Advanced Mathematics Unit 2 Limits and Continuity

Size: px
Start display at page:

Download "Advanced Mathematics Unit 2 Limits and Continuity"

Transcription

1 Advanced Mathematics 3208 Unit 2 Limits and Continuity

2 NEED TO KNOW Expanding

3 Expanding Expand the following: A) (a + b) 2 B) (a + b) 3

4 C) (a + b)4

5 Pascals Triangle:

6 D) (x + 2) 4 E) (2x -3) 5

7 Random Factoring Methods A) x 2 9 Look for Patterns B) x

8 C) 8x 3-64

9 II. Functions, Graphs, and Limits Analysis of graphs. With the aid of technology. Prelude to the use of calculus both to predict and to explain the observed local and global behaviour of a function. 9

10 Analysis of Graphs Using graphing technology: 1. Sketch the graph of y = x

11 Analysis of Graphs 1. y = x 3 27 A) Find the zeros x = 3 B) Find the local max and min points These are points that have either the largest, or smallest y value in a particular region, or neighbourhood on the graph. There are no local max or min points 11

12 C) Identify any points where concavity changes from concave up to concave down (or vice a versa). The point of inflection is (0, -27) 12

13 2. Sketch the graph of: A) y 2 x 1 x 2 B) y = x 2 What do you notice? y = x 2 is a slant (or oblique) asymptote.

14 Rational Functions f(x) is a rational function if f( x) where p(x) and q(x) are polynomials and qx ( ) 0 px ( ) qx ( ) Rational functions often approach either slant or horizontal asymptotes for large (or small) values of x

15 Rational Functions are not continuous graphs. There various types of discontinuities. There vertical asymptotes which occur when only the denominator (bottom) is zero. There are holes in the graph when there is zero/zero 0 0

16 3. Describe what happens to the x 2 function y near x = 2. The graph seems to approach the point (2, 4) What occurs at x = 2? Division by zero. The function is undefined when x = 2. In fact we get There is a hole in the graph. What occurs at x = -2? x 2 4 Division by zero however this time there is a vertical asymptote

17 4. Describe what happens to the sinx function y as x gets close to x 0. The function seems to approach 1 Does it make any difference if the calculator is in degrees or radians? Yes, it only approaches 1 in radians. 17

18 Limits of functions (including one-sided limits). A basic understanding of the limiting process. Estimating limits from graphs or tables of data. Calculating limits using algebra. Calculating limits at infinity and infinite limits

19 Zeno s Paradox Half of Halves Mathematically speaking: nn 1 lim i n i 1 2 i 1 i This is the limit of an infinite series 19

20 How many sides does a circle have? 5 sides? 18 sides? 20

21 Limit of a Function The limit of a function tells how a function behaves near a certain x- value. Suppose if I wanted to go to a certain place in Canada. We would use a map 21

22 Consider: If we have a function y = f(x) and we are trying to find out what the value of the function is for a x- value under the shaded area, we could make an estimate of what it would be by looking at the function before it goes into or leaves the shaded area. Guess what the function value is at x = 3 22

23 The smaller the shaded area can be made, the better the approximation would be. Guess what the function value is at x = 3 23

24 Guess what the function value is at x = 3 24

25 Guess what the function value is at x = 3 25

26 Mathematically speaking: As x gets close to a, f(x) gets close to a value L This can be written: lim f ( x ) L xa Note: This is not multiplication. It means The limit of f(x) as x approaches a equals L 26

27 We can get values of f(x) to be arbitrarily close to L by looking at values of x sufficiently close to a, but not equal to a. It does not matter if f(a) is defined. We are only looking to see what happens to f(x) as x approaches a 27

28 Limits using a table of values. 1. Determine the behaviour of f (x) as x approaches 2. 28

29 This is the limit from the right side of x = 2 This is the limit from the left side of x = 2 Examples: (Using a Table of Values) 2 2 x 4 x 4 lim lim 4 x 2 x 2 x 2 x x x 4 x 4 x 2. Find: x x x 4 lim 4 x 2 x 2 2 x 4 lim 4 x 2 x 2 29

30 Examples: (Using a Table of Values) sin sin lim lim Find: (radians) sin sin lim 1 (radians) sin sin lim 1 30

31 3. For the function f 1 ( x ) x, complete the table below x (x) Sketch the graph of y = f(x) 2 4 y 4 2 x 31

