Lecture 7: Indeterminate forms; L Hôpitals rule; Relative rates of growth. If we try to simply substitute x = 1 into the expression, we get
|
|
- Phillip Goodman
- 5 years ago
- Views:
Transcription
1 Lecture 7: Indeterminate forms; L Hôpitals rule; Relative rates of growth 1. Indeterminate Forms. Eample 1: Consider the it If we try to simply substitute = 1 into the epression, we get. This is a so-called indeterminate form. A it of that form could be anything. After all, every derivative f ) = h f + h) f) h is of that form that is, a it of this form could be any possible number). We have seen that in many cases algebra can be used to simplify the epressions to obtain a non-indeterminate form whose it we can evaluate. In this eample = 1) + 1) = + 1 = We can do the division by 1 in this eample since we are only considering the it as approaches 1, but not what happens at = 1. Therefore, since 1, we can divide by 1. There are many possible indeterminate forms. They are,, ), ), 1,,. The actual value of these its depends on how fast the respective numerators, denominators, basis, eponents and factors approach 1,, or. The following eamples illustrate them: Eample : sin Eample 3: 1 ln 1 In Calculus I, we went through a rather complicated geometrical argument to show that this it equals 1.) e Eample 4: 1 cos ) Eample 5: 1 Eample 6: e Notice that you can rewrite this one in the form ). 1
2 Eample 7: ln + Eample 8: + 1 1) ) We already know how to handle these its. Hint: multiply top and bottom by the conjugate. Eample 9: 1 + r n n )n 1 Trick: When the variable is in the base and in the eponent, begin by taking the it of the logarithm. Eample 1: Same here. Eample 11: 1 Eample 1: + 1 ) ) Indeterminate?. L Hôpital s Rule. L Hôpital s rule is a tool to handle the case ) and ). Theorem: L Hôpital s rule) If f) and g) then If f) and g) then f) a g) = a f ) g ) f) a g) = a f ) g ) A proof of this theorem is outlined in the book. However, it is easy to see where this comes from using linear approimations, which we will learnt about last semester: If f is continuous and continuously differentiable at = a, then one can approimate f) near = a by its linearization: f) fa) + f a) a) This approimation gets better and better as a. Same for g). Thus in the it we can replace f and g by their approimations: f) a g) = fa) + f a) a) a ga) + g a) a) If fa) = ga) = this epression simplifies to f) a g) = f a) a g a) as long as g a). The fact that we can use this rule even if g a) = requires a more careful proof. While this is only an outline of a proof for a special case it gives good intuition. It shows that L Hopitals rule is just an application of linear approimations. In class we ll apply L Hôpital s rule to solve the eamples above, where applicable.
3 3. Relative rates of growth. It is often important to determine how fast functions f) grow for very large values of, and to compare the growth rate of various functions. E 1: Any quadratic function grows faster than any linear function eventually. That is, even though for some values of the quadratic function may have smaller magnitude and grow slower than the linear function, the quadratic growth will dominate the linear one if is large enough. Compare and, for eample, as in Figure 1.) Figure Figure 1 E : We know that while the values of one linear function may be larger than those of another, any two linear functions eventually grow slower than any quadratic function. Figure compares, 1 and, as an eample.) E 3: You may have noticed that eponential functions like and e seem to grow more rapidly as gets large than polynomials and rational functions. Figure 3 compares e and with. You can see the eponentials outgrowing as increases. In fact, as, the functions and e grow faster than any power of, even 1,,. To get a feeling for how rapidly the values of y = e grow with increasing, think of graphing the function on a large blackboard, with the aes scaled in centimeters. At = 1 cm, the graph is e 1 3 cm above the - ais. At = 6 cm, the graph is e 6 43 cm 4m high probably higher than the ceiling). At = 1 cm, the graph is e 1,6 cm m high, higher than most buildings. At = 4 cm, the graph is more than halfway to the moon, and