9.4 Power Series II: Geometric Series
|
|
- Marlene Manning
- 6 years ago
- Views:
Transcription
1 9.4 Power Series II: Geometric Series A particularly important skill to develop for the AP eam, other than checking that you re in RADIAN mode, is to represent certain types of rational functions as a geometric series. Rather than producing the power series using Taylor s Rule, you will want to develop the series by manipulating a geometric series or, in some cases, using Long Division. Eample : First we ll do a quick review of geometric series. Geometric series are formed by multiplying by a common ratio r. (a) Suppose I told you to start with a and to let would the sum be? r, what geometric series would you write? What 3 (b) What if a and r? 3 (c) What if a and r? Eample : Verify your answer from Eample (c) by finding the power series for (a) using Taylor s Rule (b) using LOOOONG DIVISION. (c) Find the radius and interval of convergence. Verify by graphing. centered at c 0 by Page of 6
2 Eample 3: Find a power series for centered at c 0, then find the interval of convergence. Include the first four nonzero terms and the general term. Eample 4: Find a power series that represents centered at c 0, then find the interval of convergence. Include the first four nonzero terms and the general term. Eample 5: Find a power series for f( ) centered at c 0, then find the interval of convergence. Find the first four nonzero terms and the general term. Page of 6
3 When you replace with a multiple of, beware a change in the radius and interval of convergence... Eample 6: Find a power series that represents centered at c 0, then find the interval of convergence. Include the first four nonzero terms and the general term. Eample 7: Find a power series for g ( ) 4 centered at c 0, then find the interval of convergence. Include the first four nonzero terms and the general term. Sometimes we cannot center our function at 0. In this case, we must try to rewrite our function with the new center showing. Eample 8: Find a power series that represents centered at c, then find the interval of convergence. Include the first four nonzero terms and the general term. Page 3 of 6
4 Eample 9: 5 (A bit of a booger) Find a power series for h ( ), centered at c, then find the interval of convergence. Include the first four nonzero terms and the general term. We can integrate or differentiate a power series to obtain a new series. When we do this, the radius of convergence will be the same, but the interval may change. In this case, we must retest the endpoints. Eample 0: Find a power series that represents radius of convergence? Interval of convergence? centered at c 0. Hint: what is d d? What is the Eample : Find a power series that represents ln radius of convergence? Interval of convergence? centered at c 0. Hint: what is d? What is the Page 4 of 6
5 Eample : (Similar to 008 BC6B ) Let f be the function given by f. (a) Write the first four nonzero terms and the general term of the Taylor series for f about 0. (b) Does the series found in part (a), when evaluated at, converge to f? Eplain why or why not. (c) The derivative of arctan is arctan about 0.. Write the first four nonzero terms of the Taylor series for (d) Use the series found in part (c) to find a rational number A such that your answer. 3 A arctan. Justify 4 0 Page 5 of 6
6 Besides finding the sum of an infinite, convergent geometric series (and the telescoping series), there is one last way we are epected to find such sums: by recognizing a given infinite, convergent series as a Taylor series evaluated at a particular value of. Eample 3: By recognizing each series as a Taylor series evaluated at a particular value of, find the sum of each of the following infinite, convergent series. 4 6 (a)! 4! 6! 3 4 ln 4 ln 4 ln 4 (b) ln 4! 3! 4! (c) ! 6 5! 6 7! 6 Page 6 of 6
AP Exam Practice Questions for Chapter 3
AP Eam Practice Questions for Chapter AP Eam Practice Questions for Chapter f + 6 7 9 f + 7 0 + 6 0 ( + )( ) 0,. The critical numbers of f are and. So, the answer is B.. Evaluate each statement. I: Because
More informationDaily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 584 Mark Sparks 2012
The Second Fundamental Theorem of Calculus Functions Defined by Integrals Given the functions, f(t), below, use F( ) f ( t) dt to find F() and F () in terms of.. f(t) = 4t t. f(t) = cos t Given the functions,
More informationAP Calculus Chapter 9: Infinite Series
AP Calculus Chapter 9: Infinite Series 9. Sequences a, a 2, a 3, a 4, a 5,... Sequence: A function whose domain is the set of positive integers n = 2 3 4 a n = a a 2 a 3 a 4 terms of the sequence Begin
More informationMath 1b Sequences and series summary
Math b Sequences and series summary December 22, 2005 Sequences (Stewart p. 557) Notations for a sequence: or a, a 2, a 3,..., a n,... {a n }. The numbers a n are called the terms of the sequence.. Limit
