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
A school social group has 30 seniors, 38 juniors, 24 sophomores, and 18 freshmen. They need to choose a president, an assistant, and an accountant. The president must be a senior and the assistant must not be a freshman. How many different ways can they pick the officers?
Answers
Step-by-step explanation:
by choosing from the sophomores or by choosing from the freshmen
Answer:
303600Step-by-step explanation:
There are:
30 seniors, 38 juniors, 24 sophomores, and 18 freshmenThe president must be a senior.
Number of ways = 30The assistant must not be a freshman.
Number of ways = 30 + 38 + 24 = 92The accountant can be anyone.
Number of ways = sum of all = 110Number of ways to pick officers:
30*92*110 = 303600solve from missing angle
Answers
The area of a trapezoid is 144cm2. The height is 16cm and the length of one of the parallel sides is 6cm. Find the length of the second parallel side. Express your answer as a simplified fraction or a decimal rounded to two places.
Answers
Step-by-step explanation:
hope this will help you have a great day!
Answer:
0.12
Step-by-step explanation:
I don't know how to explain it to you, but I hope you'll understand from the picture. hope it helps.
In order to take good notes while reading, you must ask yourself ________________. a. “What is the main idea?” c. “What have I learned?” b. “What are the supporting details?” d. all of the above Please select the best answer from the choices provided A B C D
Answers
Answer: D do all of the above and you will get good grades
Step-by-step explanation:
What is the range of F?
Answers
Answer:
A should be right :)
Answer:
I think D as that's the highest point
Step-by-step explanation:
I think D as that's the highest point
If 2x⁴ + 2y⁴ + 2z⁴ = 144, what is the mean of x⁴, y⁴ and z⁴
Answers
Answer: 24
Step-by-step explanation:
Given
[tex]2x^4+2y^4+2z^4=144[/tex]
dividing whole equation by 2, we get
[tex]\Rightarrow x^4+y^4+z^4=72[/tex]
Mean is given by the sum of the value of entities divided by the total no of entities
[tex]\Rightarrow \text{Mean}=\dfrac{x^4+y^4+z^4}{3}\\\\\Rightarrow \text{Mean}=\dfrac{72}{3}\\\\\Rightarrow \text{Mean}=24[/tex]
if the submarine travels 518 kilometers in 14 hours, how many kilometers does the submarine travel in 1 hour
Answers
Answer:
37 km
Step-by-step explanation:
Answer:
37kph
Step-by-step explanation:
518/14=37
For the polynomial x^3-7x^2+12x-6, 1 is a zero. Express g(x) as a product of linear factors.
Answers
Answer:
(x–1)(x²–6x+6)
Explanation:
Given that 1 is a zero, we can use synthetic division to compute the factor.
If 1 is a zero (x–1) is a factor.
Here is the process:
| 1 | 1 -7 12 -6
↓ +(1) +(-6) +(6)
↓ ↓ ↓
1 -6 6 0
→ [1]x² + [-6]x + [6]1 + [0]/x–1 =
x² – 6x + 6.
Therefore the factors are (x–1)(x²–6x+6)
Which angle is an adjacent interior angle?
Answers
need the GCF of the following term
Answers
Answer:
x²y²
Step-by-step explanation:
Refer to image below:
Help me it’s really important
Answers
Answer:
Step-by-step explanation:
Quadratic equation has been given as,
[tex]y=-x^{2} +6x+4[/tex]
Converting this equation into vertex form,
[tex]y=-(x^{2}-6x)+4[/tex]
[tex]y=-[x^{2}-2(3x)+9-9]+4[/tex]
[tex]y=-[x^{2}-2(3x)+3^2]+4+3^2[/tex]
[tex]y=-(x-3)^2+4+9[/tex]
[tex]y=-(x-3)^2+13[/tex]
By comparing this equation with the vertex form of a quadratic equation,
y = (x - h)² + k
Here, (h, k) is the vertex of the parabola.
Vertex of the parabola → x = 3, y = 13
Axis of symmetry → x = 3
The question is attached in the photo PLEASE HELP ASAP!!!!!!
THIS IS DUE TODAY!!!!!!!!!!!!
