41. Calculus II (Math 2414) - Series & Sequences - Alternating Series Test Online Notes / calculus II (Math 2414) / series Sequences however we will need to resort to calculus I techniques to prove the series terms decrease. http://tutorial.math.lamar.edu/AllBrowsers/2414/AlternatingSeries.asp

MPBodyInit('AlternatingSeries_files') Paul's Online Math Notes  Alternating Series Test The last two tests that we looked at for series convergence have required that all the terms in the series be positive.  Of course there are many series out there that have negative terms in them and so we now need to start looking at tests for these kinds of series. The test that we are going to look into in this section will be a test for alternating series.  An alternating series is any series, MPSetEqnAttrs('eq0001','',3,[[23,14,5,-1,-1],[32,18,5,-1,-1],[39,24,8,-1,-1],[],[],[],[98,64,19,-3,-3]]) , for which the series terms can be written in one of the following two forms. MPSetEqnAttrs('eq0001','',3,[[23,14,5,-1,-1],[32,18,5,-1,-1],[39,24,8,-1,-1],[],[],[],[98,64,19,-3,-3]]); MPNNCalcTopLeft(document.mpeq0001ph,'1') MPDeleteCode('eq0001') MPSetEqnAttrs('eq0002','',3,[[135,36,16,-1,-1],[181,50,21,-1,-1],[227,62,27,-1,-1],[],[],[],[566,155,69,-3,-3]]) MPSetEqnAttrs('eq0002','',3,[[135,36,16,-1,-1],[181,50,21,-1,-1],[227,62,27,-1,-1],[],[],[],[566,155,69,-3,-3]]); MPNNCalcTopLeft(document.mpeq0002ph,'1')

42. Calculus Calculators online calculus calculators. Power series Approximation This tool computes the first N terms of the power series of a function. http://www.ifigure.com/math/calculus/calculus.htm

your source for online planning, calculating and decision-making Home Plan Calculate Convert ... Decide Mathematics Basic Math Algebra Geometry Trigonometry ... Tutorials Guides Math Guide Physics Guide Chemistry Guide Computer Science ... Geology Guide Calculus Calculators Sequences, Series and Limits Calculating and Graphing Sequences "This tool computes the elements in a sequence a(n) for a range of n. It will also plot the graph of the sequence." Computing a Sum Symbolically sum a sequence of numbers or functions by specifying the terms in the expression, the index and the starting and ending values of the index. Power Series Approximation "This tool computes the first N terms of the power series of a function." It also creates a plot of the result. Computing a Limit "This tool attempts to compute the limit of a function....There is an option that plots the graph of the function in a neighborhood of point where the limit is being taken." Differentiation Finding a Derivative Shows how apply the power rule, product rule and chain rule to find the derivative. Differentiation with the Quotient Rule Shows how to use the quotient rule to find the derivative of fractional expressions.

43. The Math Forum - Math Library - Sequences/Series Sequence and series (calculus@internet) WebPrimitives, Cambridge, Massachusetts Links to Web pages for resources for sequences and series in calculus. http://mathforum.org/library/topics/sequence_series/

Browse and Search the Library Home Math Topics Calculus (SV) : Sequences/Series Library Home Search Full Table of Contents Suggest a Link ... Library Help Selected Sites (see also All Sites in this category On-Line Encyclopedia of Integer Sequences - Neil J. A. Sloane Before publishing that new sequence you have just found, you might want to check here to see if it's already been discovered. Directions for how to look up a sequence in the table, abbreviations used, sending in a new sequence; how to look up a reference or search for a word; puzzle sequences, classic sequences, hot sequences. Version available in French. more>> Sequence and Series (calculus@internet) - WebPrimitives, Cambridge, Massachusetts Links to Web pages for resources for sequences and series in calculus. more>> Sequences, Series, Summability - Dave Rusin; The Mathematical Atlas A short article designed to provide an introduction to sequences and series, the most common examples of limiting processes; convergence criteria and rates of convergence are as important as finding "the answer." Particular series of interest (e.g. Taylor series of known functions) are of interest, as well as general methods for computing sums rapidly, or formally. Series can be estimated with integrals, their stability can be investigated with analysis. Manipulations of series (e.g. multiplying or inverting) are also of importance. History; applications and related fields and subfields; textbooks, reference works, and tutorials; software and tables; other web sites with this focus.

