You have given a function λ : R → R with the following properties (x ∈ R, n ∈ N):
λ(n) = 0 , λ(x + 1) = λ(x) , λ(n+1/2)=1
Find two functions p, q : R → R with q(x)6=0 for all x such that λ(x) = q(x)(p(x) + 1).
Answer:
One possible combination:
[tex]p(x) = \sin(2\, \pi \, x) - 1[/tex].
[tex]\displaystyle q(x) = \frac{1}{2}[/tex].
[tex]\displaystyle \lambda(x) = \frac{1}{2}\, \sin(2\, \pi\, x) - \frac{1}{2} + 1 = \frac{1}{2}\, \sin(2\, \pi\, x) + \frac{1}{2}[/tex].
Step-by-step explanation:
The requirement that [tex]\lambda(x + 1) = \lambda(x)[/tex] for all [tex]x \in \mathbb{R}[/tex] suggests that [tex]\lambda[/tex] should be a cyclic function with a period of [tex]1[/tex].
One such example is the sine function. Let [tex]\omega[/tex] denote a non-zero real number. Consider the expression [tex]\sin(\omega\, x)[/tex].
[tex]\begin{aligned}\sin(\omega\, x) &= \sin(\omega\, x + 2\, \pi) \\ &= \sin\left(\omega\, \left(x + \frac{2\, \pi}{\omega}\right)\right)\end{aligned}[/tex].
Note how replacing [tex]x[/tex] with [tex]\displaystyle \left(x + \frac{2\, \pi}{\omega}\right)[/tex] does not change the value of [tex]\sin(\omega\, x)[/tex]. Therefore, the period of [tex]\sin(\omega\, x)\![/tex] would be [tex]\displaystyle \frac{2\, \pi}{\omega}[/tex].
The question requires that replacing the [tex]x[/tex] with [tex](x + 1)[/tex] should not change the value of [tex]\lambda(x)[/tex]. In other words, the period of [tex]\lambda[/tex] should be [tex]1[/tex]. If [tex]\lambda(x)\![/tex] is indeed in the form [tex]\sin(\omega\, x)[/tex], the value of [tex]\omega[/tex] should ensure that [tex]\displaystyle \frac{2\, \pi}{\omega} = 1[/tex]. Therefore, [tex]\omega = 2\,\pi[/tex].
It would take a few more transformations to ensure that for all integers [tex]n[/tex],[tex]\displaystyle \lambda(n) = 0[/tex] and [tex]\displaystyle \lambda\left(n + \frac{1}{2}\right) = 1[/tex].
One possible approach is to make each [tex]n[/tex] a trough of this function, and each [tex]\displaystyle\left(n + \frac{1}{2}\right)[/tex] a crest of this function. The corresponding sine wave would have a range of only [tex]1[/tex]. However, without any transformation, [tex]y = \sin(2\, \pi\, x)[/tex] would have a range of two.
Therefore, it would be necessary to compress [tex]y = \sin(2\, \pi\, x)[/tex] vertically by a factor of [tex]2[/tex], so that its range meets the needs.
[tex]y = \displaystyle \frac{1}{2}\, \sin(2\, \pi\, x)[/tex] is the output of one such compression and has the correct period and range. However, its troughs would be at [tex]y = -\displaystyle \frac{1}{2}[/tex] (rather than [tex]y = 0[/tex]) whereas its crests are at [tex]\displaystyle y = \frac{1}{2}[/tex] (rather than [tex]y = 1[/tex].) It would be necessary to shift this function upwards by [tex]\displaystyle \frac{1}{2}[/tex] unit for the troughs and crests to meet match the ones in the requirements.
[tex]\begin{aligned} y &= \displaystyle \frac{1}{2}\, \sin(2\, \pi\, x) + \frac{1}{2} \\ &= \frac{1}{2}\, \sin(2\, \pi\, x) - \frac{1}{2} + 1 = \frac{1}{2}\left(\sin(2\, \pi\, x) - 1\right) + 1\end{aligned}[/tex].
i need help my people
Answer:
no
Step-by-step explanation:
If all of their coresponding angles will have the same measure (AAA Th. of symilarity) then thhey will be symilar
84.6+30.4+66=181 not 180
It takes Louis 1 2/3 hours to write 5/6 of a short story. How long will it take Louis to complete the story?
Answer:
2 hours
Step-by-step explanation:
Answer:
do it in thirds so 10/6 and 5/6 now simplify = 2/6= 1/6 of the story so full story would be 12/6=6/3=2 hours
Step-by-step explanation:
What decimal equivalent to 5/6
Answer:
0.83333333
Step-by-step explanation:
The 3 is repeating
Answer:
0.83333333...
Step-by-step explanation:
Which statement is true about the expression?
