To determine the values of $x$ for which a given linear approximation is accurate to within 0.07 at $a = 0$, we follow a general process for handling linear approximations. Let's break down the steps and key concepts involved:

General Approach:

1. Linear Approximation Formula: For a function $f(x)$, the linear approximation $L(x)$ around a point $a$ is given by: $L(x) = f(a) + f'(a)(x - a)$ In this case, $a = 0$, so: $L(x) = f(0) + f'(0) \cdot x$

2. Error Estimate: The goal is to find when the error of the approximation is within 0.07. The error is the absolute difference between the function $f(x)$ and its linear approximation $L(x)$: $|f(x) - L(x)| < 0.07$

3. Solve the Inequality: After applying the linear approximation, we solve the inequality for $x$, which will give us the range of values where the approximation is accurate within 0.07.

To proceed, I would need the function $f(x)$ for which you are applying the linear approximation. Could you provide the function?

Let me know if you'd like a detailed explanation of these steps!

Further questions to deepen understanding:

1. What is the function $f(x)$ you are working with in this problem?
2. How is the derivative $f'(x)$ used in linear approximations?
3. How does the choice of $a = 0$ affect the linear approximation?
4. Can you explain why the error bound is important in approximation problems?
5. What happens to the error as $x$ moves further away from $a$?

Tip:

Linear approximations are most accurate near the point of approximation $a$, and the error tends to grow as $x$ moves further away from $a$.