To use linear approximation, we can apply the formula for the linearization of a function near a point. The formula for the linear approximation $L(x)$ of a function $f(x)$ near a point $x = a$ is:

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

Given:

• $f(6) = 4$
• $f'(6) = 2$
• We want to estimate $f(5.85)$, so $x = 5.85$ and $a = 6$.

Now, let's substitute these values into the linear approximation formula:

$f(5.85) \approx f(6) + f'(6)(5.85 - 6)$

Calculate each term:

1. $f(6) = 4$
2. $f'(6) = 2$
3. $5.85 - 6 = -0.15$

Substitute these into the equation:

$f(5.85) \approx 4 + 2 \times (-0.15)$

$f(5.85) \approx 4 - 0.3$

$f(5.85) \approx 3.7$

So, the linear approximation estimate for $f(5.85)$ is 3.7.

Would you like more details or have any questions?

Here are some additional questions for exploration:

1. How does linear approximation differ from other methods of estimating function values?
2. What would happen to the approximation if $f'(6)$ were a negative value?
3. How could we improve accuracy if we needed a more precise estimate of $f(5.85)$?
4. Why is the point $a = 6$ chosen for approximation in this case?
5. How does changing $x$ to values closer to or farther from $6$ affect the linear approximation?

Tip: Linear approximation works best for values of $x$ close to the point $a$ because it assumes the function behaves almost like a straight line near that point.