√18 + √2
A. It is rational and equal to 4 times the square root of 2.
B. It is irrational and equal to 4 times the square root of 2.
C. It is rational and equal to 3 times the square root of 2.
D. It is irrational and equal to 3 times the square root of 2.
Answer:
Your number is (3 sqrt(2)) / sqrt(2) = 3, and is a rational number indeed. I don't know exactly how to interpret the rest of the question. If r is a positive rational number and p is some positive real number, then sqrt(r^2 p) / sqrt(p) is always rational, being equal r. Possibly your question refers to situtions in which sqrt(c) is not uniquely determined, as for c negative real number or complex non-real number. In those situations a discussion is necessary. Also, in general expressions the discussion is necesary, because the denominator must be different from 0, and so on.
Step-by-step explanation:
Answer:
it is rational and equal to 4/2
i took the test. I hope i recieve brainliest. glad i could help you today
y+6=2
Make sure all work is shown
Answer:
y=-4
Step-by-step explanation:
y+6=2
y=2-6
y=-4
Answer:
y = -4
Step-by-step explanation:
y + 6 = 2
-6 -6
subtract 6 from both sides
y = -4
The city of Joyville has 87,388 residents.
Answer:
ok?
Step-by-step explanation:
Convert 7 pounds 3 ounces to kilograms.
Conversion ratios:
1 lb = 16 oz
1 kg = 2.2 lb
HINT: Covert 3oz to lbs, then add that number to the 7lb. Then convert to kg.
Round your final answer to the nearest tenth.
Answer:
3.3
Step-by-step explanation:
7lb 3oz
3oz/16oz=.1875lb
7lb+.1875lb=7.1875lb
7.1875lb/2.2kg=3.27kg
The final answer is 3.3 kg (round to the nearest tenth).
What is ratio?Ratio basically compares quantities, that means it shows the value of one quantity with respect to the other quantity.
If a and b are two values, their ratio will be a:b,
Given that,
1 lb = 16 ounces
1 kg = 2.2 lb
To solve this type of questions, use ratio property,
Since,
16 ounces = 1 lb
⇒1 ounce = 1/16 lb
⇒3 ounces = 3/16 lb
Add 7 lb to 3/16,
= 3/16+7 = 7.1875 lb
Since,
2.2 lb = 1 kg
⇒1 lb = 1/2.2 kg
⇒7.1875 lb = 7.1875/2.2 kg = 3.267 kg
In round of the answer is 3.3 kg
To learn more about Ratio on :
https://brainly.com/question/13419413
#SPJ2
8
Mark is comparing prices of different brands before he goes shopping. Who has the best deal on ice cream?
Brand
Size & Price
Turkey Hill
$4.50 for 30 ounces
Breyers
$15.36 for 128 ounces
Ben & Jerrys $2.88 for 16 ounces
Answer:
Answer:
The best deal would be Breyers 15.36 for 128 ounces
Step-by-step explanation:
128/15.36 Is 8 an ounce.
However every other ice cream devided by he same amount of ounces is more that the price of breyers. So basically for bryers you get more for less!'
128/4.50= $28
128/2.88= $44
What is the completely factored form of 5k^2 - 46k + 77?
Answer:
The completely factored form of 5k^2 - 46k + 77 is (−7) (5−11)
Step-by-step explanation:
May I have brainliest please? :)
Answer:
(k-7)(5k-11)
Step-by-step explanation:
Peter spent $54.90 on shirts, $64.95 on shoes, and $7.55 for a tie. Which is the most
accurate estimate of how much he spent?
Answer:
I'd say about 127.
Step-by-step explanation:
54.90 + 64.95 is probably about 120, and plus the 7.55 is about 127.
Write a linear equation for each table relating X and Y
Answer:
a. y = 2 + 2x
b. y = 20 - 4x
c. y = 2 + 1.5x
d. y = 20 - 3x
Step-by-step explanation:
I Hope That This Helps! :)
Colin gets pic 'n mix when at the cinema and makes his bag up with 3 types of sweet. He pick:
3 times as many cola bottles as smarties. He also picks twice as many smarties as
marshmallows. What proportion of the bag of sweets are cola bottles?
Give your answer in its simplest form.
Answer: 2/3
Explanation: Assume that the number of smarties he picked is c.
Now, we are given that:
1- He picked 3 times as many cola bottles as smarties. This means that:
Number of cola bottles he picked = 3s
2- He picked twice as many smarties as marshmallows. This means that:
number of marshmallows he picked = 0.5s
Now to get the proportion o f cola bottles, we will divide the number of cola bottles by the total number of sweets as follows:
proportion of cola bottles =
Hope this helps :)
In the year 2005, a person bought a new car for $27500. For each consecutive year after that, the value of the car depreciated by 12%. How much would the car be worth in the year 2008, to the nearest hundred dollars?
