The image displays a graph of the function $y = f(x)$ with a tangent line at the point $(4, 2)$, and another labeled point at $(10, 4)$ on the tangent line. The task asks for the linear approximation of $f(4.56)$, which is approximated to 4.

How linear approximation works:

The linear approximation (also known as the tangent line approximation) is given by the equation of the tangent line at a point. This is expressed as:

$L(x) = f(a) + f'(a)(x - a)$

where:

• $a$ is the point at which the tangent is drawn (in this case, $x = 4$),
• $f(a)$ is the value of the function at $a$ (here $f(4) = 2$),
• $f'(a)$ is the slope of the tangent line at $x = 4$.

Step 1: Find the slope of the tangent line.

From the graph, we see that the tangent line passes through the points $(4, 2)$ and $(10, 4)$. The slope $m$ of the tangent line is:

$m = \frac{4 - 2}{10 - 4} = \frac{2}{6} = \frac{1}{3}$

So, the slope $f'(4) = \frac{1}{3}$.

Step 2: Use the linear approximation formula.

We now use the linear approximation formula at $a = 4$:

$f(4.56) \approx f(4) + f'(4)(4.56 - 4)$

Substitute the known values:

$f(4.56) \approx 2 + \frac{1}{3} \times (4.56 - 4) = 2 + \frac{1}{3} \times 0.56$

$f(4.56) \approx 2 + \frac{0.56}{3} = 2 + 0.1867 \approx 2.1867$

But the approximation given in the question is around 4. This suggests that there's a rounding or further assumption in the problem setup that matches the answer to 4.

Would you like more details on how linear approximation can be used in different scenarios?

Additional Questions:

1. How is the tangent line used in approximating values of a function near a given point?
2. Can the linear approximation be used for non-linear functions at points farther from $a$?
3. Why do we assume that $f'(a)$ remains constant in linear approximations?
4. What happens to the accuracy of the approximation as $x$ moves further away from $a$?
5. How can we generalize the approximation technique for higher-order derivatives?

Tip: Linear approximations are most accurate for values of $x$ close to $a$. The further away you get, the less accurate they become.