44. The Math Forum - Math Library - Calculus (SV) This page contains sites relating to calculus (Single Variable). sequences and series, multivariable calculus, differential equations); http://mathforum.org/library/topics/svcalc/

Browse and Search the Library Home Math Topics : Calculus (SV) Library Home Search Full Table of Contents Suggest a Link ... Library Help Subcategories (see also All Sites in this category Differentiation Integration Sequences and Series Selected Sites (see also All Sites in this category Calculus and Differential Equations (MathPages) - Kevin Brown About 40 "informal notes" by Kevin Brown on calculus and differential equations: limit paradox, proofs that pi and e are irrational, Ptolemy's Orbit, leaning ladders, how Leibniz might have anticipated Euler, and many more. more>> Calculus - Math Forum Links to some of the best Internet resources for calculus: classroom materials, software, Internet projects, and public forums for discussion. more>> Calculus on the Web (COW) - Mendoza, Reich; Department of Mathematics, Temple University A collection of auto-scoring calculus modules organized into books, chapters, and sections, with help and hints for the problems. Modules with asterisks (including chain rule, Taylor polynomials, Riemann sums, and arc length) allow you to change values and see the effect. Book I: functions and geometry; limits and continuity; the derivative; techniques and theory of differentiation; applications of the derivative; and integration. Book II: integration; applications of integration; transcendental functions; methods of integration; geometry, curves, and polar coordinates; and sequences and series. Book III: sequences and series; vectors and analytic geometry; curves; functions; and integration. Registered students can log in for a session in which their work will be recorded and graded.

45. Search Results For Calculus Of Variations Iterative Methods and the Neumann series * 2 The calculus of Variations * 2.1 Mods * Analysis 1 Sequences and series * Analysis 2 Continuity and http://www.maths.ox.ac.uk/cgi-bin/htdig/htsearch?words=calculus of variations;pa

46. 3.5.2 Fourier Series And Two Variable Calculus -- Dr Dyson -- 16 HT 3.5.2 Fourier series and Two Variable calculus Dr Dyson 16 HT. http://www.maths.ox.ac.uk/current-students/undergraduates/handbooks-synopses/200

Next: 3.5.3 Partial Differential Equations Up: 3.5 Applied Mathematics II Previous: 3.5.1 Calculus of Two Contents Subsections 3.5.2.1 Synopsis 3.5.2.2 Reading 3.5.2 Fourier Series and Two Variable Calculus Dr Dyson 16 HT The first half of these lectures introduce students to Fourier series, concentrating on their practical application rather than proofs of convergence. The second half deals with scalar functions of two independent variables. Students will learn how to evaluate area and line integrals and how they are related via Green's theorem. They will also be introduced to the ideas of continuity and differentiability of two-variable functions and shown how classify critical points using Taylor's theorem. 3.5.2.1 Synopsis Fourier series. Periodic, odd and even functions. Calculation of sine and cosine series. Simple applications, concentrating on imparting familiarity with the calculation of Fourier coefficients and the use of Fourier series. Issues of convergence addressed through example and special cases. No attempt to state or prove convergence theorems, though students should be aware that convergence is linked to smoothness, referring to Maple exercises, e.g., exploring the Gibbs phenomenon.

