Q3Mathematics I

Question

Find the asymptotes of $y^{2} (x - b) = x^{3} + a^{3}$ .

Answer 1

The mathematical formula for the Fourier coefficient $a_{n}$ requires integrating the function over its fundamental period:

a_{n} = \frac{1}{π} \int_{- π}^{π} (x + x^{2}) cos (n x) d x = \frac{1}{π} [\int_{- π}^{π} x cos (n x) d x + \int_{- π}^{π} x^{2} cos (n x) d x]

The first integrand, $x cos (n x)$ , is strictly an odd function, meaning its definite integral over the symmetric interval $[- π, π]$ is identically zero. The second integrand, $x^{2} cos (n x)$ , is strictly even. Using integration by parts on the even term definitively yields $a_{n} = \frac{4 ( - 1 ) ^{n}}{n ^{2}}$ .

Q3Mathematics I

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