Limits: How to approach them?
|
|
- David Fox
- 5 years ago
- Views:
Transcription
1 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, we will be looking at its involving piece wise functions and indefinite forms such as 0,,, 0 0 Eample 1 Given the piecewise function f() = , 2 ln( 1) + 1, > 2 Find the following : (a) 2 + f(), (b) 2 f(), (c) 2 f() Solution for (a): We have that 2 +, which means is getting close to 2 from the right. We are looking at numbers that are greater than 2 such as 2.01, etc. So we have to grab the part of the function that has these numbers in its domain. That would be the part where > 2 So we will now compute the it: 2 f() = ln( 1) + 1 = ln(2 1) + 1 = ln(1) + 1 = = Solution for (b): We have that 2, which means is getting close to 2 from the left. We are looking at numbers that are less than 2 such as 1.99, 1.999, etc. We have to grab the part of the function that has these numbers in its domain. That would be the part where < 2 Computing the it: f() = = 3(2) = = 11 Solution for (c): 2 f() eists only if 2 f() = 2 + f(). But 2 + f() = 1 and 2 f() = 11. So 2 f() = does not eist Eample 2 Find Solution: If we were to plug in 2 into our function, on both the numerator and denominator, you get the form 0. This is an indeterminate form. We will need to do some more work to know whether this it 0
2 eists or not. Typically, when you have the form 0, we will use algebra to simplify our function. Often, 0 this will lead you to cancel the factors that make the epression into the form 0 0 Let s go ahead and do this for the above problem = 2 4( 2)( + 2) ( 2)( ) = 2 4( + 2) = = 4 3 The factor of 2 cancelled from both the numerator and denominator. This was causing the form 0 0 Limits at Infinity The following eamples deal with its at infinity i.e. or Eample 3 Find Solution: This it has the form which is indeterminate. Factoring is not a feasible option. So a what we will do is use the fact that ± r = 0 for any number r > 0. How will we use this fact in this problem? I am going to use algebra and divide every term on both the numerator and denominator with the highest power or highest growing term. This step will make every term look like a ± r ( ) = ( ) 3 = = = = 4 The step where we divided by the highest power in both the numerator and denominator is a very useful step when dealing with infinite its. But beware! This technique only works well with infinite its and nothing else!! Eample 4 Find
3 2 + 4 Solution: This it has the form of. The top looks like because as goes to, is positive because 2 gets larger than 4. We will approach this problem in a similar manner as we did eample 3 but with a couple differences. So what would be the highest growing power? We might want to say 2, but 2 is inside the square root. So we are really looking at just as the highest power. So let us divide by on the numerator and denominator =. So what do we do with the numerator? I can t divide everything by because of the square root. What we can do is place inside the square root. We can try to rewrite as = 2. However, in reality, 2 =. But we have, not. But there is a fi for, 0 that. We can use the definition of =. Because, we will take the part that, < 0 corresponds to. Meaning, = 2 or = = = 2 = = = = 1 Eample 5 Find e e e + e Solution: The reason I want to do this eample so we can review the growth of e or e. In terms of computing the it, nothing will change as the previous two eamples. As, we have that e 0 and e. (If you are unsure of why this is the true, I suggest looking at the graphs of e and e ). Once again, we have the indeterminate form of the larger of the two so I will divide every term by e e e e +e = e e e e e = e +e e e 2 1 = 0 1 = 1 e e will be
4 A couple things to note: We used some rules of algebra e e = e ( ) = e 2. Also, e a 0 if for all a > 0. Lastly, if the question had been asking for, then e would be the larger of two. The last two problems deal with the indeterminate forms and 0 Other Indeterminate forms. Eample 6 Find 2 Solution: If we were to take this it we would have the form of. Many mistaken this for 0. However, all that we are saying is our two terms grow infinitely large but may not be the same number. In other words, when 2 is 500, might be 400; when 2 is 1000, is 500. All we can say is the terms grow really large but we can t say they cancel each other. The way to handle the situation is to use algebra like we have with the previous problems. When we have functions that have a square root and a binomial inside of it, we use the conjugate a lot. The conjugate consists of multiplying both the numerator and denominator by the almost same eact binomial but with an opposite sign in the middle. 