SECTION 1.3: Rates of Change

Rates of Change can be calculated in a similar way to the slope of a linear function, as analyzed earlier. The rate of change of a non-linear function is determined as an average.
If a function y = f(t),
then the average rate of change of y = change in y/ change in t = f(b)-f(a)/ (b-a),
assuming that the rate is being calculates between an interval of t=a and t=b.

Rates can be positive or negative. They help determined whether a function is increasing or decreasing. Increasing functions have positive rates and decreasing functions have negative rates. A function is said to be increasing if the values of f(x) increase as x increases. Similarly, a function f is said to be decreasing if the values of f(x) decrease as x increases.

Rates of change can be easily visualized:

The change in y refers to the vertical change of the graph while the change in x refers to horizontal change of the graph over a period of time. In the diagram below, the change in y between the two demarcated points would be approximately (90-50) while the change in x would be (0-(-10)).


The shape of a graph can also be determined by its concavity. A graphs that bends upwards from left to right is called concave up while a graph that bends downwards is considered to be concave down. The graph above is an example of a concave up graph.

Everyday rates that we encounter include velocity and speed, i.e.,

Average velocity = change in displacement/change in time while,

Average speed = change in distance/ change in time

SECTION 2.1: Instantaneous Rate of Change

To calculate velocity to exact accuracy required, smaller and smaller intervals on either side of the time taken can be determined until the average velocity agrees to the number decimal places needed. This can also be expressed as instantaneous velocity.

Instantaneous velocity is defined for an object at time t as the limit of the average velocity of the object over shorter and shorter time intervals containing t.

To look at a more general idea of the above states reasoning, instantaneous rate of change of a function at say, t, is also called the rate of change of f at t. It is defined as the limit of the average rates of change of f over shorter and shorter intervals around a. This instantaneous rate of change is also referred to as the derivative of f at point a and is written as f '(a).

The derivative of a function is essentially the slope of the function at a or the slope of the line tangent to the curve at a. Similar to the rate of change, a derivative can be positive or negative thereby determining whether the function is increasing or decreasing.

What I found challenging about these sections was deciphering whether or not having a rate of change still indicated a slope between the two points despite the fact that the graph was not linear.

The concept of average rates of change is always useful in everyday situations as not all relationships between values can be of a linear form.