\( \rm \int_{0}^1 \int_{0}^1x \bigg \{ \frac{1}{1 - xy} \bigg \}dydx \\ \)
Answers
The fractional part vanishes when the argument is an integer; in this case, for
\(\left\{\dfrac1{1-xy}\right\} = 0 \iff \dfrac1{1-xy} = n \iff xy = \dfrac{n-1}n\)
which are hyperbolas in the \((x,y)\)-plane.
Observe that between neighboring hyperbolas, we have
\(\dfrac{n-1}n < xy < \dfrac n{n+1} \\\\ ~~~~ \implies \dfrac1{n+1} < 1-xy < \dfrac1n \\\\ ~~~~ \implies n < \dfrac1{1-xy} < n+1 \\\\ ~~~~ \implies \left\{\dfrac1{1-xy}\right\} = \dfrac1{1-xy} - \left\lfloor\dfrac1{1-xy}\right\rfloor = \dfrac1{1-xy} - n\)
Split up the integral over \([0,1)^2\) along the curves \(xy=\frac{n-1}n\). The subregions somewhat resemble the layers or scales of an onion (see attached plot with the first 5 "scales").
Let \(S_n\) denote the \(n\)-th (\(n\in\Bbb N\)) "scale", starting from the blue region closest to the origin and counting diagonally upward in the direction of (1, 1).
In Cartesian coordinates, the integral over \(n\)-th "scale" is
\(\displaystyle \iint_{S_n} x \left(\frac1{1-xy} - n\right) \, dy \, dx \\\\\\ ~~~~~~~~= \int_{(n-1)/n}^{n/(n+1)} \int_{(n-1)/(nx)}^1 x \left(\frac1{1-xy} - n\right) \, dy dx \\\\\\ ~~~~~~~~~~~~~ + \int_{n/(n+1)}^1 \int_{(n-1)/(nx)}^{n/((n+1)x)} x \left(\frac1{1-xy} - n\right) \, dx\)
(see attached plot of the 2nd "scale" for reference)
The integral is trivial, so I'll leave it to you to confirm that it drastically reduces to
\(\displaystyle \iint_{S_n} x \left(\frac1{1-xy} - n\right) \, dy \, dx = \frac1{2n (n+1)^2} = \frac12 \left(\frac1n - \frac1{n+1} - \frac1{(n+1)^2}\right)\)
Now we recover the original integral by summing over \(\Bbb N\).
\(\displaystyle \int_0^1 \int_0^1 x \left\{\frac1{1-xy}\right\} \, dy \, dx = \frac12 \sum_{n=1}^\infty \left(\frac1n - \frac1{n+1} - \frac1{(n+1)^2}\right) \\\\ ~~~~~~~~ = \frac12 \left(\left(1-\frac12\right)+\left(\frac12-\frac13\right)+\left(\frac13-\frac14\right)+\cdots\right) - \frac12 \sum_{n=2}^\infty \frac1{n^2} \\\\ ~~~~~~~~ = \frac12 - \frac12 \left(\sum_{n=1}^\infty \frac1{n^2} - 1\right) \\\\ ~~~~~~~~ = \frac12 - \frac12 \left(\frac{\pi^2}6 - 1\right) = \boxed{1 - \frac{\pi^2}{12}}\)
![[tex] \rm \int_{0}^1 \int_{0}^1x \bigg \{ \frac{1}{1 - Xy} \bigg \}dydx \\ [/tex]](https://i5t5.c14.e2-1.dev/h-images-qa/answers/attachments/5OEU1jBtRkKAc9JljGw4INJxAdiAaw5S.png)
![[tex] \rm \int_{0}^1 \int_{0}^1x \bigg \{ \frac{1}{1 - Xy} \bigg \}dydx \\ [/tex]](https://i5t5.c14.e2-1.dev/h-images-qa/answers/attachments/LRp7brJAc63UMRD1TTomJHSvgJpHh1bN.png)
Related Questions
A student ran out of time on a multiple choice exam and randomly guessed the answers for two problems. Each problem had 5 answer choices – a, b, c, d, e – and only one correct answer. What is the probability that she answered both of the problems correctly? Write your answer as a fraction in simplest form
Answers
The probability that the student, who ran out of time on a multiple choice exam and randomly guessed the answers for two problems, answered both of the problems correctly is ⁹/₂₅.