2 = ( 2 ) = [ 2 + ] [ 2 + ] = [ 2 ] Stopping for a moment we have the form of which is still indeterminate. However, we know how to handle the situation by dividing by the highest growing power. I will leave the reader to work out the details. The final answer will be 1 2 Eample 7 Find ( ) If we were to look at this it by merely plugging in 0 from the right, I would have the form 0 which is indefinite. We can t say this result is going to be zero. 1 grows really large as 0+. This means can approach 500, 1000, etc. At the same time,as 0 +, (3 3 ) is approaching zero. This +1 implies that (3 3 ) can approach 1 then 1, etc. If I were to multiply these results this would mean that
5 if 1 3 is 500 and (3 ) is 1 then I would have 250. If 1 3 is 1000 and (3 ) is 1 then I would have The point that I am trying to make is that even though one part goes to and the other part goes to 0, we can t say the result will be zero. The result might be a number like 250, as the eample was showing. Again, we use algebra to solve this it: I will get a common denominator for , and multiply by = ) 3( = + 1 There are other indeterminate forms like 0 0, 1, = = 3
Let y = f (t) be a function that gives the position at time t of an object moving along the y-axis. Then
Limits From last time... Let y = f (t) be a function that gives the position at time t of an object moving along the y-ais. Then Ave vel[t, t 2 ] = f (t 2) f (t ) t 2 t f (t + h) f (t) Velocity(t) =. h!0
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 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 informationReview: Limits of Functions - 10/7/16
Review: Limits of Functions - 10/7/16 1 Right and Left Hand Limits Definition 1.0.1 We write lim a f() = L to mean that the function f() approaches L as approaches a from the left. We call this the left
More 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 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 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 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 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 informationMAT 210 Test #1 Solutions, Form A
1. Where are the following functions continuous? a. ln(x 2 1) MAT 210 Test #1 Solutions, Form A Solution: The ln function is continuous when what you are taking the log of is positive. Hence, we need x
More informationOne Solution Two Solutions Three Solutions Four Solutions. Since both equations equal y we can set them equal Combine like terms Factor Solve for x
Algebra Notes Quadratic Systems Name: Block: Date: Last class we discussed linear systems. The only possibilities we had we 1 solution, no solution or infinite solutions. With quadratic systems we have
More informationLimits Involving Infinity (Horizontal and Vertical Asymptotes Revisited)
Limits Involving Infinity (Horizontal and Vertical Asymptotes Revisited) Limits as Approaches Infinity At times you ll need to know the behavior of a function or an epression as the inputs get increasingly
More informationChapter 10 Introduction to the Derivative
Chapter 0 Introduction to the Derivative The concept of a derivative takes up half the study of Calculus. A derivative, basically, represents rates of change. 0. Limits: Numerical and Graphical Approaches
More information3.3 Limits and Infinity
Calculus Maimus. Limits Infinity Infinity is not a concrete number, but an abstract idea. It s not a destination, but a really long, never-ending journey. It s one of those mind-warping ideas that is difficult
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 informationMAT12X Intermediate Algebra
MAT12X Intermediate Algebra Workshop 3 Rational Functions LEARNING CENTER Overview Workshop III Rational Functions General Form Domain and Vertical Asymptotes Range and Horizontal Asymptotes Inverse Variation
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 informationFunctions If (x, y 1 ), (x, y 2 ) S, then y 1 = y 2
Functions 4-3-2008 Definition. A function f from a set X to a set Y is a subset S of the product X Y such that if (, y 1 ), (, y 2 ) S, then y 1 = y 2. Instead of writing (, y) S, you usually write f()
