Instantaneous Velocity

There are a number of confusing concepts you'll run into when studying calculus.
One of these is the concept of instantaneous velocity. Up until this point you probably only had to do calculations with average velocity, but this is much different. Instantaneous velocity is the velocity of the particle at a very small specific instant of time.

Let's take a look at how we can calculate the instantaneous velocity from a position function. Let's say we have some particle and its path is defined as s(t) = 2x^2 + 1 m, where t is the time in seconds.

Let's find the velocity of this particle after one second. Now this is where it is important to distinguish between average velocity and instantaneous velocity.
The average velocity is actually quite easy to calculate, but is technically less accurate.

The average velocity is the velocity of the particle over a range of time, as opposed to at a specific point.

To calculate the average velocity we must simply find the position value at two instances of time, subtract them, and then divide them by the time interval.
This is a common way of calculating velocity, but it isn't very accurate with nonlinear functions.
To calculate the instantaneous velocity we must take the derivative of the position function. Luckily this is a simple polynomial and we can find its derivative as; v(t) = 4x + 1 This derivative will be the equation for our velocity at any given point.

To find the velocity after one second we must simply sub in the value of one into this equation; v(1) = 4*1 + 1 = 5 m/s This is the exact value of the velocity after one second.

We could not come to this value using the average velocity formula.

So as you can see this is a much more accurate way of finding how fast something is moving.