evaluate the difference quotient f(x)=4x+3x-x^2 , (f(x+h)-f(x))/h

Given the function \( f(x) = 4x + 3x - x^2 \), we need to evaluate the difference quotient:

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

### Step 1: Simplify the function
First, simplify the expression for \( f(x) \):
\[
f(x) = 4x + 3x - x^2 = 7x - x^2.
\]

### Step 2: Express \( f(x+h) \)
Now calculate \( f(x+h) \) by substituting \( x+h \) into the simplified function \( f(x) \):
\[
f(x+h) = 7(x+h) - (x+h)^2.
\]
Expanding both terms:
\[
f(x+h) = 7x + 7h - (x^2 + 2xh + h^2).
\]
So:
\[
f(x+h) = 7x + 7h - x^2 - 2xh - h^2.
\]

### Step 3: Compute the difference \( f(x+h) - f(x) \)
Now subtract \( f(x) = 7x - x^2 \) from \( f(x+h) \):
\[
f(x+h) - f(x) = \left( 7x + 7h - x^2 - 2xh - h^2 \right) - \left( 7x - x^2 \right).
\]
Simplifying:
\[
f(x+h) - f(x) = 7h - 2xh - h^2.
\]

### Step 4: Form the difference quotient
Now, divide this expression by \( h \) to get the difference quotient:
\[
\frac{f(x+h) - f(x)}{h} = \frac{7h - 2xh - h^2}{h}.
\]
Simplifying:
\[
\frac{f(x+h) - f(x)}{h} = 7 - 2x - h.
\]

### Final Answer:
The difference quotient is:
\[
\frac{f(x+h) - f(x)}{h} = 7 - 2x - h.
\]

Do you want further details or have any questions?

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Here are 8 relative questions for further exploration:

1. What is the derivative of \( f(x) = 7x - x^2 \) using first principles?
2. How does the difference quotient relate to the derivative of a function?
3. How would the result change if the function was \( f(x) = 4x^2 + 3x \)?
4. Can you evaluate the limit of the difference quotient as \( h \to 0 \)?
5. What is the geometric interpretation of the difference quotient?
6. How does the difference quotient behave for linear functions?
7. What happens to the difference quotient when \( x = 0 \)?
8. How would the difference quotient change if the function had a cubic term?

**Tip:** The difference quotient is a stepping stone to understanding the derivative, which measures the instantaneous rate of change of a function at a point.

Learn how to evaluate the difference quotient for the function f(x) = 4x + 3x - x^2 with a detailed step-by-step solution. This problem covers key algebraic concepts, including the difference quotient and function evaluation. Suitable for students in Grades 11-12.

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