent
More informationCalculus. Central role in much of modern science Physics, especially kinematics and electrodynamics Economics, engineering, medicine, chemistry, etc.
Calculus Calculus - the study of change, as related to functions Formally co-developed around the 1660 s by Newton and Leibniz Two main branches - differential and integral Central role in much of modern
More informationExistence of a Limit on a Dense Set, and. Construction of Continuous Functions on Special Sets
Existence of a Limit on a Dense Set, and Construction of Continuous Functions on Special Sets REU 2012 Recap: Definitions Definition Given a real-valued function f, the limit of f exists at a point c R
More informationChapter 1 Functions and Limits
Contents Chapter 1 Functions and Limits Motivation to Chapter 1 2 4 Tangent and Velocity Problems 3 4.1 VIDEO - Secant Lines, Average Rate of Change, and Applications......................... 3 4.2 VIDEO
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 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 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 informationSection 1.4 Tangents and Velocity
Math 132 Tangents and Velocity Section 1.4 Section 1.4 Tangents and Velocity Tangent Lines A tangent line to a curve is a line that just touches the curve. In terms of a circle, the definition is very
More informationLast week we looked at limits generally, and at finding limits using substitution.
Math 1314 ONLINE Week 4 Notes Lesson 4 Limits (continued) Last week we looked at limits generally, and at finding limits using substitution. Indeterminate Forms What do you do when substitution gives you
More informationDRAFT - Math 101 Lecture Note - Dr. Said Algarni
2 Limits 2.1 The Tangent Problems The word tangent is derived from the Latin word tangens, which means touching. A tangent line to a curve is a line that touches the curve and a secant line is a line that
More informationDefinition (The carefully thought-out calculus version based on limits).
4.1. Continuity and Graphs Definition 4.1.1 (Intuitive idea used in algebra based on graphing). A function, f, is continuous on the interval (a, b) if the graph of y = f(x) can be drawn over the interval
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 informationHomework for Section 1.4, Continuity and One sided Limits. Study 1.4, # 1 21, 27, 31, 37 41, 45 53, 61, 69, 87, 91, 93. Class Notes: Prof. G.
GOAL: 1. Understand definition of continuity at a point. 2. Evaluate functions for continuity at a point, and on open and closed intervals 3. Understand the Intermediate Value Theorum (IVT) Homework for
More informationComposition of Functions
Math 120 Intermediate Algebra Sec 9.1: Composite and Inverse Functions Composition of Functions The composite function f g, the composition of f and g, is defined as (f g)(x) = f(g(x)). Recall that a function
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 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 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 information80 Wyner PreCalculus Spring 2017
80 Wyner PreCalculus Spring 2017 CHAPTER NINE: DERIVATIVES Review May 16 Test May 23 Calculus begins with the study of rates of change, called derivatives. For example, the derivative of velocity is acceleration
More informationThe main way we switch from pre-calc. to calc. is the use of a limit process. Calculus is a "limit machine".
A Preview of Calculus Limits and Their Properties Objectives: Understand what calculus is and how it compares with precalculus. Understand that the tangent line problem is basic to calculus. Understand
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 informationMon 3 Nov Tuesday 4 Nov: Quiz 8 ( ) Friday 7 Nov: Exam 2!!! Today: 4.5 Wednesday: REVIEW. In class Covers
Mon 3 Nov 2014 Tuesday 4 Nov: Quiz 8 (4.2-4.4) Friday 7 Nov: Exam 2!!! In class Covers 3.9-4.5 Today: 4.5 Wednesday: REVIEW Linear Approximation and Differentials In section 4.5, you see the pictures on
More informationChapter 5: Limits, Continuity, and Differentiability
Chapter 5: Limits, Continuity, and Differentiability 63 Chapter 5 Overview: Limits, Continuity and Differentiability Derivatives and Integrals are the core practical aspects of Calculus. They were the
More informationExam 1. (2x + 1) 2 9. lim. (rearranging) (x 1 implies x 1, thus x 1 0
Department of Mathematical Sciences Instructor: Daiva Pucinskaite Calculus I January 28, 2016 Name: Exam 1 1. Evaluate the it x 1 (2x + 1) 2 9. x 1 (2x + 1) 2 9 4x 2 + 4x + 1 9 = 4x 2 + 4x 8 = 4(x 1)(x
More informationRespect your friends! Do not distract anyone by chatting with people around you Be considerate of others in class.
