Differential Calculus

When you have two points on a line, you can calculate the slope, as you probably learned in algebra. That slope describes how the line is changing. If you only have one point, you can still describe how the line is changing - that's called a derivative.

To bridge it to the real world, say you know your position. If you take the derivative of that with respect to time, you know how your position is changing with respect to time. That is called velocity, which just means speed in a specific direction. You can take it a step further and take the derivative of velocity with respect to time, and that's called acceleration. It's how your speed is changing with time - are you going faster or slower than you were a minute ago?

The next few derivatives with respect to time are called jerk, snap, crackle, pop, and rice crispies.

Integral Calculus

Let's say you have a line again. You want to know the total area under the line. You calculate this by integration. An example would be that you're modeling a lake and want to see how many pollutants are being leeched into the ground. You'll have to study the influent/effluent concentrations and diffusion rate, but ultimately you're just taking an integral.

Multivariable Calculus

It's just like the other two, but you have more than one dimension.

Differential Equations

Sometimes we get weird equations that contain the derivative of a function (see differential calculus.) These are called differential equations. An example of how they might be used is how one differential equation (called the Navier-Stokes equation) can be used to describe how a fluid flows, or how another differential equation (a mass balance) can be used to describe how a chemical species is evolving in a reactor.

Higher Calculus

IDK, I'm an engineer, not a mathematician or physicist.