More informationTAYLOR SERIES [SST 8.8]
TAYLOR SERIES [SST 8.8] TAYLOR SERIES: Every function f C (c R, c + R) has a unique Taylor series about x = c of the form: f (k) (c) f(x) = (x c) k = f(c) + f (c) (x c) + f (c) (x c) 2 + f (c) (x c) 3
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 informationAP Calculus BC Summer Assignment 2018
AP Calculus BC Summer Assignment 018 Name: When you come back to school, I will epect you to have attempted every problem. These skills are all different tools that we will pull out of our toolbo at different
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 informationINTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS
INTEGRATION OF RATIONAL FUNCTIONS BY PARTIAL FRACTIONS. Introduction It is possible to integrate any rational function, constructed as the ratio of polynomials by epressing it as a sum of simpler fractions
More informationVII. Techniques of Integration
VII. Techniques of Integration Integration, unlike differentiation, is more of an art-form than a collection of algorithms. Many problems in applied mathematics involve the integration of functions given
More informationTroy High School AP Calculus Summer Packet
Troy High School AP Calculus Summer Packet As instructors of AP Calculus, we have etremely high epectations of students taking our courses. We epect a certain level of independence to be demonstrated by
More informationSOLVED PROBLEMS ON TAYLOR AND MACLAURIN SERIES
SOLVED PROBLEMS ON TAYLOR AND MACLAURIN SERIES TAYLOR AND MACLAURIN SERIES Taylor Series of a function f at x = a is ( f k )( a) ( x a) k k! It is a Power Series centered at a. Maclaurin Series of a function
More information5.6 Asymptotes; Checking Behavior at Infinity
5.6 Asymptotes; Checking Behavior at Infinity checking behavior at infinity DEFINITION asymptote In this section, the notion of checking behavior at infinity is made precise, by discussing both asymptotes
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 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 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 informationMathematics 132 Calculus for Physical and Life Sciences 2 Exam 3 Review Sheet April 15, 2008
Mathematics 32 Calculus for Physical and Life Sciences 2 Eam 3 Review Sheet April 5, 2008 Sample Eam Questions - Solutions This list is much longer than the actual eam will be (to give you some idea of
More informationCHAPTER 1 Systems of Linear Equations
CHAPTER Systems of Linear Equations Section. Introduction to Systems of Linear Equations. Because the equation is in the form a x a y b, it is linear in the variables x and y. 0. Because the equation cannot
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 informationLet 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 informationFinal exam (practice) UCLA: Math 31B, Spring 2017
Instructor: Noah White Date: Final exam (practice) UCLA: Math 3B, Spring 207 This exam has 8 questions, for a total of 80 points. Please print your working and answers neatly. Write your solutions in the
More informationExamples of the Accumulation Function (ANSWERS) dy dx. This new function now passes through (0,2). Make a sketch of your new shifted graph.
Eamples of the Accumulation Function (ANSWERS) Eample. Find a function y=f() whose derivative is that f()=. dy d tan that satisfies the condition We can use the Fundamental Theorem to write a function
More informationPower series and Taylor series
Power series and Taylor series D. DeTurck University of Pennsylvania March 29, 2018 D. DeTurck Math 104 002 2018A: Series 1 / 42 Series First... a review of what we have done so far: 1 We examined series
More informationAPPM 1360 Final Exam Spring 2016
APPM 36 Final Eam Spring 6. 8 points) State whether each of the following quantities converge or diverge. Eplain your reasoning. a) The sequence a, a, a 3,... where a n ln8n) lnn + ) n!) b) ln d c) arctan
More information1. Pace yourself. Make sure you write something on every problem to get partial credit. 2. If you need more space, use the back of the previous page.
***THIS TIME I DECIDED TO WRITE A LOT OF EXTRA PROBLEMS TO GIVE MORE PRACTICE. The actual midterm will have about 6 problems. If you want to practice something with approximately the same length as the
More informationWarm up: Recall that if you require real coe cients (not complex), polynomials always factor into degree 1 or 2 parts.
Today: 6.3 Partial fractions Warm up: Recall that if you require real coe cients (not comple), polynomials always factor into degree or parts. For eample, 3 + +3 5=( )( + i)( ++i) =( {z } )( + + 5) {z
More informationSolutions Quiz 9 Nov. 8, Prove: If a, b, m are integers such that 2a + 3b 12m + 1, then a 3m + 1 or b 2m + 1.
Solutions Quiz 9 Nov. 8, 2010 1. Prove: If a, b, m are integers such that 2a + 3b 12m + 1, then a 3m + 1 or b 2m + 1. Answer. We prove the contrapositive. Suppose a, b, m are integers such that a < 3m
More informationPartial Fraction Decompositions
Partial Fraction Rational Functions Partial Fraction Finding Partial Fractions Integrating Partial Fraction Eamples Rational Functions Definition Eample A function of the type P/Q, where both P and Q are
More informationSummer 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 informationChapter 9: Infinite Series Part 2
Name: Date: Period: AP Calc BC Mr. Mellina/Ms. Lombardi Chapter 9: Infinite Series Part 2 Topics: 9.5 Alternating Series Remainder 9.7 Taylor Polynomials and Approximations 9.8 Power Series 9.9 Representation
More information1. Find the domain of the following functions. Write your answer using interval notation. (9 pts.)
MATH- Sample Eam Spring 7. Find the domain of the following functions. Write your answer using interval notation. (9 pts.) a. 9 f ( ) b. g ( ) 9 8 8. Write the equation of the circle in standard form given
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 informationSo, t = 1 is a point of inflection of s(). Use s () t to find the velocity at t = Because 0, use 144.
AP Eam Practice Questions for Chapter AP Eam Practice Questions for Chapter f 4 + 6 7 9 f + 7 0 + 6 0 ( + )( ) 0,. The critical numbers of f( ) are and.. Evaluate each point. A: d d C: d d B: D: d d d
More informationRepresentation of Functions by Power Series. Geometric Power Series
60_0909.qd //0 :09 PM Page 669 SECTION 9.9 Representation of Functions b Power Series 669 The Granger Collection Section 9.9 JOSEPH FOURIER (768 80) Some of the earl work in representing functions b power
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 informationPart 1: Integration problems from exams
. Find each of the following. ( (a) 4t 4 t + t + (a ) (b ) Part : Integration problems from 4-5 eams ) ( sec tan sin + + e e ). (a) Let f() = e. On the graph of f pictured below, draw the approimating
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 information4.3 Rational Inequalities and Applications
342 Rational Functions 4.3 Rational Inequalities and Applications In this section, we solve equations and inequalities involving rational functions and eplore associated application problems. Our first
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 informationIntegration Techniques for the AB exam
For the AB eam, students need to: determine antiderivatives of the basic functions calculate antiderivatives of functions using u-substitution use algebraic manipulation to rewrite the integrand prior
More informationFocusing on Linear Functions and Linear Equations
Focusing on Linear Functions and Linear Equations In grade, students learn how to analyze and represent linear functions and solve linear equations and systems of linear equations. They learn how to represent
More informationLEARN ABOUT the Math
1.5 Inverse Relations YOU WILL NEED graph paper graphing calculator GOAL Determine the equation of an inverse relation and the conditions for an inverse relation to be a function. LEARN ABOUT the Math
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 information1 Exam 1 Spring 2007.
Exam Spring 2007.. An object is moving along a line. At each time t, its velocity v(t is given by v(t = t 2 2 t 3. Find the total distance traveled by the object from time t = to time t = 5. 2. Use 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 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 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 2300 Calculus II University of Colorado
Math 3 Calculus II University of Colorado Spring Final eam review problems: ANSWER KEY. Find f (, ) for f(, y) = esin( y) ( + y ) 3/.. Consider the solid region W situated above the region apple apple,
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 informationDifferential Equations Practice: Euler Equations & Regular Singular Points Page 1
Differential Equations Practice: Euler Equations & Regular Singular Points Page 1 Questions Eample (5.4.1) Determine the solution to the differential equation y + 4y + y = 0 that is valid in any interval
More informationChapter 11 - Sequences and Series
Calculus and Analytic Geometry II Chapter - Sequences and Series. Sequences Definition. A sequence is a list of numbers written in a definite order, We call a n the general term of the sequence. {a, a
More informationAP Calculus AB Summer Assignment
AP Calculus AB 07-08 Summer Assignment Welcome to AP Calculus AB! You are epected to complete the attached homework assignment during the summer. This is because of class time constraints and the amount
More informationUNC Charlotte Super Competition Level 3 Test March 4, 2019 Test with Solutions for Sponsors
. Find the minimum value of the function f (x) x 2 + (A) 6 (B) 3 6 (C) 4 Solution. We have f (x) x 2 + + x 2 + (D) 3 4, which is equivalent to x 0. x 2 + (E) x 2 +, x R. x 2 + 2 (x 2 + ) 2. How many solutions
More informationWarmup for AP Calculus BC
Nichols School Mathematics Department Summer Work Packet Warmup for AP Calculus BC Who should complete this packet? Students who have completed Advanced Functions or and will be taking AP Calculus BC in
More informationTAYLOR AND MACLAURIN SERIES
TAYLOR AND MACLAURIN SERIES. Introduction Last time, we were able to represent a certain restricted class of functions as power series. This leads us to the question: can we represent more general functions
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 informationSummer Packet Honors PreCalculus
Summer Packet Honors PreCalculus Honors Pre-Calculus is a demanding course that relies heavily upon a student s algebra, geometry, and trigonometry skills. You are epected to know these topics before entering
More informationBell Quiz 2-3. Determine the end behavior of the graph using limit notation. Find a function with the given zeros , 2. 5 pts possible.
