Instructions: Answer all questions completely, show your work, and clearly mark the answer. Please number all pages. Each question is worth 25 points for a total of 75 points. 1. Various functions have point singularities where the solution is undefined at a particular value x. For example, f(x)=exp(x21) is undefined at x=0. From calculus, we know that the x0limf(x)=0, however, numerically a Taylor series of the function at some x close to 0 is easier to compute. (a) Using a first order Taylor series approximation of f(x) around x=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 x=0.25. 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?.

1. Instructions: Answer all questions completely, show your work, and clearly mark the answer.
Please number all pages. Each question is worth 25 points for a total of 75 points. 1. Various
functions have point singularities where the solution is undefined at a particular value x. For
example, f(x)=exp(x21) is undefined at x=0. From calculus, we know that the x0limf(x)=0,
however, numerically a Taylor series of the function at some x close to 0 is easier to compute. (a)
Using a first order Taylor series approximation of f(x) around x=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 x=0.25. 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?