at = 43 cm, the graph is high enough to reach past the sun s closest stellar neighbor, the red dwarf star Proima Centauri. Yet with = 43 cm from the origin, the graph is still less than feet to the right of the y-ais. Figure 3 e Here we want to compare, in particular, the growth rates of the new functions we have learned about logarithms, eponentials), as well some of those we already know about polynomials, square roots, other powers). The following definition precisely states what it means for one function to grow faster than, grow slower than, or grow at the same rate as another one, eventually, that is, if is large enough. For present purposes, we restrict our attention to functions whose values eventually become and remain positive as. 3
4 Definition: Let f) and g) be positive for sufficiently large. f) 1. f) grows faster than g) as if g) =. f). f) grows slower than g) as if g) =. f) 3. f) and g) grow at the same rate as if g) = L, where L is some finite number. This definition implies that if f grows faster than g, then f will eventually be much larger than g. Similarly, if f grows slower than g, then f will eventually be much smaller than g. In order to compute the its involved we often use L Hôpital s rule. The notion of relative rates of growth will be very useful to us later on in this semester, when we talk about integrals over infinite domains and when we talk about series. In the problems below you will establish, among others, that: Any two polynomial functions of equal degree grow at the same rate. m grows slower than n if m < n. a grows slower than b if a < b. Logarithms grow slower than polynomials which grow slower than growing) eponentials. Eercises: 1. a) Show that grows faster than as. b) Show that and 1 grow at the same rate as.. a) Show that e grows faster than as. b) Show that e grows faster than as. 3. a) Show that ln grows slower than as. b) Show that ln grows at the same rate as ln ) as. 4. Show that any quadratic function f) = a + a 1 + a grows faster than any linear function g) = b 1 + b, where a,b 1 >, as. 4
5 5. Show that any two linear functions, f) = a + a 1 and g) = b + b 1, a 1,b 1 >, grow at the same rate namely linearly) as. Similarly, one can show that any two polynomial functions of equal degree grow at the same rate. 6. a) Show that log a ) and log b ) grow at the same rate as. b) Show that a grows slower than b as, if a < b. For eample, grows slower than e. c) Show that a grows slower than b if a < b. For eample, grows slower than which grows slower than. 7. Show that ln) grows at the same rate as ln ). 5. Let n be a positive integer. a) Show that ln) grows slower than n as. a) Show that n grows slower than e as. 8. Which of the following functions grow faster than? Which grow at the same rate as? Which grow slower? a) + 4 h) b) i) 1 c) + j) 1.1) d) 3 k).9) e) 4 3 l) log 1 f) ln m) g) 3 e 9. Which of the following functions grow faster than ln? Which grow at the same rate as ln? Which grow slower? a) log 3 g) 5ln b) ln h) e c) i) log ) d) 1/ j) log 1 1) e) ln) k) lnln ) f) l) 1ln + 1. Order the following functions from slowest growing to fastest growing, as. Group functions that grow at the same rate together. a) e h) b) i) c) ln) j) ln ) d) e / k) ln ) e) l) e f).9) m) g) 1/ n) 1/ 5
EXPONENT REVIEW!!! Concept Byte (Review): Properties of Exponents. Property of Exponents: Product of Powers. x m x n = x m + n
Algebra B: Chapter 6 Notes 1 EXPONENT REVIEW!!! Concept Byte (Review): Properties of Eponents Recall from Algebra 1, the Properties (Rules) of Eponents. Property of Eponents: Product of Powers m n = m
More information1 Rational Exponents and Radicals
Introductory Algebra Page 1 of 11 1 Rational Eponents and Radicals 1.1 Rules of Eponents The rules for eponents are the same as what you saw earlier. Memorize these rules if you haven t already done so.
More information3.7 Indeterminate Forms - l Hôpital s Rule
3.7. INDETERMINATE FORMS - L HÔPITAL S RULE 4 3.7 Indeterminate Forms - l Hôpital s Rule 3.7. Introduction An indeterminate form is a form for which the answer is not predictable. From the chapter on lits,
More informationDefinition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.
