Choose different step sizes h and plot a log-log graph of error vs step size to compare the numerical derivatives found using the forward, backward, and central difference formulas with the actual derivative. How accurate are the numerical derivatives at x = 0, 1, 1.5, and 2? At what step sizes do you obtain the minimum errors, and what are the minimum errors (see Sauer figure 5.1)? when x=? can get the minimum errors
Question
Choose different step sizes h and plot a log-log graph of error vs step size to compare the numerical derivatives found using the forward, backward, and central difference formulas with the actual derivative. How accurate are the numerical derivatives at x = 0, 1, 1.5, and 2? At what step sizes do you obtain the minimum errors, and what are the minimum errors (see Sauer figure 5.1)? when x=? can get the minimum errors
Solution
As an AI, I'm unable to directly plot graphs or perform computations. However, I can guide you on how to do it.
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Choose a function for which you know the derivative. For example, let's take f(x) = sin(x). The derivative of this function is f'(x) = cos(x).
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Choose different step sizes h. For example, you can choose h = 1, 0.1, 0.01, 0.001, etc.
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Calculate the numerical derivatives at x = 0, 1, 1.5, and 2 using the forward, backward, and central difference formulas.
Forward difference formula: f'(x) ≈ (f(x+h) - f(x)) / h Backward difference formula: f'(x) ≈ (f(x) - f(x-h)) / h Central difference formula: f'(x) ≈ (f(x+h) - f(x-h)) / (2h)
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Calculate the actual derivative at x = 0, 1, 1.5, and 2 using the derivative function f'(x) = cos(x).
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Calculate the error at each x for each step size h and for each numerical derivative. The error is the absolute difference between the numerical derivative and the actual derivative.
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Plot a log-log graph of error vs step size for each numerical derivative. You should have three graphs, one for each numerical derivative.
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From the graphs, you can see at what step sizes you obtain the minimum errors and what the minimum errors are.
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To find out at which x you can get the minimum errors, you would need to repeat the above steps for different x values and compare the results.
Remember, the accuracy of the numerical derivatives depends on the step size h. The smaller the step size, the more accurate the numerical derivative. However, if the step size is too small, you may encounter round-off errors.
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