Evaluate the difference quotient for the given function. Simplify your answer.

*f*(*x*) = 4 + 5*x* − *x*2,    

*f*(5 + *h*) − *f*(5)*/h*

To evaluate the difference quotient for the given function \( f(x) = 4 + 5x - x^2 \), we need to follow these steps:

1. **Find \( f(5 + h) \)**:  
   Substitute \( x = 5 + h \) into the function:
   \[
   f(5 + h) = 4 + 5(5 + h) - (5 + h)^2
   \]
   Simplify:
   \[
   f(5 + h) = 4 + 25 + 5h - (25 + 10h + h^2)
   \]
   \[
   f(5 + h) = 29 + 5h - 25 - 10h - h^2
   \]
   \[
   f(5 + h) = 4 - 5h - h^2
   \]

2. **Find \( f(5) \)**:  
   Substitute \( x = 5 \) into the function:
   \[
   f(5) = 4 + 5(5) - 5^2
   \]
   Simplify:
   \[
   f(5) = 4 + 25 - 25 = 4
   \]

3. **Compute the difference quotient**:
   \[
   \frac{f(5 + h) - f(5)}{h} = \frac{(4 - 5h - h^2) - 4}{h}
   \]
   Simplify the numerator:
   \[
   \frac{-5h - h^2}{h}
   \]
   Factor \( h \) out of the numerator:
   \[
   \frac{h(-5 - h)}{h}
   \]
   Cancel out \( h \) (assuming \( h \neq 0 \)):
   \[
   -5 - h
   \]

So, the simplified difference quotient is \(\boxed{-5 - h}\).

Would you like a detailed explanation of any part of this solution, or do you have any other questions?

Here are 5 related questions you might find interesting:
1. How do you interpret the difference quotient geometrically?
2. What is the difference quotient for the function \( f(x) = x^2 + 3x + 2 \)?
3. How does the difference quotient relate to the derivative of a function?
4. Can the difference quotient be used to find the slope of a secant line?
5. What happens to the difference quotient as \( h \) approaches zero?

**Tip:** The difference quotient is a fundamental concept in calculus, representing the average rate of change of a function over an interval. It's the basis for the derivative.

Learn how to evaluate the difference quotient for the function f(x) = 4 + 5x - x^2. Follow a step-by-step solution to simplify your answer, covering key calculus concepts such as the difference quotient and function evaluation. Suitable for college-level calculus students.

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