Solve for e: e=mc2 m=3 c=6

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

Ben needs to buy orange juice he can buy 64 fl oz at the grocery store for $2.00 or he can buy 256 fl oz at the whole sale for $

Sholpan [36]

1. 2.00/64=3.125 cents per oz
2. 7.50/256=2.930 cents per oz.
3. The best deal is at the whole sale

8 0

2 years ago

What is the y-intercept of y = 5 (6.8)^x<br> a. 5<br> b. 6.8 <br> c. 1 <br> d. x

ivolga24 [154]

The answer to this question is A, the graphs y intercept is 5

7 0

2 years ago

On a normal distribution of iq test scores, the average score would be:

spayn [35]

The average IQ test score would be 100.

What is a normal distribution?

A probability distribution that is symmetric about the mean or average is the normal distribution, sometimes referred to as the Gaussian distribution. It demonstrates that data that are close to the average value occur more frequently than data that are far from the mean. The normal distribution appears as a "bell curve" on a graph.

Average IQ Score

A score of 100 is regarded as the average IQ on various tests. The majority of results (68%) are within one standard deviation of the mean (that is, between 85 and 115). This indicates that the average result is within plus or minus 15 points for almost 70% of test takers.

Learn more about average here:

brainly.com/question/2426692

#SPJ4

4 0

1 year ago

In a coastal town, the humidity, measured in grams of water per kilogram of air, increases by 43% for every 1 degree celsius inc

larisa86 [58]

Answer:

In order to increase humidity by 49.17 grams per kilogram, the increase in temperature will be 18 degree Celsius approx.

Step-by-step explanation:

Initial humidity = 5.75 g/kg

Final humidity= 49.17 g/kg

Percentage increase in humidity= (49.17-5.75)/5.75 *100= 755.13

Increase in temperature= 755.13/43 = 17.56

Increase in temperature = 18 degree Celsius approx.

4 0

2 years ago

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