math121a-f23:hw_8
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math121a-f23:hw_8 [2023/10/21 04:57] pzhou created |
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| ====== Homework 8 ====== | ====== Homework 8 ====== | ||
| - | 1. Sine wave decomposition. | + | 1. Sine and Cosine |
| - | Suppose you are given a function on an interval, $f(x): [0, 1] \to \R$, such that $f(x)$ vanishes on both end points $f(0)=f(1)=0$. Such function $f(x)$ can be expressed as a sum of 'sine waves'. | + | Suppose you are given a function on an interval, $f(x): [0, 1] \to \R$. Such function $f(x)$ can be expressed as a sum of 'sine waves' |
| - | $$ f(x) = \sum_{n=1}^\infty c_n \sin(n \pi x). $$ | + | |
| - | Can you figure out a way to determine the coefficients $c_n$? | + | $$ f(x) = a_0 + \sum_{n=1}^\infty a_n \cos(2n \pi x) + b_n \sin(2n \pi x). $$ |
| + | |||
| + | Can you figure out a way to determine the coefficients $a_n$ and $b_n$? | ||
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| + | Test out your method for the following function | ||
| + | $$ f(x) = \begin{cases} 1 & 0 < x < 1/2 \cr | ||
| + | 0 & 1/2 \leq x \leq 1 | ||
| + | \end{cases} | ||
| + | $$ | ||
| + | |||
| + | find $a_0, a_1, b_1$ and plot the truncated Fourier series | ||
| + | $$ a_0 + a_1 \cos(2 \pi x) + b_1 \sin(2 \pi x). $$ | ||
| + | How does this resemble your original given function? | ||
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| $$ f'(t) + f(t) = 0 $$ | $$ f'(t) + f(t) = 0 $$ | ||
| And suppose $f(0) = 1$. Can you solve $f(t)$ for $t > 0$? | And suppose $f(0) = 1$. Can you solve $f(t)$ for $t > 0$? | ||
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| 3. Consider the following equation, for $t>0$, | 3. Consider the following equation, for $t>0$, | ||
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| 4. Consider the following equation, for $t>0$, | 4. Consider the following equation, for $t>0$, | ||
| + | $$ [(d/dt)^2 + 1] f(t) = 0 $$ | ||
| + | And suppose $f(0) = 1, f' | ||
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| + | 5 (bonus, optional). Consider the following equation, for $t>0$, | ||
| $$ (d/dt + 1) (d/dt + 1) f(t) = 0 $$ | $$ (d/dt + 1) (d/dt + 1) f(t) = 0 $$ | ||
| And suppose $f(0) = 1, f' | And suppose $f(0) = 1, f' | ||
math121a-f23/hw_8.1697864276.txt.gz · Last modified: 2026/02/21 14:44 (external edit)
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