State whether the sequence converges and, if it does, find the limit.
1. (n+4)/n
2. (n+8)/(n^2)
3. tan((n(pi))/(4n+3))
4. ln(3n/(n+1))
5. n^2/(sqrt(8n^4+1))
6. (1+(1/n))^(5n)

To determine which of the given quadrature formulas are of the interpolatory type, let's first understand the concept of an interpolatory quadrature formula.

An interpolatory quadrature formula is one that approximates the integral of a function using a weighted sum of the function's values at specific points, known as nodes.

Now let's analyze the given quadrature formulas:

a) Sf(dx*(2))

This formula doesn't provide any information about the nodes or weights to be used for approximation.

Therefore, we cannot determine if it is of the interpolatory type.

b) Sf(a)dx = f(-1) + f(1)

This formula approximates the integral of a function using the sum of the function's values at the nodes x = -1 and x = 1.

The weights associated with these nodes are both 1.

Since this formula uses specific nodes and weights, it can be considered an interpolatory quadrature formula. In conclusion, the second formula (Sf(a)dx = f(-1) + f(1)) is of the interpolatory type.

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the resulting matrix after performing the row operation -2R+R2 R2 on the given matrix would depend on the original matrix provided.

What is matrix?

The plural version of the word matrix is a matrix, which refers to the arrangements of numbers, variables, symbols, or phrases in a rectangular table with varying numbers of rows and columns. These arrays have a rectangular shape, and several operations like addition, multiplication, and transposition are specified for them. The components of the matrix are referred to as its entries or numbers. Vertical and horizontal entries in matrices are referred to as columns and rows, respectively. A matrix with m rows and n columns will contain m n entries. The uppercase letter 'A', which here stands for "matrix," Aij.

The row operation -2R+R2 R2 means that we take row 2 of the matrix and multiply it by -2, and then add the result to row 2. This will change the values in row 2 of the matrix, but leave the other rows unchanged.

For example, if the original matrix was:

| 2 3 4 |

| 5 6 7 |

| 8 9 10 |

And we apply the row operation -2R+R2 R2 to row 2, we would get:

| 2 3 4 |

| 1 0 -1 |

| 8 9 10 |

Notice that we took row 2, which was [5 6 7], multiplied it by -2 to get [-10 -12 -14], and then added it to row 2, which gave us [5+(-10) 6+(-12) 7+(-14)] = [ -5 -6 -7].

Therefore, the resulting matrix after performing the row operation -2R+R2 R2 on the given matrix would depend on the original matrix provided.

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To determine the theoretical atomic radius of iron with a body-centered cubic (BCC) crystal structure and a cell dimension of 2.86 Å, we will follow these steps:

1. Remember that in a BCC structure, the atoms touch along the body diagonal of the unit cell.
2. The body diagonal length (d) can be found using the formula d = √3 * a, where a is the cell dimension (2.86 Å).
3. In a BCC structure, the body diagonal is equal to 4 times the atomic radius (r), so we can write d = 4r.
4. Combine steps 2 and 3, and solve for the atomic radius (r).

Let's calculate the atomic radius of iron:

1. d = √3 * 2.86 Å ≈ 4.95 Å
2. 4.95 Å = 4r
3. r ≈ 1.24 Å

So, the theoretical atomic radius of iron in a BCC crystal structure with a cell dimension of 2.86 Å is approximately 1.24 Å (Option C).

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The missing coordinates for K, L, and M are (10, 25), (5, 20), and (30, 25), respectively.

THE LINE is a cultural resolution that main people and offers urban dwellers a unique experience while protected the natural environment. It defines the idea of  development and the design of future cities.

A coordinate system in geometry is the method for determining the location of points or other geometric objects on a manifold, In  Euclidean space, uniquely using one or more numbers or coordinates.

To find the missing coordinates for K, L, and M, we need to know the direction in which the line is going. If we assume that the line is going from left to right, then we can use the coordinates of K, L, and M to determine the missing values.

