How does instantaneous rate of change differ from average rate of change?

1 Answer
Aug 4, 2014

Instantaneous rate of change is essentially the value of the derivative at a point; in other words, it is the slope of the line tangent to that point. Average rate of change is the slope of the secant line passing through two points; it gives the average rate of change across an interval.

Below is a graph showing a function, f(x), and the secant line across an interval [2, 4]. The slope of this secant line, which is

(Deltay)/(Deltax) = (f(4) - f(2))/(4 - 2)

is the average rate of change of f(x).

http://sci.tamucc.edu/~jgiraldo/calculus1/classesguidelines/18DerivativeDefinition/DerivativeDefinitionv07.htmlhttp://sci.tamucc.edu/~jgiraldo/calculus1/classesguidelines/18DerivativeDefinition/DerivativeDefinitionv07.html

Below is a graph showing the function f(x) = x^2, as well as the line tangent at x = 2. The slope of this line is:

dy/dx = f'(2) = 2*2 = 4,

and it is the instantaneous rate of change at the point (2,4).

http://sci.tamucc.edu/~jgiraldo/calculus1/classesguidelines/18DerivativeDefinition/DerivativeDefinitionv07.htmlhttp://sci.tamucc.edu/~jgiraldo/calculus1/classesguidelines/18DerivativeDefinition/DerivativeDefinitionv07.html