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.