Various functions have point singularities where the solution is undefined at a particular value r. For...

    

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Various functions have point singularities where the solution is undefined at a particular value r. For example, f(x) 1 (¹) = exp (1) is undefined at = 0. From calculus, we know that the lim, o f(x) = 0, however, numerically a Taylor series of the function at some a close to 0 is easier to compute. (a) Using a first order Taylor series approximation of f(r) around z=0.5, calculate the f(0). What are the absolute and relative errors? (b) Repeat the calculations using a second order Taylor series approximation around x = 0.5. Does the Taylor series show convergence? Why? (c) Recompute the first and second Taylor series around 20.25. Compare the absolute errors. Based on this information, what order of magnitude do you expect the absolute error the of first and second Taylor series around x = 0.125 to be? Various functions have point singularities where the solution is undefined at a particular value r. For example, f(x) 1 (¹) = exp (1) is undefined at = 0. From calculus, we know that the lim, o f(x) = 0, however, numerically a Taylor series of the function at some a close to 0 is easier to compute. (a) Using a first order Taylor series approximation of f(r) around z=0.5, calculate the f(0). What are the absolute and relative errors? (b) Repeat the calculations using a second order Taylor series approximation around x = 0.5. Does the Taylor series show convergence? Why? (c) Recompute the first and second Taylor series around 20.25. Compare the absolute errors. Based on this information, what order of magnitude do you expect the absolute error the of first and second Taylor series around x = 0.125 to be?

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a Firstorder Taylor series approximation of fr at r05 is fr 1 r 05 f05 We can calculate f05 e 164872 ... View the full answer