To prove the statement, we need to show that the limit of the integral tends to zero as n approaches infinity:
[tex]Lim(n→∞) ∫[0,1] [f(x)]^n dx = 0[/tex]
Given that f(x) is a strictly increasing continuous function on the interval [0,1], we can make use of the properties of such functions to prove the statement.
Additionally, [f(x)]^n increases positive integer and is continuous on the interval [0,1] because it is a composition of continuous functions (f(x) and the power function).
[tex]∫[0,1] [f(x)]^n dx[/tex]
Integrating this inequality over the interval [0,1], we have:
[tex]0 ≤ ∫[0,1] [f(x)]^n dx ≤ ∫[0,1] 1 dx0 ≤ ∫[0,1] [f(x)]^n dx ≤ 1[/tex]
0 and 1 are for the positive integer n
Now, as n approaches infinity, we can apply the squeeze theorem. Since the integral is bounded between 0 and 1, and both 0 and 1 approach zero as n tends to infinity, the limit of the integral must also be zero:
[tex]Lim(n→∞) ∫[0,1] [f(x)]^n dx = 0[/tex]
Therefore, we have proven that the limit of the integral as n approaches infinity is zero:
[tex]1 lim(n→∞) ∫[0,1] [f(x)]^n dx = 0[/tex]
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One pump can fill a tank of water in 5 hours. A second tank can fill the same tank in 4 hours. If both pumps are used together, how long will it take to fill the tank?
4/5 hours
20/9 hours
9/20 hours
5/4 hours
Answer:
4/5 hours
Step-by-step explanation:
Answer:
20/9 hours
Step-by-step explanation:
I got it right on the quiz
A little boy stands on a carousel and rotates AROUND 4 times. If the distance between the little boy and the center of the carousel is 6 feet, then how many feet did the little boy travel? (C = 2πr). Use 3.14 for pi. (Hint: he is going AROUND 4 times not just once). Only put in the number do not put in the unit of measure (ft). *
Answer:
150.72
Step-by-step explanation:
C = 2*pi * r
C = 2* 3.14 * 6
C = 37.68
1 revolution = 37.68
4 revolutions = 4 * 37.68
4 revolutions = 150.72
A blue circular target is tacked onto a square corkboard. The area of the target is 75 square units. What is the area of the corkboard that is not covered by the target?
HELP ASAP
44/15 converted into a mixed number (convert a fraction into a decimal before converting it into a mixed number)
Answer:
2 14/15
Step-by-step explanation:
44/15= 2.9333333333333
2.9333333333333= 2 14/15
Find the missing length PLEASE HELP NEED IT ASAP
Answer:
c = [tex]\sqrt{58}[/tex]
Step-by-step explanation:
Using Pythagoras' identity in the right triangle
c² = 7² + 3² = 49 + 9 = 58 ( take the square root of both sides )
c = [tex]\sqrt{58}[/tex]
Answer:
58Step-by-step explanation:
According to Pythagoras Theorem:
[tex] {7}^{2} + {3}^{2} = {c}^{2} [/tex]
Hence,
[tex] = > {c}^{2} = {7}^{2} + {3}^{2} [/tex]
[tex] = > {c}^{2} = 49 + 9[/tex]
[tex] = > {c}^{2} = 58[/tex]
[tex] = > c = \sqrt{58} [/tex]
So,
[tex]c = \sqrt{58} [/tex]
Is the answer.
help me with this plsss
Hugo wanted to reorganize his superhero figure collection. He split his collection evenly
among 3 shelves. When he was done, there were 12 figures on each shelf. How many
figures are in Hugo's collection?
Answer:
36
Step-by-step explanation:
3*12=36
I need help, I dont understand
Answer:
20,19,18
Step-by-step explanation:
because you have to go backwards
Raffia went shopping for new furniture he bought a couch for $750 and a mattress for 950 dollars. When he came home he had $350 in his purse. How much money did he have when he left home. Please answer step by step will mark brainliest and thanks !!!!!!!!!!!
Answer:
$2050
Step-by-step explanation:
750+950+350
A food safety guideline is that the mercury in fish should be below 1 part per million (ppm) Listed below are the amounts of mercury (ppm) found in tuna sushi sampled at different stores in a major city Construct a 10% confidence interval estate f too much marcury in una sush? 051 0.01 0.10 0.96 128 057 056 What is the confidence interval estimate of the population mean? pppp Round to three decimal places as needed) Does it appear that there is too much mercury in una sush? OA. Yes, because it is possible that the mean is greater than 1 ppm. Also, at least one of the sample values exceeds 1 ppm, so at least some of the fish have too much mercury 1. No, because it is possible that the mean is not greater than 1 ppm Also, at least one of the sample values is less than 1 ppm, so at least some of the fish sale OC. Yes because it is possible that the mean is not greater than 1 ppm. Also, at least one of the sample values exceeds 1 ppm, so at least some of the fish have too much mercury OD. No. because it is not possible that the mean is greater than 1 ppm Also at least one of the sample values is less than 1 ppm, so at least some of the fish are safe
The confidence interval estimate of the population mean is (0.37, 0.75). The correct option is B No. Based on the confidence interval, it is not possible to say that the mean mercury level in tuna sushi is greater than 1 ppm.
How to explain the informationThe confidence interval includes 1 ppm, so the true mean could be 1 ppm or less. Additionally, at least one of the sample values is less than 1 ppm, so at least some of the fish are safe.
The sample standard deviation is calculated by finding the square root of the sum of the squared deviations from the mean for each value in the sample. The sample size is the number of values in the sample.
Confidence interval = sample mean ± 1.645 * (sample standard deviation / ✓(sample size))
Confidence interval = 0.51 ± 1.645 * (0.24 / ✓(7))
= (0.37, 0.75)
As you can see, the confidence interval includes 1 ppm, so the true mean could be 1 ppm or less.
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Plz help ASAP !!!!! Plzzz
Answer:
A box has 6 pencils. Each pencil weighs x grams. The 6 pencils weigh a combined total of 54 grams.
Step-by-step explanation:
Find the sum of the measures of the exterior angles of a convex 65-gon
Answer: 360
Step-by-step explanation: Sum of exterior angles of any convex polygon is always 360, think about it.
An article in Technometrics ("Validation of Regression Models: Methods and Examples"presented the following data (which have been rounded to the nearest whole value) on the motor fuel octane ratings of several blends of gasoline:
89 99 90 92 93 88 88 91 95 88
90 83 88 93 88 90 84 90 92 91
93 94 88 92 90 91 91 88 94 97
89 90 89 91 89 89 90 84 92 92
90 92 88 93 91 93 89 92 88 94
87 87 89 90 90 91 8591 94 89
93 90 8790 91 91 93 93 92 93
89 100 90 89 87 90 96 91 88 92
a. Construct a frequency distribution for these data by filling in the table below. Use 8 bins. Lower limit Upper limit Midpoint Frequency Relative frequency
b. Use the frequency distribution on the previous page to construct a histogram for these data.
To analyze the motor fuel octane ratings data, we can start by constructing a frequency distribution and then use it to create a histogram.
The frequency distribution will provide information about the distribution of the data across different octane rating ranges, while the histogram visually represents the distribution graphically.
a. To construct a frequency distribution, we divide the data into bins and count the frequency of values falling within each bin. In this case, we will use 8 bins. We determine the lower and upper limits for each bin and calculate the midpoint by averaging the limits. Then, we count the number of values within each bin and calculate the relative frequency by dividing the frequency by the total number of values.
b. Once the frequency distribution is constructed, we can use it to create a histogram. The histogram represents the frequency or relative frequency of values within each bin as vertical bars. The bins are plotted along the x-axis, and the height of the bars represents the frequency or relative frequency on the y-axis. The histogram allows us to visualize the distribution pattern of the motor fuel octane ratings data, showing if it is skewed, symmetric, or has other characteristics.
By constructing a frequency distribution and creating a histogram, we can gain insights into the distribution of the motor fuel octane ratings data. The frequency distribution table provides a summarized view of how the data is spread across different octane rating ranges, while the histogram visually represents the same information in a graphical form. Both the frequency distribution and the histogram aid in understanding the distribution pattern, identifying potential outliers or gaps, and informing further analysis or decision-making related to the data.
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Consider the arithmetic sequence presented in the table below. What is the first term, ay, and the 22nd term of the sequence? n 4 37 an 16 115 Hint: Q, = a1 + d(n-1), where ay is the first term and d is the common difference. O a = 7, 222 - 68 O a = 3, 222 150 O a = 3, 222 = 148 O Qı = 7, 222 = 70
The first term of the arithmetic sequence is 16 and the 22nd term is 148.
To solve for the first term, we can use the following formula:
a_n = a_1 + d(n-1)
where:
a_n is the nth term of the sequence
a_1 is the first term of the sequence
d is the common difference of the sequence
We know that the 4th term is 16 and the 37th term is 115. We can use these values to solve for d.
d = a_{37} - a_4 = 115 - 16 = 99
Now that we know d, we can solve for a_1 using the value of the 4th term.
a_1 = a_4 - d = 16 - 99 = -83
Finally, we can solve for the 22nd term using the value of a_1 and d.
a_{22} = a_1 + d(22-1) = -83 + 99(21) = 148
Therefore, the first term of the arithmetic sequence is 16 and the 22nd term is 148.
Here is a more detailed explanation of the solution:
The first step is to identify the common difference of the sequence. This can be done by subtracting any two consecutive terms in the sequence. In this case, the 4th term is 16 and the 37th term is 115. Subtracting these two terms, we get 115 - 16 = 99. This is the common difference of the sequence.
Once we know the common difference, we can solve for the first term by using any of the terms in the sequence. In this case, we will use the 4th term. The formula for the nth term of an arithmetic sequence is a_n = a_1 + d(n-1), where a_n is the nth term, a_1 is the first term, and d is the common difference. Substituting the 4th term, the common difference, and n = 4, we get a_4 = a_1 + d(4-1). Simplifying this equation, we get 16 = a_1 + 99. Solving for a_1, we get a_1 = -83.
Finally, we can solve for the 22nd term by using the first term and the common difference. The formula for the nth term of an arithmetic sequence is a_n = a_1 + d(n-1). Substituting the first term, the common difference, and n = 22, we get a_{22} = -83 + 99(22-1). Simplifying this equation, we get a_{22} = 148.
Therefore, the first term of the arithmetic sequence is 16 and the 22nd term is 148.
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Triangle ABC has a side length of 8 and is dilated to Triangle XYZ with a side length
of 4. Is this a reduction or enlargement, and what is the scale factor?
v
Enlargement; scale factor of 1/2
Reduction; scale factor of 2
Enlargement; scale factor of 2
Reduction; scale factor of 1/2
Which equation represents a line that passes through the two points in the
table?
A. y-3 = 2/3(x-3)
B. y+3 = 3/2(x+3)
C. y-3 = 3/2(x-2)
D. y+3 = 2/3(x+3)
24.1 let fn(x) = 1 2 cos2 √ nx n . prove carefully that (fn) converges uniformly to 0 on r.
We need to prove that the sequence of functions fn(x) = (1/2)cos^2(√nx) converges uniformly to 0 on the real numbers.
To prove uniform convergence, we need to show that for any ε > 0, there exists a positive integer N such that |fn(x) - 0| < ε for all n > N and for all x in the real numbers.
Given fn(x) = (1/2)cos^2(√nx), we observe that as n increases, the argument √nx inside the cosine function becomes larger, causing the cosine to oscillate more rapidly between 0 and 1. Since we are multiplying the cosine by (1/2) and taking its square, fn(x) gradually approaches 0 as n increases.
