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The image you uploaded contains a mathematical exercise, which seems to be written in Greek. Here's a translation and summary of the problem in English:

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### Problem 1 (20 points)
Consider a computer with a set of real numbers, with parameters:
\[ \beta = 10, \quad t = 4, \quad L = -10, \quad U = 10 \]
and rounding.

(i) **Calculate the value of the function**:
\[ f(x) = e^{-x} - 1 \]
for \( x = 0.01 \).

Perform the calculations using standard computer arithmetic. Find the relative error and compare it with the machine's arithmetic error.

(ii) **Suggest a more accurate method** for calculating the value of the function \( f(x) = e^{-x} - 1 \) for \( x = 0.01 \) using the same parameters. Again, calculate the relative error and compare it with the machine's arithmetic error.

(iii) **Which of these methods** leads to more accurate results and why?

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This problem deals with numerical methods, especially focusing on the accuracy and precision of calculations using different techniques, comparing the results with machine arithmetic limitations.

Would you like me to proceed with solving the first part (i), or provide insights for all parts? Let me know!

### Follow-up Questions:
1. What is the role of parameters \(\beta\), \(t\), \(L\), and \(U\) in machine arithmetic?
2. How do you compute the relative error in numerical calculations?
3. What numerical method can improve accuracy for calculating \( e^{-x} - 1 \) when \(x\) is small?
4. Why does numerical precision matter when calculating small values of \(x\)?
5. How does rounding affect the accuracy of computations in floating-point arithmetic?

### Tip:
When computing small values of a function like \( e^{-x} - 1 \), precision can be improved by using series expansions or alternative formulations to avoid cancellation errors.

Calculate the value of the function f(x) = e^{-x} - 1 for x = 0.01 using computer arithmetic. Find the relative error and compare it with the machine's arithmetic error. Suggest a more accurate method for calculating the function's value and compare relative errors. Determine which method is more accurate and why.

Explore numerical methods for calculating e^{-x} - 1 when x is small. Learn about error comparison, machine precision, and improved calculation methods using Taylor series. Suitable for first-year undergraduate students.

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