8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections.6 and.) 8. Equivalent Inequalities Definition 8. Two inequalities are equivalent
More informationMATH 1010E University Mathematics Lecture Notes (week 8) Martin Li
MATH 1010E University Mathematics Lecture Notes (week 8) Martin Li 1 L Hospital s Rule Another useful application of mean value theorems is L Hospital s Rule. It helps us to evaluate its of indeterminate
More informationMATH 250 TOPIC 11 LIMITS. A. Basic Idea of a Limit and Limit Laws. Answers to Exercises and Problems
Math 5 T-Limits Page MATH 5 TOPIC LIMITS A. Basic Idea of a Limit and Limit Laws B. Limits of the form,, C. Limits as or as D. Summary for Evaluating Limits Answers to Eercises and Problems Math 5 T-Limits
More informationExample 1: What do you know about the graph of the function
Section 1.5 Analyzing of Functions In this section, we ll look briefly at four types of functions: polynomial functions, rational functions, eponential functions and logarithmic functions. Eample 1: What
More informationSection 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 informationAPPLICATIONS OF DIFFERENTIATION
4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION 4.4 Indeterminate Forms and L Hospital s Rule In this section, we will learn: How to evaluate functions whose values cannot be found at
More informationAP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals
AP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals 8. Basic Integration Rules In this section we will review various integration strategies. Strategies: I. Separate
More informationIndeterminate Forms and L Hospital s Rule
APPLICATIONS OF DIFFERENTIATION Indeterminate Forms and L Hospital s Rule In this section, we will learn: How to evaluate functions whose values cannot be found at certain points. INDETERMINATE FORM TYPE
More informationPerforming well in calculus is impossible without a solid algebra foundation. Many calculus
Chapter Algebra Review Performing well in calculus is impossible without a solid algebra foundation. Many calculus problems that you encounter involve a calculus concept but then require many, many steps
More informationAPPLICATIONS OF DIFFERENTIATION
4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION 4.4 Indeterminate Forms and L Hospital s Rule In this section, we will learn: How to evaluate functions whose values cannot be found at
More informationExponential Functions, Logarithms, and e
Chapter 3 Starry Night, painted by Vincent Van Gogh in 1889 The brightness of a star as seen from Earth is measured using a logarithmic scale Eponential Functions, Logarithms, and e This chapter focuses
More informationAlgebra Concepts Equation Solving Flow Chart Page 1 of 6. How Do I Solve This Equation?
Algebra Concepts Equation Solving Flow Chart Page of 6 How Do I Solve This Equation? First, simplify both sides of the equation as much as possible by: combining like terms, removing parentheses using
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 informationAlgebra. Robert Taggart
Algebra Robert Taggart Table of Contents To the Student.............................................. v Unit 1: Algebra Basics Lesson 1: Negative and Positive Numbers....................... Lesson 2: Operations
More informationL Hopital s Rule. We will use our knowledge of derivatives in order to evaluate limits that produce indeterminate forms.
L Hopital s Rule We will use our knowledge of derivatives in order to evaluate its that produce indeterminate forms. Main Idea x c f x g x If, when taking the it as x c, you get an INDETERMINATE FORM..
