Wavy Surface Approximation

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Vincent Vu November 8, 2021 George Mason University MATH 401: Mathematics through 3D Printing This print represents a Taylor series approximation of a multivariable function. In single variable calculus, a Taylor series can be used to approximate a function about a point using derivatives. Higher accuracy is achieved by taking more and more derivatives, and for many functions this means that they can be represented using an infinite series of derivatives. When extended to multivariable functions, the same process is applied except extra terms are involved due to each function having partial derivatives with respect to multiple variables rather than a single variable. The function I used for my approximations was f(x,y) = sin(x^2 + y). The linear approximation given by the first partial derivatives yields a tangent plane. I did my approximations about the point f(0.5,0.5) as the approximation about f(0,0), or the origin, yielded a tangent plane that was only dependent on the variable y whereas the point I used yielded a tangent plane dependent on both x an y. For the quadratic approximation about f(0.5, 0.5), it was clear from plotting it against the function that it was much more accurate than the linear approximation due to the higher curvature of the quadratic surface being more in line with the wavy nature of the sine function. Due to the gaps between each of the surfaces, I had to generate supports and then sand down the surfaces once they were removed. Luckily it didn’t look like they made an impact on the appearance minus a few spots on the bottom, but nobody will be looking at the bottom anyway. The STL file was exported from Mathematica where the dimensions were extremely small, so the object had to be scaled to 500%. It’s still a bit difficult to determine which surface is which, so if I had a chance to reprint it I would definitely have looked for a way to make it more clear what the original surface was. Print settings and references listed below.