Difference Quotient

Definition and Purpose

The difference quotient is a fundamental concept in precalculus and calculus, used to analyze the rate of change of a function. It is defined for a function g(x) as:

• Difference Quotient Formula:

• This formula is the basis for the definition of the derivative in calculus.

• It measures the average rate of change of the function over the interval from a to a+h.

Step-by-Step Solution Example

Given the function g(x) = x^2 - 2x + 1, the difference quotient is calculated as follows:

1. Step 1: Evaluate g(a+h)

   • Substitute a+h into the function:

   • Expand and simplify:

2. Step 2: Evaluate g(a)

   • Substitute a into the function:

3. Step 3: Compute the numerator

   • Subtract g(a) from g(a+h):

   • Combine like terms:

4. Step 4: Divide by h

   • Apply the difference quotient formula:

   • Factor and simplify:

Final Simplified Expression

• The simplified difference quotient for g(x) = x^2 - 2x + 1 is:

Example Application

• This process is essential for understanding how functions change and is foundational for calculus.

• For instance, if a = 3 and h = 0.1, the difference quotient gives the average rate of change near x = 3.