More informationNONLINEAR FUNCTIONS A. Absolute Value Exercises: 2. We need to scale the graph of Qx ( )
NONLINEAR FUNCTIONS A. Absolute Value Eercises:. We need to scale the graph of Q ( ) f ( ) =. The graph is given below. = by the factor of to get the graph of 9 - - - - -. We need to scale the graph of
More informationExponential and Logarithmic Functions
Lesson 6 Eponential and Logarithmic Fu tions Lesson 6 Eponential and Logarithmic Functions Eponential functions are of the form y = a where a is a constant greater than zero and not equal to one and is
More informationAP Calculus AB Summer Assignment
AP Calculus AB Summer Assignment Name: When you come back to school, it is my epectation that you will have this packet completed. You will be way behind at the beginning of the year if you haven t attempted
More informationC. Finding roots of trinomials: 1st Example: x 2 5x = 14 x 2 5x 14 = 0 (x 7)(x + 2) = 0 Answer: x = 7 or x = -2
AP Calculus Students: Welcome to AP Calculus. Class begins in approimately - months. In this packet, you will find numerous topics that were covered in your Algebra and Pre-Calculus courses. These are
More information3.5 Continuity of a Function One Sided Continuity Intermediate Value Theorem... 23
Chapter 3 Limit and Continuity Contents 3. Definition of Limit 3 3.2 Basic Limit Theorems 8 3.3 One sided Limit 4 3.4 Infinite Limit, Limit at infinity and Asymptotes 5 3.4. Infinite Limit and Vertical
More informationSection 2.6 Limits at infinity and infinite limits 2 Lectures. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I
Section 2.6 Limits at infinity and infinite its 2 Lectures College of Science MATHS 0: Calculus I (University of Bahrain) Infinite Limits / 29 Finite its as ±. 2 Horizontal Asympotes. 3 Infinite its. 4
More informationPre-Algebra 8 Notes Exponents and Scientific Notation
Pre-Algebra 8 Notes Eponents and Scientific Notation Rules of Eponents CCSS 8.EE.A.: Know and apply the properties of integer eponents to generate equivalent numerical epressions. Review with students
More informationThe Harvard Calculus Program in a Computer Classroom
The Harvard Calculus Program in a Computer Classroom Carmen Q. Artino Department of Mathematics The College of Saint Rose Albany, NY 103 Office: (518) 454-516 Fa: (518) 458-5446 e-mail: artinoc@rosnet.strose.edu
More informationCourse. Print and use this sheet in conjunction with MathinSite s Maclaurin Series applet and worksheet.
Maclaurin Series Learning Outcomes After reading this theory sheet, you should recognise the difference between a function and its polynomial epansion (if it eists!) understand what is meant by a series
More informationCore Connections Algebra 2 Checkpoint Materials
Core Connections Algebra 2 Note to Students (and their Teachers) Students master different skills at different speeds. No two students learn eactly the same way at the same time. At some point you will
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 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 informationAlgebra Final Exam Review Packet
Algebra 1 00 Final Eam Review Packet UNIT 1 EXPONENTS / RADICALS Eponents Degree of a monomial: Add the degrees of all the in the monomial together. o Eample - Find the degree of 5 7 yz Degree of a polynomial:
More informationCALCULUS BASIC SUMMER REVIEW
NAME CALCULUS BASIC SUMMER REVIEW Slope of a non vertical line: rise y y y m run Point Slope Equation: y y m( ) The slope is m and a point on your line is, ). ( y Slope-Intercept Equation: y m b slope=
More informationIn this note we will evaluate the limits of some indeterminate forms using L Hôpital s Rule. Indeterminate Forms and 0 0. f(x)
L Hôpital s Rule In this note we will evaluate the its of some indeterminate forms using L Hôpital s Rule. Indeterminate Forms and 0 0 f() Suppose a f() = 0 and a g() = 0. Then a g() the indeterminate
More informationOctober 27, 2018 MAT186 Week 3 Justin Ko. We use the following notation to describe the limiting behavior of functions.