32 Using the table and graph as a guide, answer following questions: What value is f (x) approaching as x becomes a larger positive number? What value is f (x) approaching as x becomes a larger negative number? Will the value of f (x) ever equal zero? Explain your reasoning. 32

33 With reference to the previous graph complete the following table 33

34 One Sided Limits Consider the function below: This is a piecewise function It consists of two different functions combined together into one function What is the equation? f( x) 2 x x 1, 1 x 1, x 1 34

35 A) B) C) Find the following using the graph and function rule lim f( x) x 1 lim f( x) x 0 lim f( x) x 2 D) lim f( x) x 1 For this limit we need to find both the left and right hand limits because the function has different rules on either side of 1. 35

36 lim ( ) f x x 1 2 lim x 1 x 1 In this case we say that the limit Does Not Exist (DNE) lim ( ) f x x 1 lim x 1 x 1 = 0 = 2 NOTE: Limits do not exist if the left and right limits at a x-value are different. 36

37 Mathematically Speaking A function will have a limit L as x approaches a, if and only if as x approaches a from the left and a from the right you get the same value, L. OR: lim f ( x ) xa L ( iff ) lim f ( x ) L xa and lim f ( x ) L xa 37

38 2.A) Draw f( x) x 2, x 2 2 x 1, x 2 B) Find: lim f( x) x 2 38

39 3.A) Draw f( x) 2 x x 2, 1 x 2, x 1 B) Find: lim f( x) x 0 C) Find: lim f( x) x 1 39

40 4. Find x 1 lim f ( x ) where f ( x ) x 2, x 1 2, x 1 40

41 5. Find lim f ( x ) where f ( x ) x 3 2 x x 4, 3 x 4, x 3 41

42 Evaluate the limits using the following piecewise function: 42

43 Identify which limit statements are true and which are false for the graph shown. 43

44 Text Page , 4, 7, 9, 15, 18 44

45 Absolute Values Definition: The absolute value of a, a, is the distance a is from zero on a number line. 3 = -3 = x = 2 a a a if a a if a 0 0 Note: - a is positive if a is negative

46 EX. -5 Here the value is negative so -5 = -(-5) = 5

47 Rewrite the following without 1. absolute values symbols x

48 4. x + 2 x 2 if x 2 0 ( x 2) if x 2 0 x 2 if x 2 ( x 2) if x 2

49 5. x = 3 x 3 if x 0 x 3 if x 0 6. x < 3

50 7. x > 3

51 Find Recall: x lim x x 0 x0 x0 x, if x 0 x x, if x 0 x x lim lim 1 x x x x lim lim 1 x x x0 x0 x lim x 0 x DNE 51

52 52

53 Find x 1 lim x 1 x 1 Find x 2 lim 2 x 4 x 2 53

54 x Greatest Integer Function x is the greatest integer function. It gives the greatest integer that is less than or equal to x. Example: A) 2 B) C) D)

55 55

56 Find lim x 2 x lim x 2 x lim x 2 x lim x 2 x DNE 56

57 57

58 58

59 Solving Limits Using Algebra There are 7 limit laws which basically allow you to do direct substitution when finding limits. Examples: Evaluate and justify each step by indicating the appropriate Limit Law 1. lim 2x 1 x 3 ( 59

60 2. lim x( x 1) x lim2x 3x 1 x 1 60

61 4. x 3 2x lim x 2 5 x 2 NOTE: : Direct substitution works in many cases, so you should always try it first. 61

62 NOTE: These limit laws basically allow you to do Direct Substitution. 4. x 3 2x lim x 2 5 x 2 Direct Substitution works in many cases, so you should always try it first. 62

63 However, there are a few cases (mostly in math courses) where direct substitution does not work immediately, or at all. 63

64 A) Draw the graph of y f( x) x 2, x 3 2, x 3 x B) Find lim f( x) x 3 In this case direct substitution would give an answer of which is not correct. Remember the limit shows what the function is approaching as x approaches a value. It does not matter what the actual function value is at that x value. 64

65 1. lim x 1 x Examples Direct substitution gives is undefined. which In this case the limit will not work because the x value the limit is approaching is not in the domain of the function. 1 lim x 1 ( x DNE Does Not Exist 65

66 Examples 1 2. lim 2 x 0 x 1 Direct substitution gives which is undefined. 0 In this case direct substitution will not work because the x value the limit is approaching is not in the domain of the function. However, as we will see later this one would not be DNE. Here we say that: lim x 0 1 x 2 66

