1. Rate of change

Rate of change describes how one quantity changes in relation to another quantity. In specific, rate of change describes the changes in the dependent variable with respect to the changes in the independent variable. 

If y=f(x), then

2. Sign of rate of change

Rate of change is positive in two situations:

• when x increases, y also increases
• when x decreases, y also decreases

Rate of change is negative in two situations:

• when 
• x increases but y decreases
• when x decreases but y increases

Example

Given the above table of values for 

x and y, find the rate of change and plot the graph.

So, rate of change is 6.

3. Average rate & instantaneous rate

Average rate is the slope of the line between two points. 

Let 

y=f(x) and let x=a and x=b be on the graph of y=f(x). Therefore, we have the two points or two sets of coordinates a,f(a) and b,f(b).

Instantaneous rate is the derivative at one point. The instantaneous rate at 

x=a is f'(a).

Example

Let 

\(fx=x^2\text{ and} \text{ x}=1 \text{ and } \text{x}=3 \text{ be on the curve y}=x^2.\)

1. Find the average rate of change between the two points.

2. Find the instantaneous rate of change at x=1.