2.1 Defining Average and Instantaneous Rates of Change at a Point

Imagine you have a general function f(x), and you want to find the rate of change of that function over a certain interval [a, b]. That’s pretty basic: you would just find the slope of the line that connects those two points. 🌉

slope = [f(b)-f(a)] / [b-a]

Another way to think about this is to let a = x, representing any x-value on the domain of the function. As follows, b = x+h, where h is any constant added to the value of x. (For example, if a = 2 and b = 7, then x=2 and x+h = 2+5 = 7) 

Our formula now becomes:

slope = [f(x+h)-f(x)]/h

Example: Find the average rate of change of f(x) = x^2 + 8x-9 on the interval [1, 3].

slope = f(3)-f(1)/(3-1) = (24-0)/2 = 12

Now, what if you want to find the rate of change at a single point? Well, of course that is impossible… a function can’t be changing at a single point. But, as the points get infinitely closer and closer together, we can see that the slope of our line gets closer and closer to this value. 

In other words, as h (the difference between the points) approaches 0, the slope of our line approaches the rate of change at a singular point of the function 

A derivative represents the instantaneous rate of change of a curve at a single point, and is also represented by the slope of the tangent line to the curve at that point. ✨

Thus, the rate of change of a function would be represented by: