I have no problem with the exercises, but I can’t wrap my head around the concept
In: Mathematics
Velocity is just a term use to define change in position (movement) with respect to time. You can name it anything you want as long as you define it the same way.
“The rate of change of a function” is the definition of a derivative. You might as well ask “Why does the circumference of a circle equal the length of the edge of the circle?”
For a curve like a position equation, the slope at any given point is equal to how fast the position is changing. The speed of a change in position is called “velocity.”
Taking the derivative of a function tells you the rate of change at any given point.
The rate of change of an object’s position is its velocity
Therefore, the derivative of a position function tells you the velocity at any given point.
The derivative measures the rate of change of something with respect to something else. In the case of motion, the derivative of motion with respect to time measures the change in motion as time changes: this by definition is velocity.
When graphing your change in displacement with respect to time there is a slope for that line. The slope is the rate of change in that line. This slope is velocity.
Velocity describes the direction and change of displacement with respect to time.
So think of this: you move forward for some amount of time. Do you move forward with a constant speed? Do you run for a while? Do you stop to tie your shoe for a moment? Did you forget your phone at home and have to go back? This change is the velocity. If you were to graph this change in displacement with respect to time, the slope of the line would be your velocity.
Since derivatives are the rate of change of a function with respect to an independent variable, velocity is the derivative of displacement with respect to time.
I hope this makes sense and doesn’t sound too repetitive.
It’s not the derivative of the “movement” equation, it’s the derivative of the **position** equation.
The derivative of position with respect to time is literally the rate at which position changes over time. Which is what velocity *is*.
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