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2 years ago

Translate the following phrase to an expression. Then, evaluate the expression for n= 3.

Mathematics

1 answer:

Oksi-84 [34.3K]2 years ago

4 0

Answer:

Step-by-step explanation:

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Find the Fourier series of f on the given interval. f(x) = 1, ?7 < x < 0 1 + x, 0 ? x < 7

Zolol [24]

$f(x)=\begin{cases}1&\text{for }-7$

The Fourier series expansion of $f(x)$ is given by

$\dfrac{a_0}2+\displaystyle\sum_{n\ge1}a_n\cos\frac{n\pi x}7+\sum_{n\ge1}b_n\sin\frac{n\pi x}7$

where we have

$a_0=\displaystyle\frac17\int_{-7}^7f(x)\,\mathrm dx$
$a_0=\displaystyle\frac17\left(\int_{-7}^0\mathrm dx+\int_0^7(1+x)\,\mathrm dx\right)$
$a_0=\dfrac{7+\frac{63}2}7=\dfrac{11}2$

The coefficients of the cosine series are

$a_n=\displaystyle\frac17\int_{-7}^7f(x)\cos\dfrac{n\pi x}7\,\mathrm dx$
$a_n=\displaystyle\frac17\left(\int_{-7}^0\cos\frac{n\pi x}7\,\mathrm dx+\int_0^7(1+x)\cos\frac{n\pi x}7\,\mathrm dx\right)$
$a_n=\dfrac{9\sin n\pi}{n\pi}+\dfrac{7\cos n\pi-7}{n^2\pi^2}$
$a_n=\dfrac{7(-1)^n-7}{n^2\pi^2}$

When $n$ is even, the numerator vanishes, so we consider odd $n$ , i.e. $n=2k-1$ for $k\in\mathbb N$ , leaving us with

$a_n=a_{2k-1}=\dfrac{7(-1)-7}{(2k-1)^2\pi^2}=-\dfrac{14}{(2k-1)^2\pi^2}$

Meanwhile, the coefficients of the sine series are given by

$b_n=\displaystyle\frac17\int_{-7}^7f(x)\sin\dfrac{n\pi x}7\,\mathrm dx$
$b_n=\displaystyle\frac17\left(\int_{-7}^0\sin\dfrac{n\pi x}7\,\mathrm dx+\int_0^7(1+x)\sin\dfrac{n\pi x}7\,\mathrm dx\right)$
$b_n=-\dfrac{7\cos n\pi}{n\pi}+\dfrac{7\sin n\pi}{n^2\pi^2}$
$b_n=\dfrac{7(-1)^{n+1}}{n\pi}$

So the Fourier series expansion for $f(x)$ is

$f(x)\sim\dfrac{11}4-\dfrac{14}{\pi^2}\displaystyle\sum_{n\ge1}\frac1{(2n-1)^2}\cos\frac{(2n-1)\pi x}7+\frac7\pi\sum_{n\ge1}\frac{(-1)^{n+1}}n\sin\frac{n\pi x}7$

3 0

2 years ago

Which function is linear?

natali 33 [55]

Answer:

B) f(x)=13x+4 and C) f(x)=|3x| − 4

5 0

2 years ago

Please with me with translating words into equations

Blizzard [7]

2x - 27=56
the answer to the question

5 0

3 years ago

Sara is renting a car for x amount of days. To rent the car she must pay an initial fee of $130 and then $17 for each day. If sa

Rus_ich [418]

Answer:

283

Step-by-step explanation:

130+17(9)

130 + 17 times the days

5 0

2 years ago

If the equation of a circle is (x - 2)2 + (y-6)2 = 4, it passes through point

san4es73 [151]

(x - 2)² + (y - 6)² = 4

You can be certain about one thing just by looking at the equation (2, 6) is the center, so, obviously the circle isnt going through this point

Since the radius is 2 if we don't move from y = 6 we have points in (0, 6) and (4, 6)

So alternative b.

To be more certain just subs the point in the x and y, if its equal, it pass through

(x - 2)² + (y - 6)² = 4

To point (4, 6)

(4 - 2)² + (6 - 6)² = 4

(2²) + 0² = 4

4 = 4

Thats right

4 0

3 years ago