To estimate $f(3.1)$ using a linear approximation, we will use the given information and the linear approximation formula:

$f(x) \approx f(a) + f'(a)(x - a)$

where:

• $a = 3$
• $f(3) = 10$
• $f'(3)$ (the derivative at $x = 3$) can be estimated from the graph of $f'(x)$.

From examining the graph of the derivative $f'(x)$, it appears that $f'(3)$ is approximately 1, since the function $f'(x)$ has a positive slope around that region.

Step-by-Step Calculation:

1. Identify the point of approximation: $f(3) = 10$

2. Approximate the derivative at $x = 3$: $f'(3) \approx 1$

3. Set up the linear approximation for $f(3.1)$: $f(3.1) \approx f(3) + f'(3) \cdot (3.1 - 3)$

4. Substitute values: $f(3.1) \approx 10 + 1 \cdot (3.1 - 3)$ $f(3.1) \approx 10 + 1 \cdot 0.1$ $f(3.1) \approx 10 + 0.1$ $f(3.1) \approx 10.1$

Conclusion:

The estimated value of $f(3.1)$ is 10.1.

Determining if the Estimate is Too Big or Too Small:

Since $f(x)$ appears to be increasing and the derivative is also increasing, our linear approximation might slightly underestimate $f(3.1)$, as the actual function $f(x)$ may have a steeper slope at points greater than $x = 3$. Therefore, this estimate of 10.1 could be slightly too small.

Would you like further explanation or have any questions?

Related Questions

1. How does the choice of $a$ affect the accuracy of linear approximations?
2. Why is a linear approximation likely to underestimate when the derivative is increasing?
3. How can we improve the accuracy of approximations for functions with non-linear behavior?
4. What is the effect of choosing $x$ values further from $a$ on the linear approximation?
5. How would the approximation change if we had a higher derivative available?

Tip:

For better accuracy in approximations near a point, use higher-order derivatives if available, as they provide more information about the function's curvature.