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The problem asks to estimate the value of \( \frac{\partial f}{\partial x} (20, 8) \), where \( x \) represents the number of workers, \( y \) represents the number of machines, and the table shows the production values for different combinations of workers (\( x \)) and machines (\( y \)).

To estimate the partial derivative \( \frac{\partial f}{\partial x} \), we need to look at how the production value changes with respect to the number of workers, while keeping the number of machines constant at \( y = 8 \).

### Steps to calculate \( \frac{\partial f}{\partial x}(20, 8) \):

1. **Find the values around \( x = 20 \) when \( y = 8 \)**:
   - For \( x = 15 \), \( f(15, 8) = 181 \)
   - For \( x = 20 \), \( f(20, 8) = 192 \)
   - For \( x = 25 \), \( f(25, 8) = 201 \)

2. **Estimate \( \frac{\partial f}{\partial x}(20, 8) \) using finite differences**:
   - We can estimate the derivative by calculating the slope between two points:
     \[
     \frac{\partial f}{\partial x}(20, 8) \approx \frac{f(25, 8) - f(15, 8)}{25 - 15} = \frac{201 - 181}{25 - 15} = \frac{20}{10} = 2.0
     \]

Thus, \( \frac{\partial f}{\partial x}(20, 8) \approx 2.0 \).

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Would you like more details on this calculation? Here are five related questions for deeper understanding:

1. How do you interpret the meaning of a partial derivative in this context?
2. Can we estimate the derivative \( \frac{\partial f}{\partial y} \) from this table?
3. What would the interpretation of \( \frac{\partial f}{\partial x} \) be for \( y = 12 \)?
4. How would you calculate the second derivative \( \frac{\partial^2 f}{\partial x^2} \)?
5. What does the constant change in production with respect to \( x \) suggest about the system?

**Tip**: For more accurate results, you can use central difference formulas to estimate derivatives when you have values on both sides of the point you're analyzing.

Utilize a tabela acima para estimar o valor de \( \frac{\partial f}{\partial x} (20, 8) \). Escreva a resposta numérica com 1 dígito decimal.

Learn how to estimate the partial derivative \( \frac{\partial f}{\partial x}(20, 8) \) using finite difference methods from a table of production values. This problem involves understanding partial derivatives and applying finite difference approximations, suitable for multivariable calculus students.

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