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  • Topic entry on real numbers
  • real function
  • Limit rules of functions
  • Limit examples
  • Improper limits
  • Growth of exponential function
  • discontinuous functions
  • Continuity of natural power
  • Continuity of sine and cosine
  • Jump discontinuity
  • Intermediate value theorem
  • Derivative

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GeezerVison is showing planetmath.org — topics on calculus

topics on calculus


This entry is an overview of many calculus related entries which can be found here, at PlanetMath.org. By calculus we real analysis at the high-school level or college level, and the entries in this page should be at either level. If an entry is written at a higher level, it will be indicated with a “GL” tag.

1 Expositions

The following are books or notes on calculus:

  • •

    http://planetmath.org/?op=getobj&from=lec&id=36An Introduction to Calculus.

2 ℝ : The real line

Basic properties of the real numbers:

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    Decimal expansion

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    Positive

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    Inverse number, Opposite number

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    (GL) http://planetmath.org/node/RealNumberReal number

  • •

    Topic entry on real numbers

3 Functions of one real variable

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    Concept of functionMathworldPlanetmath and real function

4 Computing limits

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    Limit of function

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    Limit of (sin x)/x as x tends to 0 (http://planetmath.org/LimitOfDisplaystyleFracsinXxAsXApproaches0)

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    Limit rules of functions

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    Limit examples

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    Improper limits

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    L’Hôpital’s rule (http://planetmath.org/LHpitalsRule)

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    Growth of exponential function

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    Example of computing limits using Taylor expansionMathworldPlanetmath

5 Continuity

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    ContinuousMathworldPlanetmath and discontinuous functions

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    Continuity of natural power

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    Continuity of sine and cosine

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    Jump discontinuity example (http://planetmath.org/ExampleOfJumpDiscontinuity)

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    Intermediate value theorem

6 Differentiation in one variable

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    Derivative

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    Sum ruleMathworldPlanetmath

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    Power ruleMathworldPlanetmathPlanetmath

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    Product ruleMathworldPlanetmath

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    Quotient ruleMathworldPlanetmath

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    Chain ruleMathworldPlanetmath

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    Derivative of inverse function

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    Derivatives of sine and cosine

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    Bolzano’s theorem

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    Least and greatest value of function

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    Higher order derivatives

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    Fractional Differentiation

7 Integration of functions of one real variable

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    Integral sign

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    Definition of Riemann integral (http://planetmath.org/RiemannIntegral)

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    A lecture on integration by substitution

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    A lecture on integration by parts

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    A lecture on trigonometric integrals and trigonometric substitutions

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    A lecture on the partial fraction decomposition method

8 Definite integral

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    Definite integral

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    Riemann integral

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    Fundamental theorem of calculusMathworldPlanetmathPlanetmath

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    Improper integral

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    List of improper integrals

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    Approximate integration: Left Hand Rule Right Hand Rule Midpoint Rule Trapezoidal Rule Simpson’s Rule

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    Integral equation

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    Fractional Integration

9 Integral Transforms

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    Fourier transformDlmfMathworldPlanetmath

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    Laplace transformDlmfMathworldPlanetmath

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    Fourier-Mellin integralMathworldPlanetmath

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    Mellin’s transform

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    Fourier sine transformDlmfMathworld

  • •

    Fourier cosine transformDlmfMathworld

10 Multivariable Calculus

10.1 Differentiation

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    Iterated limit

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    Jacobian matrix

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    Differentiation under the integral sign

10.2 Integration

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    Stokes’ theorem

11 Differential Equations

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    Differential equationMathworldPlanetmath

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    (GL) Existence and uniqueness of solution of ordinary differential equations

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    Index of differential equations

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    Separation of variablesMathworldPlanetmath

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    Method of integrating factors

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    (GL) Examples of solving a PDE:  a) Heat equation (http://planetmath.org/ExampleOfSolvingTheHeatEquation), b) Wave equationMathworldPlanetmath (http://planetmath.org/SolvingTheWaveEquationByDBernoulli)

12 Infinite Series

12.1 Series of Numbers

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    Series

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    Topic entry on series of complex terms

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    http://planetmath.org/?op=getobj&from=lec&id=37Infinite Series: Tests for Convergence and Examples

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    (GL?) Non-existence of universal series convergence criterion

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    Cauchy general condition for convergence (http://planetmath.org/CauchyCriterionForConvergence)

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    Geometric seriesMathworldPlanetmath, Harmonic seriesMathworldPlanetmath

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    Sum of series depends on order

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    Manipulating convergent series

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    (GL?) Multiplication of series

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    Leibniz’ estimate for alternating seriesMathworldPlanetmath (http://planetmath.org/LeibnizEstimateForAlternatingSeries)

12.2 Function Sequences and Series

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    Limit of function sequence

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    The limit of a uniformly convergent sequence of continuous functions is continuous

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    Sum function of series

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    Termwise differentiation

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    Weierstrass’ criterion of uniform convergence

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    (GL) Fourier series (http://planetmath.org/FourierSineAndCosineSeries)

12.3 Power Series and Taylor Series

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    Power seriesMathworldPlanetmath and Taylor series

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    Taylor’s theorem

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    Newton’s binomial series (http://planetmath.org/BinomialFormula)

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    http://planetmath.org/node/ExampleOfTaylorPolynomialsForSinXExample of Taylor polynomials for sin⁡x

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    Example of Taylor polynomials for the exponential functionDlmfDlmfMathworldPlanetmath

  • •

    Examples on how to find Taylor series from other known series

  • •

    Getting Taylor series from differential equation

13 Additional Topic

  • •

    Non-Newtonian calculus

Title topics on calculus
Canonical name TopicsOnCalculus
Date of creation 2013-03-22 15:39:33
Last modified on 2013-03-22 15:39:33
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 65
Author alozano (2414)
Entry type Topic
Classification msc 97A80
Related topic CommonFormulasInCalculusOfDifferentialForms
Related topic NumerableSet
Related topic NonNewtonianCalculus2
Defines Calculus
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