Chapter 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
|
|
- Joy Mitchell
- 5 years ago
- Views:
Transcription
1 Chapter 2 Limits and Continuity 2.1 Rates of change and Tangents to Curves Definition : 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 1 ) = f(x 1 + h) f(x 1 ), h 0 x 2 x 1 h = slope of secant to the graph of f Definition : Slope of a curve y = f(x) at p(x 1, y 1 ) is the slope of the tangent line at p, f(x 1 + h) f(x 1 ) slope = lim. h 0 h 11
2 2.2 Limits of functions and Limits laws Definition : Limits of a function : Let f(x) defined on an open interval about x 0, except possibly at x 0 itself, if f(x) is arbitrarily close to L for all x sufficiently close to x 0, we say that f approaches the limit L as x approaches x 0. and write lim x x0 f(x) = L Theorem : The Limit Laws: Let L, M, c, k be real numbers and lim x c f(x) = L, lim x c g(x) = M then: 1) lim x c [f(x) ± g(x)] = L ± M. 2) lim x c [kf(x) = kl. 3) lim x c [f(x)g(x) = LM. 4) lim x c f(x) g(x) = L M, M 0. 5) lim x c [f(x)] n = L n, n is positive integer. 6) lim x c n f(x) = n L, nɛn. ( If n is even, we assume that lim x c f(x) = L > 0. 12
3 Theorem : Limits of Polynomials: If P (x) = a n x n + a n 1 x n a 0 Then lim x c P (x) = P (c) = a n c n + a n 1 c n a 0. Theorem : Limits of Rational functions : If P (x) and Q(x) are polynomials and Q(c) 0 then: P (x) lim x c = P (c). Q(x) Q(c) Eliminating zero Denominators Algebraically: Finding limits by multiplying both numerator and denominator by the conjugate : Theorem : The Sandwich theorem: Suppose that g(x) f(x) h(x), x in some open interval containing c, except possibly at x = c itself. Suppose also that lim x c g(x) = lim x c h(x) = L, Then: lim x c f(x) = L 13
4 Theorem Suppose that f(x) g(x), x in some open interval containing c, except possibly at x = c itself and the limits of f, g both exist as x c Then: lim x c f(x) lim x c g(x). 14
5 2.3 The precise definition of a limit Definition : Let f(x) be defined on an open interval about x 0, except possibly at x 0 itself, we say that lim x x0 f(x) = L if ɛ > 0, δ > 0 such that x, 0 < x x 0 < δ f(x) L < ɛ 15
6 2.4 One sided Limits Definition : 1) Let f(x) be defined on (c, b), let x approaches c from within that interval, then f approaches L and we say that f has a right -hand limit L at c. and write: lim x c + f(x) = L. 2) If f(x) defined on (a, c) and x approaches c from within that interval, f(x) approaches M then f has a left -hand limit M at c and write lim x c f(x) = M Find limits Theorem : A function f(x) has a limit as x c it has left -hand and righthand limits their and those one sided limits are equal. lim x c f(x) = L lim x c + f(x) = L and lim x c f(x) = L Precise Definition of one sided Limits Definition ) lim x c + f(x) = L if, ɛ > 0, δ > 0 such that x, c < x < c + δ f(x) L < ɛ 2) lim x c f(x) = L if, ɛ > 0, δ > 0 such that x, c δ < x < c f(x) L < ɛ 16
7 Theorem : lim θ 0 sin θ θ = 1, θ in radian. 17
8 2.5 Continuity Definition : Continuity at an Interior Point: A function y = f(x) is continuous at an interior point c of its domain if lim x c f(x) = f(c) Continuity at End Point: A function y = f(x) is continuous at a left end point a or continuous at a right end point b of its domain if lim x a + f(x) = f(a) or lim x b f(x) = f(b) respectively. If f is not continuous at a point c, we say f is discontinuous at c, and c is a point of discontinuity of f. Continuity Test: A function f(x) is continuous at an interior point x = c of its domain it meets the following three conditions : (1) f(c) exist. (2) lim x c f(x) exist. (3) lim x c f(x) = f(c). Types of discontinuity : 1) Removable Discontinuity : The function has a limit at a point c and we can remove the discontinuity by redefine the function lim x c f(x) = f(c). 2) Jump discontinuity : The one sided limits exist but have different values. 3) Infinite Discontinuity : the function has an infinite limit as x c 4) Oscillating Discontinuity : the function oscillate as x c 18
9 Continuity on an interval : The function is continuous on an interval it is continuous at every point of the interval. Note : A continuous function is the function that is continuous at every point of its domain. Theorem Properties of continuous functions: If the functions f, g are continuous at x = c then the following are continuous at c. f f ± g, kf, fg,, g(c) 0, f n, nɛn g and n f provided it is defined on an open interval containing c, nɛn. 19
