91Ó°ÊÓ

Use Theorem \(11.3 .5(\mathbf{b})\) or where applicable, Exercise \(11.1 .35(\mathbf{b}),\) to find the Fourier sine series of \(f\) on \([0, L]\). $$ f(x)=x\left(x^{3}-2 L x^{2}+L^{3}\right) $$

Use Theorem \(11.3 .5(\mathbf{b})\) or where applicable, Exercise \(11.1 .35(\mathbf{b}),\) to find the Fourier sine series of \(f\) on \([0, L]\). $$ f(x)=x\left(3 x^{4}-10 L^{2} x^{2}+7 L^{4}\right) $$

Use Theorem \(11.3 .5(\mathbf{b})\) or where applicable, Exercise \(11.1 .35(\mathbf{b}),\) to find the Fourier sine series of \(f\) on \([0, L]\). $$ f(x)=x\left(3 x^{4}-5 L x^{3}+2 L^{4}\right) $$

(a) Prove Theorem \(11.3 .5(\mathbf{c})\). (b) In addition to the assumptions of Theorem \(11.3 .5(\mathbf{c})\), suppose \(f^{\prime \prime}(L)=0, f^{\prime \prime}\) is continuous, and \(f^{\prime \prime \prime}\) is piecewise continuous on \([0, L] .\) Show that $$ c_{n}=\frac{16 L^{2}}{(2 n-1)^{3} \pi^{3}} \int_{0}^{L} f^{\prime \prime \prime}(x) \sin \frac{(2 n-1) \pi x}{2 L} d x, \quad n \geq 1 $$

Use Theorem 11.3.5(c) or, where applicable, Exercise 11.1.42(b), to find the mixed Fourier cosine series of \(f\) on \([0, L]\). $$ f(x)=x^{2}(L-x) $$

Use Theorem 11.3.5(c) or, where applicable, Exercise 11.1.42(b), to find the mixed Fourier cosine series of \(f\) on \([0, L]\). $$ f(x)=L^{2}-x^{2} $$

Use Theorem 11.3.5(c) or, where applicable, Exercise 11.1.42(b), to find the mixed Fourier cosine series of \(f\) on \([0, L]\). $$ f(x)=L^{3}-x^{3} $$

Use Theorem 11.3.5(c) or, where applicable, Exercise 11.1.42(b), to find the mixed Fourier cosine series of \(f\) on \([0, L]\). $$ f(x)=2 x^{3}+3 L x^{2}-5 L^{3} $$

Use Theorem 11.3.5(c) or, where applicable, Exercise 11.1.42(b), to find the mixed Fourier cosine series of \(f\) on \([0, L]\). $$ f(x)=4 x^{3}+3 L x^{2}-7 L^{3} $$

Use Theorem 11.3.5(c) or, where applicable, Exercise 11.1.42(b), to find the mixed Fourier cosine series of \(f\) on \([0, L]\). $$ f(x)=x^{4}-2 L x^{3}+L^{4} $$