Determine the values of x for which the given linear approximation is accurate to within 0.07 at a = 0

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 \( a \).

Determine the values of x for which the given linear approximation is accurate to within 0.07 at a = 0.

Explore how to determine the range of x values for which a linear approximation is accurate to within 0.07 at a = 0. This problem covers key calculus concepts like linear approximation, error bounds, and Taylor series, suitable for Grades 11-12 and college students.

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