41. Convergence Tests For Infinite Series - HMC Calculus Tutorial of an infinite series, ¥ å k = 0, a k = a 0 + a 1 + a 2 + ¼. The proofs or thesetests are interesting, so we urge you to look them up in your calculus text. http://www.math.hmc.edu/calculus/tutorials/convergence/

Convergence Tests for Infinite Series In this tutorial, we review some of the most common tests for the convergence of an infinite series k = a k = a + a + a The proofs or these tests are interesting, so we urge you to look them up in your calculus text. Let s a s a s n n k = a k n partial sums converges to a limit L, then the series is said to converge to the sum L and we write k = a k = L. For j k = a k converges if and only if k = j a k converges, so in discussing convergence we often just write a k Example Consider the geometric series k = x k The n th partial sum is s n = 1 + x + x + x n Multiplying both sides by x, xs n = x + x + x + x n+1 Subtracting the second equation from the first, (1-x)s n = 1-x n+1 so for x s n 1-x n+1 1-x For x lim n s n 1-x It is easy to see that k = x k diverges for x Thus k = x k = 1/(1-x) for x x Divergence Test If lim k a k 0, then k = a k diverges. Example The series k = k/(2k+1) diverges, since lim k k/(2k+1) = 1/2 Integral Test Let f(x) be continuous, decreasing, and positive for x Then k = 1 f(k) converges if and only if f(x)dx converges.

42. Infinite Series Presentations - Calculus infinite series presentations. Directions. Symbolically. Given series.Write out first several terms. Grading series Presentations. http://teachers.sduhsd.k12.ca.us/abrown/Resources/PresentationStuff/Calculus/Cal

infinite series presentations Directions Symbolically Given series. Write out first several terms. Choose appropriate test to apply. Justify choice Why is it the best choice? Show that the series meets the conditions of the test. Clearly demonstrate application of test to the series. If possible, find sum and/or approximation with appropriate accuracy. If another series test may be used, apply it to support your result. Do this at least once. If no other test applies, choose at least two tests that may look like possibilities and explain why those test do not actually apply. (If appropriate, you may want to do this even if multiple tests do apply to your series. For example, an alternating series where the alternating series test fails, but both the n th -term test and ratio test apply.) Numerically Series Terms Write out the first several terms. Calculate decimal approximations for these terms. (Approximations should be accurate to at least three decimal places unless you need more to see the differences in the terms.) Partial Sums Calculate at least the first TEN partial sums ( S S S S ). You may want to program your calculator, use a spreadsheet, or use

43. Connected Calculus Functions; The Fundamental Theorem of calculus the HoHum Theorem of calculus; ClassifyingEquilibrium Points in Two Dimensions; Contents. series A Guide to http://www.math.montana.edu/frankw/ccp/calculus/topic.htm

The Connected Curriculum Project Contents Models, Data, and Curve Fitting Estimation and Limits Sequences and Discrete Dynamical Systems Derivatives and Differention ... Numerical Methods Models, Data, and Curve Fitting A Guide to this Chapter. The Mean and the Median Linear Models Linear Regression Quadratic Models Exponential Models Logistic Models Periodic Models Contents Estimation and Limits A Guide to this Chapter. Introduction When is it Safe to Drink the Water? A Descriptive Look Estimation and Limits Area ... Contents Sequences and Discrete Dynamical Systems A Guide to this Chapter. Quickstart Population Models Exponential Models Exponential Models with Immigration Linear Dynamical Systems ... Contents Derivatives and Differention A Guide to this Chapter. Introduction Curved Mirrors Comparing a Function with its Derivative Computing the Derivative as a Limit Visualizing the Derivative as a Limit Sometimes There Isn't a Tangent Interpretation Powers Linearity of the Derivative Polynomials Derivatives of the Trigonometric Functions Product Rule Quotient Rule Chain Rule Inverse Function Rule Derivatives of the Inverse Trigonometric Functions The Exponential Function Logarithms Implicit Differentiation Related Rates Mean Value Theorem Higher Order Derivatives Partial Derivatives Directional Derivatives The Gradient Contents Graphing A Guide to this Chapter.

