Question

5. If \(f(x)=\left\{\begin{array}{cc}0 & -2<x<0 \\ x & 0<x<2\end{array} \quad\right.\)is periodio of period 4 , and whose Fourier series is given by \(\frac{a_{0}}{2}+\sum_{n=1}^{2}\left[a_{n} \cos \left(\frac{n \pi}{2} x\right)+b_{n} \sin \left(\frac{n \pi}{2} x\right)\right], \quad\) find \(a_{n}\)

A. \(\frac{2}{n^{2} \pi^{2}}\)

B. \(\frac{(-1)^{n}-1}{n^{2} \pi^{2}}\)

C. \(\frac{4}{n^{2} \pi^{2}}\)

D. \(\frac{2}{n \pi}\)

\(\mathbf{E}_{1} \frac{2\left((-1)^{n}-1\right)}{n^{2} \pi^{2}}\)

F. \(\frac{4}{n \pi}\)

6. Let \(f(x)-2 x-l\) on \([0,2]\). The Fourier sine series for \(f(x)\) is \(\sum_{w}^{n} b_{n} \sin \left(\frac{n \pi}{2} x\right)\), What is \(b, ?\)

A. \(\frac{4}{3 \pi}\)

B. \(\frac{2}{\pi}\)

C. \(\frac{4}{\pi}\)

D. \(\frac{-4}{3 \pi}\)

E. \(\frac{-2}{\pi}\)

F. \(\frac{-4}{\pi}\)

7. Let \(f(x)\) be periodic of period 4, where \(f(x)=x^{2}-2 x-2\) on \((-2,2]\).

The Fourier series for \(f(2)\) will converge to

A. \(-1\)

B. 1

C. \(-2\)

D. 2

E. \(-3\)

F. 3

8. Consider the heat conduction problem

$$ \begin{aligned} &\frac{\partial^{2} u}{\partial x^{2}}=\frac{1}{4} \frac{\partial u}{\partial t}, 0<x<3, t="">0 \\ &u(0, t)=2, u(3, t)=5, t>0 \\ &u(x, 0)=5,0 \leq x \leq 3 \end{aligned} $$

Then the steady state temperature as \(t \rightarrow \infty\) is

A. 2

B. 5

C. \(5-\mathrm{x}\)

D. \(x-2\)

E. \(x+2\)

F. \(x+3\)

Answer 1