Differential and integral calculus

I decided to not follow the same order of topics that is followed in a regular course or a textbook. There isn't any particular reason for that, except that some concepts such as the domain of a function of one variable can be easily extended to multiple variables. Why not explain for a single variable and then naturally extend it to multiple variables?

I'm making the assumption that if you are studying calculus, you already know how to plot a graph and the types of functions. I'm also skipping the proofs regarding real numbers. I'm skipping writing historical notes because the idea of this site is not to be a textbook. The goal of this site is also not to be rigorous text.

Much of the difficulty to learn calculus can be attributed to reading. When we learn math at school it's usually nothing more than a series of rules and meaningless calculations. When you have equations to solve, ask yourself "What does the equation mean? Is there a physical meaning? Can volume be negative? Can area be negative? Can the independent variable assume any value? Is the domain of the function any real value or are there forbidden values? What does the root of an equation mean?". When reading a problem, ask yourself "Did I understand the conditions? Is there any piece of information missing from the exercise?". Also, it often helps to read some numbers or letters with a different point of view. Is that letter a special quantity with a name? Is that number just a number or does it bear some meaning? Sometimes, if you read "times two" as "double the value", it can help you understand what is going on in some exercise.

For example: read [math]\displaystyle{ x = 3 }[/math] One way to read it is literal "[math]\displaystyle{ x }[/math] is equal to [math]\displaystyle{ 3 }[/math]". The other way is this "we have the intersection point of [math]\displaystyle{ f(x) = x }[/math] and [math]\displaystyle{ g(x) = 3 }[/math]". This slightly change in perspective can lead to the solution of many problems and exercises. Sometimes we have equations and doing the calculation ends at some [math]\displaystyle{ 0 = 3 }[/math] equality. It either means that there is some mistake or that the equality is comparing functions that never intersect at any point. Another case. [math]\displaystyle{ f(x) \gt 0 }[/math]. One way to read is literal "the function's value at [math]\displaystyle{ x }[/math] is greater than zero", the other is this "we are considering all points where the function's graph is above the [math]\displaystyle{ x }[/math] axis".

In calculus we have lots of expressions that have many terms and they can be derivatives, integrals, limits or just plain numbers. When reading the proofs of properties of limits, derivatives and integrals, be aware that you are not just reading [math]\displaystyle{ f(a) = b }[/math] as "[math]\displaystyle{ f }[/math] of [math]\displaystyle{ a }[/math] is equal to [math]\displaystyle{ b }[/math]" but "the function [math]\displaystyle{ f }[/math], at [math]\displaystyle{ x = a }[/math], has a value equal to [math]\displaystyle{ b }[/math]". To properly read expressions is important everywhere, not just mathematics. When we have proofs about the associative property for example, we aren't just writing the same thing in two different ways. One expression is equal to the other, but we read each one differently.

There is something about asking questions that I noticed in some classes. There are questions that people make that the teacher answers with "What?" or "What you are asking doesn't make sense". When a question doesn't make sense for the teacher it means that you are probably not understanding some concept and the question really doesn't make sense from the point of view of someone who understands that concept. For example: is the tangent function continuous? Because tangent of the right angle doesn't exist. Yes, the tangent of 90° yields a vertical line which extends to infinity. However, the right angle is not part of the domain of said function. When we plot the graph of tan(x) we don't consider the right angle.

When reading the solution for an exercise, try to understand what you were missing or what you were doing wrong. Just reading and then copying it won't make you really learn it. Some authors like to leave proofs to the reader, which is often annoying for some people because they were expecting the author to do it. If you are going to do the proof don't overdo it, they aren't meant to be burdens.

Bibliography

(textbooks from Brazil are usually meant for the local students, not having translations in other languages

• Guidorizzi. H. L.; Um curso de cálculo volumes 1 - 4. 2001.
• Stewart J.; Calculus. 2013.
• Ávila G. S.; Cálculo das funções de uma variável volume 1. 2003.
• Spivak. M.; Calculus. 2008.
• Apostol T. M.; Calculus vol I - II. 1967.