10 Theorem : If f is continuous at c and g is continuous on f(c) then g f is continuous at c. Theorem :Limits of continuous functions: If g is continuous at b and lim x c f(x) = b then lim x c g(f(x)) = g(lim x c f(x)) = g(b). Theorem Intermediate value theorem (IV T ) : If f is continuous function on a closed interval [a, b], and if y 0 f(a), f(b) then y 0 = f(c) for some c in [a, b]. is any value between Note that : 1) If f is not continuous at a point between a, b then we can find y 0 f(c) 2) IV T tells us that if f is continuous then any interval on which f changes sign, contains a zero of the function. That is, y 0 = 0 between f(a), f(b) cɛ[a, b] such that f(c) = 0 We call this a solution or root of the equation f(x) = 0. 20
11 2.6 Limits involving infinity ; Asymptotes of graphs Definition ) We say that f(x) has the limit L as x and write lim x f(x) = L if ɛ > 0, a corresponding number M such that x > M, f(x) L < ɛ. 2) We say that f(x) has the limit L as x and write lim x f(x) = L if ɛ > 0, a corresponding number N such that x < N, f(x) L < ɛ. Show that : Theorem : All the limit laws are true when we replace x c by x ± Limits at infinity of rational functions : To determine the limit of rational function as x ±, we first divide the numerator and denominator by the highest power of x in the denominator 21
12 Horizontal Asymptotes :(H.A) A line y = b is a H.A of the graph of a function y = f(x) if either lim x f(x) = b or lim x f(x) = b If both limits hold we say that f has a H.A from both sides right and left. 22
13 Oblique Asymptotes :(O.A) If the degree of numerator of rational function is 1 greater than the degree of the denominator, the graph has an oblique asymptote. We find an equation for the asymptote by dividing numerator by denominator to express f as linear function plus a remainder that goes to zero as x ± Infinite limits : lim x x 2 =, lim x 0 1 x 2 = in each case the limit does not exist. Precise Definition of limits : 1) lim x x0 f(x) = if B > 0, δ > 0 such that x, 0 < x x 0 < δ f(x) > B. 2) lim x x0 f(x) = if B < 0, δ > 0 such that x, 0 < x x 0 < δ f(x) < B. 1 Prove that lim x 0 = x 2 Vertical Asymptotes :(V.A) A Line x = a is vertical asymptote of the graph of f if either lim x a + f(x) = ± or lim x a f(x) = ± Find H.A and V.A of the following functions: 23
14 Dominant Terms : To know how f behaves near a point or when x large f(x) = x2 +1 x 1 = (x + 1) + 2 x 1 We see that f behaves like (x + 1) as x and it behaves like 2 x 1 as x 1. 1) Graph the rational function Include the graphes equations of asymptotes and Dominant terms. y = 2x x+1 2) Graph giving the following conditions: f(1) = 1, f( 1) = 0, lim x f(x) = 0, lim x 0 + f(x) =, lim x 0 f(x) =, lim x f(x) = 1. 24
Chapter 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 information2.2 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationLecture Notes for Math 1000
Lecture Notes for Math 1000 Dr. Xiang-Sheng Wang Memorial University of Newfoundland Office: HH-2016, Phone: 864-4321 Office hours: 13:00-15:00 Wednesday, 12:00-13:00 Friday Email: swang@mun.ca Course
More information3 Polynomial and Rational Functions
3 Polynomial and Rational Functions 3.1 Polynomial Functions and their Graphs So far, we have learned how to graph polynomials of degree 0, 1, and. Degree 0 polynomial functions are things like f(x) =,
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 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 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 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 information1.1 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 informationLesson 2.1: Quadratic Functions
Quadratic Functions: Lesson 2.1: Quadratic Functions Standard form (vertex form) of a quadratic function: Vertex: (h, k) Algebraically: *Use completing the square to convert a quadratic equation into standard
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 informationLimit Theorems. MATH 464/506, Real Analysis. J. Robert Buchanan. Summer Department of Mathematics. J. Robert Buchanan Limit Theorems
Limit s MATH 464/506, Real Analysis J. Robert Buchanan Department of Mathematics Summer 2007 Bounded Functions Definition Let A R, let f : A R, and let c R be a cluster point of A. We say that f is bounded
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 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 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 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 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 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 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 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 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. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.
