How to Learn Calculus

Do you want to learn Calculus, but don't know where to start? In this article we will explain to you how to approach studying Calculus: how to prepare for learning it and exactly which resources - textbooks and video lectures - to use. After reading this article you will be ready to start learning everything on your own.

In order to learn Calculus from the scratch, we will be using free resources available on Real Not Complex. In fact, we have a whole section dedicated to Calculus.

So we basically could leave you with that list and end the article. Everything that you need to succeed is on that list. But of course it might be a bit daunting and confusing for a newcomer to see so many different textbooks and video lectures. Calculus section is one of the bigger sections on our website, so it's understandable that it might not be immediately clear where to start.

That's exactly why we wrote this article.

Prerequisites

The first question that probably pops up in your mind is - am I even ready to study Calculus?

Calculus might seem like a challenging subject at a first glance, but that's mostly because it relies quite heavily on other topics. In fact, we believe that from the initial subjects that you typically study during the first semesters of the university, Calculus requires the highest amount of prerequisite knowledge.

But don't worry. Even if you didn't feel comfortable with mathematics in school, you can still very easily catch up. In fact, we have created a guide just like this one, which explains how to learn Algebra, Trigonometry and Precalculus, which are the topics needed to get into Calculus easily.

Although usually we advocate learning stuff on a "need to know" basis, here we encourage you to spend a solid amount of time learning prerequisites listed in the link and making sure you feel comfortable with them. Contrary to on of the names - Precalculus - they are not necessary only for Calculus. They will appear in almost all mathematical subjects in some capacity.

If you learn Algebra, Trigonometry and Precalculus very well, studying Calculus itself will be much easier. Calculus introduces a lot of ideas which are brand new for someone who never studied university-level math. It will be much less challenging to understand those ideas, if you don't have to constantly remind yourself how to deal with basics like, say, trigonometric functions or adding fractions.

Relation to Real Analysis

Depending on from which country you are, you might have heard Calculus and Real Analysis used interchangeably. This comes from the fact that - at least in the English literature - Real Analysis is basically Calculus, but seen from a more theoretical perspective.

Real Analysis courses (and therefore also textbooks and videos) will be usually full of more abstract definitions, theorems and proofs.

We say that precisely because we want to warn you. If you are just beginning your journey with mathematics, don't use English Real Analysis books! What you will see in them will terrify you to the point where you will most likely give up. We don't want that!

Calculus is basically designed as a softer, more accessible introduction to Real Analysis and its concepts. It also focuses more on computations and applications. You might encounter occasional proofs here and there, but usually Calculus leans more toward introducing new concepts in a intuitive way and using those concepts to actually calculate some stuff.

Subfields of Calculus

As a beginner you also will need to know that Calculus is divided into two subfields: Single Variable Calculus and Multivariable Calculus.

It's simple - Multivariable Calculus builds on ideas from Single Variable Calculus. So you don't want to study Multivariable Calculus first, ever.

If you are just beginning, either looks for materials that focus on Single Variable Calculus only, or the ones that simply cover both topics in one package.

Textbooks

Uff. That was a lot to digest.

But now you are ready to actually start studying. We will recommend you some resources, but by any means don't feel limited and forced to use precisely the ones that we will mention here.

Not by accident on Real Not Complex we post a lot of free resources. If you find yourself disliking the textbook you study, or maybe you like it, but one of the chapters suddenly feels incomprehensible - don't be afraid to look for a different textbook.

In fact it's good to use multiple resources concurrently when studying. This will enhance your understanding of the subject, show you how different authors approach it and maybe even teach you different notation styles.

Ok, without further ado. What can you use to ensure smooth start of your Calculus journey?

A safe choice would definitely be a Calculus: Early Transcendentals by Lyryx Learning and D. Guichard. It's very well edited and written with beginners in mind.

Calculus: Early Transcendentals by Lyryx Learning and D. Guichard

If you enjoy physics and prefer to learn on some physical examples, you will like Active Calculus by Matthew Boelkins a lot. This one has also an additional benefit of having a nicely designed online version. After reading that book make sure to read Active Calculus Multivariable, which obviously covers Multivariable section of Calculus.

Active Calculus by Matthew Boelkins

One last resource that you might like is Calculus One. It is not only a textbook but also a full blown online course, with YouTube videos, which may be a big help in your studies.

Why we have chosen those textbooks in particular? Obviously because they have gained some popularity and appreciation in the math community. But also because all of them have problems and solutions provided. It's crucial, because, since you will be self studying, you want to do as many problems as possible and be sure to check your answers every time, to make sure you are going in the right direction.

Also all of those textbooks have tables of contents with links to particular sections. This makes navigating the textbooks easier. You will need that when you will be confused on some topic when reading one book and you will want to read on the same topic in the other.

Video Lectures

We already mentioned that Calculus One has videos included.

Although we recommend learning from textbooks, you absolutely can - and even should - supplement your learning with video lectures. Especially when you are lost, seeing someone explain the confusing topic will be invaluable.

Well, you are in luck, because Calculus is one of the topics that have the best video lectures selections.

You can't go wrong with Khan Academy - Calculus and Multivariable Calculus sections - or Professor Leonards Calculus series (covering both Single and Multivariable Calculus). They are extremely approachable and friendly for the newcomers. Whenever you feel that the textbook you are reading is losing you, immediately watch their lectures on the specific subjects.

Khan Academy Calculus playlist

You might also want to check out MIT lectures - Single Variable Calculus and Multivariable Calculus. The thought of studying from the lectures of one of the most acclaimed tech universities in the world might intimidate you, but give it a try. Those lectures are also surprisingly accessible and are a pleasure to watch.

And that's it!

Learning Calculus can be stressful for many people. It introduces a lot of new concepts and heavily relies on the students previous knowledge.

But it is also a subject that has a huge library of free resources available online.

By making sure you have solid fundamentals and by using all those amazing resources you will have much easier time studying.

If you encounter any problems when following this guide please let us know. We will continue improving it, so that future readers have an even easier time studying.