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 informationand lim lim 6. The Squeeze Theorem
Limits (day 3) Things we ll go over today 1. Limits of the form 0 0 (continued) 2. Limits of piecewise functions 3. Limits involving absolute values 4. Limits of compositions of functions 5. Limits similar
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 informationCalculus 2 - Examination
Calculus - Eamination Concepts that you need to know: Two methods for showing that a function is : a) Showing the function is monotonic. b) Assuming that f( ) = f( ) and showing =. Horizontal Line Test:
More informationPhysics 116A Solutions to Homework Set #2 Winter 2012
Physics 6A Solutions to Homework Set #2 Winter 22. Boas, problem. 23. Transform the series 3 n (n+ (+ n determine the interval of convergence to a power series and First we want to make the replacement
More informationSOLVING QUADRATICS. Copyright - Kramzil Pty Ltd trading as Academic Teacher Resources
SOLVING QUADRATICS Copyright - Kramzil Pty Ltd trading as Academic Teacher Resources SOLVING QUADRATICS General Form: y a b c Where a, b and c are constants To solve a quadratic equation, the equation
More information4. (6 points) Express the domain of the following function in interval notation:
Eam 1-A L. Ballou Name Math 131 Calculus I September 1, 016 NO Calculator Allowed BOX YOUR ANSWER! Show all work for full credit! 1. (4 points) Write an equation of a line with y-intercept 4 and -intercept
More information4.3 Division of Polynomials
4.3 Division of Polynomials Learning Objectives Divide a polynomials by a monomial. Divide a polynomial by a binomial. Rewrite and graph rational functions. Introduction A rational epression is formed
More informationPre-Calculus Notes Section 12.2 Evaluating Limits DAY ONE: Lets look at finding the following limits using the calculator and algebraically.
Pre-Calculus Notes Name Section. Evaluating Limits DAY ONE: Lets look at finding the following its using the calculator and algebraicall. 4 E. ) 4 QUESTION: As the values get closer to 4, what are the
More informationSection 5.5 Complex Numbers
Name: Period: Section 5.5 Comple Numbers Objective(s): Perform operations with comple numbers. Essential Question: Tell whether the statement is true or false, and justify your answer. Every comple number
More informationNotes 3.2: Properties of Limits
Calculus Maimus Notes 3.: Properties of Limits 3. Properties of Limits When working with its, you should become adroit and adept at using its of generic functions to find new its of new functions created
More information5 3w. Unit 2 Function Operations and Equivalence Standard 4.1 Add, Subtract, & Multiply Polynomials
Unit Function Operations and Equivalence This document is meant to be used as an eample guide for each of the skills we will be holding students accountable for with Standard 4.1. This document should
More informationHow to Find Limits. Yilong Yang. October 22, The General Guideline 1
How to Find Limits Yilong Yang October 22, 204 Contents The General Guideline 2 Put Fractions Together and Factorization 2 2. Why put fractions together..................................... 2 2.2 Formula
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 informationDefine a rational expression: a quotient of two polynomials. ..( 3 10) (3 2) Rational expressions have the same properties as rational numbers:
1 UNIT 7 RATIONAL EXPRESSIONS & EQUATIONS Simplifying Rational Epressions Define a rational epression: a quotient of two polynomials. A rational epression always indicates division EX: 10 means..( 10)
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 informationMath 154 :: Elementary Algebra
Math 4 :: Elementary Algebra Section. Additive Property of Equality Section. Multiplicative Property of Equality Section.3 Linear Equations in One-Variable Section.4 Linear Equations in One-Variable with
More informationLesson #9 Simplifying Rational Expressions
Lesson #9 Simplifying Rational Epressions A.A.6 Perform arithmetic operations with rational epressions and rename to lowest terms Factor the following epressions: A. 7 4 B. y C. y 49y Simplify: 5 5 = 4
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 information1 Exponential Functions Limit Derivative Integral... 5
Contents Eponential Functions 3. Limit................................................. 3. Derivative.............................................. 4.3 Integral................................................