Bell Quiz 2-3 2 pts Determine the end behavior of the graph using limit notation. 5 2 1. g( ) = 8 + 13 7 3 pts Find a function with the given zeros. 4. -1, 2 5 pts possible Ch 2A Big Ideas 1 Questions
More informationOhio s Learning Standards-Extended. Mathematics. The Real Number System Complexity a Complexity b Complexity c
Ohio s Learning Standards-Extended Mathematics The Real Number System Complexity a Complexity b Complexity c Extend the properties of exponents to rational exponents N.RN.1 Explain how the definition of
More informationLESSON 9.1 ROOTS AND RADICALS
LESSON 9.1 ROOTS AND RADICALS LESSON 9.1 ROOTS AND RADICALS 67 OVERVIEW Here s what you ll learn in this lesson: Square Roots and Cube Roots a. Definition of square root and cube root b. Radicand, radical
More informationAlgebra I Notes Unit Nine: Exponential Expressions
Algera I Notes Unit Nine: Eponential Epressions Syllaus Ojectives: 7. The student will determine the value of eponential epressions using a variety of methods. 7. The student will simplify algeraic epressions
More informationSUMMATION TECHNIQUES
SUMMATION TECHNIQUES MATH 53, SECTION 55 (VIPUL NAIK) Corresponding material in the book: Scattered around, but the most cutting-edge parts are in Sections 2.8 and 2.9. What students should definitely
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 informationUnit 11 - Solving Quadratic Functions PART ONE
Unit 11 - Solving Quadratic Functions PART ONE PREREQUISITE SKILLS: students should be able to add, subtract and multiply polynomials students should be able to factor polynomials students should be able
More informationTaylor and Maclaurin Series. Copyright Cengage Learning. All rights reserved.
11.10 Taylor and Maclaurin Series Copyright Cengage Learning. All rights reserved. We start by supposing that f is any function that can be represented by a power series f(x)= c 0 +c 1 (x a)+c 2 (x a)
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 information3.2 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 information8.5 Taylor Polynomials and Taylor Series
8.5. TAYLOR POLYNOMIALS AND TAYLOR SERIES 50 8.5 Taylor Polynomials and Taylor Series Motivating Questions In this section, we strive to understand the ideas generated by the following important questions:
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 informationHarbor Creek School District
Unit 1 Days 1-9 Evaluate one-sided two-sided limits, given the graph of a function. Limits, Evaluate limits using tables calculators. Continuity Evaluate limits using direct substitution. Differentiability
More informationSolutions to Math 41 Exam 2 November 10, 2011
Solutions to Math 41 Eam November 10, 011 1. (1 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 is or.
More informationM098 Carson Elementary and Intermediate Algebra 3e Section 11.3
Objectives. Solve equations by writing them in quadratic form.. Solve equations that are quadratic in form by using substitution. Vocabulary Prior Knowledge Solve rational equations: Clear the fraction.
More informationSection 8.2: Integration by Parts When you finish your homework, you should be able to
Section 8.2: Integration by Parts When you finish your homework, you should be able to π Use the integration by parts technique to find indefinite integral and evaluate definite integrals π Use the tabular
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 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 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 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 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 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 informationMat104 Fall 2002, Improper Integrals From Old Exams
Mat4 Fall 22, Improper Integrals From Old Eams For the following integrals, state whether they are convergent or divergent, and give your reasons. () (2) (3) (4) (5) converges. Break it up as 3 + 2 3 +
More informationAdvanced Calculus BC Summer Work Due: 1 st Day of School
Dear Calculus BC student, I hope that ou re all enjoing our first few das of summer! Here s something that will make it a little more fun! Enclosed ou will find a packet of review questions that ou should
More informationProblem 1 Oh Snap... Look at the Denominator on that Rational
Problem Oh Snap... Look at the Denominator on that Rational Previously, you learned that dividing polynomials was just like dividing integers. Well, performing operations on rational epressions involving
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 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 information5x 3x x 5(2x 4) 11. x Cumulative Exam #2 Review Guide. Adding & Subtracting: Simplify radicals Add/subtract like radicals
Foundations of Algebra Name: Date: Block: Cumulative Eam # Review Guide What you need to know & be able to do 1. Adding and Subtracting Radicals Things to remember Adding & Subtracting: Simplify radicals
More informationINFINITE SEQUENCES AND SERIES
11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES In section 11.9, we were able to find power series representations for a certain restricted class of functions. INFINITE SEQUENCES AND SERIES
More informationRepresentation of Functions as Power Series.
