If you really think they'll work for everybody, I'm suspicious. It is troubling how widespread misunderstanding of calculus is 150 years later. They got rid of the infinitesimal business once and for all, replacing infinitesimals with limits. Some want their mathematics pure and others find it dry as dust if there isn't real world motivation.It would help, I think, if you indicate who your recommendations are for. During the 1800s, mathematicians, and especially Cauchy, finally got around to rigorizing calculus. Some want an answer key and others find it too tempting and prefer it doesn't exist. It can provide detailed step-by-step solutions to given differentiation problems in a tutorial-like format. Some like exercises aplenty, and others prefer a few well-chosen problems. Solve any calculus differentiation problem with this calculus tutorial software.Calculus Problem Solver can solve differentiation of any arbitrary equation and output the result. Some need rigor and others prefer intuition. I've found that people have different styles and need different things. However, the impression I get is that you think these textbook suggestions are right for everybody. (controversial at best)But I agree those are not representative of the whole piece. (highly controversial)So it is very beneficial to learn the nonstandard approach. It's a meritorious effort and will be helpful to many, I'm sure.My impression regarding it being presented as the "one true way" came from these statements:The best calculus book is undoubtly…. Lest I be misunderstood in offering criticism, let me say thank you for doing this.
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G) Maximization problems including Lagrange multipliers Prerequisites: single-variable calculusĪ) Basic spatial geometry, like parametrization of lines and curvesī) Limits and continuity in multivariable functionsĬ) Differentiation of multivariable functionsĭ) Integration of multivariable functions Prerequisites: Differentiation and integrationį) Integration and differentiation with seriesĮverything in single variable calculus of course also works in multiple dimensions. For example, the sine and the logarithm functions can be approximated very nicely with series. For example, to find dy/dx for y x 2: Let x change by ±dx ☑/n. Can one 'do calculus' with high school algebra Yes.
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In a sense, you can think of them as quantities of the form. These are quantities so small that they are smaller than any positive real number. It is based on the concept of infinitesimal quantities, or just infinitesimals, for short.
Does calculus need infinitesimals series#
Sequences and series are important for approximating certain functions. The fact that my 'lesser extension' of R than Robinsons R does not include such infinitesimals as 10 n does not make the infinitesimal polynomial ratios any less useful. Today, this intuitive method is called infinitesimal calculus. Integration is used to find areas, lengths, and so much more.į) Application of integration to finding areas, volumes and lengths Integration is the opposite process of differentiation. You can deduce surprisingly many facts from this procedure. Differentiation is finding the tangent line to a specific function.