Use the ideas of Richardson's extrapolation and Romberg's method to evaluate the first derivative of f...

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Use the ideas of Richardson's extrapolation and Romberg's method to evaluate the first derivative of f = ex at x = 0. To construct the first column of the Romberg's table, use the formula (exth - ex)/h and h = 10-1, and proceed by halving h. Write a code to construct the Romberg's table and provide its output. Create a table as large as necessary to calculate all 15 decimal places of the integral correctly, if possible. If not, how low does the error get with this approach? Is subtractive cancellation still a concern? Compare the results with that of the previous question. b) Repeat part (a) starting with the formula (exth ex-h)/(2h) and h = 10-¹. Use the ideas of Richardson's extrapolation and Romberg's method to evaluate the first derivative of f = ex at x = 0. To construct the first column of the Romberg's table, use the formula (exth - ex)/h and h = 10-1, and proceed by halving h. Write a code to construct the Romberg's table and provide its output. Create a table as large as necessary to calculate all 15 decimal places of the integral correctly, if possible. If not, how low does the error get with this approach? Is subtractive cancellation still a concern? Compare the results with that of the previous question. b) Repeat part (a) starting with the formula (exth ex-h)/(2h) and h = 10-¹.

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