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.

I had a teacher of atmospheric chemistry that told that during her undergraduate years, in a time that there weren't advanced computers and people didn't have cellphones with processors capable of calculating anything in a matter of seconds, she learned integrals with papers and scissors. What? She told me that there was a procedure that involved tracing graphs on papers, then cutting the paper and using a high precision balance to measure the paper's weight. That's something in mathematics that I don't understand. Some teachers try their very best to make graphical associations or to make associations with applications in other fields. But some teachers may even refuse to see mathematics beyond the equations themselves.

There is something regarding intuition that I have no idea how to explain. In calculus we are faced with the rigour of having to prove properties. I once heard a calculus teacher say that some students do calculations based on faith. With calculus we have the concept of limits and I think this is where faith is brought in. People believe in something out of faith, because it feels the right thing. There are easy limits of functions that result in infinity or zero. Often those limits are easy because they match intuition. But that's the whole problem of using faith to make assumptions regarding calculus. There are many exercises that defy intuition. Mathematical intuition is not faith! I'm not discussing religion, but the fact that often people believe that the result of something should be greater than, less than, out of faith. In regards to probability this is really problematic as it's easy to regard high and low probabilities as an exercise of faith. You can't guess the results of calculations with faith! That's something I've seen in numerical methods. Sometimes people go in a "trial and error" mode, expecting that the answer is going to come after many trials and errors. Which is an exercise of faith in disguise. I have to admit that I did it too!

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.