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Problem in Calculus related to the minimalization of a definite integral

I could not post this note elsewhere, hence am posting it here

This relates to the problem involving minimization of the integral (between 0 to pi/2) of the expression (sin(x) - (px^2+qx_)^2

Conceptualizing the problem geometrically, the above could be translated as =minimizing the distance between the sine curve and the parabola y=px^2 +qx

My feeling is this can happen if and only if the zero crossings and the peaks of the curves match, ie if px^2+qx at x=pi = 0 or if px^2+qx has its maximum at pi/2

Both these result in the following expression between p and q q = -p*pi

One idea seems to be to simplify the expression by expressing it as a difference between the integrals of sin(x) and (px^2+qx) separately (which can be interpreted as the difference between areas under curves) which would also minimize the squared difference , since their peaks match (possible links to orthogonality etc)

The other idea seems to be let the maximum of the parabola be equal to 1 at pi/2

One could possibly get more sophisticated by requiring tangency between the curves at pi/2 but in any case, the four numbers seem to be of equal magnitude but opposite in sign which seems to give zero

One could of course also look at taylors expansion of sin(x)

The above seems to be related to least squares of a sin curve by a parabola or vice-versa in a least squares approximation

It would have made more sense to look at polynomials involving sin(x) and ,more generally sin(nx) which is related to the fourier series expansion / fourier transform of px^2+qx

Note by Sundar R
3 years, 2 months ago

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