Math120 - Precalculus. Final Review. Fall, 2011 Prepared by Dr. P. Babaali 1 Algebra 1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.
More 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 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 informationDepartment of Mathematics, University of Wisconsin-Madison Math 114 Worksheet Sections (4.1),
Department of Mathematics, University of Wisconsin-Madison Math 114 Worksheet Sections (4.1), 4.-4.6 1. Find the polynomial function with zeros: -1 (multiplicity ) and 1 (multiplicity ) whose graph passes
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 informationSection 1.2 Combining Functions; Shifting and Scaling Graphs. (a) Function addition: Given two functions f and g we define the sum of f and g as
Section 1.2 Combining Functions; Shifting and Scaling Graphs We will get new functions from the ones we know. Tow functions f and g can be combined to form new functions by function addition, substraction,
More informationChapter 2 NAME
QUIZ 1 Chapter NAME 1. Determine 15 - x + x by substitution. 1. xs3 (A) (B) 8 (C) 10 (D) 1 (E) 0 5-6x + x Find, if it exists. xs5 5 - x (A) -4 (B) 0 (C) 4 (D) 6 (E) Does not exist 3. For the function y
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 informationAnna D Aloise May 2, 2017 INTD 302: Final Project. Demonstrate an Understanding of the Fundamental Concepts of Calculus
Anna D Aloise May 2, 2017 INTD 302: Final Project Demonstrate an Understanding of the Fundamental Concepts of Calculus Analyzing the concept of limit numerically, algebraically, graphically, and in writing.
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 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 3: September 12, 2017 Section A2. Professor Jennifer Balakrishnan,
What is on today Professor Jennifer Balakrishnan, jbala@bu.edu 1 Techniques for computing limits 1 1.1 Limit laws..................................... 1 1.2 One-sided limits..................................
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 informationInfinite Limits. By Tuesday J. Johnson
Infinite Limits By Tuesday J. Johnson Suggested Review Topics Algebra skills reviews suggested: Evaluating functions Graphing functions Working with inequalities Working with absolute values Trigonometric
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 informationThis Week. Professor Christopher Hoffman Math 124
This Week Sections 2.1-2.3,2.5,2.6 First homework due Tuesday night at 11:30 p.m. Average and instantaneous velocity worksheet Tuesday available at http://www.math.washington.edu/ m124/ (under week 2)
More informationPreCalculus Notes. MAT 129 Chapter 5: Polynomial and Rational Functions. David J. Gisch. Department of Mathematics Des Moines Area Community College
PreCalculus Notes MAT 129 Chapter 5: Polynomial and Rational Functions David J. Gisch Department of Mathematics Des Moines Area Community College September 2, 2011 1 Chapter 5 Section 5.1: Polynomial Functions
More informationAP Calculus AB. Limits & Continuity.
1 AP Calculus AB Limits & Continuity 2015 10 20 www.njctl.org 2 Table of Contents click on the topic to go to that section Introduction The Tangent Line Problem Definition of a Limit and Graphical Approach
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 informationLecture 2 (Limits) tangent line secant line
Lecture 2 (Limits) We shall start with the tangent line problem. Definition: A tangent line (Latin word 'touching') to the function f(x) at the point is a line that touches the graph of the function at
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 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 informationMath 150 Midterm 1 Review Midterm 1 - Monday February 28
Math 50 Midterm Review Midterm - Monday February 28 The midterm will cover up through section 2.2 as well as the little bit on inverse functions, exponents, and logarithms we included from chapter 5. Notes
More informationMath 1120 Calculus, sections 3 and 10 Test 1
October 3, 206 Name The problems count as marked The total number of points available is 7 Throughout this test, show your work This is an amalgamation of the tests from sections 3 and 0 (0 points) Find
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 information1 Limits Finding limits graphically. 1.3 Finding limits analytically. Examples 1. f(x) = x3 1. f(x) = f(x) =
Theorem 13 (i) If p(x) is a polynomial, then p(x) = p(c) 1 Limits 11 12 Fining its graphically Examples 1 f(x) = x3 1, x 1 x 1 The behavior of f(x) as x approximates 1 x 1 f(x) = 3 x 2 f(x) = x+1 1 f(x)
More informationChapter 1/3 Rational Inequalities and Rates of Change
Chapter 1/3 Rational Inequalities and Rates of Change Lesson Package MHF4U Chapter 1/3 Outline Unit Goal: By the end of this unit, you will be able to solve rational equations and inequalities algebraically.
