Difference Quotient
What is the difference quotient in calculus? We start with the definition and then we calculate the difference quotient for different functions as examples with detailed explanations.
Let f be a function whose graph is shown below.
A and B are points on the graph of f. A line passing trough 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 = \dfrac{f (x + h) - f(x)}{(x + h) - x}
Simplify the denominator to obtain
m = \dfrac{f (x + h) - f(x)}{h}
This slope is very important in calculus where it is used to define the derivative of function f which in fact defines the local variation of a function in mathematics. It is called the difference quotient. In the examples below, we calculate and simplify the difference quotients of different functions.
Example 1: Find the difference quotient of function f defined by
Solution to Example 1
Example 2: Find the difference quotient of the following function
Solution to Example 2
Example 3: Find the difference quotient of function f given by
Solution to Example 3
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