Question

Calculate the rate of change. 0    1   2    3    4   5 15 25 40 60 85 115

Answer 1

Dy/dx=10,15,20,25,30

d2y/dx2=5,5,5,5,5  (there is no "rate of change" that is constant because acceleration is positive meaning that velocity keeps increasing...)

Since d2y/dx2 is constant, acceleration is constant, and this means that it is a quadratic equation of the form y=ax^2+bx+c

There are only three unknowns so we can use just three data points.

9a+3b+c=60
4a+2b+c=40
a+b+c=25  getting differences between these...

5a+b=20
3a+b=15  and again...

2a=5, so a=2.5, making 3a+b=15 become:

7.5+b=15, so b=7.5, making a+b+c=25 become:

2.5+7.5+c=25, so c=15

So the quadratic which produced the table is:

y=2.5x^2+7.5b+15

This is the regular old position function you see with any object moving with constant acceleration (like gravity when simplified).

The acceleration is 2.5 it has an initial velocity of 7.5 and an initial position of 15.