To formally prove uniform convergence, we can start by fixing ε > 0 and then choose N such that √Nx > 1/ε. By selecting this N, we can show that for all n > N, |fn(x) - 0| < ε for any x in the real numbers. This demonstrates that the sequence of functions fn(x) converges uniformly to 0 on the real numbers.
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What are the variables in this expression?
f + 2g + h/4 - 7
A) f and g only
B) g and h only
C) f, g, and -7 only
D) f, g, and h only
2. Find the solution to the recurrence relation bn = 4bn-1-4bn-2 with initial values bo = 1 and b₁ = 2.
The resultant of the recurrence relation bn = 4bn-1-4bn-2 with initial values bo = 1 and b₁ = 2 is bn = (1 + (n/2)) 2ⁿ.
The given recurrence relation is
bn = 4bn-1 - 4bn-2 with initial values bo = 1 and b₁ = 2. Now, we have to find the solution to the recurrence relation. It can be observed that the given recurrence relation is a second-order recurrence relation. The characteristic equation of the recurrence relation is given by:
r² = 4r - 4, which can be simplified as r² - 4r + 4 = 0. We need to solve this equation. It can be solved as r = 2
The characteristic equation of the given recurrence relation has two equal roots r1 = r2 = 2. Therefore, the general result of the recurrence relation is given by:
bn = (A + Bn) 2ⁿ
For the given initial values b₀ = 1, and b₁ = 2 we can write:
b₀ = (A + B(0)) 2⁰ => A = 1b₁ = (A + B(1)) 2¹ => A + 2B = 2
On solving these equations, we get A = 1 and B = 1/2
The resultant to the recurrence relation is:
bn = (1 + (n/2)) 2ⁿ
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hi please help i’ll give brainliest
Answer:
Ellipse. It is irregular at some points.
Step-by-step explanation:
Answer:
b
Step-by-step explanation:
someone help me please like right now lol of and don’t put any links pls!
Answer:
8 shirts
Step-by-step explanation:
Please help!!
A circle has a circumference of 1007 cm. What is the radius of the circle?
Answer:
3/2 I'm not sure I think it's that
Help me plzzz with rectangle
Answer:
26
Step-by-step explanation:
Let X be a RV (note that we don’t assume anything about what type of a RV it is) and let H(x) = P(X < x). Rigorously discuss the continuity properties of H: is it continuous / left-continuous / right-continuous / not guaranteed to be any of those?
The continuity property of H is that it is:
c. right-continuous
Continuity property: A function is continuous if it has no jumps, i.e., the graph can be drawn without lifting the pencil from the paper. For any random variable X, the function H is right-continuous.
Property of left-continuity: A function is left-continuous if the limit of the function from the left side of x exists and is the same as the function value at x. In the case of the function H, it is not continuous from the left side. In fact, there is a jump that occurs in the limit as x approaches any value.Property of right-continuity: A function is right-continuous if the limit of the function from the right side of x exists and is the same as the function value at x. In the case of the function H, it is continuous from the right side.This is due to the fact that the limit of H(x) as x approaches any value from the right side is equivalent to H(x) as x approaches that value from the right side, so H is continuous from the right side.
A random variable does not have to be any particular type of random variable to have a continuous H. For any random variable X, the continuity property applies to the function H, as demonstrated by the continuity of H from the right side.To know more about continuity property, visit the link : https://brainly.com/question/30328478
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Take the sample mean of this data series: 15, 26, 25, 23, 26, 28, 20, 20, 31, 45, 32, 41, 54, 23, 45, 24, 90, 19, 16, 75, 29 And the population mean of this data series: 15, 26, 25, 23, 26, 28, 20, 20, 31, 45, 32, 41, 54, 23, 45, 24, 90, 19, 100, 75, 29 Calculate the difference between the two quantities (round to two decimal places)
The sample mean of the given data series is 33.7 and the population mean is 35.86. The difference between the two quantities, rounded to two decimal places, is -2.16.
