Question 143398
I'm going to give you the first example that comes to mind:

Say you are driving down the road and the speed limit changes to +10 mph so you accelerate in order to approach the speed limit. 

Let's make an equation for this. 

a=a(x)=x; This is your acceleration at time x.

If we integrate this function we come up with a different formula:

{{{int( x, dx )=(1/2)(x^2)+c}}}. This gives you the velocity, per some constant c, of the acceleration function. We can continue to integrate and get the position function of the vehicle. I will omit this, simply because I've already put too much calculus into this answer.

As you can see, however, given a directly varied acceleration (that is, the acceleration is the same as the time), we end up with a function to describe the velocity of the vehicle that is a quadratic equation with a=1/2 b=0 and c a constant.

Let's do an example. Say you want to know what your acceleration is at x=5. According to the formula, f(5)=5 implies that the velocity will be 25/2+c mph. The constant is of course the speed at which you were traveling before accelerating. In any case, this certainly applies to many peoples' day to day lives.