67 3. lim x 2 2 x 4 x Direct Substitution Whenever you get 0, this means there is some simplification you can do to the function before you do the direct substitution. 0 What would you do here?? lim 2 x x x x2 x2 Factor 4 ( 2)( 2) lim x 2 x 2 lim x x 2 67

68 4. x 2 x 2 lim 2 x 1 2 x x (1) Direct Substitution What would you do here?? Factor 68

69 5. lim h 0 ( h h ( Direct Substitution What would you do here?? More work!! 69

70 6. lim x 2 x 2 2 x Direct Substitution What would you do here?? Rationalize the Numerator How do we rationalize a square root? We multiply top and bottom by the conjugate. The conjugate is the other factor of the difference of squares 70

71 lim x 2 x 2 2 x 2 71

72 7. lim h 0 ( 2 h 1 h 1 2 ( What would you do here?? Simplify the rational expression 72

73 8. Find lim x 4 ( x 2( x 2 x 4 x 4 73

74 9. Find x 1 x x x 3 x 3 lim x 1 74

75 lim 10. Find x 2 3x 6 x 2 x 2 75

76 A) lim x 3 2 x 2x 3 2 x x 6 Practice: 76

77 B) lim x 1 x x Practice: 77

78 C) x 9 9 lim 3 Practice: x x 78

79 D) x 13 4 lim x 3 7 x 2 Practice: 79

80 Page # 3, 11, 14, 15, 17-19,21-28, 44,45 80

81 Continuity

82 What is meant by a continuous function? A curve that can be drawn without taking your pencil from the paper. Which letters of the alphabet are the result of continuous lines?

83 What functions are continuous? Polynomials These are continuous everywhere f( x) gx ( ) These are continuous for all values of x except for the roots of g(x) = 0. Rational Functions In other words it is continuous for all values in the domain

84 Exponential and Logarithmic Functions Sine and Cosine graphs Absolute Value Graphs

85 What type of discontinuities are there?

86 We need a way of defining continuity to know whether or not a function is discontinuous or continuous at a point. Definition: A function y = f(x) is continuous at a number b, if lim f ( x ) f ( b) xb

87 This can be broken into 3 parts 1. f(b) is defined (It exists) b is in the domain of f(x) 2. lim f( x) exists. x lim ( ) lim f ( x ) b In other words f x x b x b 3. Part 1 = Part 2 lim f ( x ) f ( b) xb

88 Describe why each place was discontinuous

89 Discuss the continuity of the following 1. f(x) = x 3 + 2x gx ( ) 3. hx ( ) x 3 This is continuous everywhere because it is a polynomial. x x 1 Discontinuous at x = 1 (VA) 1 is not in the Domain x 4, x 3 2, x 3 Not continuous at x = 3. WHY? lim f ( x ) f (3) 2

90 1, x 1 x x 1, x f ( x ) x, 1 x 1 x = -1 We need to check x = -1 and x = 1. Do we need to check x = 0? NO! In 1/x, x=0 is not in x < -1 lim f( x) lim x 1 x 1 lim f( x) x 1 lim x 1 1 x Thus f(x) is continuous at x = 1 x 2 1 1

91 1, x 1 x x 1, x f ( x ) x, 1 x 1 x = 1 lim ( ) f x x 1 lim x 1 x 2 1 lim f( x) x 1 x 1 lim x 1 2 Thus f(x) is discontinuous at x = 1 since the left and right limits are not the same.

92 5. y = sinx Continuous everywhere 6. y = cos x Not continuous at VA x k, k 2 7. y = 2 x Continuous everywhere

93 Examples What value of k would make the following functions continuous? 1. 2 x 4, x 2 f( x) x 2 kx, 2

94 2. hx ( ) 2 x x x 2, 2 5 x k, x 2

95 3. f( x) 2 x kx x kx, 1 3, x 1

96 4. For what value of the constant c is the function f( x) x c, x 2 2 cx 1, x 2 continuous at every number?

97 Page 54 # 1, 4, 7,15-18,31, 33,34

98 There is one other type of discontinuity Graph 1 y sin x This is known as an Oscillating Discontinuity y x

99 The function sin(1/x) is not defined at x = 0 so it is not continuous at x = 0. The function also oscillates between -1 and 1 as x approaches 0. Therefore, the limit does not exist.