Answer:
Getting my point back lol
Step-by-step explanation:
To make orange fizz, Noah mixes 2 scoops of powder with 6 cups of water. Andre mixes 4 scoops of powder with 18 cups of water. Which mixture is more orangey flavored?
i need it ASAP PLEASE HELP ME WITH THIS PROBLEM!
AnswerThe tables are given below and
since,
so, Noah's mixture will taste more like orange, than Andre's mixture.
Step-by-step explanation:
To make orange fizz, Noah mixes 4 scoops of powder with 6 cups of water, Andre mixes 5 scoops of powder with 8 cups of water.
so, can construct the following tables -
A) table that shows different amounts of powder and water that taste the same as Noah's mixture :-
Serial no. Powder in scoops cups of water
1 8 12
2 12 18
3 16 24
4. 20 30
5. 24 36
B) table that shows different amounts of powder and water that taste the same as Andre 's mixture :-
Serial no. Powder in scoops cups of water
1 10 16
2 15 24
3 20 32
4. 25 40
5. 30 48
C) Since,
so, Noah's mixture wili taste more like orange, than Andre's mixture.
Step-by-step explanation:
What percent of 41.7 is 39?
the descriptive method:
{4, 8, 12, 16, 20)
Answer:
The set of the first five positive numbers multiples of 4.
Step-by-step explanation:
Which equations have a value less than 6,766?
A. one fourth x 6,766 = ________
B. 6 x 6,766 = ________
C. one half x 6,766 = ________
D. 1 x 6,766 = ________
A and C
D and B
A and B
C and D
please help giving brainlyest
A and C are the correct answers.
I hope this helps.
144:128 convert into simplest form
Mr. Scott want to tip his taxicab driver 18%. If his commute costs $25, what is the tip?
Answer:
4.5$
Step-by-step explanation:
25 times .18 = 4.5$
Find the area of a circle with diameter 14 cm use 3.14
Answer:
153.94
Step-by-step explanation:
50pounds to 35 pounds
Answer:
I am assuming you mean the difference, In that case it is 15 pounds.
50- 35= 15
Find an equation parallel to x = 0 and passing through (5. - 1).
Answer:
x=5
Both are vertical lines and parallel to each other.
8/15 divided by ( -0.35) hurry pls it’s for a test
Answer:
-1.52380952381
Step-by-step explanation:
Answer:
-1.5238...
Step-by-step explanation:
I just did it on my calculator. Hope it helps!
in point slope form: passes through (-2,-1), slope = 4?
Answer:
y+1= 4(x+2)
You could also do: y - -1= 4(x- -2)
They are the same so it shouldnt matter
Estimate. Tell whether the estimate is greater than or less than the actual product
23
x 24
A. O 25 10 = 250; less than
B. O 25 15 = 375; greater than
C. O 20 20 = 400, less than
D. O 30 20 = 600; greater than
Answer:
don't know
Step-by-step explanation:
PLEASE HELP ME WITH THIS QUESTION ASAP
Answer: $801
Step-by-step explanation:
to find the slope, you can divide 445÷5=89. (now we know that the rate is $89/1 month)
to find the cost of 9 months, multiply 89•9=801
so since it's asking the cost after 9 months, i think the answer should be $801.
Are the 2 triangles congruent? If so, what postulate did you use?
+
++
O AAS
O ASA
SAS
SSS
OHL
Not congruent
Are the 2 triangles congruent? If so, what postulate did you use?"
Answer:
Step-by-step explanation:
4
The art teacher bought 6 boxes of colored pencils. Each box contains 12 colored pencils. She wants to give each of her 10 students as many colored pencils as possible and keep the rest. Based on this information, which statement is true
Answer:
60
Step-by-step explanation:
(6x12)-(6x2)
72-12
60
Answer:
I need the statement before I answer
Step-by-step explanation:
(i'll be giving brainliest on this one!! :) )
Solve seven times four and five eighths.
A) twenty eight and three eighths
B) twenty eight and five eighths
C) thirty two and three eighths
D) thirty two and five eighths
(Here is an image if needed :D) :
Answer:
Third option
Step-by-step explanation:
In order to find the answer to this question you must change the mixed number into a improper fraction and then multiply the numerators to the denominators.
[tex]7\times4\frac{5}{8}[/tex]
Remember that all numbers have a invisible one as the denominator:
[tex]7=\frac{7}{1}[/tex]
[tex]\frac{7}{1} \times4\frac{5}{8}[/tex]
Turn the mixed number into a improper fraction:
[tex]8\times4=32+5=37[/tex]
(Remember to keep the denominator the same)
[tex]=\frac{37}{8}[/tex]
Now multiply:
[tex]7\times37=259[/tex]
[tex]1\times8=8[/tex]
[tex]=\frac{259}{8}[/tex]
Turn the improper fraction into a mixed number:
[tex]\frac{259}{8} =32\frac{3}{8}[/tex]
[tex]=32\frac{3}{8}[/tex]
Hope this helps.