Taylor's Theorem

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This is a Taylor's Theorem approximation created in Mathematica version 10 for George Mason University Math 493: Mathematics Through 3D Printing. The project demonstrates Taylor's theorem by 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!. For the function f(x) = cos(x), and a = π, 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. Then, 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. The thickness of design was originally 0.1, but when taking off supports damaged the design. The thickness was increased to 0.5, which made it easier to remove support. How I Designed This: Instructions: In Mathematica, the following functions 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}]