find the difference quotient, f(x+h) f(x)/h for f(x) = 4x - x²

The difference quotient formula for a function \( f(x) \) is given by:

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

For \( f(x) = 4x - x^2 \), let's compute \( f(x+h) \) first:

\[
f(x+h) = 4(x+h) - (x+h)^2
\]

Now, let's simplify \( f(x+h) \):

\[
f(x+h) = 4x + 4h - (x^2 + 2xh + h^2)
\]

\[
f(x+h) = 4x + 4h - x^2 - 2xh - h^2
\]

Now, substitute \( f(x+h) \) and \( f(x) = 4x - x^2 \) into the difference quotient formula:

\[
\frac{f(x+h) - f(x)}{h} = \frac{(4x + 4h - x^2 - 2xh - h^2) - (4x - x^2)}{h}
\]

Simplify the expression:

\[
= \frac{4x + 4h - x^2 - 2xh - h^2 - 4x + x^2}{h}
\]

Cancel out terms:

\[
= \frac{4h - 2xh - h^2}{h}
\]

Now, divide each term by \( h \):

\[
= 4 - 2x - h
\]

Thus, the difference quotient is:

\[
\frac{f(x+h) - f(x)}{h} = 4 - 2x - h
\]

Would you like further clarification or details?

Here are 5 related questions to expand on this topic:
1. How does the difference quotient relate to the derivative of a function?
2. Can you find the derivative of \( f(x) = 4x - x^2 \) using first principles?
3. What happens to the difference quotient as \( h \) approaches 0?
4. How do you apply the difference quotient to more complex functions?
5. Can you explain how to interpret the slope of a function geometrically?

**Tip:** The difference quotient is the foundation for understanding derivatives in calculus, which measure the instantaneous rate of change of a function.

find the difference quotient, f(x+h) - f(x)/h for f(x) = 4x - x²

Learn how to find the difference quotient for the function f(x) = 4x - x² with a detailed step-by-step explanation. This problem covers key algebraic and calculus concepts such as the difference quotient, simplifying expressions, and preparing for derivative calculations. Suitable for high school students in Grades 10-12.

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