Principles of Numerical Methods First Name Last Name 1. Use the centered-difference formula for numerical first derivative 1 f'(x)=(f( Num 2h f(x-h)) a) to approximate f'(x) at x = 1, where f(x)=2x-, for h=10", with n = -1, -2, ..., -17; b) prove that the error of the approximation, i.e., f'(x)- f'(x) - O(h). Num Note: The results of a) should be presented in the form of a table, an example of which is given below: Num (1)) 2. h AbsError 10-1 0.50166750019844 0.00166750019844 10-2 10-3 0.50001666674999 0.00001666674999 0.50000013368668 0.00000016666668 10-17 Final Exa Suppose we need to find the sum of the following ten numbers given on the right. First, addition is performed starting from the smallest number in ascending order. Then we add the same 10 numbers in reverse order, from largest to smallest. Assume that our "hypothetical" computer we are using works with decimal numbers, the mantissa of which can be represented by four significant digits. Thus, all numbers whose mantissas have more than four significant figures must be rounded. Compare the results of these two additions with the exact value of the sum of the given ten numbers. In each case, calculate the absolute and relative errors. Comparing these errors, what is your conclusion? a = 0.2897-100 a2 = 0.4976 100 a3 = 0.2488 10' a4 = 0.7259 10' as = 0.1638 102 a6 = 0.6249.102 a7 = 0.2162-103 as = 0.5233-10 a9 = 0.1403 104 a10= 0.5291-104