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Define the Error between two functions, f(x) and g(x) evaluated at n is given by∫−nn(f(x)−g(x))2dx.
Find the Error in terms of n between the functions cos(x) and sin(x).
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Suppose that we would like to fit a linear model, ax+b, to the function y=2x2+4x+1 close to x=0.
Show that the error between 2x2+4x+1 and ax+b evaluated at n is given by 58n5+32n3(a2−8a+20−4b)+2n(1−b)2.
Hence, determine the value of a minimising the Error.
Similarly, find the value of b minimising the error in terms of n.
By considering n→0, find the line best approximating the function in the neighborhood of x=0.
Verify that this gives the same answer as the tangent line at x=0.
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