October 27, 208 MAT86 Week 3 Justin Ko Limits. Intuitive Definitions of Limits We use the following notation to describe the iting behavior of functions.. (Limit of a Function A it is written as f( = L
More informationChapter 5: Limits, Continuity, and Differentiability
Chapter 5: Limits, Continuity, and Differentiability 63 Chapter 5 Overview: Limits, Continuity and Differentiability Derivatives and Integrals are the core practical aspects of Calculus. They were the
More informationWith topics from Algebra and Pre-Calculus to
With topics from Algebra and Pre-Calculus to get you ready to the AP! (Key contains solved problems) Note: The purpose of this packet is to give you a review of basic skills. You are asked not to use the
More information9.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED LESSON
CONDENSED LESSON 9.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations solve
More informationMidterm 1 Solutions. Monday, 10/24/2011
Midterm Solutions Monday, 0/24/20. (0 points) Consider the function y = f() = e + 2e. (a) (2 points) What is the domain of f? Epress your answer using interval notation. Solution: We must eclude the possibility
More information3.8 Limits At Infinity
3.8. LIMITS AT INFINITY 53 Figure 3.5: Partial graph of f = /. We see here that f 0 as and as. 3.8 Limits At Infinity The its we introduce here differ from previous its in that here we are interested in
More informationWest Essex Regional School District. AP Calculus AB. Summer Packet
West Esse Regional School District AP Calculus AB Summer Packet 05-06 Calculus AB Calculus AB covers the equivalent of a one semester college calculus course. Our focus will be on differential and integral
More informationAim: How do we prepare for AP Problems on limits, continuity and differentiability? f (x)
Name AP Calculus Date Supplemental Review 1 Aim: How do we prepare for AP Problems on limits, continuity and differentiability? Do Now: Use the graph of f(x) to evaluate each of the following: 1. lim x
More informationAlgebra 2-2nd Semester Exam Review 11
Algebra 2-2nd Semester Eam Review 11 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Determine which binomial is a factor of. a. 14 b. + 4 c. 4 d. + 8
More informationarb where a A, b B and we say a is related to b. Howdowewritea is not related to b? 2Rw 1Ro A B = {(a, b) a A, b B}
Functions Functions play an important role in mathematics as well as computer science. A function is a special type of relation. So what s a relation? A relation, R, from set A to set B is defined as arb
More informationL Hopital s Rule. We will use our knowledge of derivatives in order to evaluate limits that produce indeterminate forms.
L Hopital s Rule We will use our knowledge of derivatives in order to evaluate its that produce indeterminate forms. Indeterminate Limits Main Idea x c f x g x If, when taking the it as x c, you get an
More informationPre-Calculus Summer Packet
Pre-Calculus Summer Packet Name ALLEN PARK HIGH SCHOOL Summer Assessment Pre-Calculus Summer Packet For Students Entering Pre-Calculus Summer 05 This summer packet is intended to be completed by the FIRST
More informationAP Calculus AB Summer Assignment
AP Calculus AB Summer Assignment Name: When you come back to school, you will be epected to have attempted every problem. These skills are all different tools that you will pull out of your toolbo this
More informationSolution Sheet 1.4 Questions 26-31
Solution Sheet 1.4 Questions 26-31 26. Using the Limit Rules evaluate i) ii) iii) 3 2 +4+1 0 2 +4+3, 3 2 +4+1 2 +4+3, 3 2 +4+1 1 2 +4+3. Note When using a Limit Rule you must write down which Rule you
More informationA. Incorrect! Apply the rational root test to determine if any rational roots exist.
College Algebra - Problem Drill 13: Zeros of Polynomial Functions No. 1 of 10 1. Determine which statement is true given f() = 3 + 4. A. f() is irreducible. B. f() has no real roots. C. There is a root
More informationSection 4.5 Graphs of Logarithmic Functions
6 Chapter 4 Section 4. Graphs of Logarithmic Functions Recall that the eponential function f ( ) would produce this table of values -3 - -1 0 1 3 f() 1/8 ¼ ½ 1 4 8 Since the arithmic function is an inverse
More informationa = B. Examples: 1. Simplify the following expressions using the multiplication rule
Section. Monomials Objectives:. Multiply and divide monomials.. Simplify epressions involving powers of monomials.. Use epressions in scientific notation. I. Negative Eponents and Eponents of Zero A. Rules.
More informationCalculus I Practice Test Problems for Chapter 2 Page 1 of 7
Calculus I Practice Test Problems for Chapter Page of 7 This is a set of practice test problems for Chapter This is in no way an inclusive set of problems there can be other types of problems on the actual
More informationMATH 116, LECTURES 10 & 11: Limits
MATH 6, LECTURES 0 & : Limits Limits In application, we often deal with quantities which are close to other quantities but which cannot be defined eactly. Consider the problem of how a car s speedometer
More information4.5 Rational functions.
