# Calculus/Introduction

Calculus is the math pertaining to slopes (or 'steepness') and areas under curves. I will define that a slope of, say 2 means that if you move 1 unit forwards, you move 2 units up, and that the same applies for any other number

## Prerequisites

• The four basic operations (+,-,*,/)
• Maybe algebra (subtle introduction in Section 2)

## The Derivative

The derivative is the math name for the math of slopes. To give an example, the word 'parabola' means that, for example, if you walk 4 meters away from it, the parabola is 4*4=16 meters above your head, and this holds true if you walked any other distance from it (and replaced the 4 with the actual distance!). Now, if you got a friend to walk up the parabola directly overhead, they would feel a slope of 2*4=8, and the same would be true if you replace 4 with anything else, and you kept the 2 because you think that 2 matters more than 8 to you...

To write this down, we say: ${\displaystyle {dy \over dx}=2x}$, and in this case, ${\displaystyle {dy \over dx}}$ means 'slope' and x means how far you walked away from the center of the parabola.

## The integral

The integral is the math name for the math of areas under curves. To give an example, if you gave me a number, which I'll call x, I can give you the integral of y=x (the height, y is the same as the horizontal distance, x), which is the area of the blue triangle. Notice that the blue triangle is half of a square with side length x. This implies that the blue triangle's area is x*x/2, and thus, the integral of x is x*x/2.

To write this down, we say: ${\displaystyle {\frac {x*x}{2}}=\int xdx}$. ${\displaystyle \int dx}$ means that you're taking the area under a curve and the x between the sign and the dx represents the y value. Had y=x*x, it would have looked like this: ${\displaystyle \int x*xdx}$.