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 informationQUADRATIC EQUATIONS. + 6 = 0 This is a quadratic equation written in standard form. x x = 0 (standard form with c=0). 2 = 9
QUADRATIC EQUATIONS A quadratic equation is always written in the form of: a + b + c = where a The form a + b + c = is called the standard form of a quadratic equation. Eamples: 5 + 6 = This is a quadratic
More informationEpsilon Delta proofs
Epsilon Delta proofs Before reading this guide, please go over inequalities (if needed). Eample Prove lim(4 3) = 5 2 First we have to know what the definition of a limit is: i.e rigorous way of saying
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 informationMath 1314 Lesson 4 Limits
Math 1314 Lesson 4 Limits What is calculus? Calculus is the study of change, particularly, how things change over time. It gives us a framework for measuring change using some fairly simple models. In
More informationSummer Packet Geometry PAP
Summer Packet Geometry PAP IMPORTANT INSTRUCTIONS FOR STUDENTS!!! We understand that students come to Geometry with different strengths and needs. For this reason, students have options for completing
More informationSANDY CREEK HIGH SCHOOL
SANDY CREEK HIGH SCHOOL SUMMER REVIEW PACKET For students entering A.P. CALCULUS AB I epect everyone to check the Google classroom site and your school emails at least once every two weeks. You should
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 informationIntermediate Algebra Section 9.3 Logarithmic Functions
Intermediate Algebra Section 9.3 Logarithmic Functions We have studied inverse functions, learning when they eist and how to find them. If we look at the graph of the eponential function, f ( ) = a, where
More information1 DL3. Infinite Limits and Limits at Infinity
Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 78 Mark Sparks 01 Infinite Limits and Limits at Infinity In our graphical analysis of its, we have already seen both an infinite
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 informationSANDY CREEK HIGH SCHOOL
SANDY CREEK HIGH SCHOOL SUMMER REVIEW PACKET For students entering A.P. CALCULUS BC I epect everyone to check the Google classroom site and your school emails at least once every two weeks. You will also
More informationLecture 7: Indeterminate forms; L Hôpitals rule; Relative rates of growth. If we try to simply substitute x = 1 into the expression, we get
Lecture 7: Indeterminate forms; L Hôpitals rule; Relative rates of growth 1. Indeterminate Forms. Eample 1: Consider the it 1 1 1. If we try to simply substitute = 1 into the epression, we get. This is
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 information(ii) y = ln 1 ] t 3 t x x2 9
Study Guide for Eam 1 1. You are supposed to be able to determine the domain of a function, looking at the conditions for its epression to be well-defined. Some eamples of the conditions are: What is inside
More informationAP CALCULUS SUMMER REVIEW WORK
AP CALCULUS SUMMER REVIEW WORK The following problems are all ALGEBRA concepts you must know cold in order to be able to handle Calculus. Most of them are from Algebra, some are from Pre-Calc. This packet
More informationReview of Rational Expressions and Equations
Page 1 of 14 Review of Rational Epressions and Equations A rational epression is an epression containing fractions where the numerator and/or denominator may contain algebraic terms 1 Simplify 6 14 Identification/Analysis
More information7.3 Adding and Subtracting Rational Expressions
7.3 Adding and Subtracting Rational Epressions LEARNING OBJECTIVES. Add and subtract rational epressions with common denominators. 2. Add and subtract rational epressions with unlike denominators. 3. Add
More information( ) ( ) ( ) 2 6A: Special Trig Limits! Math 400
2 6A: Special Trig Limits Math 400 This section focuses entirely on the its of 2 specific trigonometric functions. The use of Theorem and the indeterminate cases of Theorem are all considered. a The it
More informationMath 1320, Section 10 Quiz IV Solutions 20 Points
Math 1320, Section 10 Quiz IV Solutions 20 Points Please answer each question. To receive full credit you must show all work and give answers in simplest form. Cell phones and graphing calculators are
More informationThe trick is to multiply the numerator and denominator of the big fraction by the least common denominator of every little fraction.
