https://file.aiquickdraw.com/mathbot/file/17274164242364vjxbi9e.jpg

The image shows a graph depicting a function \( f(x) \) along with its linear approximation at \( x = a \), given as \( y = 3x - 1 \) (shown in blue). There are several questions based on this graph:

(a) **What is the value of \( a \)?**
   - From the graph, it appears that the linear approximation intersects the curve at around \( x = 5 \). Therefore, \( a = 5 \).

(b) **What is the value of \( f(a) \)?**
   - Since the linear approximation is evaluated at \( a = 5 \), we substitute \( x = 5 \) into the linear equation \( y = 3x - 1 \) to get:
     \[
     f(5) = 3(5) - 1 = 15 - 1 = 14
     \]
   So, \( f(a) = 14 \).

(c) **Use the linear approximation to approximate the value of \( f(5.2) \).**
   - The linear approximation is \( y = 3x - 1 \). To approximate \( f(5.2) \), substitute \( x = 5.2 \) into the equation:
     \[
     f(5.2) \approx 3(5.2) - 1 = 15.6 - 1 = 14.6
     \]

(d) **Is the approximation an under- or overestimate?**
   - From the graph, the actual curve of \( f(x) \) lies above the linear approximation near \( x = 5.2 \). This indicates that the linear approximation underestimates the true value of \( f(5.2) \).

Would you like more details or explanations on any part? Here are five related questions to explore further:

1. How is the tangent line equation derived for a function at a given point?
2. Why is linear approximation useful in calculus?
3. What is the difference between a linear approximation and the actual function in terms of concavity?
4. How can we check whether a linear approximation overestimates or underestimates the function?
5. How would we compute a higher-order approximation, such as a quadratic one?

**Tip:** The accuracy of linear approximation improves when used near the point of tangency but decreases further away from the point of interest.

The figure below shows f(x) and its linear approximation at x = a, y = 3x - 1. What is the value of a, f(a), and the linear approximation at f(5.2)?

Explore how to find the linear approximation of a function at x = a, using y = 3x - 1 as the tangent line. This includes finding f(a), approximating f(5.2), and determining if the result is an under- or overestimate. Suitable for high school calculus students.

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