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.