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.