evaluate the difference quotient for the given function f(x)=5+5x-x^2, f(4/h)-f(4)/h

To evaluate the difference quotient \(\frac{f(4+h) - f(4)}{h}\) for the function \(f(x) = 5 + 5x - x^2\), follow these steps:

### Step 1: Find \(f(4+h)\)
Substitute \(x = 4 + h\) into the function \(f(x)\):

\[
f(4+h) = 5 + 5(4+h) - (4+h)^2
\]

Expanding this:

\[
f(4+h) = 5 + 20 + 5h - (16 + 8h + h^2)
\]

Simplify the expression:

\[
f(4+h) = 25 + 5h - 16 - 8h - h^2
\]
\[
f(4+h) = 9 - 3h - h^2
\]

### Step 2: Find \(f(4)\)
Substitute \(x = 4\) into the function \(f(x)\):

\[
f(4) = 5 + 5(4) - 4^2
\]
\[
f(4) = 5 + 20 - 16
\]
\[
f(4) = 9
\]

### Step 3: Calculate the Difference Quotient
Now, use the difference quotient formula:

\[
\frac{f(4+h) - f(4)}{h} = \frac{(9 - 3h - h^2) - 9}{h}
\]

Simplify the numerator:

\[
\frac{9 - 3h - h^2 - 9}{h} = \frac{-3h - h^2}{h}
\]

Factor the numerator:

\[
\frac{-h(3+h)}{h}
\]

Simplify by canceling out \(h\) (assuming \(h \neq 0\)):

\[
- (3 + h)
\]

### Final Answer:
The difference quotient \(\frac{f(4+h) - f(4)}{h}\) simplifies to:

\[
-(3 + h)
\]

Would you like further details or have any questions?

### Related Questions:
1. How would the difference quotient change if the function were \(f(x) = x^2 + 2x\)?
2. Can you find the derivative of \(f(x) = 5 + 5x - x^2\) using the definition of the derivative?
3. What is the significance of the difference quotient in calculus?
4. How does the difference quotient relate to the concept of the slope of a tangent line?
5. How would the result change if the function were quadratic with a different leading coefficient?

### Tip:
The difference quotient is a fundamental concept that leads directly into the definition of the derivative, which represents the slope of the tangent line to the curve at a point.

Learn how to evaluate the difference quotient \(\frac{f(4+h) - f(4)}{h}\) for the function \(f(x) = 5 + 5x - x^2\) with a step-by-step solution. This problem involves calculus concepts, including the difference quotient formula and function evaluation. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation