These benefits are not something that come so naturally to people trained to be pure mathematicians.Д¬alculus 1, Lecture 16A: Continuous Growth Rates, Errors for Tangent Line Approximations, and NewtonвЂ™s Method Continuous (Instantaneous) Relative Growth Rates Indeed, this approach results in some of the main benefits that scientists, engineers, economists, etcвЂ¦ get out of learning calculus. I discussed this at the end of my previous blog post, вЂњDifferentiable Functions and Local LinearityвЂќ.ДЄ more serious benefit of Approach #2 is that it can give you insight into many applications of calculus.

You could call this approach Calculus Sans Limits. Part of the fun that arises from this approach is that calculus formulas can be derived without resorting to the use of limits. Approach #2 also has the benefit of being a lot of fun! - once you get used to it, at least. Most people donвЂ™t have the stomach for approach #1. They can be kept at an intuitive, but non-rigorous, level.They can be made rigorous through the arduous process of studying the subject of non-standard analysis.So if there are no such real numbers, how can they possibly be used? There are two approaches to the answer this question. Moreover, if, then, , etcвЂ¦ can be вЂњmuch smaller thanвЂќ itself (by many вЂњorders of magnitudeвЂќ). Given any real number, the number, but. There is also no smallest positive real number! Д«ut, first things first: there are no such real numbers! You cannot divide by infinity! 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 вЂњinfinitesimalsвЂќ, for short. Today, this intuitive method is called infinitesimal calculus. Instead, they approached calculus in an intuitive way. Did you know that Newton and Leibniz did not know the precise definition of a limit?

Are infinitesimals real