Answers
(a) Extend the definition of f(x) to make it an even function f*(x),
[tex]f^*(x)=\begin{cases}0&\text{if }-2\le x<-1\\1&\text{if }0\le x\le1\\0&\text{if }1<x\le2\end{cases}[/tex]
and we take f* to be periodic over an interval of length P = 4. We compute the coefficients of the cosine series:
[tex]a_n = \displaystyle\frac2P \int_{-2}^2 f^*(x)\cos\left(\frac{2n\pi}Px\right)\,\mathrm dx = \frac{2\sin\left(\frac{n\pi}2\right)}{n\pi}[/tex]
Note that a₀ = 1 (you can compute the integral again without the cosine, or just take the limit as n -> 0). For all other even integers n, the numerator vanishes, so we split off the odd case for n = 2k - 1 :
[tex]a_{2k-1} = \dfrac{2\sin\left(\frac{(2k-1)\pi}2\right)}{(2k-1)\pi}[/tex]
Then the cosine series of f(x) is
[tex]\displaystyle \frac{a_0}2 + \sum_{k=1}^\infty a_{2k-1}\cos\left(\frac{(2k-1)\pi}2x\right)[/tex]
(b) For the sine series, you instead extend f(x) to an odd function f*(x),
[tex]f^*(x)=\begin{cases}-1&\text{if }-2\le x\le-1\\x&\text{if }-1<x<1\\1&\text{if }1\le x\le2\end{cases}[/tex]
Again, P = 4, and the coefficient of the sine series are given by
[tex]b_n = \displaystyle\frac2P\int_{-2}^2f^*(x)\sin\left(\frac{2n\pi}Px\right)\,\mathrm dx[/tex]
which we can again split into the even/odd cases,
[tex]b_{2k} = -\dfrac1{k\pi}[/tex]
[tex]b_{2k-1} = \dfrac{4(-1)^{k+1}+2(2k-1)\pi}{(2k-1)^2\pi^2}[/tex]
So the sine series is
[tex]\displaystyle \sum_{k=1}^\infty\left(b_{2k}\sin\left(k\pi x\right) + b_{2k-1}\sin\left(\frac{(2k-1)\pi}2x\right)\right)[/tex]
I've attached plots of the extended versions of f(x) along with the corresponding series up to degree 4.
Related Questions
4. Dominick sent the following number of text messages each day over the last week:
SUN
MON
TUES
WED
THURS
FRI
SAT
64
58
63
51
60
61
70
What is the mean absolute deviation?
Explain what the mean absolute deviation represents in the situation:
Answers
Simplify and write answer in standard form.
Answers
Step-by-step explanation:
2x^3 - 10x^2 + 8x -3
hope this helps
HELP ASAPP! WILL GIVE MOST BRAINILIST
Answers
Answer:
the probability is 6/7
4b + 8c + 12 – 8b – 2c + 6.
Answers
Jason bought 12 new CDs to add to his
collection. The next day, his baby
brother went into his room and
scratched half of his CDs. He now only
has 56 working CDs left. How many
CDs did Jason start with?
Answers
Answer:
He had origanlly had 68 altogether
Step-by-step explanation:
Because if u count the scrateched ones which was 6 is now 62 then add the other six because there was 12 and his brother scrateched half which is basically 12 / 2 = 6 so if u add the six is 68
What Im basically saying is 56 + 12 = 68
If you live in Portland, you live in Oregon. Cara does not live in Oregon. Therefore, Cara does not live in Portland.
a.) Overlapping Sets
b.) Subsets
c.) Disjoint Sets
Please somebody help me
Answers
Calculate the exact value of cos(a−b) given that sin a=12/13 with π/2
Answers
Answer:
Step-by-step explanation:
[tex]sin ~a=\frac{12}{13} ,\frac{\pi}{2} <a<\pi\\a ~is~in~2nd~quadrant.\\sin~b=\frac{3}{5} ,0<b<\frac{\pi}{2} \\b~is~in~1st~quadrant.\\cos~b=\sqrt{1-sin ^2b} \\=\sqrt{1-(\frac{3}{5} )^2} \\=\sqrt{\frac{25-9}{25} } \\=\sqrt{\frac{16}{25} } \\=\frac{4}{5} ,\\cos~a=-\sqrt{1-sin^2 a} \\=-\sqrt{1-(\frac{12}{13} )^2} \\=-\sqrt{\frac{169-144}{169} } \\=-\sqrt{\frac{25}{169} } \\=-\frac{5}{13} \\cos(a-b)=cos~acos~b+sin~a sin~b\\=\frac{-5}{13}* \frac{4}{5} +\frac{12}{13}* \frac{3}{5} \\=?[/tex]
Simplify :
4a(6) + 3a
1. 24 a2 + 3 a
2. 13 a
3. 27 a
4. 27 a2
Answers
Answer:
Hi! The answer to your question is 3. 27a
Step-by-step explanation:
☆*: .｡..｡.:*☆☆*: .｡..｡.:*☆☆*: .｡..｡.:*☆☆*: .｡..｡.:*☆
☆Brainliest is greatly appreciated!☆
Hope this helps!!
- Brooklynn Deka
Answer:
3
Step-by-step explanation:
Bodmas brakets first
Take out Common factor
Multiple
a. find the area
b. find the perimeter
Answers
Answer:
Area= 4y^2 + 14y + 12
Perimeter = 10y + 16
Step-by-step explanation:
Area of a reactangle = length × width
= (4y+6) × (y+2)
= 4y^2 + 6y + 8y + 12
= 4y^2 + 14y + 12
Perimeter of a rectangle = 2 × (a +b)
= 2 × ( 4y + 6 + y+ 2)
= 2 × ( 5y + 8)
= 10y + 16
433.54 in terms of pi
Answers
Answer:
138.0700636942675[tex]\pi[/tex]
Step-by-step explanation:
Is the answer
433.54 x π/180.
How do u convert kilometres into minutes
Example 0.8kilometer what till ut b in minutes woukd u times by 70
Answers
Answer:
It is not possible
Step-by-step explanation:
Given
0.8km
Required
Convert to minutes
In conversion, only units of the same metric can be converted.
i.e.
You can convert meters to miles (they both measure distance)
You can convert seconds to hours (they both measure time)
You can convert kilogram to pounds (they both measure weight)
But in the given question, you attempt to convert from distance to time.
This is not possible, because time and distance are different measurements
Situation: Find the substance's half-life, in days. Round your answer to the nearest tenth. A 50 gram sample of a substance that's used to treat thyroid disorders has a k- value of 0.1133. Enter the correct answer. N = Noe -kt DONE No = initial mass (at time t = 0) 1? N = mass at time t k = a positive constant that depends on the substance itself and on the units used to measure time t = time, in days
Find the substance's half-life , in days.
Round your answer to the nearest tenth.
Answers
Answer:
The substance's half-life is 6.1 days.
Step-by-step explanation:
The half-life of the substance can be calculated by knowing the constant decay:
[tex] t_{1/2} = \frac{ln(2)}{k} [/tex]
k: is the decay constant = 0.1133 d⁻¹
Hence, the half-life is:
[tex] t_{1/2} = \frac{ln(2)}{k} = \frac{ln(2)}{0.1133 d^{-1}} = 6.1 d [/tex]
Therefore, the substance's half-life is 6.1 days.
I hope it helps you!
A tree in Tom's backyard grows 0.6 meters each month. If the tree is 2.82 meters tall, for how many months has the tree been growing?
Answers
Answer:
4.7months
Step-by-step explanation:
2.82÷0.6 = 4.7
The effect of the time of day a science class is taught on test scores is being
studied in an experiment. Which of these is most likely to be an extraneous
factor that could also affect test scores?
A. Student clothing
B. Color of the school hallways
C. Principal of the school
D. Length of class period
Answers
Answer:D
Step-by-step explanation:
The effect of the time of day a science class is taught on test scores being studied in an experiment will be the length of the class period. The correct option is D.
What is a test analysis?
A test score is a piece of data, typically a number, that describes how well a test-taker performed. It is "a summary of the evidence contained in an examinee's replies to the test items that are connected to the construct or constructs being tested," according to one formal definition.
Given that the effect of the time of day a science class is taught on test scores is being studied in an experiment.
The time of the class periods is affected by the length of the class period. The length of the class period of one subject should be optimum so that students get the concepts easily without getting bored.
Therefore, an effect of the time of day a science class is taught on test scores being studied in an experiment will be the length of the class period. The correct option is D.
To know more about a test analysis follow
https://brainly.com/question/22516544
#SPJ2
Sally owns 145 shares of stock in a property management company. The current price of the stock is $42.17 per share. If the price of the stock decreases to $35.94 in the next month, how much did the value of Sally's shares decrease over the month?
○ $903.35
○ $947.68
○ $925.78
○ $968.55
Answers
Please check it more
Answer:
903.35
Step-by-step explanation:
sorry if im wrong
(hope this helps can i plz have brainlist :D hehe)
Find x. Round to the nearest tenth if necessary.
Answers
Answer:
your answers are in the attachment
hope it helps...
Step-by-step explanation:
Camp counselors kept track of the number of granola bars they served each day.
What is the median of the data set?
Answers
Answer:
the median is 43
Step-by-step explanation:
Determine any data values that are missing from the table, assuming that the data represent a linear function.
X Y
7 16
9 17
18
13
a Missing x:13
Missing y:20
b. Missing x:13
Missing y.19
C. Missing x:11
Missing y:20
d. Missing x:11
Missing y:19
Answers
Answer:
well y is 19 and x is 11
Step-by-step explanation:
all I did is add one for a while then after that I go and I put 9 + 4 which equals 13 then in between 13 is 11 so I divided 13 by 2 basically and it's just 11
PLEASE HELP ME PLEASEE
Answers
*☆*――*☆*――*☆*――*☆*――*☆*――*☆*――*☆*――*☆**☆*――*☆*――*☆*――*☆
Answer: A= 1 b= -8 c= -20
Explanation:
I hope this helped!