What is the probability?Probability refers to the likelihood of an expected outcome occurring out of many possible outcomes.
Probability is computed as the quotient of expected outcome over total outcomes.
The number of questions left = 2
The number options of for each question = 5
Since there are 5 options for each question, 4 of which are wrong, the probability of getting both wrongs is:
⁴/₅ x ⁴/₅ = ¹⁶/₂₅ = 0.64 or 64%
Therefore, probability of getting both questions correct is 36% (1 - 64%) or ⁹/₂₅ (1 - ¹⁶/₂₅).
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if elephants grass grows 3 3/4 inches a day, how many inches will it grow in 7 days
Answers
The elephant grass will grow 105/4 inches in 7 days
The elephant grass grows 3 3/4 inches in a day
The number of inches it can grow in 7 hours can be calculated as follows
15/4= 1
x= 7
cross multiply both sides
x= 15/4 × 7
x= 105/4
Hence the elephant grass will grow 105/4 inches in 7 days
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Geometry question. Please help fast.
Answers
The correct answers are 1st and 4th i.e translate 2 units down and reflect the triangle.
We can easily deduce that the triangle A'B'C' is 2 units higher than the triangle ABC so if we want to map the both the triangles we have to make the triangle A'B'C' lower in the graph.
So, both the 1st and 4th points translate and reflect down the triangle A'B'C' which will be mapped according to the triangle ABC.
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Cheese Or Pickles? WILL GIVE BRAINLIEST
Answers
Answer:
Pickles. My friends are allergic to cheese.
If the average yield of cucumber acre is 800 kg, with a variance 1600 kg, and that the amount of the cucumber follows the normal distribution. 1- What percentage of a cucumber give the crop amount between 778 and 834 kg? 2- What the probability of cucumber give the crop exceed 900 kg ?
Answers
Answer:
a
The percentage is
\(P(x_1 < X < x_2 ) = 51.1 \%\)
b
The probability is \(P(Z > 2.5 ) = 0.0062097\)
Step-by-step explanation:
From the question we are told that
The population mean is \(\mu = 800\)
The variance is \(var(x) = 1600 \ kg\)
The range consider is \(x_1 = 778 \ kg \ x_2 = 834 \ kg\)
The value consider in second question is \(x = 900 \ kg\)
Generally the standard deviation is mathematically represented as
\(\sigma = \sqrt{var (x)}\)
substituting value
\(\sigma = \sqrt{1600}\)
\(\sigma = 40\)
The percentage of a cucumber give the crop amount between 778 and 834 kg is mathematically represented as
\(P(x_1 < X < x_2 ) = P( \frac{x_1 - \mu }{\sigma} < \frac{X - \mu }{ \sigma } < \frac{x_2 - \mu }{\sigma } )\)
Generally \(\frac{X - \mu }{ \sigma } = Z (standardized \ value \ of \ X)\)
So
\(P(x_1 < X < x_2 ) = P( \frac{778 - 800 }{40} < Z< \frac{834 - 800 }{40 } )\)
\(P(x_1 < X < x_2 ) = P(z_2 < 0.85) - P(z_1 < -0.55)\)
From the z-table the value for \(P(z_1 < 0.85) = 0.80234\)
and \(P(z_1 < -0.55) = 0.29116\)
So
\(P(x_1 < X < x_2 ) = 0.80234 - 0.29116\)
\(P(x_1 < X < x_2 ) = 0.51118\)
The percentage is
\(P(x_1 < X < x_2 ) = 51.1 \%\)
The probability of cucumber give the crop exceed 900 kg is mathematically represented as
\(P(X > x ) = P(\frac{X - \mu }{\sigma } > \frac{x - \mu }{\sigma } )\)
substituting values
\(P(X > x ) = P( \frac{X - \mu }{\sigma } >\frac{900 - 800 }{40 } )\)
\(P(X > x ) = P(Z >2.5 )\)
From the z-table the value for \(P(Z > 2.5 ) = 0.0062097\)
what is the value sin(?)= cos 28
Answers
Answer: 62
Step-by-step explanation:
Using the fact that cos(90-x)=sin(x) we get that 90-x=28, so x=62 and the answer is simply 62.
Hope that helped,
-sirswagger21
activity c has an early start time of 8, an early finish time of 12, a latest start time of 13, and a latest finish time of 17. its slack is:
Answers
The slack for activity C is 5 hours.
What is the Slack?Slack is the amount of time that a task can be delayed without affecting the overall project schedule. In this case, the slack for activity C is calculated as follows:
Slack = Latest start time - Early start time = 13 - 8 = 5 hours
Alternatively, slack can also be calculated as the difference between the latest finish time and the early finish time:
Slack = Latest finish time - Early finish time
Slack = 17 - 12
Apply the subtraction operation, and we get
Slack = 5 hours
Therefore, activity C has a slack of 5 hours.
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Determine whether 548 is greater than or less than 373. Then write the expression showing this using < or >.
Answers
Answer:
548 > 373
Step-by-step explanation:
548 is greater than 373 because when we compare the digits from left to right, we find that the first digit of 548 (5) is greater than the first digit of 373 (3). Therefore, we can conclude that 548 is greater than 373.
The ">" symbol is used to represent "greater than" in mathematical comparisons.
Hope this helps!
which car can travel furthur on one gallon of gas/ green car 18.4 miles on 0.8 gallons or black car 17.4 miles on 0.75 gallons
Answers
Answer:
The black car
Step-by-step explanation:
First, we need to fin the mile to gallon ratio.
To get this we would divide the number of miles by the gallon for each car
Green Car:
18.4/0.8
23 miles per gallon
Black car:
17.4/0.75
23.2 miles per gallon
In conclusion, The black car would travel further on one gallon of gas than the green car would.
Hope this helps!!
Please tell me if I have messed up!!
I enjoy learning from my mistakes:)
Can anyone help me with this?
Answers
Answer:
\(\frac{7}{2}\)
Step-by-step explanation:
\(2 \frac{5}{8} = \frac{21}{8}\)
Divide \(\frac{21}{8}\) by \(\frac{3}{4}\)
\(\frac{21}{8}\) × \(\frac{4}{3}\) = \(\frac{7}{2}\)
1. The prism below measures 6 inches x 3 inches x 10 inches. A diagonal
line, d, is drawn inside the prism. Find the length of the diagonal, d.
I will give brainliest answer for correct and fastest answer!!
Answers
9514 1404 393
Answer:
√145 ≈ 12.04
Step-by-step explanation:
The length of the space diagonal is the root of the sum of the squares of the prism edge lengths:
d² = a² +b² +c² = 6² +3² +10² = 145
d = √145 ≈ 12.04 . . . inches
__
The face diagonal is found using the Pythagorean theorem in the usual way:
f² = a² + b²
The space diagonal is the hypotenuse of the right triangle whose sides are the face diagonal and the remaining edge:
d² = f² +c²
d² = a² +b² +c²
__
Here is a diagram.
The question is the mean
Answers
Answer:
I think it's 8 and 12
Answer:
12 and 8
Step-by-step explanation:
We can find the mean by adding all of the numbers and then dividing the total by the ammount of numbers we added. In this case we are adding 12 and 8. This gives us 20. We then divide 20 by 2 and get our mean which is 10. The range is the ammount that needs to be subtracted from the highest number to the lowest number. In this case 12 subtracted by 4 gives us 8 while still allowing for there to be a mean 10.
:
What is 475.189 rounded to the nearest hundredth?
Answers
Answer:
It is:475.19.
HOPE THIS HELPED
Answer:
475.19.
Step-by-step explanation:
The hundredths place is two places to the right of the decimal, or (0.01).
In the number given, the number to the right of the hundredths place is larger than 5, meaning we will round up giving us:
475.189 --> 475.19.
a productive approach to identifying a research topic, and subsequent research question, is to utilize which of the following sources?