44. Before Calculus PWS OnLine series calculus Modules OnLine Before calculus. The material SequencesPWS OnLine series - calculus Modules OnLine PWS http://www.scit.wlv.ac.uk/university/scit/maths/calculus/modules/topics/precalc/

PWS OnLine Series - Calculus Modules OnLine Before Calculus The material in this section is a brief review of the concepts you will need in order to understand more complicated topics in calculus, such as functions in calculus derivatives and integration. You are strongly advised to review the material in this section before starting elsewhere in the modules. Algebra: Solving polynomials using algebraic, numerical and graphical techniques, and observing the results of the Fundamental Theorem of Algebra Trigonometry: Basic manipulation of angles, triangles and circles, and the functions which describe these relationships, as well as their use in defining polar coordinates. Complex Numbers: Complex numbers allow us to express timing differences, square roots of negative numbers, and geometric properties in a compact notation. Conic Sections: The functions which result from planar sections through three-dimensional spaces serve a variety of purposes in calculus. Here you'll see how those functions are generated. Functions: An introduction the definition and graphs of polynomials, sinusoids, exponenents, and bell curves which can be used to describe physical and mathematical phenomena.

45. Functions PWS OnLine series calculus Modules OnLine FUNCTIONS. PWS OnLine series -calculus Modules OnLine Newton s Law * Optics * Supply and Demand http://www.scit.wlv.ac.uk/university/scit/maths/calculus/modules/topics/function

PWS OnLine Series - Calculus Modules OnLine FUNCTIONS An introduction to functions , and how they are used in calculus. Functions allow us to express a mathematical relationship between two quantities, which makes them very powerful tools for modeling and predicting. In the module on sequences , you'll see other ways to express mathematical relationships. Exponents and Logarithms: Transendental functions are useful in the study of physical and economic phenomena which grow and decay over time. Trigonometric Functions: Trigonometric functions are periodic relationships. They appear over and over in the study of calculus, since they describe a wide variety of physical phenomena. Plotting functions from data: Data is our evidence of real world phenomena. Fitting data with functions allows us to predict what will happen in new situations. This process is known as modelling. Properties and Manipulation of Functions: Functions have characteristics which allow us to predict their mathematical behavior, such as continuity and asymptotes, and we can perform functions on each other to generate new functions, take inverses, and other useful transformations. Where Am I?

46. Stewart Series Website Cole Stewart series Website. Click on the book you are using. Visit the AdvancedPlacement site to find valuable resources for teaching calculus in Advanced http://www.stewartcalculus.com/

@import url(http://www.brookscole.com/stylesheets/bonus.css); Home Contact Us Find Your Rep BookShop ... Brooks/Cole : Stewart Series Website Click on the book you are using. Visit the Advanced Placement site to find valuable resources for teaching calculus in Advanced Placement classes. Privacy Policy Thomson Learning is a division of The Thomson Corporation

47. UBC Calculus Online Course Notes value; The definite integral; The Fundamental Theorem of calculus; How to use Taylorpolynomials and series Introduction to series; Convergence of series; Finding http://www.ugrad.math.ubc.ca/coursedoc/math101/notes/

Course Notes Here is an interactive text to accompany the course. We invite you to take advantage of the many demonstrations to explore Calculus more deeply. Definite Integrals and the Fundamental Theorem of Calculus Introduction Overview of calculus from a historical perspective Some special sums How Archimedes found the area of a circle ... How to use the Fundamental Theorem Applications of the Definite Integral Acceleration, Velocity, and Position Average value of a function Calculating Simple Volumes Arc Length and Surface Area ... Work Techniques for Antidifferentiation Indefinite Integrals Change of Variables Integration by Parts More Applications Differential Equations Separable Differential Equations The Logistic Equation and Integration by Partial Fractions The Logistic Equation with Harvesting and Qualitative Methods More about Qualitative Methods for Differential Equations ... An Application: Can cats control a rat population? Weighted Averages Centers of Mass Probability Densities Means, Medians, and Averages Moments of a Probability Distribution ... The Normal distribution More Techniques for Integration Numerical Integration Improper Integrals Taylor polynomials and series Introduction to series Convergence of series Finding Taylor Series Why do Taylor series work?