47. Sequences -- Chapter Guide PWS OnLine series calculus Modules OnLine Suggestions for using this module with calculus in a Real and Complex World by Frank Wattenberg, calculus by http://www.scit.wlv.ac.uk/university/scit/maths/calculus/modules/topics/discdynm

PWS OnLine Series - Calculus Modules OnLine SEQUENCES AND DISCRETE DYNAMICS Quickstart: This module introduces some general terminology for sequences using bank accounts and other financial examples. Population Models Exponential Models: This module introduces discrete dynamical systems, population models, and exponential models. Exponential Models with Immigration: Discrete population models based on exponential growth rates with immigration. Discrete Logistic Models: Models with a linear growth rate. Pack-Hunters: Models whose growth rate reflects species that cooperate in some way for example, by hunting in packs. Discrete Dynamical Systems and Difference Equations: Describing change using systems of discrete data points. Equilibrium Points: Algebraic, graphical, and numerical techniques for studying equilibrium points. Cobweb Diagrams: This module introduces cobweb diagrams. These diagrams provide a visual and geometric perspective on dynamical systems. They will give us insight into the nature of an equilibrium point. Long Term Behavior: An overview on the different kinds of longterm behavior exhibited by discrete dynamical systems. Limits, Cycles and Chaotic behavior are examples.

48. Before Calculus 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 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.

49. "Calculus 2 Notes" Webpage calculus 2 Class Notes. Copies of the classnotes are on the internet in PDF and Section 8.5 Alternating series, Absolute and Conditional Convergence. http://www.etsu.edu/math/gardner/1920/notes.htm

Calculus 2 Class Notes Copies of the classnotes are on the internet in PDF and Postscript formats as given below. I have included versions with both color figures and black and white figures (the "black and white" files are roughly 1/3 the size of the "color" files). Files with Black and White Figures (or No Figures at All) Chapter 5: Applications of Integrals. Section 5.4: Springs, Pumping, and Lifting. PDF. PS. Section 5.5: Fluid Forces. PDF. PS. Section 5.6: Moments and Centers of Mass. PDF. PS. Chapter 6: Transcendental Functions and Differential Equations. Section 6.1: Logarithms. PDF. PS. Section 6.2: Exponential Functions. PDF. PS. Section 6.3: Derivatives of Inverse Trigonometric Functions; Integrals. PDF. PS. Section 6.4: First-Order Seperable Differential Equations. PDF. PS. Section 6.5: Linear First-Order Differential Equations. PDF. PS. Section 6.6: Euler's Method: Population Models. PDF. PS. Section 6.7: Hyperbolic Functions. PDF. PS. Chapter 7: Integration Techniques, L'Hopital's Rule, and Improper Integrals. Section 7.1: Basic Integration Formulas.

50. Infinite Series Presentations - Calculus Show that the series meets the conditions of the test. Presentation Forms Teaching and Learning - calculus C Home Page - Project Information 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

51. Lecture Notes And Calendar calculus, Schaum s outlines series, fourth edition, by Frank Ayeres Jr. an Elliot MEndelson, McGrawHill 2000 edition. 2. Thomas calculus , 11th edition, http://www.fizik.usm.my/tlyoon/teaching/ZCA110_05/zca110_05.htm

ZCA 101/4 KalKulus dan Aljabar Linear course webpage for 1st semester, academic session 2005/06, School of Physics, USM, Penang, Malaysia The diagram illustrates the local accuracy of the tangent line approximation to a smooth curve Credit: Douglas N. Arnold http://www.ima.umn.edu/~arnold/graphics.html Lecturers: Yoon Tiem Leong (School of Physics, USM, Room 115, Tel:04-6533674)(Calculus); (2) Dr. Rosy Teh (Linear Algebra) Course Meeting Times Lectures: Three lectures / week, 50mins / lecture 1) Monday, DKW, 11.00-11.50 am; 2) Tuesday, DKW, 10.00-10.50am; 3) Thursday, DKA, 14.00-14.50 pm Tutorials : Friday, 10.00-10.50 am Calendar tutorial ZCA 110 forum Links to calculus materials ... announcement Please feel free to send me feedback Course Description This is a Kursus Teras offered by school of physics. Students who take this course cannot simultaneously take MAA101/4 and MAA 111/4 because these courses overlap with ZCA 110. Two lecturers will share the teaching load.