Answers
Answer:
Step-by-step explanation:
A productive approach to identifying a research topic and subsequent research question can involve utilizing multiple sources, including:
Literature review: Reviewing existing research in the field can help identify gaps in knowledge and areas that need further investigation.
Personal interests and experiences: Consider your own experiences, interests, and expertise to identify a topic that you are passionate about and can contribute to.
News articles and current events: Stay up-to-date on current events and identify topics that are relevant and pressing in your field.
Professional associations and organizations: These organizations often have conferences and publications that highlight current research trends and topics.
Government reports and data sources: These sources can provide data and information on a wide range of topics and can serve as a starting point for identifying a research question.
Ultimately, a combination of these sources can be useful in identifying a research topic and subsequent research question. It's important to consider the feasibility, importance, and originality of the research topic, and to choose a topic that you can research thoroughly and effectively.
. Ten weeks of data on the Commodity Futures Index are 7.35, 7.40, 7.55, 7.56, 7.60, 7.52,
7.52, 7.70, 7.62, and 7.55.
a. Construct a time series plot. What type of pattern exists in the data?
b. Use trial and error to find a value of the exponential smoothing coefficient that results in a relatively small MSE.
Answers
By iterating through different alpha values, you can identify the optimal alpha for this data set that results in the smallest MSE.
How to solvea. To construct a time series plot, you would typically use software like Excel or programming languages like Python or R.
However, I can describe the general trend based on the given data points.
Week 1: 7.35
Week 2: 7.40
Week 3: 7.55
Week 4: 7.56
Week 5: 7.60
Week 6: 7.52
Week 7: 7.52
Week 8: 7.70
Week 9: 7.62
Week 10: 7.55
Based on the data, there seems to be a slight overall upward trend in the Commodity Futures Index over the ten weeks, but there are also small fluctuations up and down throughout the period.
b. To find the best exponential smoothing coefficient (alpha) using the trial and error method, you would typically use software or programming languages to calculate the Mean Squared Error (MSE) for each alpha value.
Choose an initial value for alpha (e.g., 0.1).
Apply the exponential smoothing formula to the data series: St = α * Xt + (1 - α) * St-1, where St is the smoothed value at time t, α is the smoothing coefficient, Xt is the observed value at time t, and St-1 is the smoothed value at time t-1.
Calculate the forecast errors (Et) as the difference between the observed value and the smoothed value for each time period: Et = Xt - St-1.
Compute the Mean Squared Error (MSE) by taking the average of the squared forecast errors.
Repeat steps 2-4 with different values of alpha (e.g., 0.2, 0.3, 0.4, etc.) until you find the alpha value that minimizes the MSE.
By iterating through different alpha values, you can identify the optimal alpha for this data set that results in the smallest MSE.
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Carl bought 24 daisies and roses. Daisies cost $0.25 each and roses cost $0.90 each. If
Carl spends a total of $16.40, how many daisies did he get and how many roses did he
get?
Answers
Answer:
Roses= $8.4
Daises= $15.6
Step-by-step explanation:
Let represent the daises and let r represent d roses
r + d= 24....equation 1
0.25r + 0.90d= 16.40.......equation 2
r= 25-d
Substitute 25-d for r in equation 2
0.25(25-d) + 0.90d= 16.40
6.25-0.25d+0.90d= 16.40
6.25+0.65d= 16.40
0.65d= 16.40-6.25
0.65d= 10.15
d= 10.15/0.65
d = 15.6
Sub 15.6 for d in equation 1
r+d= 24
r+15.6= 24
r= 24-15.6
= 8.4
Price of daises is $15.6
Roses is $8.4
pls help
find f(p-5)
Answers
\(\text{Given that,}~f(j) = -5j-4\\\\f(p-5) = -5(p-5) -4\\\\~~~~~~~~~~~~=-5p+25-4\\\\~~~~~~~~~~~~=-5p+21\)
vectors
u = (-3,2,1,0), v = (4,7,-3,2) and w = (5,-2,8,1)
a) 2u +7v - w
b) |(1/|w|)w|
Answers
Answer:
its a im pretty sure
Step-by-step explanation:
Meg found another treasure between −13
4
feet and
-21
2
feet. Where was that treasure?
