Taylor's Theorem

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This is Taylor's Theorem approximation created in Mathematica version 10. George Mason University Math 493 for Mathematics Through 3D Printing. This project demonstrates Taylor’s theorem giving an approximation to the cosine function. The Taylor series is given by f(x)= ∑_(n=0) to ∞ cn (x-a)^n, where cn = f^n(a)/n!. where f(x)=cos(x) and a=pi. Using polynomials functionality to compute the nth degree, the nth degree is increased by even numbers which brings us to a better approximation. Let x^n/n! , where n equals a positive even integer starting at n=0 cos⁡(x)=1-x^2/2!+x^4/4!-x^6/6!+x^8/8!-x^10/10!… Note: For x^n/n! , where n equals a positive odd integer starting at n=1 will create an approximation to the sine function. Print Settings Printer Brand: MakerBot Printer: MakerBot Replicator (5th Generation) Rafts: Yes Supports: Yes Notes: Raft not required. Thickness of design was originally 0.1, but when taking off supports damaged the design. Thickness was increased to 0.5, which made it easier to remove support. How I Designed This Instructions In Mathematica the following function were created and then put together towards the end in output5. f[x,y]:={x,y,Cos[x]}; Output1 = ParametricPlot3D[f[x,y],{x,-8,8}, {y,2,7} PlotStyle->{Green,Thickness[0.5]}, Mesh->False] h[x,y]:={x,y,(1-x^2/2!)}; Output2 = ParametricPlot3D[h[x,y],{x,-4,4}, {y,2,7} PlotStyle->{Red,Thickness[0.5]}, Mesh->False] g[x,y]:={x,y,(1-x^2/2!+x^4/4!)}; Output3 = ParametricPlot3D[g[x,y],{x,-4,4}, {y,2,7} PlotStyle->{Blue,Thickness[0.5]}, Mesh->False] k[x,y]:={x,y,(1-x^2/2!+x^4/4!-x^6/6!+x^8/8!-x^10/10!)}; Output4 = ParametricPlot3D[k[x,y],{x,-6,6}, {y,2,7} PlotStyle->{Yellow,Thickness[0.5]}, Mesh->False] output5=Show[{output1, output2, output3, output4}]