What is the difference between rates of change you found in Algebra and rates of change you find in Calculus?

Question

What is the difference between rates of change you found in Algebra and rates of change you find in Calculus?

Answer 1

The rate of change you found in algebra is the AVERAGE rate of change, which is the classic (y2-y1)/(x2-x1). The rate of change in calculus is usually the INSTANTANEOUS rate of change, i.e. how fast the function changes *at a specific x value* rather than the average value of change during some interval.

 

The two rates become the same value when you make the space between the two x values approach 0. In fact, that is the entire purpose of Calculus. It is the study of what happens when the steps (spacing) gets smaller and smaller.

 

This is how we derive the formal definition of the derivative.

 

    lim h->0 [ f(x+h) - f(x) ] / [ (x+h) - x] 

=  lim h->0 [ f(x+h) - f(x) ] / h 

 

Ignoring the limit part, the is the definition of the slope for any two points. Now using the limit part, we decrease h until it has the same x coordinate as our original point. So we have found the slope at our specific x value - the INSTANTANEOUS slope.

 

I hope this helps!