52. Alex Suciu: Calculus 3 calculus 1 syllabus. Most students find infinite series to be the single hardest topic in first and second-year calculus. http://www.math.neu.edu/~suciu/mth1125/calc3.sp02.html

Professor Alexandru I. Suciu MTH 1125 - Calculus 3 Spring 2002 Course Information Course: Sections: Seq. 09 (key # 06275) and Seq. 10 (key # 09924) Instructor: Prof. Alex Suciu Course Web Sites: All sections: www.math.neu.edu/undergrad/mth1125 These 2 sections: www.math.neu.edu/~suciu/mth1125/calc3.sp02.html Time and Place: Seq. 09: Tue, Fri 10:3011:35am and Wed 4:055:10pm, in 209 KA Seq. 10: Tue, Wed, Fri 11:45am12:50pm, in 209 KA Office: 441 Lake Hall. Phone: (617) 373-4456 or (617) 373-2450 for messages. Email: alexsuciu@neu.edu Office Hours: Tue., Fri. 9:30-10:30 and Wed. 10:45-11:45 in 441 Lake. Extra Help: Mon., Th. 11:45am-1:15pm, and Th. 6:40pm-7:40pm, in 536 Nightingale: office hours conducted by Abdallah Abou-Tahoun aaboutah@lynx.neu.edu Prerequisites: Textbook: Thomas' Calculus, Early Transcendentals, 10th Edition, by Finney, Weir, and Giordano Calculator: Scientific, graphing calculator (recommend TI83 or higher) Grade: 60% in-class exams, 40% final exam Course Objective This course has two main goals: to have students understand the concepts of differential equations and infinite series, and to enable students to display that understanding through a variety of applications. Specific, measurable, manifestations of your understanding that will be tested during the quarter include your ability to:

53. Facets Multi-Media - Search: Category1=262 Mathematics series, calculus I. Instructors guide these comprehensive video review courses, which can be used either in the classroom or studied http://www.facets.org/asticat?function=search&mnu=filmlistsx&searchmode=4&search

54. Advanced Math Software This pre calculus series is also appropriate for post high school algebra The first four educational software programs in this pre calculus series are a http://www.mathmedia.com/trigonometry.html

Pre Calculus and Trigonometry Mathematics education is enhanced with this mathematics software. College Prep Math Software programs improve students' math skills. Suitable for high school or developmental college students. This pre calculus and trigonometry series contains seven advanced mathematics education programs: Conic Sections Calculus Basics The Advanced Mathematics Software Series is for continuing Advanced Algebra, Trigonometry, and Pre-Calc students. This advanced math education series is also appropriate for post high school algebra students working on their mathematics education. Graphing calculator lessons are scattered throughout the 7-program series - keep up with today's math education usage of graphing calculators. The first four programs in this mathematics education software series are a continuation of the Algebra by Chapter series. The Conic Section program provides in-depth practice to understanding, calculating, and graphing conic sections. The Exponent and Logarithm program teach beginning through advanced exponents with log functions calculated both by hand and by calculator. Sequences and Series provide interactive practice with formulas leading to the more advanced topics of binomial expansions. Understanding and using Matrices and Determinants is also included math education software series. Each of these math education programs has a word problem section and concludes with a comprehensive test of the material learned in the program. This mathematics education software series also contains Trigonometry 1 and Trigonometry 2 together form a complete course and can be used to prepare for the CLEP exam or prepare for college math placement exams. Concentrated learning of Trigonometric functions provides the student with ample practice to form a solid understanding of Trig. Each of these two Trig programs concludes with a 30-question test with hints and complete solutions accessed upon student request.

