PennCalc | Main / Calculus Single Variable

Time 2020-09-29 12:03:59

Web Name: PennCalc | Main / Calculus Single Variable

WebSite: http://calculus.seas.upenn.edu

ID:63949

Keywords:

Main,PennCalc,Calculus,

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About the PennCalcWiki - PennCalc SVC Table of ContentsIntroductionFunctionsDifferentiationIntegrationApplicationsDiscretizationUPDATE: August 2018: The PennCalcWiki is broken, thanks to a server upgrade to PFP-7 which renders the skins and latex plugin unusable. Sadly, this wiki cannot easily be converted to be PHP-7 compatible. You might want to try the wayback machine. IntroductionIntroduction - Prerequisites and general info FunctionsFunctions - Definition and examples of functionsThe Exponential - The exponential function definedTaylor series - The Taylor series defined and appliedComputing Taylor series - Using composition to compute Taylor seriesConvergence - Problems with some Taylor seriesExpansion points - Taylor series expansions about other pointsLimits - Definition of limit and continuityL'Hopital's Rule - Statement and examples in a Taylor series contextOrders of growth - Relative growth of the most common functions DifferentiationDerivatives -Definition and interpretations of the derivativeDifferentiation rules -Rules for differentiating combinations of functionsLinearization -First order Taylor approximationsHigher derivatives -Definition and interpretation of higher derivativesOptimization -Classifying critical points and finding extremaDifferentials -Implicit differentiation and related ratesDifferentiation as an operator -Using operators to compute other derivatives IntegrationAntidifferentiation -The indefinite integral and separable differential equationsExponential growth examples -More examples of exponential growth and decayMore differential equations -Linear first order differential equationsODE Linearization -Solving harder differential equationsIntegration by Substitution -Substitution as an integration techniqueIntegration by parts -Using the product rule as an integration techniqueTrigonometric substitution -Integration using trigonometric substitutionsPartial fractions -Integration of rational functions using algebraDefinite integrals -Definition and interpretation of the definite integralFundamental Theorem of Integral Calculus -Connecting definite and indefinite integralsImproper integrals -Computing definite integrals when FTIC does not applyTrigonometric integrals -Products and powers of trigonometric functionsTables and computers -Using tables of integrals and mathematics software ApplicationsSimple Areas -Finding the area of regions in the planeComplex Areas -Areas of more complex regions in the planeVolumes -Using the volume element to compute volumeVolumes of revolution -Volumes from revolving a region about an axisVolumes in arbitrary dimension -Fourth dimension and beyondArclength -Finding the length along a curveSurface area -Surface area of a solid of revolutionWork -Computing work with integrationElements -Pressure, force, and other applicationsAverages -The average value of a function over an intervalCentroids and centers of mass -Finding centroid and center of mass with integrationMoments and gyrations -Moment of inertia and radius of gyrationFair probability -Uniform distributionProbability densities -Using the density function to compute probabilitiesExpectation and variance -Properties and interpretations of probability distributions DiscretizationSequences -Discrete-input functionsDifferences -Derivatives of sequencesDiscrete calculus -How to integrate discrete functionsNumerical ODEs -Using sequences to solve ODEsNumerical integration -Using sequences to solve definite integralsSeries -Infinite series as improper discrete integralsConvergence tests 1 -Comparison-type testsConvergence tests 2 -Geometric series-type testsAbsolute and conditional -Two types of series convergencePower series -Interval and radius of convergenceTaylor series redux -Details about Taylor series convergenceApproximation and error -How to estimate an infinite seriesCalculus -What we can and cannot do...for now Wrap-UpForeshadowing -Towards multivariable calculusIntroduction loop trail

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