Math 1431 Dr. Melahat Almus almus@math.uh.edu http://www.math.uh.edu/~almus Visit CASA regularly for announcements and course material! If you e-mail me, please mention your course (1431) in the subject
More informationAim: How do we prepare for AP Problems on limits, continuity and differentiability? f (x)
Name AP Calculus Date Supplemental Review 1 Aim: How do we prepare for AP Problems on limits, continuity and differentiability? Do Now: Use the graph of f(x) to evaluate each of the following: 1. lim x
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 informationf ( x) = L ( the limit of f(x), as x approaches a,
Math 1205 Calculus Sec. 2.4: The Definition of imit I. Review A. Informal Definition of imit 1. Def n : et f(x) be defined on an open interval about a except possibly at a itself. If f(x) gets arbitrarily
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 informationSBS Chapter 2: Limits & continuity
SBS Chapter 2: Limits & continuity (SBS 2.1) Limit of a function Consider a free falling body with no air resistance. Falls approximately s(t) = 16t 2 feet in t seconds. We already know how to nd the average
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 informationAP Calculus BC. Chapter 2: Limits and Continuity 2.4: Rates of Change and Tangent Lines
AP Calculus BC Chapter 2: Limits and Continuity 2.4: Rates of Change and Tangent Lines Essential Questions & Why: Essential Questions: What is the difference between average and instantaneous rates of
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 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 informationMATH 409 Advanced Calculus I Lecture 16: Mean value theorem. Taylor s formula.
MATH 409 Advanced Calculus I Lecture 16: Mean value theorem. Taylor s formula. Points of local extremum Let f : E R be a function defined on a set E R. Definition. We say that f attains a local maximum
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 informatione x = 1 + x + x2 2! + x3 If the function f(x) can be written as a power series on an interval I, then the power series is of the form
Taylor Series Given a function f(x), we would like to be able to find a power series that represents the function. For example, in the last section we noted that we can represent e x by the power series
More informationNotes on uniform convergence
Notes on uniform convergence Erik Wahlén erik.wahlen@math.lu.se January 17, 2012 1 Numerical sequences We begin by recalling some properties of numerical sequences. By a numerical sequence we simply mean
More informationC-N M151 Lecture Notes (part 1) Based on Stewart s Calculus (2013) B. A. Starnes
Lecture Calculus is the study of infinite Mathematics. In essence, it is the extension of algebraic concepts to their limfinity(l). What does that even mean? Well, let's begin with the notion of functions
More informationAP Calculus. Derivatives.
1 AP Calculus Derivatives 2015 11 03 www.njctl.org 2 Table of Contents Rate of Change Slope of a Curve (Instantaneous ROC) Derivative Rules: Power, Constant, Sum/Difference Higher Order Derivatives Derivatives
More informationPart 2 Continuous functions and their properties
Part 2 Continuous functions and their properties 2.1 Definition Definition A function f is continuous at a R if, and only if, that is lim f (x) = f (a), x a ε > 0, δ > 0, x, x a < δ f (x) f (a) < ε. Notice
More informationMATH CALCULUS I 1.5: Continuity
MATH 12002 - CALCULUS I 1.5: Continuity Professor Donald L. White Department of Mathematical Sciences Kent State University D.L. White (Kent State University) 1 / 12 Definition of Continuity Intuitively,
More informationContinuity. The Continuity Equation The equation that defines continuity at a point is called the Continuity Equation.
Continuity A function is continuous at a particular x location when you can draw it through that location without picking up your pencil. To describe this mathematically, we have to use limits. Recall
More informationGraphing Rational Functions
Unit 1 R a t i o n a l F u n c t i o n s Graphing Rational Functions Objectives: 1. Graph a rational function given an equation 2. State the domain, asymptotes, and any intercepts Why? The function describes
More informationSince the two-sided limits exist, so do all one-sided limits. In particular:
SECTION 3.6 IN-SECTION EXERCISES: EXERCISE 1. The Intermediate Value Theorem 1. There are many correct graphs possible. A few are shown below. Since f is continuous on [a, b] and π is between f(a) = 3
More informationMATH 151 Engineering Mathematics I
MATH 151 Engineering Mathematics I Fall, 2016, WEEK 4 JoungDong Kim Week4 Section 2.6, 2.7, 3.1 Limits at infinity, Velocity, Differentiation Section 2.6 Limits at Infinity; Horizontal Asymptotes Definition.
More informationHoles in a function. Even though the function does not exist at that point, the limit can still obtain that value.