More informationSection 3.7. Rolle s Theorem and the Mean Value Theorem
Section.7 Rolle s Theorem and the Mean Value Theorem The two theorems which are at the heart of this section draw connections between the instantaneous rate of change and the average rate of change of
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 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 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 informationMath 320: Real Analysis MWF 1pm, Campion Hall 302 Homework 8 Solutions Please write neatly, and in complete sentences when possible.
Math 320: Real Analysis MWF pm, Campion Hall 302 Homework 8 Solutions Please write neatly, and in complete sentences when possible. Do the following problems from the book: 4.3.5, 4.3.7, 4.3.8, 4.3.9,
More information2.2. Limits Involving Infinity. Copyright 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall
2.2 Limits Involving Infinity Copyright 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Finite Limits as x ± What you ll learn about Sandwich Theorem Revisited Infinite Limits as x a End
More information(x + 3)(x 1) lim(x + 3) = 4. lim. (x 2)( x ) = (x 2)(x + 2) x + 2 x = 4. dt (t2 + 1) = 1 2 (t2 + 1) 1 t. f(x) = lim 3x = 6,
Math 140 MT1 Sample C Solutions Tyrone Crisp 1 (B): First try direct substitution: you get 0. So try to cancel common factors. We have 0 x 2 + 2x 3 = x 1 and so the it as x 1 is equal to (x + 3)(x 1),
More informationAnnouncements. Topics: Homework:
Topics: Announcements - section 2.6 (limits at infinity [skip Precise Definitions (middle of pg. 134 end of section)]) - sections 2.1 and 2.7 (rates of change, the derivative) - section 2.8 (the derivative
More informationChapter 2: Limits & Continuity
Name: Date: Period: AP Calc AB Mr. Mellina Chapter 2: Limits & Continuity Sections: v 2.1 Rates of Change of Limits v 2.2 Limits Involving Infinity v 2.3 Continuity v 2.4 Rates of Change and Tangent Lines
More informationA function is actually a simple concept; if it were not, history would have replaced it with a simpler one by now! Here is the definition:
1.2 Functions and Their Properties A function is actually a simple concept; if it were not, history would have replaced it with a simpler one by now! Here is the definition: Definition: Function, Domain,
More informationSolutions to Math 41 First Exam October 18, 2012
Solutions to Math 4 First Exam October 8, 202. (2 points) Find each of the following its, with justification. If the it does not exist, explain why. If there is an infinite it, then explain whether it
More informationMath 117: Honours Calculus I Fall, 2002 List of Theorems. a n k b k. k. Theorem 2.1 (Convergent Bounded) A convergent sequence is bounded.
Math 117: Honours Calculus I Fall, 2002 List of Theorems Theorem 1.1 (Binomial Theorem) For all n N, (a + b) n = n k=0 ( ) n a n k b k. k Theorem 2.1 (Convergent Bounded) A convergent sequence is bounded.
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 information1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.
Math120 - Precalculus. Final Review Prepared by Dr. P. Babaali 1 Algebra 1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents. (a) 5
More information1 Functions and Graphs
1 Functions and Graphs 1.1 Functions Cartesian Coordinate System A Cartesian or rectangular coordinate system is formed by the intersection of a horizontal real number line, usually called the x axis,
More information= lim. (1 + h) 1 = lim. = lim. = lim = 1 2. lim
Math 50 Exam # Solutions. Evaluate the following its or explain why they don t exist. (a) + h. h 0 h Answer: Notice that both the numerator and the denominator are going to zero, so we need to think a
More informationHorizontal and Vertical Asymptotes from section 2.6
Horizontal and Vertical Asymptotes from section 2.6 Definition: In either of the cases f(x) = L or f(x) = L we say that the x x horizontal line y = L is a horizontal asymptote of the function f. Note:
More information