100 Page 27 # 1-5, 7, 9, 10

101

102

103

104

Advanced Mathematics Unit 2 Limits and Continuity

Advanced 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 information

UNIT 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 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 information

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

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 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 information

Math 115 Spring 11 Written Homework 10 Solutions

Math 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 information

UNIT 3. Recall From Unit 2 Rational Functions

UNIT 3. Recall From Unit 2 Rational Functions UNIT 3 Recall From Unit Rational Functions f() is a rational function if where p() and q() are and. Rational functions often approach for values of. Rational Functions are not graphs There various types

More information

Rational Functions. Elementary Functions. Algebra with mixed fractions. Algebra with mixed fractions

Rational 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 information

Limits at Infinity. Horizontal Asymptotes. Definition (Limits at Infinity) Horizontal Asymptotes

Limits 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 information

ter. on Can we get a still better result? Yes, by making the rectangles still smaller. As we make the rectangles smaller and smaller, the

ter. 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 information

Chapter 2: Functions, Limits and Continuity

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 information

2.1 Limits, Rates of Change and Slopes of Tangent Lines

2.1 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 information

Section Properties of Rational Expressions

Section 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 information

MAT137 Calculus! Lecture 20

MAT137 Calculus! Lecture 20 official website http://uoft.me/mat137 MAT137 Calculus! Lecture 20 Today: 4.6 Concavity 4.7 Asypmtotes Next: 4.8 Curve Sketching Indeterminate Forms for Limits Which of the following are indeterminate

More information

MTH4100 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 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 information

LIMITS AT INFINITY MR. VELAZQUEZ AP CALCULUS

LIMITS 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 information

DRAFT - Math 101 Lecture Note - Dr. Said Algarni

DRAFT - 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 information

Limits, Continuity, and the Derivative

Limits, 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 information

CH 2: Limits and Derivatives

CH 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 information

Topic 3 Outline. What is a Limit? Calculating Limits Infinite Limits Limits at Infinity Continuity. 1 Limits and Continuity

Topic 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 information

2.1 The Tangent and Velocity Problems

2.1 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 information

Chapter 2. Limits and Continuity 2.6 Limits Involving Infinity; Asymptotes of Graphs

Chapter 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 information

Section 3.1 Extreme Values

Section 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 information

MAT 1339-S14 Class 4

MAT 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 information

Section 3.3 Limits Involving Infinity - Asymptotes

Section 3.3 Limits Involving Infinity - Asymptotes 76 Section. Limits Involving Infinity - Asymptotes We begin our discussion with analyzing its as increases or decreases without bound. We will then eplore functions that have its at infinity. Let s consider

More information

This Week. Professor Christopher Hoffman Math 124

This 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 information

1.5 Inverse Trigonometric Functions

1.5 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 information

Continuity. The Continuity Equation The equation that defines continuity at a point is called the Continuity Equation.

Continuity. 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 information

Math 12 Final Exam Review 1

Math 12 Final Exam Review 1 Math 12 Final Exam Review 1 Part One Calculators are NOT PERMITTED for this part of the exam. 1. a) The sine of angle θ is 1 What are the 2 possible values of θ in the domain 0 θ 2π? 2 b) Draw these angles

More information

10/22/16. 1 Math HL - Santowski SKILLS REVIEW. Lesson 15 Graphs of Rational Functions. Lesson Objectives. (A) Rational Functions

10/22/16. 1 Math HL - Santowski SKILLS REVIEW. Lesson 15 Graphs of Rational Functions. Lesson Objectives. (A) Rational Functions Lesson 15 Graphs of Rational Functions SKILLS REVIEW! Use function composition to prove that the following two funtions are inverses of each other. 2x 3 f(x) = g(x) = 5 2 x 1 1 2 Lesson Objectives! The

More information

Math 473: Practice Problems for Test 1, Fall 2011, SOLUTIONS

Math 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 information

1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.

1. 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 information

Calculus. Weijiu Liu. Department of Mathematics University of Central Arkansas 201 Donaghey Avenue, Conway, AR 72035, USA

Calculus. Weijiu Liu. Department of Mathematics University of Central Arkansas 201 Donaghey Avenue, Conway, AR 72035, USA Calculus Weijiu Liu Department of Mathematics University of Central Arkansas 201 Donaghey Avenue, Conway, AR 72035, USA 1 Opening Welcome to your Calculus I class! My name is Weijiu Liu. I will guide you

More information

To get horizontal and slant asymptotes algebraically we need to know about end behaviour for rational functions.