48. Calculus Online: Lab 6 calculus Online Lab 6. Welcome to Lab 6 of Math 101, Sections 203, 204, 207and 209. Instructions. . which we call the Taylor series of f at x = 0. http://www.ugrad.math.ubc.ca/coursedoc/math101/labs/lab6/

Calculus Online: Lab 6 Welcome to Lab 6 of Math 101, Sections 203, 204, 207 and 209 Instructions Answer all the questions below. At the bottom of the page you will find a tool for saving your work or reloading it if you would like to change or add to it at a later time. Enter your login id and student number into the fields, and then use the Save and Load buttons. The last version that you save is the one that will be marked. Your work must be submitted by 11:59 P.M. on 9 April 1999. Warning! If you leave this page or reload it, your work will be lost. Always save your work before leaving or reloading. Overview You can't really do that many different kinds of numerical computations. In grade one, you learned how to add and subtract and in grade three how to multiply and divide. Since then, you really haven't learned much new. The reason why is that pretty much all numerical operations can be derived from these simple four. Even computers don't do much more than this. So how do we perform computations involving functions like sin cos , and tan ? In this lab we are going to investigate how to approximate functions like these using polynomial functions. Remember that polynomials are functions made by adding, subtracting, and multiplying numbers, all computations you could do by hand if you had to.

49. Calculus History this work contains the first clear statement of the Fundamental Theorem of the calculus. Newton swork on Analysis with infinite series was written in 1669 and http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/The_rise_of_calculus.html

A history of the calculus Analysis index History Topics Index The main ideas which underpin the calculus developed over a very long period of time indeed. The first steps were taken by Greek mathematicians. To the Greeks numbers were ratios of integers so the number line had "holes" in it. They got round this difficulty by using lengths, areas and volumes in addition to numbers for, to the Greeks, not all lengths were numbers. Zeno of Elea , about 450 BC, gave a number of problems which were based on the infinite. For example he argued that motion is impossible:- If a body moves from A to B then before it reaches B it passes through the mid-point, say B of AB. Now to move to B it must first reach the mid-point B of AB . Continue this argument to see that A must move through an infinite number of distances and so cannot move. Leucippus Democritus and Antiphon all made contributions to the Greek method of exhaustion which was put on a scientific basis by Eudoxus about 370 BC. The method of exhaustion is so called because one thinks of the areas measured expanding so that they account for more and more of the required area.

50. Difference Equations To Differential Equations The fundamental theorem of calculus, Cumulative area, 0407-02, 267 Polynomial approximationsand Taylor series, Polynomial approximations, Taylor polynomials, 04-07 http://math.furman.edu/~dcs/book/

Difference Equations to Differential Equations An introduction to calculus Difference Equations to Differential Equations is distributed under the GNU General Public License ("copyleft"). See below for details. Each section of the text is available in both PostScript and Portable Document Format (PDF) formats. If you require a PostScript viewer, click here for information on obtaining and installing a PostScript viewer. If you require a PDF viewer, click here or here for information on obtaining and installing a PDF viewer. Difference Equations to Differential Equations was written with the help of Tex DVIPS xdvi PDFTeX ... Mathematica A companion multi-variable calculus text, The Calculus of Functions of Several Variables is available here Send e-mail to Dan Sloughter to report any errors. Chapter Section Applet Date PostScript PDF Sequences, limits, and difference equations Calculus: areas and tangents Area of a circle Tangent line for a parabola 154 kb 103 kb Sequences 235 kb 128 kb The sum of a sequence 128 kb 109 kb Difference equations 210 kb 113 kb Nonlinear difference equations Inhibited population growth 291 kb 106 kb Functions and and their properties Functions and their graphs 406 kb 132 kb Trigonometric functions Square wave approximation Sound wave approximation 413 kb 138 kb Limits and the notion of continuity 355 kb 141 kb Continuous functions 188 kb 102 kb Some consequences of continuity 202 kb 105 kb Best affine approximations Best affine approximations Affine approximations 206 kb 111 kb Best affine approximations, derivatives and rates of change

