Question

Compare and contrast the notion of average rate of change with instantaneous rate of change. Full credit answers will incorporate the concept of tangent lines and tangent lines and secant lines into their explanation. Please provide full detail with drawing as needed.

Answer 1

It shall be noted that average rate of change , say of function y=f(x) over an interval [x1, x2] is given by:

Whereas, the instantaneous rate of change of the function y=f(x) is given by:

Graphically, it can be shown that average rate of change is the slope of the straight line (also known as secant line) between any two points on a function y=f(x), this function being linear, non-linear, polynomial, quadratic or from a family of functions.

Whereas, the instantaneous rate of change is the slope of the tangent to a point on a given functional form of the graph.

It is only for a linear function y=f(x)=mx+c, where m=slope of the linear line, the average rate of change and instantaneous rate of change are same and equal to m.

This can be shown as follows: