Determine the value of the difference quotient (the average rate of change) near the point ( 3 , f ( 3 ) ) when: f ( x )

To determine the value of the difference quotient (which is the average rate of change of the function $f(x)$ near the point $(3, f(3))$), we'll use the formula for the difference quotient:

$\text{Difference Quotient} = \frac{f(3 + h) - f(3)}{h}$

Given:

• $f(x) = x^2 - 4x$
• $h = 0.5$

Step 1: Calculate $f(3)$

$f(3) = 3^2 - 4(3) = 9 - 12 = -3$

Step 2: Calculate $f(3 + h)$

$f(3 + 0.5) = f(3.5)$ $f(3.5) = (3.5)^2 - 4(3.5) = 12.25 - 14 = -1.75$

Step 3: Compute the difference quotient

$\frac{f(3.5) - f(3)}{0.5} = \frac{-1.75 - (-3)}{0.5} = \frac{-1.75 + 3}{0.5} = \frac{1.25}{0.5} = 2.5$

So, the value of the difference quotient (average rate of change) near the point $(3, f(3))$ is 2.5.

If you have any questions or would like further details, feel free to ask!

Here are some related questions:

1. What is the difference quotient for $f(x) = x^2 - 4x$ at $x = 4$ and $h = 0.5$?
2. How does the value of $h$ affect the difference quotient?
3. What happens to the difference quotient as $h$ approaches zero?
4. Can the difference quotient be used to approximate the derivative?
5. How do you interpret the difference quotient in the context of a graph?

Tip: The difference quotient is a fundamental concept in calculus, as it leads to the definition of the derivative, representing the instantaneous rate of change at a point.