For K (10, ||), we know that the x-coordinate is 10, but we don't know the y-coordinate. Since K is between L and M, we can assume that its y-coordinate is somewhere between 20 and 30. If we take the average of 20 and 30, we get 25. Therefore, the missing coordinate for K is (10, 25).

For L (||, 20), we know that the y-coordinate is 20, but we don't know the x-coordinate. Since L is to the left of K, we can assume that its x-coordinate is somewhere between 0 and 10. If we take the average of 0 and 10, we get 5. Therefore, the missing coordinate for L is (5, 20).

For M (30, ||), we know that the x-coordinate is 30, but we don't know the y-coordinate. Since M is to the right of L, we can assume that its y-coordinate is somewhere between 20 and 30. If we take the average of 20 and 30, we get 25. Therefore, the missing coordinate for M is (30, 25).

Therefore, the missing coordinates for K, L, and M are (10, 25), (5, 20), and (30, 25), respectively.

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The proportion of stay-at-home residents in arkansas is approximately 0.0166. Please note that the numbers used in this example are hypothetical

To estimate the proportion of stay-at-home residents in Arkansas, you can follow these steps:

1. Find relevant data: Look for a reliable source that provides the necessary information about stay-at-home residents in Arkansas. This could be government reports, research studies, or online databases.

2. Identify the total population: Determine the total number of residents in Arkansas. According to the U.S. Census Bureau, the population of Arkansas in 2020 was around 3,011,524.

3. Identify the number of stay-at-home residents: From the data source, find the number of stay-at-home residents in Arkansas.

4. Calculate the proportion: To find the proportion, divide the number of stay-at-home residents by the total population of Arkansas. For example, if there are 50,000 stay-at-home residents in Arkansas, the proportion would be:

Proportion = (Number of stay-at-home residents) / (Total population)
Proportion = 50,000 / 3,011,524

5. Round to four decimal places: If required, round the resulting proportion to four decimal places. In our example:

Proportion ≈ 0.0166

Please note that the numbers used in this example are hypothetical, and you will need to find the actual number of stay-at-home residents in Arkansas from a reliable source to get the correct proportion.

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The unique solution to the given system is x(t) = [tex]e^t[/tex][1; -1] + [tex]e^5^t[/tex][1; 1].

To solve the given 2x2 system of differential equations x'(t) = Ax(t) with the initial condition x(0) = [2; 0], we find the eigenvalues and eigenvectors, and then express the solution as x(t) = [tex]e^(^d^1^t^)[/tex]v1 + [tex]e^(^d^2^t^)[/tex]v2.

1. Find the matrix A: A = [3, 2; 2, 3]
2. Find eigenvalues (d1, d2) and eigenvectors (v1, v2) of A.
3. Calculate the matrix exponential using the eigenvalues and eigenvectors.
4. Apply the initial condition x(0) = [2; 0] to find the coefficients.

Following these steps, we find d1 = 1, d2 = 5, v1 = [1; -1], and v2 = [1; 1].

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According to the information, the jet has traveled 4400 miles.

To find how many miles the jet has traveled, we need to know the total time it has been in the air. Since the jet left the airport 4 hours ago, and assuming it has been flying at a constant speed of 600 mph ever since, we have:

Total time in air = 4 hours + time since reaching top speed

We can convert this total time to minutes by multiplying by 60:

Total time in air = 4 × 60 + time since reaching top speed

Total time in air = 240 + time since reaching top speed (in minutes)

Now, we can use the equation:

distance = speed × time

to find the distance traveled by the jet. The speed is 600 mph, but we need to convert it to miles per minute by dividing by 60:

speed = 600 mph ÷ 60 = 10 miles per minute

The time is the total time in air we just calculated. Therefore:

distance = 10 miles per minute × (240 + time since reaching top speed)

We don't know the exact value of the time since reaching top speed, but we know it is less than 4 hours (since the jet reached top speed 7 minutes after takeoff and has been flying at a constant speed of 600 mph ever since). Therefore, we can assume it is less than 240 minutes. Let's take a conservative estimate and assume it is 200 minutes:

distance = 10 miles per minute × (240 + 200) = 4400 miles

Therefore, the jet has traveled 4400 miles.

 600|            .__

     |          .     \

     |        .        \

     |      .            \

     |    .                \

     |  .                    \

     |________________________

         0       7 min     t

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The events a and b in a sample space with  positive probability shows that  p(b|a) > p(b)when p(a|b) > p(a).

We want to prove that P(B|A) > P(B) if and only if P(A|B) > P(A) whic are positive probability.

First, let's recall the definition of conditional probability: P(B|A) = P(A ∩ B) / P(A) and P(A|B) = P(A ∩ B) / P(B).

Now, let's prove both directions of the statement:

(1) If P(B|A) > P(B), then P(A|B) > P(A):

Given that P(B|A) > P(B), we have:

P(A ∩ B) / P(A) > P(B)

Now, multiply both sides by P(A):

P(A ∩ B) > P(A) * P(B)

Now, divide both sides by P(B):

P(A ∩ B) / P(B) > P(A)

Thus, P(A|B) > P(A).

(2) If P(A|B) > P(A), then P(B|A) > P(B):

Given that P(A|B) > P(A), we have:

P(A ∩ B) / P(B) > P(A)

Now, multiply both sides by P(B):

P(A ∩ B) > P(A) * P(B)

Now, divide both sides by P(A):

P(A ∩ B) / P(A) > P(B)

Thus, P(B|A) > P(B).

Therefore, we have proven that P(B|A) > P(B) if and only if P(A|B) > P(A).

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The side b is 24 cm, angle A is approximately 16.3 degrees, and angle B is approximately 73.7 degrees using Pythagorean theorem.

Using the Pythagorean theorem, we can solve for b:

[tex]a^2 + b^2 = c^2 \\7^2 + b^2 = 25^2 \\49 + b^2 = 625 \\b^2 = 576[/tex]
b = 24 cm

Now, to find angle B:

[tex]sin(B)[/tex] = opposite/hypotenuse = a/c = 7/25
[tex]B = sin^-1(7/25) = 16.3 degrees[/tex]

To find angle A:

A = 90 degrees - B = 73.7 degrees

Therefore, the angles are:
A ≈ 73.7 degrees
B ≈ 16.3 degrees
C = 90 degrees

To solve the given right triangle with a = 7 cm and c = 25 cm, we will first find the missing side b using the Pythagorean theorem, then find the angles A and B using trigonometric functions.

Step 1: Find side b using the Pythagorean theorem.
In a right triangle, a² + b² = c²
Given, a = 7 cm and c = 25 cm, so:
[tex]7² + b² = 25²49 + b² = 625\\b² = 625 - 49\\b² = 576\\b = \sqrt{576}[/tex]
b = 24 cm

Step 2: Find angle A using sine or cosine.
Using sine, we have sin(A) = a/c
[tex]sin(A) = 7/25\\A = arcsin(7/25)[/tex]
A ≈ 16.3 degrees (rounded to the nearest tenth)

Step 3: Find angle B using the fact that the sum of angles in a triangle is 180 degrees.
Since it's a right triangle, angle C is 90 degrees. Thus:
A + B + C = 180 degrees
16.3 + B + 90 = 180
B ≈ 180 - 16.3 - 90
B ≈ 73.7 degrees (rounded to the nearest tenth)

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Answer:

98.59006

Step-by-step explanation:

simplify to what it asks, since you didn't give us what to simplify to

We have proven that ∠JKM = ∠NKL.

To prove that ∠JKM = ∠NKL,

       J

      / \

     /   \

    /     \

   K-------N

  / \       / \

 /    \     /     \

/      \   /         \

M-------K-------L

To prove that JKM is congruent to NKL, we need to show that all corresponding sides and angles are equal.

First, we know that KM bisects JKN, so angle JKM is congruent to angle NKM (by the angle bisector theorem). Similarly, KN bisects MKL, so angle LKN is congruent to angle MKN.

Also, we know that JK is equal to KN (by the definition of a bisector), and KM is equal to ML (since KM bisects the side KL).

we will use the given information that KM bisects ∠JKN and KN bisects ∠MKL.
Since KM bisects ∠JKN, it means that it divides ∠JKN into two equal angles.