*** I will make you brainiest if the answer is correct ***
Answers
What is the distance between the two points?
(0.5, 1.3), (-0.4, -1.2)
Answers
Answer:
2.66 (rounded to the nearest hundredth)
What is 3x2 [-2 3 0. -4 2 6. 6 -5 6.]
Answers
Answer:
The correct answer is choice B. Have a good day!
If the sum of six consecutive odd integers is 252, what is the smallest of the six integers?
Answers
Answer:
37 smallest number
Step-by-step explanation:
If your integer is x
The next 5 odd integers are:
x+2
x+4
x+6
x+8
x+10
x+x+2+x+4+x+6+x+8+x+10=252
6x+30=252
6x=252-30
6x=222
x=222/6
x=37
Solve the system of equations by substitution.
5x + y =18
Y=4x
Answers
Answer:
x =2 y= 8
Step-by-step explanation:
Can someone help me pls
Answers
Answer:
57°
Step-by-step explanation:
180-56=114
114÷2=57
Answer:
57°
Step-by-step explanation:
AO and BO are both the radius of the circle, so they are equal. Since it makes an iscosoles triangle, angles A and B are equal.
180-66=114
114/2=57
Find the value of x. Round to the nearest tenth
Answers
9514 1404 393
Answer:
42.3
Step-by-step explanation:
The appropriate trig ratio is ...
Tan = Opposite/Adjacent
tan(78°) = x/9
x = 9×tan(78°)
x ≈ 42.3
A yard contains 16 ounces. 9 ounces is what fraction of a pound?
Answers
9514 1404 393
Answer:
9/16
Step-by-step explanation:
We presume an error in the problem statement. A yard is a measure of distance, not weight.
A pound contains 16 ounces. Then 9 ounces is 9/16 of a pound.
__
As a decimal fraction, that is 0.5625 lb.
determine which of the following pairs of fractions are equivalent.
1)4/5 and 16/20
no links or photo
Answers
We know that because if you multiply the denominator AND numerator by 4, you get 16/20.
Answer:
explanation
Step-by-step explanation:
just multiply the numerator and the detonator by 4 and u get 16/20
it took bryan one hour to ride 6 1/4 miles on his bike. how far will bryan be able to ride in 3 1/2 hours?
A 21 7/8
B 9 3/4
C 5 1/2
D 9 1/4
Answers
21. Sandra is 4 feet tall. Pablo is 10% taller than Sandra, and Michaela is 8% taller than Pablo. a. Explain how to find Michaela's height with the given information.
Answers
Answer:
Step-by-step explanation:
Find 10% of Sandra’s height then find 8% of Pablo’s height, then combine. 4 feet 9 inches.
A market sells coffee from Brazil for $3 per pound and coffee from Colombia for $4 per pound. How
many pounds of each should be used in order to sell the blend of 100 pounds for $3.50 per pound?
Answers
Answer:
It would be 28 brazilian and 4 Colombian
Step-by-step explanation:
I multiplied alot and please forgive me if this is wrong
plz help will give crown
Answers
Answer:
C. [tex] a \leqslant 4[/tex]
Step-by-step explanation:
[tex]a \leqslant 4[/tex]
Option C is the correct answer.
A bowl contains 10 red balls and 10 blue balls.A woman selects balls at random without looking at them. How many balls must she select to be sure of having at least three balls of the same color?
Answers
Answer:
5 balls
Step-by-step explanation:
Sorry if it's wrong.
What number goes on top?
Answers
Answer:
33
Step-by-step explanation:
16+17 math that is why
Name the solid that is formed from the net.
a. Square Pyramid
b. Triangular Pyramid
c. Triangular Prism
d. Cylinder
e. Rectangular Prism
f. Cone
g. Cube
Answers
The diagonal length of a TV is 32 inches, and the height is 17 inches. How wide is the TV?
Pls
Answers
Answer:
7√15
Step-by-step explanation:
32²-17²=735
√735
joption
[tex]x \sqrt{?} [/tex]
Answers
Answer:
???
Step-by-step explanation:
A school librarian ordered new books
Answers
Answer:
35
Step-by-step explanation:
Answer:
8232
Step-by-step explanation:
