Factor completely 121x^2

Olin [163]

With $2L=\pi$ , the Fourier series expansion of $f(x)$ is

$\displaystyle f(x)\sim\frac{a_0}2+\sum_{n\ge1}a_n\cos\dfrac{n\pi x}L+\sum_{n\ge1}b_n\sin\dfrac{n\pi x}L$
$\displaystyle f(x)\sim\frac{a_0}2+\sum_{n\ge1}a_n\cos2nx+\sum_{n\ge1}b_n\sin2nx$

where the coefficients are obtained by computing

$\displaystyle a_0=\frac1L\int_0^{2L}f(x)\,\mathrm dx$
$\displaystyle a_0=\frac2\pi\int_0^\pi f(x)\,\mathrm dx$

$\displaystyle a_n=\frac1L\int_0^{2L}f(x)\cos\dfrac{n\pi x}L\,\mathrm dx$
$\displaystyle a_n=\frac2\pi\int_0^\pi f(x)\cos2nx\,\mathrm dx$

$\displaystyle b_n=\frac1L\int_0^{2L}f(x)\sin\dfrac{n\pi x}L\,\mathrm dx$
$\displaystyle b_n=\frac2\pi\int_0^\pi f(x)\sin2nx\,\mathrm dx$

You should end up with

$a_0=0$
$a_n=0$
(both due to the fact that $f(x)$ is odd)
$b_n=\dfrac1{3n}\left(2-\cos\dfrac{2n\pi}3-\cos\dfrac{4n\pi}3\right)$

Now the problem is that this expansion does not match the given one. As a matter of fact, since $f(x)$ is odd, there is no cosine series. So I'm starting to think this question is missing some initial details.

One possibility is that you're actually supposed to use the even extension of $f(x)$ , which is to say we're actually considering the function

$\varphi(x)=\begin{cases}\frac\pi3&\text{for }|x|\le\frac\pi3\\0&\text{for }\frac\pi3$

and enforcing a period of $2L=2\pi$ . Now, you should find that

$\varphi(x)\sim\dfrac2{\sqrt3}\left(\cos x-\dfrac{\cos5x}5+\dfrac{\cos7x}7-\dfrac{\cos11x}{11}+\cdots\right)$

The value of the sum can then be verified by choosing $x=0$ , which gives

$\varphi(0)=\dfrac\pi3=\dfrac2{\sqrt3}\left(1-\dfrac15+\dfrac17-\dfrac1{11}+\cdots\right)$
$\implies\dfrac\pi{2\sqrt3}=1-\dfrac15+\dfrac17-\dfrac1{11}+\cdots$

as required.

5 0

2 years ago

A rectangular prism measures 225 inches by 315 inches by 2 inch.

Bas_tet [7]

Answer:

No I don't agree with Priya

The reason why I don't agree with Priya is because 25 Inch side blocks are BIGGER than 15 inch side blocks. That means that it will take less amount of 25 inch side blocks to fill up the volume of the rectangular prism rather than 15 inch side blocks.

(I'm sorry, I don't know what was your previous question, so I can't answer that part. If you comment it below, I will do it!)

5 0

1 year ago

How to solve d - 5 = 7 algebraically

adelina 88 [10]

Answer:

d = 12

Step-by-step explanation:

d - 5= 7

( d - 5 ) + 5 = 7 + 5

d - 5 + 5 = 12

d = 12

6 0

2 years ago

Read 2 more answers

I Need help with this please it’s Algebra and it’s 6 days late

Gnom [1K]

1. (x+8)(x-5)
3. 15.2

5 0

2 years ago

50 pts and brainliest for first answer

gavmur [86]

Answer:

An acute angle ("acute" meaning "small") is an angle smaller than a right angle. The range of an acute angle is between 0 and 90 degrees.

An obtuse triangle (or obtuse-angled triangle) is a triangle with one obtuse angle (greater than 90°) and two acute angles. Since a triangle's angles must sum to 180° in Euclidean geometry, no Euclidean triangle can have more than one obtuse angle.

Protractor: an instrument for measuring angles, typically in the form of a flat semicircle marked with degrees along the curved edge.

Degrees: a unit of measurement of angles, one three-hundred-and-sixtieth of the circumference of a circle.

Right Angel: an angle of 90°, as in a corner of a square or at the intersection of two perpendicular straight lines.

Straight Angle: an angle of 180°.

Step-by-step explanation:

8 0

2 years ago

Read 2 more answers

Factor completely 121x^2 - 16.