51. Thomas' Calculus Skill Mastery Quizzes Skill Mastery Quizzes Chapter 8 Infinite series Choose a Quiz Please choosefrom the following five quizzes. Quiz 1, Quiz 2, Quiz 3, Quiz 4, Quiz 5. http://occawlonline.pearsoned.com/bookbind/pubbooks/thomas_awl/chapter1/medialib

Skill Mastery Quizzes Chapter 8 Infinite Series Choose a Quiz Please choose from the following five quizzes. Quiz 1 Quiz 2 Quiz 3 Quiz 4 ... Quiz 5

52. Some Calculus With Power Series. Previous Power series. Contents Some calculus with power series. Theorem10.4.1 Suppose that the power series is convergent on the interval . http://ndp.jct.ac.il/tutorials/Infitut1/node112.html

Next: Taylor series. Up: Series of functions. Previous: Power series. Contents Some calculus with power series. Theorem 10.4.1 Suppose that the power series is convergent on the interval . Its sum defines on that interval a function of the variable The function has derivatives of any order , and these derivatives are obtained by term-by-term differentiation, namely: These power series have the same interval of convergence as thge original power series. Theorem 10.4.2 Suppose that the power series is convergent on the interval and denote its sum by . Then the power series is convergent on the same interval and we have: Example 10.4.3 (A series for ) Let . This series is abolutely convergent for (v.s. Thm prop geometric series cv and Def def absolute cv). thus it is convergent; its sum is We integrate the function: Integrating the series term-by-term and equating both sides to for , we have: Theorem 10.4.4 (The product of two power series) Suppose that the two power series and are absolutely convergent for . Denote Then the series is absolutely convergent for and its sum is equal to Example 10.4.5

53. Some Calculus With Power Series. Some calculus with power series. Theorem 10.4.1 Suppose that the powerseries is convergent on the interval (x 0 R, x 0 +R). Its http://ndp.jct.ac.il/tutorials/Infitut1-win/node107.htm

Some calculus with power series. Theorem 10.4.1 Suppose that the power series is convergent on the interval x R x R ). Its sum defines on that interval a function f of the variable x The function f has derivatives of any order n , and these derivatives are obtained by term-by-term differentiation, namely: These power series have the same interval of convergence as the original power series. Theorem 10.4.2 Suppose that the power series is convergent on the interval x R x R ) and denote its sum by f x ). Then the power series is convergent on the same interval and we have: Example 10.4.3 (A series for tex2htm_wrap_inline$x$) Let x and Def ). Thus it is convergent; its sum is We integrate the function: Integrating the series term-by-term and equating both sides to for x =0, we have: Theorem 10.4.4 (The product of two power series) Suppose that the two power series and x R . Denote Then the series x R and its sum is equal to F x G x Example 10.4.5 Consider the two geometric series and , i.e. The product of these series is given by: i.e.