Complex Fractions A complex fraction is an expression that features fractions within fractions. To simplify complex fractions, we only need to master one very simple method. Simplify 7 6 +3 8 4 3 4 The
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 informationEXPONENT 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 informationMATH 408N PRACTICE MIDTERM 1
02/0/202 Bormashenko MATH 408N PRACTICE MIDTERM Show your work for all the problems. Good luck! () (a) [5 pts] Solve for x if 2 x+ = 4 x Name: TA session: Writing everything as a power of 2, 2 x+ = (2
More informationAll work must be shown in this course for full credit. Unsupported answers may receive NO credit.
AP Calculus. Worksheet Day All work must be shown in this course for full credit. Unsupported answers may receive NO credit.. The only way to guarantee the eistence of a it is to algebraically prove it.
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 informationPre-Calculus Mathematics Limit Process Calculus
NOTES : LIMITS AND DERIVATIVES Name: Date: Period: Mrs. Nguyen s Initial: LESSON.1 THE TANGENT AND VELOCITY PROBLEMS Pre-Calculus Mathematics Limit Process Calculus The type of it that is used to find
More informationFeedback D. Incorrect! Exponential functions are continuous everywhere. Look for features like square roots or denominators that could be made 0.
Calculus Problem Solving Drill 07: Trigonometric Limits and Continuity No. of 0 Instruction: () Read the problem statement and answer choices carefully. () Do your work on a separate sheet of paper. (3)
More information3 Applications of Derivatives Instantaneous Rates of Change Optimization Related Rates... 13
Contents Limits Derivatives 3. Difference Quotients......................................... 3. Average Rate of Change...................................... 4.3 Derivative Rules...........................................
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 informationUniversity of Colorado at Colorado Springs Math 090 Fundamentals of College Algebra
University of Colorado at Colorado Springs Math 090 Fundamentals of College Algebra Table of Contents Chapter The Algebra of Polynomials Chapter Factoring 7 Chapter 3 Fractions Chapter 4 Eponents and Radicals
More informationSolutions to Math 41 First Exam October 12, 2010
Solutions to Math 41 First Eam October 12, 2010 1. 13 points) Find each of the following its, with justification. If the it does not eist, eplain why. If there is an infinite it, then eplain whether it
More informationPart Two. Diagnostic Test
Part Two Diagnostic Test AP Calculus AB and BC Diagnostic Tests Take a moment to gauge your readiness for the AP Calculus eam by taking either the AB diagnostic test or the BC diagnostic test, depending
More informationCOUNCIL ROCK HIGH SCHOOL MATHEMATICS. A Note Guideline of Algebraic Concepts. Designed to assist students in A Summer Review of Algebra
COUNCIL ROCK HIGH SCHOOL MATHEMATICS A Note Guideline of Algebraic Concepts Designed to assist students in A Summer Review of Algebra [A teacher prepared compilation of the 7 Algebraic concepts deemed
More informationSolution. Using the point-slope form of the equation we have the answer immediately: y = 4 5 (x ( 2)) + 9 = 4 (x +2)+9
Chapter Review. Lines Eample. Find the equation of the line that goes through the point ( 2, 9) and has slope 4/5. Using the point-slope form of the equation we have the answer immediately: y = 4 5 ( (
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 informationAP Calculus AB Summer Assignment
Name: AP Calculus AB Summer Assignment Due Date: The beginning of class on the last class day of the first week of school. The purpose of this assignment is to have you practice the mathematical skills
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 informationCALC1 SUMMER REVIEW WORK
CALC SUMMER REVIEW WORK The following problems are all ALGEBRA concepts you must know cold in order to be able to handle Calculus. Most of them are from Algebra, some are from Pre-Calc. This packet is
More informationChapter 7 Class Notes. Intermediate Algebra, MAT1033C. SI Leader Joe Brownlee. Palm Beach State College
Chapter 7 Class Notes Intermediate Algebra, MAT033C Palm Beach State College Class Notes 7. Professor Burkett 7. Rational Expressions and Functions; Multiplying and Dividing Chapter 7 takes factoring to
More informationSECTION 2.5: THE INDETERMINATE FORMS 0 0 AND
(Section 2.5: The Indeterminate Forms 0/0 and / ) 2.5. SECTION 2.5: THE INDETERMINATE FORMS 0 0 AND LEARNING OBJECTIVES Understand what it means for a Limit Form to be indeterminate. Recognize indeterminate
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 informationConcept Category 5. Limits. Limits: graphically & algebraically Rate of Change
Concept Category 5 Limits Limits: graphically & algebraically Rate of Change Skills Factoring and Rational Epressions (Alg, CC1) Behavior of a graph (Alg, CC1) Sketch a graph:,,, Log, (Alg, CC1 & ) 1 Factoring
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 informationChapter 1 Review of Equations and Inequalities
Chapter 1 Review of Equations and Inequalities Part I Review of Basic Equations Recall that an equation is an expression with an equal sign in the middle. Also recall that, if a question asks you to solve
More information[Limits at infinity examples] Example. The graph of a function y = f(x) is shown below. Compute lim f(x) and lim f(x).
