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Ghella [55]
2 years ago
12

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

Mathematics
1 answer:
Zolol [24]2 years ago
3 0
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
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Answer:

Algebraically, f is even if and only if f(-x) = f(x) for all x in the domain of f. A function f is odd if the graph of f is symmetric with respect to the origin. Algebraically, f is odd if and only if f(-x) = -f(x) for all x in the domain of f.

Step-by-step explanation:

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if a truck travel 248 miles in 4 hours then the truck traveled an average of how many miles each hour write an equation to solve
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https://www.slader.com/discussion/question/if-a-truck-traveled-248-miles-in-4-hours-then-the-truck-9c45464f/

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The numerator and denominator of a fraction are in the ratio of 3 to 5. If the numerator and denominator are both increased by 2
ivolga24 [154]

Question:

The numerator and denominator of a fraction are in the ratio of 3 to 5. If the numerator and denominator are both decreased by 2, the fraction is now equal to  \frac{1}{2}.

If n = the numerator and d = the denominator, which of the following systems of equations could be used to solve the problem?

5n = 3d and n - 2 = 2d - 4

5n = 3d and 2n - 4 = d - 2

3n = 5d and 2n - 4 = d - 2

Answer:

5n = 3d and 2n – 4 = d – 2

Solution:

Let n be the numerator of the fraction and d be the denominator of the fraction.

Given the numerator and denominator of a fraction are in the ratio of 3 to 5.

This can be written as n : d = 3 : 5.

⇒ \frac{n}{d}= \frac{3}{5} – – – – (1)

Do cross multiplication, we get

⇒ 5n = 3d

When the numerator and denominator are decreased by 2, the fraction is equal to \frac{1}{2}.

⇒ \frac{n-2}{d-2}= \frac{1}{2}

Do cross multiplication, we get

⇒ 2(n –2)=1(d – 2)

⇒ 2n – 4 = d – 2

Hence, 5n = 3d and 2n – 4 = d – 2 can be used to solve the problem.

6 0
3 years ago
Read 2 more answers
Lily is a botanist who works for a garden that many tourists visit. The function f(s) = 2s + 30 represents the number of flowers
Vesnalui [34]

Answer:

  A. b(w) = 80w +30

  B. input: weeks; output: flowers that bloomed

  C. 2830

Step-by-step explanation:

<h3>Part A:</h3>

For f(s) = 2s +30, and s(w) = 40w, the composite function f(s(w)) is ...

  b(w) = f(s(w)) = 2(40w) +30

  b(w) = 80w +30 . . . . . . blooms over w weeks

__

<h3>Part B:</h3>

The input units of f(s) are <em>seeds</em>. The output units are <em>flowers</em>.

The input units of s(w) are <em>weeks</em>. The output units are <em>seeds</em>.

Then the function b(w) above has input units of <em>weeks</em>, and output units of <em>flowers</em> (blooms).

__

<h3>Part C:</h3>

For 35 weeks, the number of flowers that bloomed is ...

  b(35) = 80(35) +30 = 2830 . . . . flowers bloomed over 35 weeks

7 0
2 years ago
A group of 5 families at Baileys Middle School sold magazine subscriptions to raise money for the school. ​ $6.00, $7.00, $4.00,
mash [69]

Answer:

The median amount of money raised is $5.00

Step-by-step explanation:

To find the median amount of money raised, we will follow the steps below:

First, arrange the data giving in ascending order. That is

$6.00, $7.00, $4.00, $5.00, $3.00

We will arrange it in ascending order, hence

$3.00, $4.00, $5.00,  $6.00, $7.00

The median is the middle number, we will now proceed to take the middle number.

The middle number is $5.00

Therefore, the median amount of money raised is $5.00

6 0
3 years ago
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