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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?

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#### 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.

Use linear approximation to estimate f(3.1) given that f(3) = 10 and the graph of the derivative f'(x) is provided.

Learn how to estimate f(3.1) using linear approximation when given f(3) = 10 and the derivative graph f'(x). This example covers concepts like linear approximation and derivatives, suitable for high school calculus students in Grades 11-12.

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