Calculus. Ah, the best thing ever invented by human in recent mathematics. Without Calculus, there would be no cell phones, computers, or even airplanes.


AP Calculus - in a nutshell.

Let’s talk more on Calculus. Recently I have just taken the AP Calculus AB test in May. It didn’t really give me a big headache studying (unlike someone else), since if you understand the basics and the fundamental stuff behind it you are good to go. Here’s some main topics discussed in the course:

Notice: This article is not a Calculus tutorial.

A Little History


That Greek guy

Some concepts in Calculus actually existed way back before Issac Newton was hit by an apple. Ancient Greek mathematicians had been using  infinitesimals to calculate areas and volumes in about 200 BC. Until 17th century, modern calculus started to appear. Several European mathematicians worked on the idea of derivative, published books including Methodus ad disquirendam maximam et minima. Issac Newton later wrote De analysi per aequationes numero terminorum infinitas based on their ideas. Then, calculus started booming.

Did I tell you that calculus was actually two different subjects before? I thought I did… but anyway, deferential calculus and integral calculus were totally different things. Until the 1600′s, the Fundamental Theorem of Calculus was finally created and proven by James Gregory, a classy Scottish mathematician; it linked the two parts together, and boom, modern calculus was born.

Later on, infinite series was created to find the sum of a infinite amount of numbers. (It is not included in AP Calculus AB.)


Limits sound weird when you first heard it. Basically it is just filling holes in a function (curve) and finding its y-coordinate. Imagine a line f(x) = x with a hole at 0. At x = 0, y would be not exist. However, applying the concepts of limits and now it would be 0 regardless of there is a hole or not.


Now this is the fun part: you can find a line that exactly touches  f(x) once!… okay I guess it is not that exciting at all. Derivative is just a function that will give you the slopes of tangent lines, and it comes out that you can also find the instantaneous rate of the original function. (how amazing is that!)

Definition of derivative

Combining the concept of limit and slope gives you this master piece.


Ever thought about how you can find the area of an irregular shape? (say yes) Well before calculus you can’t. Tries to combine all those formulas of finding areas of squares and circles, and headaches will come to haunt you. Introducing the integral! Integrals can be used to find the area under a curve, such as x2. It turns out that that area between the curve x2 and the x-axis from 0 to 2 is exactly 8/3 unit2. How incredible is that!

Related Rates

Related rates is a cleverly-created tool in mathematics used to determine different rates based on known rates. In other words, you can find out how fast your money is disappearing by knowing how much snacks you bought in a day.

Related rates

Want to find how fast a triangle is growing over time? Don't worry related rates got you covered.


Thank me because now you know how fun and useful calculus is. You’re welcome. ;)

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