Continuity. To handle complicated functions, particularly those for which we have a reasonable formula or formulas, we need a more precise definition.
|
|
- Berniece Webb
- 6 years ago
- Views:
Transcription
1 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 functions, particularly those for which we have a reasonable formula or formulas, we need a more precise definition. Definition. A function f is continuous at a point a if 1. f(a) is defined, that is a is in the domain of f. 2. The function f has a limit as x approaches a. 3. lim fx () = fa () x a
2 a Limit exists at a, but value does not.
3 a Here f(a) exists, but the limit does not.
4 Here f(a) exists, but the limit does not. a
5 a Here the limit exists at a, and the value also exists, but they are not equal.
6 Definition: f is continuous on an interval (a, b) if it is continuous at all points of that interval. Example 1. (a) Polynomials are continuous everywhere (b) Rational functions are continuous at all points where the denominator is not 0. Example 2. f(x) = x is continuous everywhere. The piecewise formula definition on f is as follows: xx 0 f() x = xx < 0 Thus if a is positive, then in some interval around a, f(x) is identical with x, and so lim f() x = lim x = a = f(a). x a x a So f is continuous at a.
7 If a is negative, then in some interval around a, f(x) is identical with x, and so lim f() x = lim x = a = f(a). x a x a So f is continuous at a. Finally, at 0, we have (for reasons similar to those mentioned above) lim f() x = lim x 0 + x 0 + x = 0, and lim f() x = lim x x 0 x 0 = 0, therefore lim () 0 x 0 f x = Since f(x) = 0, the function is also continuous at 0, and so is continuous everywhere.
8 It is clear that the graph is not broken, nor does it have holes.
9 The following theorem follows immediately from the main theorem on limits. Theorem. Suppose that functions f and g are continuous at c. Then (a) f + g is continuous at c, (b) f g is continuous at c, (c) f g is continuous at c, (d) f / g is continuous at c, if g(c) 0, and it has a discontinuity at c if g(c) = 0. We show how to prove part (c), and leave the others. Part (d) is proved in the text. lim f()() xgx = lim f() x lim gx () x a x a x a = f()() aga
10 Composition g(x) f(x) ( f g)( x) = f(()) gx Example: 2 gx () = x+ 1 f() x = 2x f gx () = f( gx ( ) = 2( x+ 1) 2= 2x+ 2 g f() x = g( f()) x = f()1 x + = 2x 2 + 1
11 Example. Let f(x) = sin(x), g(x) = x 2. Compute: f g, g f, f f, g g. ( f g) x = f gx = gx = x2 ( ) = = 2= 2= 2 g f () x g( f()) x f() x (sin( x)) sin (). x ( ) () (()) sin( ( )) sin( ) f f () x = f ( f ()) x = sin( f ()) x = sin(sin( x )) ( g g )() x = ggx ( ( )) = gx () 2= ( x 22 ) = x 4
12 Continuity of Compositions The following result is useful for calculating limits of composite functions. Theorem. Let lim stand for any of the following: lim lim x a x a + lim lim lim x a x + x Then if lim g(x) = L, and if f is continuous at L, we have lim f(g(x)) = f(l).
13 Example. Let f(x) = x. We know that f is continuous everywhere. It follows that if lim g(x) = L, then lim g(x) = L. Thus lim 4 + x3 = lim (4 + x3) x 2 x 2 = (4 + ( 2) 3) = 4 = 4 Also lim sin( x) + 1 = 1 = 1 x 0
14 Example. Find any points of discontinuity for the functions below. 1. f() x = 2 3x + x 5x Continuous everywhere, since it is a polynomial. 2. f() x = x+ 3 = x+ 3 2 x + 3x ( x+ 3) x Thus f(x) = 1 x> 3 x 1 x< 3 x Possible problems at 0 and at 3. At 0 it tends to infinity from both sides, so it does not have a finite limit, or a value.
15 At 3 it has different limits from the right and left and no value. Thus the points of discontinuity are at 0 and 3.
16 Continuity from the left and right. If we only have right hand or left hand limits, we can define a similarly one-sided version of continuity. Definition: (a) We say that a function f is continuous from the right at a number c if lim f () x = fc x c () + (b) We say that a function f is continuous from the left at a number c if lim f() x = fc () x c Note that f is continuous at c if and only if it is continuous from the right at c and from the left at c.
