Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modernmathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus. Calculus is the study of change, in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. Calculus has widespread applications in science, economics, and engineering and can solve many problems for which algebra alone is insufficient.
Calculus has historically been called “the calculus of infinitesimals”, or “infinitesimal calculus”.
MIT offers the course called Single Variable Calculus which offers a thorough introduction to the subject of calculus. It is taught by David Jerison. The course consists of 39 lectures each being about 45 minutes as far as I know, plus a lot of exercises and answers for you to get it under your finger nails. I have seen the first couple of videos and can recommend this course as something which will give you a gentle but thorough introduction to the topic.
The only thing I would recommend is that rather than you watching the tidbits of videos for each lecture, you watch the whole lecture which has been taught. This gives a much more coherent understanding of the topic.
This is another course taught at MIT which targets high school students and aims to give an introduction to Calculus and why it is important. From the course website it gives the description “Highlights of Calculus is a series of short videos that introduces the basics of calculus—how it works and why it is important. The intended audience is high school students, college students, or anyone who might need help understanding the subject.”
As with most other basic math topics Khan academy has a long series on calculus. This is more focused on how to a apply the different formulas and what the different elements does than actually deriving the formulas and proving that they are true. I have used this resource to gain a better understanding of some topics and not least refresh some topics that I had nearly forgotten. I wouldn’t use this as a stand alone source if you are looking into calculus, but I would definitely use it as a supplement for most other sources to pick up on things that are difficult to understand when you are first introduced to them.
Khan Academy covers topics in both single and multivariable calculus.
Even though this book is very expensive it is held in high esteem by very many people with quotes like “The author strikes a perfect balance between theory and technique by explaining the “why” of calculus in addition to the “how”. ” as well as “This is definitely not a high school book, but I’m using it in high school. Surprisingly, it’s working out far better than the official book that belongs to the school. The reason is that it’s a really good book.” So if you need a good calculus book I would consider finding the money as it seems that you can’t find much better, and I have to agree.
The book is focused on single variable calculus.