Holes in a function For rational functions, factor both the numerator and the denominator. If they have a common factor, you can cancel the factor and a zero will exist at that x value. Even though the
More information2.1 The Tangent and Velocity Problems
2.1 The Tangent and Velocity Problems Tangents What is a tangent? Tangent lines and Secant lines Estimating slopes from discrete data: Example: 1. A tank holds 1000 gallons of water, which drains from
More informationNotes Chapter 5 Systems of Linear Equations. Section Number and Topic 5.1 Solve Systems by Graphing
Notes Chapter 5 Systems of Linear Equations Section Number and Topic 5.1 Solve Systems by Graphing Standards REI.3.6 REI.4.11 CED.1.2 MAFS.912.N-Q.1.2 MAFS. 912.A- SSE.1.1 MAFS. 912.A- CED.1.2 CED.1.3
More informationAP Calculus AB Summer Assignment
AP Calculus AB 017-018 Summer Assignment Congratulations! You have been accepted into Advanced Placement Calculus AB for the next school year. This course will count as a math credit at Freedom High School
More information+ i sin. + i sin. = 2 cos
Math 11 Lesieutre); Exam review I; December 4, 017 1. a) Find all complex numbers z for which z = 8. Write your answers in rectangular non-polar) form. We are going to use de Moivre s theorem. For 1, r
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 information2.4. Characterising Functions. Introduction. Prerequisites. Learning Outcomes
Characterising Functions 2.4 Introduction There are a number of different terms used to describe the ways in which functions behave. In this Section we explain some of these terms and illustrate their
More informationTopology Homework Assignment 1 Solutions
Topology Homework Assignment 1 Solutions 1. Prove that R n with the usual topology satisfies the axioms for a topological space. Let U denote the usual topology on R n. 1(a) R n U because if x R n, then
More informationLimits and Their Properties
Chapter 1 Limits and Their Properties Course Number Section 1.1 A Preview of Calculus Objective: In this lesson you learned how calculus compares with precalculus. I. What is Calculus? (Pages 42 44) Calculus
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 informationDetermine whether the formula determines y as a function of x. If not, explain. Is there a way to look at a graph and determine if it's a function?
1.2 Functions and Their Properties Name: Objectives: Students will be able to represent functions numerically, algebraically, and graphically, determine the domain and range for functions, and analyze
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 informationChapter 1 Limits and Their Properties
Chapter 1 Limits and Their Properties Calculus: Chapter P Section P.2, P.3 Chapter P (briefly) WARM-UP 1. Evaluate: cot 6 2. Find the domain of the function: f( x) 3x 3 2 x 4 g f ( x) f ( x) x 5 3. Find
More informationTopic Subtopics Essential Knowledge (EK)
Unit/ Unit 1 Limits [BEAN] 1.1 Limits Graphically Define a limit (y value a function approaches) One sided limits. Easy if it s continuous. Tricky if there s a discontinuity. EK 1.1A1: Given a function,
More informationMath 106 Calculus 1 Topics for first exam
Chapter 2: Limits and Continuity Rates of change and its: Math 06 Calculus Topics for first exam Limit of a function f at a point a = the value the function should take at the point = the value that the
More informationMath 1431 Final Exam Review
Math 1431 Final Exam Review Comprehensive exam. I recommend you study all past reviews and practice exams as well. Know all rules/formulas. Make a reservation for the final exam. If you miss it, go back
More informationMATH The Derivative as a Function - Section 3.2. The derivative of f is the function. f x h f x. f x lim
MATH 90 - The Derivative as a Function - Section 3.2 The derivative of f is the function f x lim h 0 f x h f x h for all x for which the limit exists. The notation f x is read "f prime of x". Note that
More information3 rd class Mech. Eng. Dept. hamdiahmed.weebly.com Fourier Series
Definition 1 Fourier Series A function f is said to be piecewise continuous on [a, b] if there exists finitely many points a = x 1 < x 2
More information(Riemann) Integration Sucks!!!
(Riemann) Integration Sucks!!! Peyam Ryan Tabrizian Friday, November 8th, 2 Are all functions integrable? Unfortunately not! Look at the handout Solutions to 5.2.67, 5.2.68, we get two examples of functions
More information9 Precise definition of limit
9 Precise definition of limit In 5.1, the limit lim x a f(x) was informally defined to be the value (if any) the height of the graph of f approaches as x gets ever closer to a. While this definition conveys
More informationBob Brown Math 251 Calculus 1 Chapter 4, Section 4 1 CCBC Dundalk
Bob Brown Math 251 Calculus 1 Chapter 4, Section 4 1 A Function and its Second Derivative Recall page 4 of Handout 3.1 where we encountered the third degree polynomial f(x) = x 3 5x 2 4x + 20. Its derivative
More informationEconomics 204 Summer/Fall 2017 Lecture 7 Tuesday July 25, 2017
Economics 204 Summer/Fall 2017 Lecture 7 Tuesday July 25, 2017 Section 2.9. Connected Sets Definition 1 Two sets A, B in a metric space are separated if Ā B = A B = A set in a metric space is connected
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 informationReview for Cumulative Test 2
Review for Cumulative Test We will have our second course-wide cumulative test on Tuesday February 9 th or Wednesday February 10 th, covering from the beginning of the course up to section 4.3 in our textbook.
