Observe the errors presented by different numerical derivation formulas using numerical derivation formulas in the file attached. Forward and backward approximations of the first derivative should be explored and centered approximations of the first derivative and and from the second derivative .
How do you get the formulas? (See: Ascher & Greif, A First Course in Numerical Methods, SIAM, 2011. Chapter 14 Numerical Differentiation.) Which formulas have fewer errors? How do they depend on h? Does the error always decrease as ? What other factor influences the error? Starting with h=1, the by a factor of 10. Plot the error against the step size h on a .
What h should they use to minimize the error in each formula?
Each derivation formula must be in a separate file. They must have as input parameters the function f to be derived and a point x at which the derivative is to be calculated. The outputs of each function are vectors with the values of h, the approximate value of the derivative, and the absolute error. To obtain the exact value of the derivative, they must calculate it symbolically.
Test your script with the functions
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