4.5 Rational functions. We have studied graphs of polynomials and we understand the graphical significance of the zeros of the polynomial and their multiplicities. Now we are ready to etend these eplorations
More informationDIFFERENTIATION RULES
3 DIFFERENTIATION RULES DIFFERENTIATION RULES 3.6 Derivatives of Logarithmic Functions In this section, we: use implicit differentiation to find the derivatives of the logarithmic functions and, in particular,
More informationSection 6.2 Long Division of Polynomials
Section 6. Long Division of Polynomials INTRODUCTION In Section 6.1 we learned to simplify a rational epression by factoring. For eample, + 3 10 = ( + 5)( ) ( ) = ( + 5) 1 = + 5. However, if we try to
More informationBasic methods to solve equations
Roberto s Notes on Prerequisites for Calculus Chapter 1: Algebra Section 1 Basic methods to solve equations What you need to know already: How to factor an algebraic epression. What you can learn here:
More information6.2 Properties of Logarithms
6. Properties of Logarithms 437 6. Properties of Logarithms In Section 6.1, we introduced the logarithmic functions as inverses of eponential functions and discussed a few of their functional properties
More informationUNIT 4A MATHEMATICAL MODELING OF INVERSE, LOGARITHMIC, AND TRIGONOMETRIC FUNCTIONS Lesson 2: Modeling Logarithmic Functions
Lesson : Modeling Logarithmic Functions Lesson A..1: Logarithmic Functions as Inverses Utah Core State Standards F BF. Warm-Up A..1 Debrief In the metric system, sound intensity is measured in watts per
More informationConceptual Explanations: Radicals
Conceptual Eplanations: Radicals The concept of a radical (or root) is a familiar one, and was reviewed in the conceptual eplanation of logarithms in the previous chapter. In this chapter, we are going
More informationR3.6 Solving Linear Inequalities. 3) Solve: 2(x 4) - 3 > 3x ) Solve: 3(x 2) > 7-4x. R8.7 Rational Exponents
Level D Review Packet - MMT This packet briefly reviews the topics covered on the Level D Math Skills Assessment. If you need additional study resources and/or assistance with any of the topics below,
More informationSection 1.2 A Catalog of Essential Functions
Chapter 1 Section Page 1 of 6 Section 1 A Catalog of Essential Functions Linear Models: All linear equations have the form rise change in horizontal The letter m is the of the line, or It can be positive,
More informationUnit 6: 10 3x 2. Semester 2 Final Review Name: Date: Advanced Algebra
Semester Final Review Name: Date: Advanced Algebra Unit 6: # : Find the inverse of: 0 ) f ( ) = ) f ( ) Finding Inverses, Graphing Radical Functions, Simplifying Radical Epressions, & Solving Radical Equations
More informationACCUPLACER MATH 0311 OR MATH 0120
The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 0 OR MATH 00 http://www.academics.utep.edu/tlc MATH 0 OR MATH 00 Page Factoring Factoring Eercises 8 Factoring Answer to Eercises
More informationCore Connections Algebra 2 Checkpoint Materials
Core Connections Algebra 2 Note to Students (and their Teachers) Students master different skills at different speeds. No two students learn eactly the same way at the same time. At some point you will
More informationAlgebra Review C H A P T E R. To solve an algebraic equation with one variable, find the value of the unknown variable.
C H A P T E R 6 Algebra Review This chapter reviews key skills and concepts of algebra that you need to know for the SAT. Throughout the chapter are sample questions in the style of SAT questions. Each
More informationHorizontal asymptotes
Roberto s Notes on Differential Calculus Chapter : Limits and continuity Section 5 Limits at infinity and Horizontal asymptotes What you need to know already: The concept, notation and terminology of its.