MATH 0 - A - Spring 009 Representation of Functions as Power Series. Our starting point in this section is the geometric series: x n = + x + x + x 3 + We know this series converges if and only if x
More informationAnswer Key. ( 1) n (2x+3) n. n n=1. (2x+3) n. = lim. < 1 or 2x+3 < 4. ( 1) ( 1) 2n n
Math Midterm Eam #3 December, 3 Answer Key. [5 Points] Find the Interval and Radius of Convergence for the following power series. Analyze carefully and with full justification. Use Ratio Test. L lim a
More information10.7 Trigonometric Equations and Inequalities
0.7 Trigonometric Equations and Inequalities 79 0.7 Trigonometric Equations and Inequalities In Sections 0., 0. and most recently 0., we solved some basic equations involving the trigonometric functions.
More informationExam 2 Solutions, Math March 17, ) = 1 2
Eam Solutions, Math 56 March 7, 6. Use the trapezoidal rule with n = 3 to approimate (Note: The eact value of the integral is ln 5 +. (you do not need to verify this or use it in any way to complete this
More informationCore Mathematics 3 A2 compulsory unit for GCE Mathematics and GCE Pure Mathematics Mathematics. Unit C3. C3.1 Unit description
Unit C3 Core Mathematics 3 A2 compulsory unit for GCE Mathematics and GCE Pure Mathematics Mathematics C3. Unit description Algebra and functions; trigonometry; eponentials and logarithms; differentiation;
More informationPractice problems from old exams for math 132 William H. Meeks III
Practice problems from old exams for math 32 William H. Meeks III Disclaimer: Your instructor covers far more materials that we can possibly fit into a four/five questions exams. These practice tests are
More informationMonroe Township High School Mathematics Department
To: AP Calculus AB Re: Summer Project 017 Date: June 017 Monroe Township High School Mathematics Department To help begin your study of Calculus, you will be required to complete a review project this
More information7.1 One-to-One Functions
514 transcendental functions 7.1 One-to-One Functions You ve seen that some equations have only one solution (for eample, 5 2 = 3 and 3 = 8), while some have two solutions ( 2 + 3 = 7) and some even have
More informationMcKinney High School AP Calculus Summer Packet
McKinne High School AP Calculus Summer Packet (for students entering AP Calculus AB or AP Calculus BC) Name:. This packet is to be handed in to our Calculus teacher the first week of school.. ALL work
More informationChapter 5: Introduction to Limits. Chapter 5 Recommendations
Chapter 5: Introduction to Limits Chapter 5 Topics: Inverse and Direct Variation Transformations of Rational Functions Graphing Reciprocals of Functions Introduction to Limits Working With One-Sided Limits
More informationMath 1B Final Exam, Solution. Prof. Mina Aganagic Lecture 2, Spring (6 points) Use substitution and integration by parts to find:
Math B Final Eam, Solution Prof. Mina Aganagic Lecture 2, Spring 20 The eam is closed book, apart from a sheet of notes 8. Calculators are not allowed. It is your responsibility to write your answers clearly..
More informationMATH 1207 R02 FINAL SOLUTION
MATH 7 R FINAL SOLUTION SPRING 6 - MOON Write your answer neatly and show steps. Except calculators, any electronic devices including laptops and cell phones are not allowed. () Let f(x) = x cos x. (a)
More informationSolution to Review Problems for Midterm #1
Solution to Review Problems for Midterm # Midterm I: Wednesday, September in class Topics:.,.3 and.-.6 (ecept.3) Office hours before the eam: Monday - and 4-6 p.m., Tuesday - pm and 4-6 pm at UH 080B)
More informationMultiplying and Dividing Rational Expressions
6.3 Multiplying and Dividing Rational Epressions Essential Question How can you determine the ecluded values in a product or quotient of two rational epressions? You can multiply and divide rational epressions
More information