https://file.aiquickdraw.com/mathbot/file/1726453448424eytez4al.jpg

The image provided asks to approximate \( f'(8) \) by using the values of \( f(8) \) and \( f(8.1) \), and the function is given as \( f(x) = x(3 - x) \).

We can solve this problem step by step:

### Step 1: Evaluate \( f(8) \)
Given \( f(x) = x(3 - x) \), substitute \( x = 8 \):
\[
f(8) = 8(3 - 8) = 8(-5) = -40
\]

### Step 2: Evaluate \( f(8.1) \)
Similarly, substitute \( x = 8.1 \) into the function:
\[
f(8.1) = 8.1(3 - 8.1) = 8.1(-5.1) = -41.31
\]

### Step 3: Approximate \( f'(8) \)
To approximate \( f'(8) \), use the difference quotient:
\[
f'(8) \approx \frac{f(8.1) - f(8)}{8.1 - 8}
\]
Substitute the values of \( f(8.1) \) and \( f(8) \):
\[
f'(8) \approx \frac{-41.31 - (-40)}{0.1} = \frac{-41.31 + 40}{0.1} = \frac{-1.31}{0.1} = -13.1
\]

### Conclusion:
Thus, the approximation for \( f'(8) \) is \( -13.1 \).

Would you like further details on any step? Let me know if you have any questions! Here are some related questions you might find useful:

1. What is the difference between a derivative and a difference quotient?
2. How do you use limits to find the exact derivative?
3. How would you approximate a derivative for different step sizes?
4. Can the approximation of the derivative be improved by using more points?
5. What is the graphical meaning of the derivative at a point?

**Tip:** To better approximate derivatives, use smaller step sizes for the difference quotient, which reduces error in estimation.

Evaluate f(8) and f(8.1) and use the results to approximate f'(8). Round your answer to one decimal place.

Learn how to approximate the derivative of f(x) = x(3 - x) at x = 8 using the difference quotient. This problem explores key calculus concepts and techniques. Suitable for students in Grades 11-12 or early college-level.

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