More informationSection 1.2 A Catalog of Essential Functions
Page 1 of 6 Section 1. A Catalog of Essential Functions Linear Models: All linear equations have the form y = m + b. rise change in horizontal The letter m is the slope of the line, or. It can be positive,
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 informationExtra Fun: The Indeterminate Forms 1, 0, and 0 0
math 30, day 38 its: l hôpital s rule, part 7 Etra Fun: The Indeterminate Forms, 0, and 0 0 Some of the most interesting its in elementary calculus have the indeterminate forms,0 0, or 0 0. All of these
More informationMATH 108 REVIEW TOPIC 6 Radicals
Math 08 T6-Radicals Page MATH 08 REVIEW TOPIC 6 Radicals I. Computations with Radicals II. III. IV. Radicals Containing Variables Rationalizing Radicals and Rational Eponents V. Logarithms Answers to Eercises
More informationMath M111: Lecture Notes For Chapter 10
Math M: Lecture Notes For Chapter 0 Sections 0.: Inverse Function Inverse function (interchange and y): Find the equation of the inverses for: y = + 5 ; y = + 4 3 Function (from section 3.5): (Vertical
More informationLecture 5: Finding limits analytically Simple indeterminate forms
Lecture 5: Finding its analytically Simple indeterminate forms Objectives: (5.) Use algebraic techniques to resolve 0/0 indeterminate forms. (5.) Use the squeeze theorem to evaluate its. (5.3) Use trigonometric
More informationIn last semester, we have seen some examples about it (See Tutorial Note #13). Try to have a look on that. Here we try to show more technique.
MATH202 Introduction to Analysis (2007 Fall and 2008 Spring) Tutorial Note #4 Part I: Cauchy Sequence Definition (Cauchy Sequence): A sequence of real number { n } is Cauchy if and only if for any ε >
More informationCalculus 1 (AP, Honors, Academic) Summer Assignment 2018
Calculus (AP, Honors, Academic) Summer Assignment 08 The summer assignments for Calculus will reinforce some necessary Algebra and Precalculus skills. In order to be successful in Calculus, you must have
More informationFunction Gallery: Some Basic Functions and Their Properties
Function Gallery: Some Basic Functions and Their Properties Linear Equation y = m+b Linear Equation y = -m + b This Eample: y = 3 + 3 This Eample: y = - + 0 Domain (-, ) Domain (-, ) Range (-, ) Range
More informationRecall from our discussion of continuity in lecture a function is continuous at a point x = a if and only if
Computational Aspects of its. Keeping te simple simple. Recall by elementary functions we mean :Polynomials (including linear and quadratic equations) Eponentials Logaritms Trig Functions Rational Functions
More informationAdministrivia. Matrinomials Lectures 1+2 O Outline 1/15/2018
Administrivia Syllabus and other course information posted on the internet: links on blackboard & at www.dankalman.net. Check assignment sheet for reading assignments and eercises. Be sure to read about
More information4.8 Partial Fraction Decomposition
8 CHAPTER 4. INTEGRALS 4.8 Partial Fraction Decomposition 4.8. Need to Know The following material is assumed to be known for this section. If this is not the case, you will need to review it.. When are
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 informationAssignment 16 Assigned Weds Oct 11
Assignment 6 Assigned Weds Oct Section 8, Problem 3 a, a 3, a 3 5, a 4 7 Section 8, Problem 4 a, a 3, a 3, a 4 3 Section 8, Problem 9 a, a, a 3, a 4 4, a 5 8, a 6 6, a 7 3, a 8 64, a 9 8, a 0 56 Section
More informationA Quick Algebra Review
1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals
More informationName. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
REVIEW Eam #3 : 3.2-3.6, 4.1-4.5, 5.1 Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Use the Leading Coefficient Test to determine the end behavior
More informationSection 4.3: Quadratic Formula
Objective: Solve quadratic equations using the quadratic formula. In this section we will develop a formula to solve any quadratic equation ab c 0 where a b and c are real numbers and a 0. Solve for this
More informationA BRIEF REVIEW OF ALGEBRA AND TRIGONOMETRY
A BRIEF REVIEW OF ALGEBRA AND TRIGONOMETR Some Key Concepts:. The slope and the equation of a straight line. Functions and functional notation. The average rate of change of a function and the DIFFERENCE-
More information8-1 Exploring Exponential Models
8- Eploring Eponential Models Eponential Function A function with the general form, where is a real number, a 0, b > 0 and b. Eample: y = 4() Growth Factor When b >, b is the growth factor Eample: y =
More information3.2 Logarithmic Functions and Their Graphs
96 Chapter 3 Eponential and Logarithmic Functions 3.2 Logarithmic Functions and Their Graphs Logarithmic Functions In Section.6, you studied the concept of an inverse function. There, you learned that
More informationSolutions to Math 41 Final Exam December 9, 2013
Solutions to Math 4 Final Eam December 9,. points In each part below, use the method of your choice, but show the steps in your computations. a Find f if: f = arctane csc 5 + log 5 points Using the Chain
More informationLog1 Contest Round 2 Theta Logarithms & Exponents. 4 points each
5 Log Contest Round Theta Logarithms & Eponents Name: points each Simplify: log log65 log6 log6log9 log5 Evaluate: log Find the sum:... A square has a diagonal whose length is feet, enclosed by the square.
More informationThis problem set is a good representation of some of the key skills you should have when entering this course.
Math 4 Review of Previous Material: This problem set is a good representation of some of the key skills you should have when entering this course. Based on the course work leading up to Math 4, you should
More informationLimits: How to approach them?
Limits: How to approach them? The purpose of this guide is to show you the many ways to solve it problems. These depend on many factors. The best way to do this is by working out a few eamples. In particular,
More informationHorizontal asymptotes
Roberto s Notes on Differential Calculus Chapter 1: Limits and continuity Section 5 Limits at infinity and Horizontal asymptotes What you need to know already: The concept, notation and terminology of
More informationUNIT 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 information2017 AP Calculus AB Summer Assignment
07 AP Calculus AB Summer Assignment Mrs. Peck ( kapeck@spotsylvania.k.va.us) This assignment is designed to help prepare you to start Calculus on day and be successful. I recommend that you take off the
More informationMA 114 Worksheet #01: Integration by parts
Fall 8 MA 4 Worksheet Thursday, 3 August 8 MA 4 Worksheet #: Integration by parts. For each of the following integrals, determine if it is best evaluated by integration by parts or by substitution. If
More informationAP CALCULUS AB - Name: Summer Work requirement due on the first day of class
AP CALCULUS AB - Name: Summer Work For students to successfully complete the objectives of the AP Calculus curriculum, the student must demonstrate a high level of independence, capability, dedication,
More informationOutline. 1 Integration by Substitution: The Technique. 2 Integration by Substitution: Worked Examples. 3 Integration by Parts: The Technique
MS2: IT Mathematics Integration Two Techniques of Integration John Carroll School of Mathematical Sciences Dublin City University Integration by Substitution: The Technique Integration by Substitution:
More informationSolutions to Problem Sheet for Week 6
THE UNIVERSITY OF SYDNEY SCHOOL OF MATHEMATICS AND STATISTICS Solutions to Problem Sheet for Week 6 MATH90: Differential Calculus (Advanced) Semester, 07 Web Page: sydney.edu.au/science/maths/u/ug/jm/math90/
More informationdegree -6x 3 + 5x 3 Coefficients:
Date P3 Polynomials and Factoring leading coefficient degree -6 3 + 5 3 constant term coefficients Degree: the largest sum of eponents in a term Polynomial: a n n + a n-1 n-1 + + a 1 + a 0 where a n 0
More informationCalculus - Chapter 2 Solutions
Calculus - Chapter Solutions. a. See graph at right. b. The velocity is decreasing over the entire interval. It is changing fastest at the beginning and slowest at the end. c. A = (95 + 85)(5) = 450 feet
More informationO.K. But what if the chicken didn t have access to a teleporter.
The intermediate value theorem, and performing algebra on its. This is a dual topic lecture. : The Intermediate value theorem First we should remember what it means to be a continuous function: A function
More information