Negative 1 and one-half
Negative 2 and three-fourths
Negative 1 and one-fourth
Negative 2 and one-fourth
Answers
Answer:
Step-by-step explanation:
if it's -1 and 3/4 and -2 1/2 then it would be D, it's the only number to fit in that range
PLEASE HELP ME
The function f(x) = -2(4)^x+1 +140
represents the number of tokens a child has x hours after arriving at an arcade.
What is the practical domain and range of the function?
If necessary, round to the nearest hundredth.
The practical domain of the situation is ?
The practical range of the situation is ?
PLEASE SEE PHOTO FOR FUNCTION
Answers
The function f(x) = -2(4)ˣ⁺¹ +140 represents the number of tokens a child has x hours after arriving at an arcade. The practical domain and range of the function are x ≥ 0 and The practical range of the situation is [140, ∞).
The given function is f(x) = -2(4)ˣ⁺¹+ 140, which represents the number of tokens a child has x hours after arriving at an arcade.
To determine the practical domain and range of the function, we need to consider the constraints and limitations of the situation.
For the practical domain, we need to identify the valid values for x, which in this case represents the number of hours the child has been at the arcade. Since time cannot be negative in this context, the practical domain is x ≥ 0, meaning x is a non-negative number or zero.
Therefore, the practical domain of the situation is x ≥ 0.
For the practical range, we need to determine the possible values for the number of tokens the child can have. Looking at the given function, we can see that the term -2(4)ˣ⁺¹represents a decreasing exponential function as x increases. The constant term +140 is added to shift the graph upward.
Since the exponential term decreases as x increases, the function will have a maximum value at x = 0 and approach negative infinity as x approaches infinity. However, due to the presence of the +140 term, the actual range will be shifted upward by 140 units.
Therefore, the practical range of the situation will be all real numbers greater than or equal to 140. In interval notation, we can express it as [140, ∞).
To summarize:
- The practical domain of the situation is x ≥ 0.
- The practical range of the situation is [140, ∞).
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Write two expressions to represent the following situation. Then, find the answer. When Nathan went to bed, the outside temperature was 28°F. When he woke up the next morning, the temperature had decreased to -13°F. By how many degrees did the temperature change during the overnight hours? i know the answer i just need 2 expressions to do I
Answers
Answer:
Step-by-step explanation:
28-13
and 28+-13
What is the range of the function shown on the graph
Answers
Answer:
-2<=y<=-7
Step-by-step explanation:
If a central intercepts an arc of 6.3 cm. in a circle whose
radius is 4.2 cm., find the measure of the central in radians.
Answers
Yeah hellooooo
7x^7+10x^4+4x^3-5x^11-10x^6-6x^7
Answers
The sum of the terms is x^7+10x^4+4x^3-5x^11-10x^6
How to determine the valueTo determine the value, we need to know that algebraic expressions are described as expressions that are composed of factors, constants, variables, terms and coefficients.
These algebraic expressions are also identified with the presence of arithmetic operations,
These operations are;
AdditionBracketParenthesesSubtractionMultiplicationDivisionFrom the information given, we have that;
7x^7+10x^4+4x^3-5x^11-10x^6-6x^7
collect the like terms
7x^7 - 6x^7+10x^4+4x^3-5x^11-10x^6
add the like terms
x^7+10x^4+4x^3-5x^11-10x^6
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The width of a rectangle is fixed at 7 cm. Determine (in terms of an inequality) those lengths for which the area will be
less than 175 cm²
Answers
The lengths for which the area will be less than 175 cm² are less than 25 cm
How to determine the lengths of the areasFrom the question, we have the following parameters that can be used in our computation:
Width = 7 cm
Area = Less than 175 cm²
The formula of area is represented as
Area = Length * Width
Using the above as a guide, we have the following:
Length * Width < 175
substitute the known values in the above equation, so, we have the following representation
Length * 7 < 175
Divide by 7
Length < 25
Hence, the length is less than 25
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HELP PLEASE AS SOON AS POSSIBLE WILL GIVE U BRAINLIST
Answers
Answer:
The table represents a nonlinear function because the rate is not constant.