To 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 information

Unit IV Derivatives 20 Hours Finish by Christmas

Unit IV Derivatives 20 Hours Finish by Christmas Unit IV Derivatives 20 Hours Finish by Christmas Calculus There two main streams of Calculus: Differentiation Integration Differentiation is used to find the rate of change of variables relative to one

More information

Unit IV Derivatives 20 Hours Finish by Christmas

Unit IV Derivatives 20 Hours Finish by Christmas Unit IV Derivatives 20 Hours Finish by Christmas Calculus There two main streams of Calculus: Differentiation Integration Differentiation is used to find the rate of change of variables relative to one

More information

Making Connections with Rational Functions and Equations

Making Connections with Rational Functions and Equations Section 3.5 Making Connections with Rational Functions and Equations When solving a problem, it's important to read carefully to determine whether a function is being analyzed (Finding key features) or

More information

6.1 Polynomial Functions

6.1 Polynomial Functions 6.1 Polynomial Functions Definition. A polynomial function is any function p(x) of the form p(x) = p n x n + p n 1 x n 1 + + p 2 x 2 + p 1 x + p 0 where all of the exponents are non-negative integers and

More information

MATH 151 Engineering Mathematics I

MATH 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 information

MATH CALCULUS I 1.5: Continuity

MATH 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 information

LIMITS AND DERIVATIVES

LIMITS AND DERIVATIVES 2 LIMITS AND DERIVATIVES LIMITS AND DERIVATIVES 2.2 The Limit of a Function In this section, we will learn: About limits in general and about numerical and graphical methods for computing them. THE LIMIT

More information

2.2 The Limit of a Function

2.2 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 information

Summer Review Packet for Students Entering AP Calculus BC. Complex Fractions

Summer Review Packet for Students Entering AP Calculus BC. Complex Fractions Summer Review Packet for Students Entering AP Calculus BC Comple Fractions When simplifying comple fractions, multiply by a fraction equal to 1 which has a numerator and denominator composed of the common

More information

2. If the values for f(x) can be made as close as we like to L by choosing arbitrarily large. lim

2. 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 information

AP Calculus AB Summer Math Packet

AP Calculus AB Summer Math Packet Name Date Section AP Calculus AB Summer Math Packet This assignment is to be done at you leisure during the summer. It is meant to help you practice mathematical skills necessary to be successful in Calculus

More information

GUIDED NOTES 5.6 RATIONAL FUNCTIONS

GUIDED 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 information

MATH 113: ELEMENTARY CALCULUS

MATH 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 information

Solutions to Math 41 First Exam October 18, 2012

Solutions 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 information

Polynomial Expressions and Functions

Polynomial Expressions and Functions Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit FOUR Page - 1 - of 36 Topic 32: Polynomial Expressions and Functions Recall the definitions of polynomials and terms. Definition: A polynomial

More information

Mathematic 108, Fall 2015: Solutions to assignment #7

Mathematic 108, Fall 2015: Solutions to assignment #7 Mathematic 08, Fall 05: Solutions to assignment #7 Problem # Suppose f is a function with f continuous on the open interval I and so that f has a local maximum at both x = a and x = b for a, b I with a

More information

Chapter 1: Limits and Continuity

Chapter 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 information

AP CALCULUS AB Study Guide for Midterm Exam 2017

AP CALCULUS AB Study Guide for Midterm Exam 2017 AP CALCULUS AB Study Guide for Midterm Exam 2017 CHAPTER 1: PRECALCULUS REVIEW 1.1 Real Numbers, Functions and Graphs - Write absolute value as a piece-wise function - Write and interpret open and closed

More information

Concepts of graphs of functions:

Concepts of graphs of functions: Concepts of graphs of functions: 1) Domain where the function has allowable inputs (this is looking to find math no-no s): Division by 0 (causes an asymptote) ex: f(x) = 1 x There is a vertical asymptote

More information

C-N M151 Lecture Notes (part 1) Based on Stewart s Calculus (2013) B. A. Starnes

C-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 information

Exponential Functions:

Exponential Functions: Exponential Functions: An exponential function has the form f (x) = b x where b is a fixed positive number, called the base. Math 101-Calculus 1 (Sklensky) In-Class Work January 29, 2015 1 / 12 Exponential

More information

Infinite Limits. Infinite Limits. Infinite Limits. Previously, we discussed the limits of rational functions with the indeterminate form 0/0.