<!> Brainliest is appreciated! <!>
- Zack Slocum
*☆*――*☆*――*☆*――*☆*――*☆*――*☆*――*☆*――*☆**☆*――*☆*――*☆*――*☆
Answer:
a = 1
b = -8
c = -20
Step-by-step explanation:
Need help with linear function
Answers
Answer:
y = 1/2x + 2
Step-by-step explanation:
help me dont put link when answering question
Answers
Answer:
b
Step-by-step explanation:
x2 - 3|x - 2| - 4x = - 6
Solve for x
Answers
Answer:
if that is x^2 -3|x-2|-4x=-6, then this should be the answer other wise it would be a biiiit diffrent
Step-by-step explanation:
What is the maximum value and at what point
Answers
9514 1404 393
Answer:
maximum z: 43point: (x, y) = (25/7, 9)Step-by-step explanation:
In the attached, we have graphed the inequalities and the objective function line. As we might surmise by looking at the constraints,* the maximum occurs at the intersection of y=9 and 7x-y = 16, where (x, y) = (25/7, 9).
The maximum value of z is 7(25/7) +2(9) = 25 +18 = 43.
__
The graph shows all inequalities reversed. This has the effect of making the solution space white, rather than buried under 4 levels of shading. As we expect with most "maximize" problems, the feasible region vertex farthest from the origin is the one of interest.
_____
* The only upper bound on y is y ≤ 9. The only upper bound on x is 7x-y≤16. When we're maximizing some objective function with positive coefficients on the variables, these will be the boundaries of interest.
can someone please help me
Answers
Answer:
B
Step-by-step explanation:
minimum = 1
first quartile = 12
median = 6
third quartile = 16
maximum = 19
Your family is driving to the beach. You travel 200 kilometers in the first two hours. During the next hour, you stop for lunch and only travel 25 kilometers. What was tour average speed?
A) 60 km/h
B)75 km/h
C)100km/h
D)225 km/h
Answers
Answer: B) 75 km/h (from this answer my brother told and he good at math so I hope this is Correct.)
Amanda covers the outside of a box, shown in the diagram below, with
decorative paper for a school project. The top of the box is open.
3 in
7 in.
8 in
If the paper costs $0.03 per square inch, what is Amanda's total cost?
Answers:
4.38
6.06
5.04
2.70
Answers
i neeed help and quick its due at 3:15
Answers
Answer:
it c
Step-by-step explanation:
A box in the shape of a rectangular prism has the dimensions shown. What is the length of the interior diagonal of the box? Round to the nearest hundredth. Enter your answer in the box.
Answers
Answer:
Step-by-step explanation:
44
What is 1/2 plus 1/3 plus 5/6 ??
Answers
Answer:
10/6 or 5/3
Step-by-step explanation:
1/2 ---> 3/6
1/3 ---> 2/6
5/6 ---> 5/6
Add those new fractions together:
(3/6) + (2/6) + (5/6) = 10/6 or 5/3
Answer:
10/6 or 5/3
Step-by-step explanation:
Hope it help's :)
A random sample of 388 married couples found that 294 had two or more personality preferences in common. In another random sample of 566 married couples, it was found that only 38 had no preferences in common. Let p1 be the population proportion of all married couples who have two or more personality preferences in common. Let p2 be the population proportion of all married couples who have no personality preferences in common.
Answers
Solution :
Given :
[tex]$n_1=388$[/tex] , [tex]$x_1 = 294, \ n_2 = 566, \ x_2 = 38$[/tex]
Sample proportion, [tex]$(\hat p_1)=\frac{x_1}{n_1}$[/tex]
[tex]$=\frac{294}{388}$[/tex]
= 0.7577
Sample proportion, [tex]$(\hat p_2)=\frac{x_2}{n_2}$[/tex]
[tex]$=\frac{38}{566}$[/tex]
= 0.0671
For 95 % CI, z = 1.96
The confidence interval for the population proportion,
[tex]$p_1-p_2=(\hat p_1-\hat p_2)\pm z \left\{\sqrt{\frac{\hat p_1 \times (1- \hat p_1)}{n_1}+\frac{\hat p_2 \times (1- \hat p_2)}{n_2}}\right\}$[/tex]
[tex]$=(0.7577-0.0671)\pm 1.96 \left\{\sqrt{\frac{0.7577 \times (1- 0.7577)}{388}+\frac{0.0671 \times (1- 0.0671)}{566}}\right\}$[/tex]
[tex]$=0.6906 \pm 1.96 \times \left 0.0241$[/tex]
= [tex]$0.6906 \pm 0.0472 $[/tex]
Lower limit : 0.6906 - 0.0472 = 0.6434
Upper limit : 0.6906 + 0.0472 = 0.7378