54. Soo Tan Series Website the author and the series, click on the appropriate Student Resource, InstructorResource, or Information links below Soo Tan Titles. calculus for the http://series.brookscole.com/tan/

@import url(http://www.brookscole.com/stylesheets/bonus.css); Home Contact Us Find Your Rep BookShop ... Brooks/Cole : Soo Tan Series Website Welcome to the Tan Applied Math Resource center, site of a wealth of resources, including online tutoring, practice tests and chapter reviews, as well as downloadable instructions for graphing calculators. To access these and other resources, or to see more information regarding the author and the series, click on the appropriate Student Resource, Instructor Resource, or Information links below Soo Tan Titles Calculus for the Managerial, Life, and Social Sciences (w/ InfoTrac) 6th Ed Student Resources ] [Instructor Resources] [ More Information Finite Mathematics for the Managerial, Life, and Social Sciences (w/ InfoTrac) 7th Ed Student Resources ] [Instructor Resources] More Information Applied Calculus for the Managerial, Life, and Social Sciences (with InfoTrac) Student Resources ] [Instructor Resources] More Information College Mathematics for the Managerial, Life, and Social Sciences (with InfoTrac) Student Resources ] [Instructor Resources] More Information Applied Mathematics for the Managerial, Life, and Social Sciences (with InfoTrac)

55. Leaving Cert. Higher Level Maths - Further Calculus And Series Index of applets from the Further calculus And series section of the TCDleaving cert higher level maths website. Further calculus And series. http://www.netsoc.tcd.ie/~jgilbert/maths_site/applets/further_calculus_and_serie

Search for: in Entire website Algebra Complex Numbers Matrices Sequences and series Differentiation Integration Circle Vectors Linear Transformations Line Geometry Trigonometry Probability Further Calculus and Series Website Home Algebra Complex Numbers Matrices ... Integration You are here: Home Category Index / Further Calculus And Series Further Calculus And Series Max/min Volume Problems - By Fergal Reid This Applet Demonstrates How Differnetial Calculus Can Quickly Solve Max/min Problems. It Provides An Illustration Of What It Is That Calculus Actually Helps Us Do, And The Power Of Calculus As A Tool For Finding Quantities Maximum Or Minimum Values, And The Conditions Under Width These Occur. [ View Applet Approximation Of Pi - By Birgit Boeckeler There Are Several Ways To Approximate Pi Mathematically, This Applet Uses The Most Illustrative One: It Draws Two Polygons And A Circle On The Screen, One Polygon Touching The Circle From The Outside, One From The Inside. The Polygons Double The Number Of Their Sides With Each Approximation, The Values Of Their Perimeters And Areas Getting More And More Close To Those Of The Circle. N.b.: The Applet Shows Just A Quarter Of The Drawing And Thus Uses A "default Zoom", As The Polygons Are Increasingly Difficult To Distinguish From The Circle. [ View Applet Maximum Area Problems - By Annie Bedford The Applet Shows The Effect Of Changing The Length And Width Of A Rectangle On The Area Of A Rectangle. The Graph Shows The Current Area's Position In Relation To The Maximum Area's Position. [

56. Karl's Calculus Tutor: Table Of Contents Improper Integrals; Areas and Volumes; Convergent series; More on Taylor Maclaurinseries; Sample Exam Questions on Integral calculus. PC Users To aid your http://www.karlscalculus.org/calculus.html

Karl's Calculus Tutor Table of Contents last update 19-Aug-03 IMPORTANT VIEWING NOTE: To properly view this page, open your view port out to at least->>>>here If you can't see the word " here " in the line above, place your mouse-cursor on the right-hand edge of the frame, hold the left mouse button, and drag the right-hand edge of the screen to the right until you can see it. See browser notes for more detials, especially if you don't see the Greek letter, omega (looks like a horseshoe), just below W click here Table of Contents Please note that Karl's Calculus Tutor is still a work in progress. Expect a new unit to come on line every month or so. Currently being drafted: Midterm Practice Exam Problems And vastly improved! I have completely revamped the Online Calculator . It is now much more user friendly and powerful. Be sure to read the Read me notes before trying any calculations. Please let me know if it helps with doing any of the on line problems (clicking will spawn a separate browser window). You can also try Online Complex Calculator , which can do arithmetic of complex numbers (I have not yet made the same improvements to the complex calculator, but that's coming).

