Difference quotient

In this step we define the difference quotient. We will learn how to determine a new value of a quantity from the old value and the amount of change and how to apply rates of change to displacement and velocity of an object moving along a path.

• The different quotient is a formula that finds the average rate of change of any function between two points. It is used to calculate the slope of the secant line between two points on the graph of a function:

A and B are points on the graph of (f(x)). A line passing through the two points (A (x,f(x))) and (B(x+h, f(x+h))) is called a secant line. The slope (m) of the secant line may be calculated as follows:

[m=frac{f(x+h)-f(x)}{x+h-x}=frac{f(x+h)-f(x)}{h}]

This slope is very important in calculus, where it is used to define the derivative of a function (f) which in fact defines the local variation of a function in mathematics. It is called the difference quotient. The difference quotient is used in the definition of the derivative.

• One application for derivatives is to estimate an unknown value of a function at a point by using a known value of a function at some given point together with its rate of change at the given point.
• The average rate of change is equal to the total change in position divided by the total change in time.
• The instantaneous rate of change measures the rate of change or slope, of a curve at a certain instant. Thus, the instantaneous rate of change is given by the derivative.