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 informationLimits and Continuous Functions. 2.2 Introduction to Limits. We first interpret limits loosely. We write. lim f(x) = L
2 Limits and Continuous Functions 2.2 Introduction to Limits We first interpret limits loosel. We write lim f() = L and sa the limit of f() as approaches c, equals L if we can make the values of f() arbitraril
More informationIncreasing/Decreasing Test. Extreme Values and The First Derivative Test.
Calculus 1 Lia Vas Increasing/Decreasing Test. Extreme Values and The First Derivative Test. Recall that a function f(x) is increasing on an interval if the increase in x-values implies an increase in
More information1.2 Functions and Their Properties Name:
1.2 Functions and Their Properties Name: Objectives: Students will be able to represent functions numerically, algebraically, and graphically, determine the domain and range for functions, and analyze
More informationMATH 113: ELEMENTARY CALCULUS
MATH 3: ELEMENTARY CALCULUS Please check www.ualberta.ca/ zhiyongz for notes updation! 6. Rates of Change and Limits A fundamental philosophical truth is that everything changes. In physics, the change
More informationx y More precisely, this equation means that given any ε > 0, there exists some δ > 0 such that
Chapter 2 Limits and continuity 21 The definition of a it Definition 21 (ε-δ definition) Let f be a function and y R a fixed number Take x to be a point which approaches y without being equal to y If there
More information5. Some theorems on continuous functions
5. Some theorems on continuous functions The results of section 3 were largely concerned with continuity of functions at a single point (usually called x 0 ). In this section, we present some consequences
More informationAnalysis II: Basic knowledge of real analysis: Part V, Power Series, Differentiation, and Taylor Series
.... Analysis II: Basic knowledge of real analysis: Part V, Power Series, Differentiation, and Taylor Series Kenichi Maruno Department of Mathematics, The University of Texas - Pan American March 4, 20
More information(a) For an accumulation point a of S, the number l is the limit of f(x) as x approaches a, or lim x a f(x) = l, iff
Chapter 4: Functional Limits and Continuity Definition. Let S R and f : S R. (a) For an accumulation point a of S, the number l is the limit of f(x) as x approaches a, or lim x a f(x) = l, iff ε > 0, δ
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 informationMath 10b Ch. 8 Reading 1: Introduction to Taylor Polynomials
Math 10b Ch. 8 Reading 1: Introduction to Taylor Polynomials Introduction: In applications, it often turns out that one cannot solve the differential equations or antiderivatives that show up in the real
More informationSection 2: Limits and Continuity
Chapter 2 The Derivative Business Calculus 79 Section 2: Limits and Continuity In the last section, we saw that as the interval over which we calculated got smaller, the secant slopes approached the tangent
More informationrhe* v.tt 2.1 The Tangent and Velocity Problems Ex: When you jump off a swing, where do you go?
2.1 The Tangent and Velocity Problems Ex: When you jump off a swing, where do you go? lf± # is.t *t, Ex: Can you approximate this line with another nearby? How would you get a better approximation? rhe*
More informationMA Lesson 12 Notes Section 3.4 of Calculus part of textbook
MA 15910 Lesson 1 Notes Section 3.4 of Calculus part of textbook Tangent Line to a curve: To understand the tangent line, we must first discuss a secant line. A secant line will intersect a curve at more
More informationx x 1 x 2 + x 2 1 > 0. HW5. Text defines:
Lecture 15: Last time: MVT. Special case: Rolle s Theorem (when f(a) = f(b)). Recall: Defn: Let f be defined on an interval I. f is increasing (or strictly increasing) if whenever x 1, x 2 I and x 2 >
More informationO.K. But what if the chicken didn t have access to a teleporter.
The intermediate value theorem, and performing algebra on its. This is a dual topic lecture. : The Intermediate value theorem First we should remember what it means to be a continuous function: A function
More informationFebruary 13, Option 9 Overview. Mind Map
Option 9 Overview Mind Map Return tests - will discuss Wed..1.1 J.1: #1def,2,3,6,7 (Sequences) 1. Develop and understand basic ideas about sequences. J.2: #1,3,4,6 (Monotonic convergence) A quick review:
More informationLimits and Continuity
Limits and Continuity Philippe B. Laval Kennesaw State University January 2, 2005 Contents Abstract Notes and practice problems on its and continuity. Limits 2. Introduction... 2.2 Theory:... 2.2. GraphicalMethod...
More information