Sample mean and population mean are important terms in statistics. The sample mean is the average of a set of data taken from a larger population, while the population mean is the average of an entire population.
To find the sample mean of the given data series, we add up all the values and divide by the total number of values. Therefore,
Sample mean = (15+26+25+23+26+28+20+20+31+45+32+41+54+23+45+24+90+19+16+75+29) / 21
Sample mean = 33.7
To find the population mean, we use the same formula but with all the values of the population included. Therefore,
Population mean = (15+26+25+23+26+28+20+20+31+45+32+41+54+23+45+24+90+19+100+75+29) / 21
Population mean = 35.86
Finally, to find the difference between the two quantities, we subtract the sample mean from the population mean. Therefore,
Difference = Population mean - Sample mean
Difference = 35.86 - 33.7
Difference = -2.16 (rounded to two decimal places)
Therefore, the difference between the sample mean and population mean is -2.16.
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Consider the curve defined by the equation y = 47° +10x. Set up an integral that represents the length of curve from the point (-3,6) to the point (2,36).
The integral that represents the length of the curve from the point (-3,6) to the point (2,36) is L = ∫[-3 to 2] √101 dx.
To find the length of the curve defined by the equation y = 47° + 10x from the point (-3,6) to the point (2,36), we can use the arc length formula for a curve:
L = ∫[a to b] √(1 + (dy/dx)^2) dx
First, let's find dy/dx by taking the derivative of the given equation:
dy/dx = d/dx(47° + 10x) = 10
Now we can substitute this value into the arc length formula:
L = ∫[-3 to 2] √(1 + (10)^2) dx
Simplifying:
L = ∫[-3 to 2] √(1 + 100) dx
L = ∫[-3 to 2] √101 dx
Thus, the integral that represents the length of the curve from the point (-3,6) to the point (2,36) is:
L = ∫[-3 to 2] √101 dx
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find the profile or loss s.p= 575
Answer:
more info?
Step-by-step explanation:
Use the Singapore Bar Method, including drawings, to solve the following problem. Identify the unit value when appropriate, including labels. The sides of the triangle are in the ratio 5:7:8 and the longest side is 36 cm longer than the shortest side. Find the perimeter of the triangle.
When the sides of the triangle are in the ratio 5:7:8 and the longest side is 36 cm longer than the shortest side, the perimeter is 240cm.
How to calculate the perimeterLet's assume the shortest side of the triangle has a length of x cm. According to the given ratio, the sides of the triangle are in the ratio 5:7:8. Therefore, the lengths of the sides can be expressed as:
Shortest side: 5x
Second side: 7x
Longest side: 8x
We are also given that the longest side is 36 cm longer than the shortest side. So we can set up the following equation:
8x = 5x + 36
Now, let's solve this equation to find the value of x:
8x - 5x = 36
3x = 36
x = 36 / 3
x = 12
Now we can substitute this value back into the expressions for the side lengths to find their actual lengths:
Shortest side: 5x = 5 * 12 = 60 cm
Second side: 7x = 7 * 12 = 84 cm
Longest side: 8x = 8 * 12 = 96 cm
Finally, we can calculate the perimeter of the triangle by adding the lengths of all three sides:
Perimeter = Shortest side + Second side + Longest side
= 60 cm + 84 cm + 96 cm
= 240 cm
Therefore, the perimeter of the triangle is 240 cm.
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The expression 3/6 kids the weight of an object onto the moon in pounds in weight
Someone please please help me on this problem
Answer:
81 Pesos
Step-by-step explanation:
Use a proportion.
$3 is to 2 Pesos as $121.10 is to x Pesos.
3/2 = 121.1/x
3x = 2 * 121.1
3x = 242.20
x = 80.7333...
Answer: 81 Pesos