TaylorSeries - dent Note that the first term on the right...

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The Taylor Series Expansion and An Introduction to Mathematica Purpose: Our first excursion into Mathematica will be simple, for more detail on Mathematica I suggest Wolfram, S. Mathematica: A System for Doing Mathematics on Computer or Blachman, N. Mathematica: A Practical Approach. The purpose of this notebook is to show the remainder term of a second order Taylor series expansion of a quadratic funciton. The Mathematics The first step is to declare a third order function. In this function x1 is the variable and a,b, and c are parameters. In[1]:= g1 = x1 3 + ax1 2 + b x1 + c Out[1]= c + b x1 + a x1 2 + x1 3 Given the original function, the second order Taylor expansion for g(x1) at x0 is: In[2]:= fg1 = I x0 3 + a x0 2 + b x0 + c M + I 3 x0 2 + 2 a x0 + b M H x1 - x0 L + 1 2 H 6 x0 + 2 a L H x1 - x0 L 2 Out[2]= c + b x0 + a x0 2 + x0 3 + H b + 2 a x0 + 3 x0 2 L H - x0 + x1 L + 1 2 H 2 a + 6 x0 L H - x0 + x1 L 2 A useful exercise to learn Mathematica is to automate the derivation from the first equation, but this will be left to the stu-
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Unformatted text preview: dent. Note that the first term on the right hand side of the equation is simply g(x0). Instructing Mathematica to simplify this expression yields: In[3]:= Simplify @ fg1 D Out[3]= c + x0 3-3 x0 2 x1 + 3 x0 x1 2 + x1 H b + a x1 L Subtracting this result from the orginal expression g(x1) yields the error of approximation: In[4]:= g1-fg1 Out[4]=-b x0-a x0 2-x0 3 + b x1 + a x1 2 + x1 3-H b + 2 a x0 + 3 x0 2 L H-x0 + x1 L-1 2 H 2 a + 6 x0 L H-x0 + x1 L 2 In[5]:= Simplify @ % D Out[5]=-H x0-x1 L 3 TaylorSeries.nb 1 Note that if x1-x0<1 then the error term converges to zero faster than x1 approaches x0. TaylorSeries.nb 2...
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This note was uploaded on 11/08/2011 for the course AEB 6533 taught by Professor Moss during the Fall '08 term at University of Florida.

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TaylorSeries - dent Note that the first term on the right...

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