Calculus and Analysis in Mathematics

Calculus entails the study of change. It is commonly divided into two main branches: differential calculus and integral calculus. The two branches are related by the fact the integration and differentiation are inverse.
Mathematical analysis is the branch of pure mathematics that not only covers integral and differential calculus but also covers measure, infinite series, analytical functions, and limits. If you begin to study calculus, your success depends on your current knowledge and previous experience of both geometry and algebra.
When studying calculus and analysis, student’s ideas and knowledge about functions and their ability to work with algebraic expressions are important, as are their ideas of similarity, ratio, gradient, right-angled triangle, measure, and circle geometry. Students should not only be able to interpret graphs of functions but also know about trigonometric function, rational functions, and the relationship between logarithms and powers.
If students are taught calculus starting with the epsilon–delta definition of limits, they encounter difficulties. This has led to the development of other teaching approaches. One approach to differentiation referred to as ‘locally-straight,’ is based on the idea of magnifying a part of a graph of a function to see it approximate to a straight with a slope that can be measured. The ‘accumulation’ idea (a quantity described by its rate of change) is recommended for use when teaching integral calculus. These two approaches exploit computer environment to tackle multiple representations (symbolic, numeric, and graphical) of mathematical functions.
References
http://www.nuffieldfoundation.org/key-ideas-teaching-mathematics/calculus-and-analysis
http://www.math.harvard.edu/~shlomo/docs/Advanced_Calculus.pdf
http://mathworld.wolfram.com/Analysis.html

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