Step-by-step explanation:
As shown in the picture below, the x side of the table has the same rate of change of +1. However, due to the fact that the y side does not have the same rate of change, +6 and +3, the table represents a non linear function. If the rate on the y side of the table were all the same, then this would be a Linear function.
Instructions: Use the ratio of a 30-60-90 triangle to solve for the variables. Leave your answers as radicals in simplest form.
If you are using a screen-reader, please consult your instructor for assistance.
x=
y=
Answers
Using the ratio of the sides, we have:$x\sqrt{3} = 12\sqrt{3}$ (opposite the $60^{\circ}$ angle is 12$\sqrt{3}$)$x = 2\sqrt{3}\cdot6$ (the hypotenuse is $2x = 12\sqrt{3}$)Simplifying, we have:$x = 12\sqrt{3}$. Therefore $x=y=12\sqrt{3}$, which is our answer
In a 30-60-90 triangle, the sides have the ratio of $1: \sqrt{3}: 2$. Let's apply this to solve for the variables in the given problem.
Instructions: Use the ratio of a 30-60-90 triangle to solve for the variables. Leave your answers as radicals in simplest form. x=y=Let's first find the ratio of the sides in a 30-60-90 triangle.
Since the hypotenuse is always twice as long as the shorter leg, we can let $x$ be the shorter leg and $2x$ be the hypotenuse.
Thus, we have: Shorter leg: $x$Opposite the $60^{\circ}$ angle: $x\sqrt{3}$ Hypotenuse: $2x$
Now, let's apply this ratio to solve for the variables in the given problem. We know that $x = y$ since they are equal in the problem.
Using the ratio of the sides, we have:$x\sqrt{3} = 12\sqrt{3}$ (opposite the $60^{\circ}$ angle is 12$\sqrt{3}$)$x = 2\sqrt{3}\cdot6$ (the hypotenuse is $2x = 12\sqrt{3}$)Simplifying, we have:$x = 12\sqrt{3}$
Therefore, $x=y=12\sqrt{3}$, which is our answer.
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A bag contains red and green marbles. The ratio of red marbles to green marbles is 3 to 2. If the total of marbles is 20, how many of each color is there?
Answers
Answer:
12 red and 8 green
Step-by-step explanation:
Multiply
Answer:
12:8
Step-by-step explanation:
Slope is 5, (5,4) is on the line
The equation of the line is y=
Answers
To get b
I got the equation by plugging in the slope and point into the y=Mx+b formula.
Hope this helps:)
3. Sketch a system of equations (one quadratic and one linear) that has no solution. Justify why this system does not have a solution.
Answers
We can solve this problem using two equations, one quadratic, and one linear, such as the graphs have no point in common, that is, both graphs have no common points or intersections between them. Therefore, we can use the following equations:
1. Quadratic equation: we can have one equation shifted to the right 2 units:
\((x-2)^2=x^2-4x+4\)2. Now, we can use a line equation so that it does not pass through the quadratic equation. Since most of the quadratic equation is in the first quadrant, we can use a line with a negative slope. We can use the x-intercept equal to (-2, 0), and (0, -2). For this, we can find the slope for this line:
x1 = -2
y1 = 0
x2 = 0
y2 = -2
\(m=\frac{y_2-y_1}{x_2-x_1}=\frac{-2-0}{0-(-2)}=-\frac{2}{2}\Rightarrow m=-1\)Applying the point-slope form of the line, we have:
\(y-y_1=m\cdot(x-x_1)\Rightarrow y-0=-1\cdot(x-(-2)=y=-(x+2)=-x-2\)\(y=-x-2\)Then, we have that the system of equation is:
\(y=x^2-4x+4,y=-x-2\)If we graph both equations, we have the following graph:
Since we can see that both graphs will never intersect each other, we can conclude that this system does not have solutions in the Real set of numbers.
The linear equation is:
\(y=-x-2\)The quadratic equation is:
\(y=x^2-4x+4\)