Rate of Change

We are all familiar with many physical rates of change. The rate of change of distance is called velocity. If distance is being measured in meters and time is being measured in seconds, the rate of change of distance (velocity) is measured in units meters per second (m/sec). Water filling a bucket can be measured in liters. If time is measured in minutes, then the rate of change of the amount of water in the bucket has units liters per minute (ltr/min). Or we cold measure the height of the water in the bucket in centimeters. The rate of change would be in centimeters per minute (cm/min).

Graphically, the rate of change of a function indicates how fast it increases or decreases as you move along the x-axis. From your study of linear equations, the standard form of an equation of a line is y = mx + c, where m is the gradient or slope of the line. For example, the line y = 2x + 5 has a gradient of 2 which means that it rises 2 units for every unit you move to the right along the x-axis. If this was the equation of the distance of a particle moving from some reference point, where the distance (y-axis) was measured in centimeters and the time (x-axis) was measured in seconds, the velocity of the particle would be 2 cm/sec which is the same as the gradient of the graph. However, the gradient (velocity) is constant since the gradient is 2 anywhere on the graph. All linear graphs have a constant gradient (rate of change). What about non-linear graphs?

Look at the graph of a non-linear function below:

You would say that the function is increasing (positive rate of change) up to about x = -0.6, decreases (negative rate of change) between about -0.6 and 0.6, and increases after 0.6. The rate of change is different depending on where you are on the graph. For many physical problems that have been modelled with an equation, we want to know what the rate of change is at different values of x. A very common problem to solve is to find where the rate of change is zero. The solution to this would find the maximum and/or minimum points because these are the points where the rate of change goes from positive to negative or vice versa.

What does this have to do with calculus? The mathematical term for the rate of change of a function is the derivative of the function and finding the derivative of a function will be the first thing I will define in my next post. Finding the derivative of a function is an operation in calculus, and this is usually the first topic developed in a calculus subject.