17 c Here f is continuous from the right, but not from the left.
18 c Here f is continuous from the left, but not from the right.
19 Here f is not continuous from the left or the right. c
20 Definition. A function f is said to be continuous on a closed interval [a, b] if the following conditions are satisfied: 1. f is continuous at each point of (a, b). 2. f is continuous from the right at a. 3. f is continuous from the left at b. The next slide shows a typical picture of a function defined and continuous on a closed interval.
21 f(x) a b
22 Example. 2 f() x = 4 x The natural domain of this function is the closed interval [ 2, 2]. For any point c in the open interval ( 2, 2) (that is 2<c<2) we have: 2 lim f() x = lim 4 x = lim 4 x2 = 4 c 2 = fc () x c x c x c At the left end point 2, we have 2 lim () lim 4 x = lim 4 x 2 f x = + x x 2 + x = 0 = f ( 2) A similar computation holds at the right endpoint 2.
23 This is the graph.
24 The Importance of Continuity in a Closed Interval The intermediate Value Theorem. If f is continuous in a closed interval [a, b], and k is any number between f(a) and f(b), then there is at least one number x in the interval [a, b], so that f(x) = k. f(a) k f(b) a b
25 f(a) k f(b) a b Possible choices for the number x in the intermediate value theorem.
26 Corollary: If f is continuous in a closed interval and its values at the end points take opposite signs, then f(x) = 0 for some x in the interval. Example. Every polynomial of odd degree has at least one real zero.
27 Of course the intermediate value theorem is not necessarily true without continuity. k a b
1.10 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 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 informationMA 123 (Calculus I) Lecture 6: September 19, 2016 Section A3. Professor Joana Amorim,
Professor Joana Amorim, jamorim@bu.edu What is on today 1 Continuity 1 1.1 Continuity checklist................................ 2 1.2 Continuity on an interval............................. 3 1.3 Intermediate
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 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 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 informationLimits, Continuity, and the Derivative
Unit #2 : Limits, Continuity, and the Derivative Goals: Study and define continuity Review limits Introduce the derivative as the limit of a difference quotient Discuss the derivative as a rate of change
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 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 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 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 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 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 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 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 informationChapter 2. Polynomial and Rational Functions. 2.3 Polynomial Functions and Their Graphs. Copyright 2014, 2010, 2007 Pearson Education, Inc.
Chapter Polynomial and Rational Functions.3 Polynomial Functions and Their Graphs Copyright 014, 010, 007 Pearson Education, Inc. 1 Objectives: Identify polynomial functions. Recognize characteristics
More informationChapter 1: Limits and Continuity
Chapter 1: Limits and Continuity Winter 2015 Department of Mathematics Hong Kong Baptist University 1/69 1.1 Examples where limits arise Calculus has two basic procedures: differentiation and integration.
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 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 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 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 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 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 informationDefinition: For nonempty sets X and Y, a function, f, from X to Y is a relation that associates with each element of X exactly one element of Y.
Functions Definition: A relation is a correspondence between two sets. If x and y are two elements in these sets and if a relation exists between x and y, then we say that x corresponds to y or that y
More informationAdvanced Mathematics Unit 2 Limits and Continuity
Advanced Mathematics 3208 Unit 2 Limits and Continuity NEED TO KNOW Expanding Expanding Expand the following: A) (a + b) 2 B) (a + b) 3 C) (a + b)4 Pascals Triangle: D) (x + 2) 4 E) (2x -3) 5 Random Factoring
More informationAdvanced Mathematics Unit 2 Limits and Continuity
Advanced Mathematics 3208 Unit 2 Limits and Continuity NEED TO KNOW Expanding Expanding Expand the following: A) (a + b) 2 B) (a + b) 3 C) (a + b)4 Pascals Triangle: D) (x + 2) 4 E) (2x -3) 5 Random Factoring
More 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 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 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 informationContinuity. Chapter 4
Chapter 4 Continuity Throughout this chapter D is a nonempty subset of the real numbers. We recall the definition of a function. Definition 4.1. A function from D into R, denoted f : D R, is a subset of
More informationContinuity. Chapter 4
Chapter 4 Continuity Throughout this chapter D is a nonempty subset of the real numbers. We recall the definition of a function. Definition 4.1. A function from D into R, denoted f : D R, is a subset of
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 informationSuppose that f is continuous on [a, b] and differentiable on (a, b). Then
Lectures 1/18 Derivatives and Graphs When we have a picture of the graph of a function f(x), we can make a picture of the derivative f (x) using the slopes of the tangents to the graph of f. In this section
More informationMATH 409 Advanced Calculus I Lecture 10: Continuity. Properties of continuous functions.
