Answer
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Work Step by Step
When provided with a table of a function's values, it is possible to approximate the derivative of the function using finite difference methods. One common approach is to use the finite difference formula for the derivative, which approximates the derivative at a given point by calculating the slope between nearby points in the table.
Specifically, to approximate the derivative of a function \( f(x) \) at a point \( x_i \), where \( x_i \) is one of the values in the table, you can use the finite difference formula:
\[ f'(x_i) \approx \frac{f(x_{i+1}) - f(x_i)}{x_{i+1} - x_i} \]
This formula calculates the average rate of change between the point \( x_i \) and the next point \( x_{i+1} \) in the table, providing an approximation of the derivative at \( x_i \).
However, it's important to note that this method provides only an approximation of the derivative, and the accuracy of the approximation depends on the spacing between the points in the table and the smoothness of the function. Additionally, using more advanced interpolation techniques or fitting a curve to the data points can yield more accurate results.
