To approximate $f(1.9)$ using the function's local linear approximation at $x = 2$, we use the formula for the linear approximation:

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

where:

• $a = 2$
• $f(2) = -3$
• $f'(2) = -4$

We want to find $L(1.9)$. Plugging in the values:

$L(1.9) = f(2) + f'(2)(1.9 - 2)$

Now substitute the given values:

$L(1.9) = -3 + (-4)(1.9 - 2)$

Calculate the difference $1.9 - 2$:

$1.9 - 2 = -0.1$

Then multiply by $f'(2) = -4$:

$-4 \times (-0.1) = 0.4$

Now, add this result to $f(2) = -3$:

$L(1.9) = -3 + 0.4 = -2.6$

Thus, the value of the approximation of $f(1.9)$ is:

$\boxed{-2.6}$

The correct choice is:

(Choice D) $-2.6$

Would you like more details on this calculation or have any questions?

Here are 5 related questions that might help further your understanding:

1. What is the significance of using a linear approximation for estimating function values?
2. How would you use the same method to approximate $f(2.1)$?
3. What does the derivative $f'(2) = -4$ tell us about the function $f(x)$ at $x = 2$?
4. How would the linear approximation change if $f'(2)$ were positive instead of negative?
5. What is the error involved in using a linear approximation versus using the actual function?

Tip: When using linear approximations, remember that they are more accurate when $x$ is close to the point $a$ where the function is being approximated.