MATH 409 Advanced Calculus I Lecture 10: Continuity. Properties of continuous functions. Continuity Definition. Given a set E R, a function f : E R, and a point c E, the function f is continuous at c if
More informationDo now as a warm up: Is there some number a, such that this limit exists? If so, find the value of a and find the limit. If not, explain why not.
Do now as a warm up: Is there some number a, such that this limit exists? If so, find the value of a and find the limit. If not, explain why not. 1 Continuity and One Sided Limits To say that 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 informationMath 473: Practice Problems for Test 1, Fall 2011, SOLUTIONS
Math 473: Practice Problems for Test 1, Fall 011, SOLUTIONS Show your work: 1. (a) Compute the Taylor polynomials P n (x) for f(x) = sin x and x 0 = 0. Solution: Compute f(x) = sin x, f (x) = cos x, f
More informationTest 3 Review. y f(a) = f (a)(x a) y = f (a)(x a) + f(a) L(x) = f (a)(x a) + f(a)
MATH 2250 Calculus I Eric Perkerson Test 3 Review Sections Covered: 3.11, 4.1 4.6. Topics Covered: Linearization, Extreme Values, The Mean Value Theorem, Consequences of the Mean Value Theorem, Concavity
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 informationMATH 114 Calculus Notes on Chapter 2 (Limits) (pages 60-? in Stewart)
Still under construction. MATH 114 Calculus Notes on Chapter 2 (Limits) (pages 60-? in Stewart) As seen in A Preview of Calculus, the concept of it underlies the various branches of calculus. Hence we
More informationIntermediate Value Theorem
Stewart Section 2.5 Continuity p. 1/ Intermediate Value Theorem The intermediate value theorem states that, if a function f is continuous on a closed interval [a,b] (that is, an interval that includes
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 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 informationPolynomial functions right- and left-hand behavior (end behavior):
Lesson 2.2 Polynomial Functions For each function: a.) Graph the function on your calculator Find an appropriate window. Draw a sketch of the graph on your paper and indicate your window. b.) Identify
More informationContinuity at a Point
Continuity at a Point When we eplored the limit of f() as approaches c, the emphasis was on the function values close to = c rather than what happens to the function at = c. We will now consider the following
More informationINTERMEDIATE VALUE THEOREM
INTERMEDIATE VALUE THEOREM Section 1.4B Calculus AP/Dual, Revised 017 viet.dang@humbleisd.net 7/30/018 1:36 AM 1.4B: Intermediate Value Theorem 1 PROOF OF INTERMEDIATE VALUE THEOREM Can you prove that
More informationSection 4.2: The Mean Value Theorem
Section 4.2: The Mean Value Theorem Before we continue with the problem of describing graphs using calculus we shall briefly pause to examine some interesting applications of the derivative. In previous
More informationMath 141: Section 4.1 Extreme Values of Functions - Notes
Math 141: Section 4.1 Extreme Values of Functions - Notes Definition: Let f be a function with domain D. Thenf has an absolute (global) maximum value on D at a point c if f(x) apple f(c) for all x in D
More informationCaculus 221. Possible questions for Exam II. March 19, 2002
Caculus 221 Possible questions for Exam II March 19, 2002 These notes cover the recent material in a style more like the lecture than the book. The proofs in the book are in section 1-11. At the end there
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 informationSection 3.1 Extreme Values
Math 132 Extreme Values Section 3.1 Section 3.1 Extreme Values Example 1: Given the following is the graph of f(x) Where is the maximum (x-value)? What is the maximum (y-value)? Where is the minimum (x-value)?
More informationCalculus 221 worksheet
Calculus 221 worksheet Graphing A function has a global maximum at some a in its domain if f(x) f(a) for all other x in the domain of f. Global maxima are sometimes also called absolute maxima. A function
More informationB553 Lecture 1: Calculus Review
B553 Lecture 1: Calculus Review Kris Hauser January 10, 2012 This course requires a familiarity with basic calculus, some multivariate calculus, linear algebra, and some basic notions of metric topology.
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 informationLimit. Chapter Introduction
Chapter 9 Limit Limit is the foundation of calculus that it is so useful to understand more complicating chapters of calculus. Besides, Mathematics has black hole scenarios (dividing by zero, going to
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 informationMath 10850, Honors Calculus 1
Math 0850, Honors Calculus Homework 0 Solutions General and specific notes on the homework All the notes from all previous homework still apply! Also, please read my emails from September 6, 3 and 27 with
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 informationSection 2.6: Continuity
Section 2.6: Continuity Problem 1 (a) Let f(x) = x 1 x 2 5x. Then f(2) = 1 6 and f(6) = 5, but there is no value of c between 2 6 and 6 for which f(c) = 0. Does this fact violate the Intermediate Value
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 informationFamilies of Functions, Taylor Polynomials, l Hopital s
Unit #6 : Rule Families of Functions, Taylor Polynomials, l Hopital s Goals: To use first and second derivative information to describe functions. To be able to find general properties of families of functions.
More informationThe Intermediate Value Theorem If a function f (x) is continuous in the closed interval [ a,b] then [ ]
Lecture 2 5B Evaluating Limits Limits x ---> a The Intermediate Value Theorem If a function f (x) is continuous in the closed interval [ a,b] then [ ] the y values f (x) must take on every value on the
More information6.1 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 of Functions (a, L)
Limits of Functions f(x) (a, L) L f(x) x a x x 20 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
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 informationINTERMEDIATE VALUE THEOREM
INTERMEDIATE VALUE THEOREM Section 1.4B Calculus AP/Dual, Revised 017 viet.dang@humbleisd.net 7/30/018 1:9 AM 1.4B: Intermediate Value Theorem 1 DEFINITION OF CONTINUITY A function is continuous at the
More informationMATH 409 Advanced Calculus I Lecture 11: More on continuous functions.
MATH 409 Advanced Calculus I Lecture 11: More on continuous functions. Continuity Definition. Given a set E R, a function f : E R, and a point c E, the function f is continuous at c if for any ε > 0 there
More informationMAT 1339-S14 Class 4
MAT 9-S4 Class 4 July 4, 204 Contents Curve Sketching. Concavity and the Second Derivative Test.................4 Simple Rational Functions........................ 2.5 Putting It All Together.........................
More informationCalculus I. 1. Limits and Continuity
2301107 Calculus I 1. Limits and Continuity Outline 1.1. Limits 1.1.1 Motivation:Tangent 1.1.2 Limit of a function 1.1.3 Limit laws 1.1.4 Mathematical definition of a it 1.1.5 Infinite it 1.1. Continuity
More informationConsequences of Continuity and Differentiability
Consequences of Continuity and Differentiability We have seen how continuity of functions is an important condition for evaluating limits. It is also an important conceptual tool for guaranteeing the existence
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 informationPolynomial and Rational Functions. Chapter 3
Polynomial and Rational Functions Chapter 3 Quadratic Functions and Models Section 3.1 Quadratic Functions Quadratic function: Function of the form f(x) = ax 2 + bx + c (a, b and c real numbers, a 0) -30
More informationV. Graph Sketching and Max-Min Problems
V. Graph Sketching and Max-Min Problems The signs of the first and second derivatives of a function tell us something about the shape of its graph. In this chapter we learn how to find that information.
More informationChapter 5 Integrals. 5.1 Areas and Distances
Chapter 5 Integrals 5.1 Areas and Distances We start with a problem how can we calculate the area under a given function ie, the area between the function and the x-axis? If the curve happens to be something
More informationJim Lambers MAT 460 Fall Semester Lecture 2 Notes
Jim Lambers MAT 460 Fall Semester 2009-10 Lecture 2 Notes These notes correspond to Section 1.1 in the text. Review of Calculus Among the mathematical problems that can be solved using techniques from
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A) 6 B) 14 C) 10 D) Does not exist
Assn 3.1-3.3 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the limit, if it exists. 1) Find: lim x -1 6x + 5 5x - 6 A) -11 B) - 1 11 C)
More informationLemma 15.1 (Sign preservation Lemma). Suppose that f : E R is continuous at some a R.
15. Intermediate Value Theorem and Classification of discontinuities 15.1. Intermediate Value Theorem. Let us begin by recalling the definition of a function continuous at a point of its domain. Definition.