57. Archive Of Reform Calculus Resources Some Good Cartoons (MAA Film series) for teaching Volumes (Stewart 6.2). A samplesecondsemester calculus course with in-class groupwork activities, projects http://barzilai.org/archive/

Reform Calculus Resources Compiled by Harel Barzilai As featured on UTK Math Archives and CollegeBoard.com's APCentral (* and authored, where not otherwise noted) Activities Projects Capsules Resources [See also College Algebra (in preparation)] [Under construction: Black Mathematicians Listings indexed by section in Calculus: Concepts and Contexts by James Stewart. If there is enough interest, I will create other indexes. Planned: on-the-fly indexes by keyword. In-Class Activities (Stewart 0.0) Graph Your Story A Week in the Life of Sue Estimating the Mile Record Guidelines and Solutions (Preview of Calculus)(Functions) (Stewart 1.1) Estimating the Mile Record Guidelines and Solutions (Preview of Calculus)(Functions) (Stewart 1.2) Composing Functions (Composition of Functions) (Stewart 1.4) Air Traffic Controller (Polar)(Parametric) A Crash Course in Calculus (Polar)(Parametric) (Stewart 1.5) Generation E: Modeling Exponential Growth (Exponential Functions)(Exponential Identities)(Exponential Growth) (Stewart 2.1) The Car Trip Solutions to The Car Trip Designing a Speedometer and A Second Train (Tangents)(Velocities)(Rates of Change)(Derivative) (Stewart 2.5)

58. MAA Calculus Film Series MAA calculus Film series. We use The Volume of a Solid of Revolution(Chandler Davis, George Leger), which runs 8 minutes, and Volume http://barzilai.org/cr/maa-films.html

MAA Calculus Film Series We use The Volume of a Solid of Revolution Chandler Davis, George Leger ), which runs 8 minutes, and Volume by Shells George Leger ), which runs about 8 and a half minutes. These are the third-to-last and second-to-last films on Tape 3 , respectively, starting at 0:29:00 and 0:36:55 respectively on that tape. If there is extra time, the last film on Tape 3, Infinite Acres Melvin Henriksen ) is both thought-provoking for the students, and humorous. Note however that the film was produced several decades ago, and that there a couple of instances of "humor" between the husband and wife, and boss and employee for that matter, that would not be considered so funny today. Finally, you can put some the tapes on reserve in the library for the students to review before exams and during projects, even suggesting they can view some of the other videos to brush up on other material such as the Definite Integral, Fundamental Theorem of Calculus, etc.

59. Syllabus For MAC2312, Spring 2000 10.8 Power series and Their Regions of Convergence 10.9 Representation of Functionsas Power series, calculus of Power series 10.10 Taylor and MacLaurin series http://www.math.princeton.edu/~bgb/mac2312/mac2312.html

Calculus II MAC 2312 (3104) Spring 2000 233 Little Hall, MTWF 6 Instructor Bernhard Bodmann, 459 Little Hall, 392 0281 x310, bgb@math.ufl.edu Textbook Calculus, Early Transcendentals (3rd edition), by James Stewart (1995) Course objective The main goals of this course are to master the basic integration techniques and apply them, familiarize yourself with infinite sequences and series, and enhance your skills in thinking, formulating, and writing more clearly, critically, and logically, which is useful in general problem solving and your future studies. Homework Selected problems will be worked out in class. In addition, you may consult the solutions key at the end of the book to check your answers. On several occasions, the even numbered problems given in in the course calendar below (in boldface) will be collected. Grading Your total score is composed of exam and quiz scores 3 x 100 pt. exams 300 points 7 x 30 pt. quizzes 150 (drop lowest two) Final exam TOTAL 600 points Final grades will be based on the total of 600 points. You are

60. ESAIM: Control, Optimisation And Calculus Of Variations calculus of variations minimization problems, existence and regularity propertiesof minimizers and This journal is part of the European series in Applied and http://www.edpsciences.org/journal/index.cfm?edpsname=cocv

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