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3 years ago

Use the graph below to answer the question that follows:

Mathematics

1 answer:

Strike441 [17]3 years ago

3 0

By graphing the given points, we will get the attached figure
which represents:
f(x) = 3 sin x
but sin x = cos (x - π/2)
∴ f(x) = 3 cos ( x - π/2 )

The correct choice is option ⇒⇒⇒ f(x) = 3 cos(x − pi over 2 )

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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

3 years ago

A blueprint of a barn is shown.The scale is 5 inches =8 feet. Find the actual length of the wall that is labeled 20 inches.

kenny6666 [7]

5=8
20/5=4
8x4=32
The answer is 32 feet

4 0

2 years ago

Find the missing angle in a triangle with the given sides.

Yuliya22 [10]

Answer:

Step-by-step explanation:

44 + 99 + x = 180

143 + x = 180

x = 37

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3 years ago

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Share successful learning strategies that you have applied in previous mathematic courses

marshall27 [118]

Sure, I have this strategy to make a sheet with questions of that chapter and do that before I study the chapter so I can see if I know a little or a lot of that chapter

5 0

2 years ago

How can you divide 58÷0.65

Orlov [11]

You can definitly use a calculator for this
The answer would be
89.2307692308
What you do is find out how many times 0.65 can go into 58 to get your answer:)

7 0

3 years ago