[Limits at infinity eamples] Eample. The graph of a function y = f() is shown below. Compute f() and f(). y -8 As you go to the far right, the graph approaches y =, so f() =. As you go to the far left,
More informationAP CALCULUS AB SUMMER ASSIGNMENT
AP CALCULUS AB SUMMER ASSIGNMENT 06-07 Attached is your summer assignment for AP Calculus (AB). It will probably take you - hours to complete depending on how well you know your material. I would not do
More information2.5 Absolute Value Equations and Inequalities
5 Absolute Value Equations Inequalities We begin this section by recalling the following definition Definition: Absolute Value The absolute value of a number is the distance that the number is from zero
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 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 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 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 informationPractice Calculus Test without Trig
Practice Calculus Test without Trig The problems here are similar to those on the practice test Slight changes have been made 1 What is the domain of the function f (x) = 3x 1? Express the answer in interval
More informationFox Lane High School Department of Mathematics
Fo Lane High School Department of Mathematics June 08 Hello Future AP Calculus AB Student! This is the summer assignment for all students taking AP Calculus AB net school year. It contains a set of problems
More informationACCUPLACER MATH 0310
The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 00 http://www.academics.utep.edu/tlc MATH 00 Page Linear Equations Linear Equations Eercises 5 Linear Equations Answer to
More informationAll work must be shown in this course for full credit. Unsupported answers may receive NO credit.
AP Calculus.1 Worksheet Day 1 All work must be shown in this course for full credit. Unsupported answers may receive NO credit. 1. The only way to guarantee the eistence of a it is to algebraically prove
More informationYOU CAN BACK SUBSTITUTE TO ANY OF THE PREVIOUS EQUATIONS
The two methods we will use to solve systems are substitution and elimination. Substitution was covered in the last lesson and elimination is covered in this lesson. Method of Elimination: 1. multiply
More informationPolynomial comes from poly- (meaning "many") and -nomial (in this case meaning "term")... so it says "many terms
Polynomials Polynomial comes from poly- (meaning "many") and -nomial (in this case meaning "term")... so it says "many terms Polynomials A polynomial looks like this: Term A number, a variable, or the
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 information6.5 Trigonometric Equations
6. Trigonometric Equations In this section, we discuss conditional trigonometric equations, that is, equations involving trigonometric functions that are satisfied only by some values of the variable (or
More informationLesson 3-2: Solving Linear Systems Algebraically
Yesterday we took our first look at solving a linear system. We learned that a linear system is two or more linear equations taken at the same time. Their solution is the point that all the lines have
More informationPart 2 - Beginning Algebra Summary
Part - Beginning Algebra Summary Page 1 of 4 1/1/01 1. Numbers... 1.1. Number Lines... 1.. Interval Notation.... Inequalities... 4.1. Linear with 1 Variable... 4. Linear Equations... 5.1. The Cartesian
More information