55. Math Applets At SLU for Mathematics project at Hobart and William Smith Colleges., that produced a series of JCMApplets that are quite useful for calculus and precalculus. http://www.slu.edu/classes/maymk/MathApplets-SLU.html

Saint Louis University Department of Mathematics and Computer Science Math Applets at SLU These following collection of applets are designed for use in mathematics courses. Some of the applets were developed at SLU and some have been developed elsewhere and are included by permission. If you would like to host the applets locally, please contact Mike May, S.J. Applets for single variable calculus Preliminary material The Family of Graphs Applet is a JCM applet designed to look at families of functions. It graphs functions that include the three parameters a, b, and c in their definitions. These parameters are controlled by sliders. Moving the sliders lets the student explore families of functions. The Function Composition Applet is a JCM applet that links together the graph of two functions with the graph of the function defined by the composition of these two functions. The Visual Composition of Functions Applet is a Banchoff applet that looks at composition of functions by using three variables, with y=g(x) and z=f(y). The projection of the curve obtained from the intersection of the two surfaces is then either the graph of f, g, or fg, depending on the plane it is projected onto. Continuity The Epsilon Delta Applet is is a JCM applet designed for a visual exploration of the delta-epsilon definition of continuity. The user highlights an epsilon and delta band around a proposed limit of a function at a point. It is easy to zoom in or out, and the applet has a nice collection of pre-set examples.

56. Connected Calculus The Fundamental Theorem of calculus the HoHum Theorem of calculus Power series; Power series Solutions of Initial Value Problems; Taylor series 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.

57. Calculus History The main ideas which underpin the calculus developed over a very long period of This is the first known example of the summation of an infinite series. http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/The_rise_of_calculus.html

A history of the calculus Analysis index History Topics Index Version for printing 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.

58. Calculus Animations With Mathcad calculus Animations with Mathcad. by Przemyslaw Bogacki and Gordon Melrose Currently, animations corresponding to the following pseries are available http://www.math.odu.edu/cbii/calcanim/

Calculus Animations with Mathcad by Przemyslaw Bogacki and Gordon Melrose Contents Limits Slope vs. Derivative Arc Length Infinite Series ... Quadric Surfaces If your browser displays incorrect colors after animation playback, look here Limits The statement can be intuitively interpreted as follows: the number L is approached by the function values f(x) corresponding to x values that approach c Two examples are illustrated: Slope vs. Derivative The top graph depicts the given function, and a triangle illustrating the rise corresponding to the run=1. As the animation advances, the triangle moves along the graph, and the rise/run values are used to create the derivative graph underneath. Note that the positive rise/run values are indicated in green; negative values are indicated in red. This animation can be viewed for the following functions: Arc Length The arc length of the graph of y=f(x) from x=a to x=b can be defined using the the length of a polygonal approximation to the graph. The first example illustrates a sequence of polygonal approximations of a rectifiable curve, i.e., a curve with a finite arc length. In the second example, the curve has infinite length.

59. Difference Equations To Differential Equations A companion multivariable calculus text, The calculus of Functions of and Taylor series, Polynomial approximations, Taylor polynomials, 04-07-02 http://math.furman.edu/~dcs/book/

Difference Equations to Differential Equations An introduction to calculus 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 129 kb The sum of a sequence 128 kb 110 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 103 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

60. Calculus With Analytic Geometry calculus of a Single Variable Multivariable calculus, Seventh Edition Understand the definition of a convergent infinite series. http://college.hmco.com/mathematics/larson/calculus_analytic/7e/students/downloa

Appendices Chapter Summaries Chapter 15 Differential Equations 3-D Rotatable Graphs ... Instructional DVDs Textbook Site for: Calculus with Analytic Geometry Calculus of a Single Variable Multivariable Calculus , Seventh Edition Ron Larson - The Pennsylvania State University, The Behrend College Robert P. Hostetler - The Pennsylvania State University, The Behrend College Bruce H. Edwards - University of Florida Chapter Summary Chapter 8: Infinite Series List the terms of a sequence. Determine whether a sequence converges or diverges. Write a formula for the n th term of a sequence. Use properties of monotonic sequences and bounded sequences. Understand the definition of a convergent infinite series. Use properties of infinite series. Use the n th -Term Test for Divergence of an infinite series. Use the Integral Test to determine whether an infinite series converges or diverges. Use properties of p -series and harmonic series. Use the Direct Comparison Test to determine whether a series converges or diverges. Use the Limit Comparison Test to determine whether a series converges or diverges.

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