Infinite 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 information

1.3 Limits and Continuity

1.3 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 information

Aim: How do we prepare for AP Problems on limits, continuity and differentiability? f (x)

Aim: 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 information

Continuity. To handle complicated functions, particularly those for which we have a reasonable formula or formulas, we need a more precise definition.

Continuity. 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 information

What makes f '(x) undefined? (set the denominator = 0)

What makes f '(x) undefined? (set the denominator = 0) Chapter 3A Review 1. Find all critical numbers for the function ** Critical numbers find the first derivative and then find what makes f '(x) = 0 or undefined Q: What is the domain of this function (especially

More information

Department 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), 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 information

AP Calculus Summer Assignment Summer 2017 Expectations for Summer Assignment on the first day of the school year.

AP Calculus Summer Assignment Summer 2017 Expectations for Summer Assignment on the first day of the school year. Welcome to AP Calculus!!! For you to be successful in the fall when you come back to school you will need to complete this summer homework assignment. This will be worth grades when you get back to class

More information

October 27, 2018 MAT186 Week 3 Justin Ko. We use the following notation to describe the limiting behavior of functions.

October 27, 2018 MAT186 Week 3 Justin Ko. We use the following notation to describe the limiting behavior of functions. October 27, 208 MAT86 Week 3 Justin Ko Limits. Intuitive Definitions of Limits We use the following notation to describe the iting behavior of functions.. (Limit of a Function A it is written as f( = L

More information

Secondary Math 3 Honors Unit 10: Functions Name:

Secondary Math 3 Honors Unit 10: Functions Name: Secondary Math 3 Honors Unit 10: Functions Name: Parent Functions As you continue to study mathematics, you will find that the following functions will come up again and again. Please use the following

More information

1. Which one of the following points is a singular point of. f(x) = (x 1) 2/3? f(x) = 3x 3 4x 2 5x + 6? (C)

1. Which one of the following points is a singular point of. f(x) = (x 1) 2/3? f(x) = 3x 3 4x 2 5x + 6? (C) Math 1120 Calculus Test 3 November 4, 1 Name In the first 10 problems, each part counts 5 points (total 50 points) and the final three problems count 20 points each Multiple choice section Circle the correct

More information

Fundamental Theorem of Algebra (NEW): A polynomial function of degree n > 0 has n complex zeros. Some of these zeros may be repeated.

Fundamental Theorem of Algebra (NEW): A polynomial function of degree n > 0 has n complex zeros. Some of these zeros may be repeated. .5 and.6 Comple Numbers, Comple Zeros and the Fundamental Theorem of Algebra Pre Calculus.5 COMPLEX NUMBERS 1. Understand that - 1 is an imaginary number denoted by the letter i.. Evaluate the square root

More information

a x a y = a x+y a x a = y ax y (a x ) r = a rx and log a (xy) = log a (x) + log a (y) log a ( x y ) = log a(x) log a (y) log a (x r ) = r log a (x).

a x a y = a x+y a x a = y ax y (a x ) r = a rx and log a (xy) = log a (x) + log a (y) log a ( x y ) = log a(x) log a (y) log a (x r ) = r log a (x). You should prepare the following topics for our final exam. () Pre-calculus. (2) Inverses. (3) Algebra of Limits. (4) Derivative Formulas and Rules. (5) Graphing Techniques. (6) Optimization (Maxima and

More information

Objectives List. Important Students should expect test questions that require a synthesis of these objectives.

Objectives List. Important Students should expect test questions that require a synthesis of these objectives. MATH 1040 - of One Variable, Part I Textbook 1: : Algebra and Trigonometry for ET. 4 th edition by Brent, Muller Textbook 2:. Early Transcendentals, 3 rd edition by Briggs, Cochran, Gillett, Schulz s List

More information

Suppose that f is continuous on [a, b] and differentiable on (a, b). Then

Suppose 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 information

Learning Objectives for Math 165

Learning Objectives for Math 165 Learning Objectives for Math 165 Chapter 2 Limits Section 2.1: Average Rate of Change. State the definition of average rate of change Describe what the rate of change does and does not tell us in a given

More information

Calculus (Math 1A) Lecture 5

Calculus (Math 1A) Lecture 5 Calculus (Math 1A) Lecture 5 Vivek Shende September 5, 2017 Hello and welcome to class! Hello and welcome to class! Last time Hello and welcome to class! Last time We discussed composition, inverses, exponentials,

