Calculus I

Calculus is a branch of mathematics that deals with the finding and properties of derivatives and integrals of functions, by methods originally based on the summation of infinitesimal differences. The two main types are differential calculus and integral calculus.

In single-variable calculus, we deal with the following topics:

  • One-variable functions and their graphs

  • Polynomial, trigonometric, radical, Inverse, and implicit functions

  • Limit of a function and limit laws

  • The precise definition of a limit

  • One-sided limits

  • Continuity of functions

  • Limits involving infinity and asymptotes in graphs

  • Tangent lines and the derivative at a point

  • Differentiation rules

  • Derivatives of trigonometric, exponential, and logarithmic functions

  • The chain rule and implicit differentiation

  • Applications of derivatives including extreme values of functions, turning points and concavity, the mean value theorem, and so on.

  • The definite integral and the fundamental theorem of calculus

  • Antiderivatives and indefinite integrals

  • The area between two curves

  • Applications of definite integrals including volume of a solid of revolution, area of a surface of revolution, arc length, and so on.

  • Techniques of integrations such as integration by parts, integration by substitution, integration of rational functions by partial fractions, and so on.

  • Definition of infinite sequences and series

  • Convergence tests such as comparison tests, integral test, ratio and root tests, Leibniz test for alternating series, and so on.

  • Power series and the radius of convergence, Taylor and Maclaurin series

  • Parametric equations and polar coordinates system

  • Complex numbers

For more details, refer to the following book:

Thomas, G. B., Weir, M. D., Hass, J., & Giordano, F. R. (2005). Thomas' calculus. Addison-Wesley.

I also recommend the following book:

Differential and Integral Calculus by Grigorii Mikhailovich Fichtenholz

As one of the founders of the Leningrad (today known as Saint Petersburg) school of "real analysis", Grigorii Mikhailovich Fichtenholz (1888-1959) was a Soviet mathematician. In addition to several books on mathematical analysis, Fichtenholz (also known as Fikhtengolts) wrote a three-volume textbook on differential and integral calculus. This textbook discusses the mathematical analysis of functions of one real variable, functions of many real variables, and complex functions. This well-presented book on mathematical analysis has not yet been translated into English, although it has been translated into Chinese, German, Persian, Polish, and Vietnamese.

The story behind its Persian translation is quite interesting. The late "Bagher Emami باقر امامی" who was a university lecturer in mathematics started a translation of this volumous book. However, the translation was later completed by another person. "While translating almost two volumes of the book, Emami gave his whole work to me and asked me to translate the rest and publish this complete course on mathematical analysis", Ostad "Parviz Shahriari پرویز شهریاری" explains in the preface of the book. Ostad Parviz Shahriari finished the work, but it took some time to find a publisher because producing such a book was expensive. Finally, Ferdos Publications published the book and this significant book on mathematical analysis (differential and integral calculus) became available for Persian speaking people.

Multi-variable Calculus

Ordinary Differential Equations

Engineering Mathematics