OKay, Understanding calculus with only 1 blog.
So that's the topic of this blog.
Notice my topic isn't our title is "learn calculus with only 1 blog" or "master calculus" or "totally", you know, "understand it completely".
Just basically understanding calculus with only 1 blog like understanding what it is you know that's what the whole point of this and really my kind of main goal here is to to figure out what calculus is.
You know there is a lot to learn on this topic, Just talk about where you would use calculus in general:
- Calculus is generally used to calculate area & volume
Suppose we want to calculate a rectangle, we use the formula: \(A=\iota*\omega\)
Suppose we want to calculate a circle, we use the formula: \(A={\pi}r^2\)
Suppose we want to calculate a triangle, we use the formula: \(A={1\over2}bh\)
So you find these formulas can calculate the area of these regular figures.
But what if you want to calculate the area of such crazy figures[1]?
You might look for the right formula, and then you actually can't find the formula at all.
And this is exactly why calculus exists.
Without calculus, you can only try to estimate the size of this area. But with calculus, you can know the exact size of this area.
So let me give you a basic example:
(Sorry about this image is not accurate, it is only used to express meaning.)
If you try to calculate the area marked in green, calculus can help you.
So we express the area as:
\[{\int_2^5}{x^2}dx\]
You may be curious about how to calculate such a formula, don't worry, I will show you step by step:
\[{\int_2^5}{x^2}dx\]
\[={x^3\over 3} \]
\[={5^3\over 3}-{2^3\over 3} \]
The symbol dx, called the differential of the variable x, indicates that the variable of integration is x. The function f(x) to be integrated is called the integrand. A function is said to be integrable if the integral of the function over its domain is finite[2].
You may be curious about how I got it in the second step. Here is a formula:
\[{\int}{x^n}dx={x^{n+1} \over n+1}+c\]
"c"means a constant, because your f(x) often has a constant. So I can give another example[3]:
\[{\int}{4x^3}dx={4x^{3+1} \over 3+1}+c={4x^4 \over 4}+c={x^4}+c\]
So as far as you know the mathematics, it is not difficult. Now what makes calculus more complicated is these curves, I mean the f(x).
References:
[1] Understand Calculus in 10 Minutes