More information

3 Polynomial and Rational Functions

3 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 information

Limits and Continuity

Limits and Continuity Limits and Continuity MATH 151 Calculus for Management J. Robert Buchanan Department of Mathematics Fall 2018 Objectives After this lesson we will be able to: Determine the left-hand and right-hand limits

More information

WEEK 7 NOTES AND EXERCISES

WEEK 7 NOTES AND EXERCISES WEEK 7 NOTES AND EXERCISES RATES OF CHANGE (STRAIGHT LINES) Rates of change are very important in mathematics. Take for example the speed of a car. It is a measure of how far the car travels over a certain

More information

DuVal High School Summer Review Packet AP Calculus

DuVal High School Summer Review Packet AP Calculus DuVal High School Summer Review Packet AP Calculus Welcome to AP Calculus AB. This packet contains background skills you need to know for your AP Calculus. My suggestion is, you read the information and

More information

AP Calculus Summer Packet

AP Calculus Summer Packet AP Calculus Summer Packet Writing The Equation Of A Line Example: Find the equation of a line that passes through ( 1, 2) and (5, 7). ü Things to remember: Slope formula, point-slope form, slopeintercept

More information

Calculus. Central role in much of modern science Physics, especially kinematics and electrodynamics Economics, engineering, medicine, chemistry, etc.

Calculus. 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 information

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:

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: 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 information

5.4 - Quadratic Functions

5.4 - Quadratic Functions Fry TAMU Spring 2017 Math 150 Notes Section 5.4 Page! 92 5.4 - Quadratic Functions Definition: A function is one that can be written in the form f (x) = where a, b, and c are real numbers and a 0. (What

More information

Calculus I. 1. Limits and Continuity

Calculus 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 information

Section 2.5. Evaluating Limits Algebraically

Section 2.5. Evaluating Limits Algebraically Section 2.5 Evaluating Limits Algebraically (1) Determinate and Indeterminate Forms (2) Limit Calculation Techniques (A) Direct Substitution (B) Simplification (C) Conjugation (D) The Squeeze Theorem (3)

More information

Chapter 1 Limits and Their Properties

Chapter 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 information

Section 4.3 Concavity and Curve Sketching 1.5 Lectures. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I

Section 4.3 Concavity and Curve Sketching 1.5 Lectures. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I Section 4.3 Concavity and Curve Sketching 1.5 Lectures College of Science MATHS 101: Calculus I (University of Bahrain) Concavity 1 / 29 Concavity Increasing Function has three cases (University of Bahrain)

More information

Albertson AP Calculus AB AP CALCULUS AB SUMMER PACKET DUE DATE: The beginning of class on the last class day of the first week of school.

Albertson AP Calculus AB AP CALCULUS AB SUMMER PACKET DUE DATE: The beginning of class on the last class day of the first week of school. Albertson AP Calculus AB Name AP CALCULUS AB SUMMER PACKET 2015 DUE DATE: The beginning of class on the last class day of the first week of school. This assignment is to be done at you leisure during the

More information

Calculus : Summer Study Guide Mr. Kevin Braun Bishop Dunne Catholic School. Calculus Summer Math Study Guide

Calculus : Summer Study Guide Mr. Kevin Braun Bishop Dunne Catholic School. Calculus Summer Math Study Guide 1 Calculus 2018-2019: Summer Study Guide Mr. Kevin Braun (kbraun@bdcs.org) Bishop Dunne Catholic School Name: Calculus Summer Math Study Guide After you have practiced the skills on Khan Academy (list

More information

Mission 1 Simplify and Multiply Rational Expressions

Mission 1 Simplify and Multiply Rational Expressions Algebra Honors Unit 6 Rational Functions Name Quest Review Questions Mission 1 Simplify and Multiply Rational Expressions 1) Compare the two functions represented below. Determine which of the following

More information

Induction, sequences, limits and continuity

Induction, 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 information

Chapter 4: More Applications of Differentiation

Chapter 4: More Applications of Differentiation Chapter 4: More Applications of Differentiation Autumn 2017 Department of Mathematics Hong Kong Baptist University 1 / 68 In the fall of 1972, President Nixon announced that, the rate of increase of inflation

