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Let's go over these problems step by step.

### Problem 1: Approximation of \(\sin x \approx x\)
We are asked to determine for what range of \(x\), the approximation \(\sin x \approx x\) is valid to a relative accuracy of \(\frac{1}{2} \times 10^{-14}\).

#### Approach:
1. For small \(x\), \(\sin x\) can be approximated by its Taylor series expansion:
   \[
   \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots
   \]
2. The relative error of the approximation \(\sin x \approx x\) comes from neglecting the higher-order terms of this series.
3. The next term in the series after \(x\) is \(-\frac{x^3}{6}\), so the relative error is approximately:
   \[
   \text{Relative error} = \frac{\left| \frac{x^3}{6} \right|}{x} = \frac{x^2}{6}
   \]
4. To ensure the approximation is good to the given relative accuracy, we set:
   \[
   \frac{x^2}{6} \leq \frac{1}{2} \times 10^{-14}
   \]
5. Solve this inequality for \(x\):
   \[
   x^2 \leq 3 \times 10^{-14}
   \]
   \[
   x \leq \sqrt{3 \times 10^{-14}} = \sqrt{3} \times 10^{-7} \approx 1.732 \times 10^{-7}
   \]

Thus, the approximation \(\sin x \approx x\) is valid for \(x \leq 1.732 \times 10^{-7}\).

### Problem 2: Maclaurin Series for \(e^x\) Approximation
We are asked to find how many terms of the Maclaurin series for \(e^x\) are needed to achieve an absolute truncation error of less than \(10^{-10}\) for \(x = 0.5\).

#### Approach:
1. The Maclaurin series for \(e^x\) is:
   \[
   e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots
   \]
2. The error in truncating the series after the \(n\)-th term is given by the next term in the series:
   \[
   \text{Error} \approx \frac{x^{n+1}}{(n+1)!}
   \]
3. We want this error to be less than \(10^{-10}\), so we need to solve:
   \[
   \frac{0.5^{n+1}}{(n+1)!} < 10^{-10}
   \]
4. We can try successive values of \(n\) to find when this inequality holds.

Let's calculate this numerically.

#### Calculation:
For \(x = 0.5\), find the smallest \(n\) such that:
\[
\frac{0.5^{n+1}}{(n+1)!} < 10^{-10}
\]

I'll compute the terms.For \(x = 0.5\), at least 10 terms of the Maclaurin series are needed to ensure that the truncation error is less than \(10^{-10}\).

Would you like more details on the reasoning or further questions?

Here are 5 related questions that could deepen your understanding:

1. How would the approximation for \(\sin x \approx x\) change if the required relative accuracy were \(\frac{1}{2} \times 10^{-12}\)?
2. Can we derive a general formula for the relative error of any Taylor series approximation?
3. How would you modify the method in Problem 2 for a different value of \(x\), say \(x = 1\)?
4. How does the relative error in approximating \(\sin x\) compare to other common approximations, like \(\cos x \approx 1\)?
5. How would the truncation error behave if we were approximating \(e^{-x}\) instead of \(e^x\)?

**Tip:** For problems involving series approximations, always check both the absolute and relative error depending on the context to ensure precision in your calculations.

Problem 1: For small x, the approximation sin(x) ≈ x is often used. For what range of x is this good to a relative accuracy of (1/2) × 10^(-14)?
Problem 2: Suppose you approximate e^x by its truncated Maclaurin series. For x = 0.5, derive how many terms of the series are needed to achieve an (absolute) truncation error of less than 10^(-10).

Explore how to approximate sin(x) ≈ x with high relative accuracy and how many terms of the Maclaurin series for e^x are needed to achieve less than 10^(-10) truncation error. Includes step-by-step solutions for both problems.

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