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 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 informationInstitute of Computer Science
Institute of Computer Science Academy of Sciences of the Czech Republic Calculus Digest Jiří Rohn http://uivtx.cs.cas.cz/~rohn Technical report No. V-54 02.02.202 Pod Vodárenskou věží 2, 82 07 Prague 8,
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 informationChapter Five Notes N P U2C5
Chapter Five Notes N P UC5 Name Period Section 5.: Linear and Quadratic Functions with Modeling In every math class you have had since algebra you have worked with equations. Most of those equations have
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 informationThe Mean Value Theorem and the Extended Mean Value Theorem
The Mean Value Theorem and the Extended Mean Value Theorem Willard Miller September 21, 2006 0.1 The MVT Recall the Extreme Value Theorem (EVT) from class: If the function f is defined and continuous on
More informationLimits Student Study Session
Teacher Notes Limits Student Study Session Solving limits: The vast majority of limits questions can be solved by using one of four techniques: SUBSTITUTING, FACTORING, CONJUGATING, or by INSPECTING A
More informationMTAEA Continuity and Limits of Functions
School of Economics, Australian National University February 1, 2010 Continuous Functions. A continuous function is one we can draw without taking our pen off the paper Definition. Let f be a real-valued
More information56 CHAPTER 3. POLYNOMIAL FUNCTIONS
56 CHAPTER 3. POLYNOMIAL FUNCTIONS Chapter 4 Rational functions and inequalities 4.1 Rational functions Textbook section 4.7 4.1.1 Basic rational functions and asymptotes As a first step towards understanding
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 informationPartial Derivatives October 2013
Partial Derivatives 14.3 02 October 2013 Derivative in one variable. Recall for a function of one variable, f (a) = lim h 0 f (a + h) f (a) h slope f (a + h) f (a) h a a + h Partial derivatives. For a
More informationIntroduction to Limits
MATH 136 Introduction to Limits Given a function y = f (x), we wish to describe the behavior of the function as the variable x approaches a particular value a. We should be as specific as possible in describing
More information1. State the informal definition of the limit of a function. 4. What are the values of the following limits: lim. 2x 1 if x 1 discontinuous?
Review for Limits Key Concepts 1 State the informal definition of the limit of a function 2 State 9 limit theorems 3 State the squeeze theorem 4 What are the values of the following limits: lim sin x x
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 informationTaylor and Maclaurin Series. Approximating functions using Polynomials.
Taylor and Maclaurin Series Approximating functions using Polynomials. Approximating f x = e x near x = 0 In order to approximate the function f x = e x near x = 0, we can use the tangent line (The Linear
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 informationCHAPTER 3: CONTINUITY ON R 3.1 TWO SIDED LIMITS
CHAPTER 3: CONTINUITY ON R 3.1 TWO SIDED LIMITS DEFINITION. Let a R and let I be an open interval contains a, and let f be a real function defined everywhere except possibly at a. Then f(x) is said to
More informationTRIG REVIEW NOTES. Co-terminal Angles: Angles that end at the same spot. (sines, cosines, and tangents will equal)
TRIG REVIEW NOTES Convert from radians to degrees: multiply by 0 180 Convert from degrees to radians: multiply by 0. 180 Co-terminal Angles: Angles that end at the same spot. (sines, cosines, and tangents
More information1.1 Functions and Their Representations
Arkansas Tech University MATH 2914: Calculus I Dr. Marcel B. Finan 1.1 Functions and Their Representations Functions play a crucial role in mathematics. A function describes how one quantity depends on
More informationFunctions. Chapter Continuous Functions
Chapter 3 Functions 3.1 Continuous Functions A function f is determined by the domain of f: dom(f) R, the set on which f is defined, and the rule specifying the value f(x) of f at each x dom(f). If f is
More information8.7 Taylor s Inequality Math 2300 Section 005 Calculus II. f(x) = ln(1 + x) f(0) = 0
8.7 Taylor s Inequality Math 00 Section 005 Calculus II Name: ANSWER KEY Taylor s Inequality: If f (n+) is continuous and f (n+) < M between the center a and some point x, then f(x) T n (x) M x a n+ (n
More informationContinuity. Warm-up. Suppose the graph of y = f (x) looks like
Continuity Warm-up Suppose the graph of y = f x) looks like answers to 4. a x! a x! a + 0 4 5 6 7. What is the domain of f x)?. What is the range of f x)?. For which values a in [, 7] does lim f x) t exist?
More information