More information

Section 0.2 & 0.3 Worksheet. Types of Functions

Section 0.2 & 0.3 Worksheet. Types of Functions MATH 1142 NAME Section 0.2 & 0.3 Worksheet Types of Functions Now that we have discussed what functions are and some of their characteristics, we will explore different types of functions. Section 0.2

More information

Pre-calculus 12 Curriculum Outcomes Framework (110 hours)

Pre-calculus 12 Curriculum Outcomes Framework (110 hours) Curriculum Outcomes Framework (110 hours) Trigonometry (T) (35 40 hours) General Curriculum Outcome: Students will be expected to develop trigonometric reasoning. T01 Students will be expected to T01.01

More information

MAXIMA AND MINIMA CHAPTER 7.1 INTRODUCTION 7.2 CONCEPT OF LOCAL MAXIMA AND LOCAL MINIMA

MAXIMA AND MINIMA CHAPTER 7.1 INTRODUCTION 7.2 CONCEPT OF LOCAL MAXIMA AND LOCAL MINIMA CHAPTER 7 MAXIMA AND MINIMA 7.1 INTRODUCTION The notion of optimizing functions is one of the most important application of calculus used in almost every sphere of life including geometry, business, trade,

More information

MATH 1040 Objectives List

MATH 1040 Objectives List MATH 1040 Objectives List Textbook: Calculus, Early Transcendentals, 7th edition, James Stewart Students should expect test questions that require synthesis of these objectives. Unit 1 WebAssign problems

More information

Pre-Calculus: Functions and Their Properties (Solving equations algebraically and graphically, matching graphs, tables, and equations, and

Pre-Calculus: Functions and Their Properties (Solving equations algebraically and graphically, matching graphs, tables, and equations, and Pre-Calculus: 1.1 1.2 Functions and Their Properties (Solving equations algebraically and graphically, matching graphs, tables, and equations, and finding the domain, range, VA, HA, etc.). Name: Date:

More information

1. Use the properties of exponents to simplify the following expression, writing your answer with only positive exponents.

1. 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 information

REQUIRED MATHEMATICAL SKILLS FOR ENTERING CADETS

REQUIRED MATHEMATICAL SKILLS FOR ENTERING CADETS REQUIRED MATHEMATICAL SKILLS FOR ENTERING CADETS The Department of Applied Mathematics administers a Math Placement test to assess fundamental skills in mathematics that are necessary to begin the study

More information

MATH 1902: Mathematics for the Physical Sciences I

MATH 1902: Mathematics for the Physical Sciences I MATH 1902: Mathematics for the Physical Sciences I Dr Dana Mackey School of Mathematical Sciences Room A305 A Email: Dana.Mackey@dit.ie Dana Mackey (DIT) MATH 1902 1 / 46 Module content/assessment Functions

More information

Math 117: Calculus & Functions II

Math 117: Calculus & Functions II Drexel University Department of Mathematics Jason Aran Math 117: Calculus & Functions II Contents 0 Calculus Review from Math 116 4 0.1 Limits............................................... 4 0.1.1 Defining

More information

MATH 1910 Limits Numerically and Graphically Introduction to Limits does not exist DNE DOES does not Finding Limits Numerically

MATH 1910 Limits Numerically and Graphically Introduction to Limits does not exist DNE DOES does not Finding Limits Numerically MATH 90 - Limits Numerically and Graphically Introduction to Limits The concept of a limit is our doorway to calculus. This lecture will explain what the limit of a function is and how we can find such

More information

Chapter 1. Functions 1.1. Functions and Their Graphs

Chapter 1. Functions 1.1. Functions and Their Graphs 1.1 Functions and Their Graphs 1 Chapter 1. Functions 1.1. Functions and Their Graphs Note. We start by assuming that you are familiar with the idea of a set and the set theoretic symbol ( an element of

More information

function independent dependent domain range graph of the function The Vertical Line Test

function independent dependent domain range graph of the function The Vertical Line Test Functions A quantity y is a function of another quantity x if there is some rule (an algebraic equation, a graph, a table, or as an English description) by which a unique value is assigned to y by a corresponding

More information

Calculus I. George Voutsadakis 1. LSSU Math 151. Lake Superior State University. 1 Mathematics and Computer Science

Calculus 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 information

Chapter 5B - Rational Functions

Chapter 5B - Rational Functions Fry Texas A&M University Math 150 Chapter 5B Fall 2015 143 Chapter 5B - Rational Functions Definition: A rational function is The domain of a rational function is all real numbers, except those values

More information