So, ∠JKM = ∠KJN. (Definition of angle bisector)
Similarly, since KN bisects ∠MKL, it divides ∠MKL into two equal angles.

So, ∠KJN = ∠NKL. (Definition of angle bisector)
Now, we can use the transitive property of equality: if ∠JKM = ∠KJN and ∠KJN = ∠NKL, then ∠JKM = ∠NKL.

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The requried measure of the angle H is m∠H = 26°.

Since the sum of the angles in a triangle is always 180 degrees, we can write:

m∠F + m∠G + m∠H = 180

Substituting the given values, we get:

(5x-6) + (3x+16) + (x+8) = 180

Simplifying and solving for x, we get:

9x + 18 = 180

9x = 162

x = 18

Now, we can use the value of x to find the measures of the angles:

m∠F = (5x-6)° = (5(18)-6)° = 84°

m∠G = (3x+16)° = (3(18)+16)° = 70°

m∠H = (x+8)° = (18+8)° = 26°

Therefore, m∠H = 180 - m∠F - m∠G = 180 - 84° - 70° = 26°

And m∠H = 26°.

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The product of zeros of cubic polynomial x³ - 3x² - x + 3 is 3

The zeroes of a cubic polynomial are the values of x at which the polynomial equals zero.

Given the cubic polynomial x³ - 3x² - x + 3, we desire to find the product of the zeroes of the polynomial. We proceed as follows.

For a cubic polynomial ax³ + bx² + cx + d with factors (x - l)(x - m)(x - n), and zeroes, l, m and n respectively, we have the the product of the zeroes are

lmn = d/a

So, comparing this with x³ - 3x² - x + 3 where a = 1 and d = 3.

So, the product of the zeroes is d/a = 3/1 = 3

So, the product of the zeroes is 3

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Hope, Casey, and Daniela are the players in order from lowest to highest points per game average.

To find the points per game for each player, we can divide their total points by the number of games they played:

Casey: 154 points ÷ 8 games = 19.25 points per game

Hope: 132 points ÷ 7 games = 18.86 points per game

Daniela: 101 points ÷ 5 games = 20.2 points per game

Casey mistakenly ordered the players as follows: Hope, Casey, and then Hope. This ordering is incorrect because Casey had a higher point-per-game average than Hope.

The correct ordering from least to greatest points per game average is: Hope, Casey, and then Daniela.

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Complete question:

Daniela scored

101 points in 5 basketball games. Casey scored 154 points in 8 games. Hope scored 132 points in 7 games. Casey tried to order the players by their points per game from least to greatest, but he made a mistake. Here's his work:

Answer:

x=−2+2sqrt6 or x=−2−2sqrt6. C, D

Step-by-step explanation:

For this equation: a=1, b=4, c=-20

1x2+4x+−20=0

Step 1: Use quadratic formula with a=1, b=4, c=-20.

x=−b±b2−4ac/2a

[tex]\tan(x )=\cfrac{\stackrel{opposite}{50}}{\underset{adjacent}{36}} \implies \tan(x)=\cfrac{25}{18}\implies x =\tan^{-1}\left( \cfrac{25}{18} \right)\implies x \approx 54.2^o[/tex]

Make sure your calculator is in Degree mode.

In the interoccurrence distribution function F(x), the conditional probability that the excess life exceeds y is the same as the probability that the interoccurrence time is less than or equal to y. And E[γt | δt = x] = ∫ x to ∞ y dF(y) / (1 - F(x)) - x expresses the conditional mean.

In a renewal process with interoccurrence distribution function F(x), the excess life γt and age δt are related by the equation γt = T - δt, where T is the time of the next renewal after time t. We can then express the conditional probability Pr{γt > y | δt = x} in terms of the interoccurrence distribution function F(x).

Pr{γt > y | δt = x} = Pr{T - δt > y | δt = x} = Pr{T > x + y} = 1 - F(x+y)

where the last step follows from the definition of the interoccurrence distribution function.

Therefore, the conditional probability that the excess life exceeds y given the age is 1 minus the probability that the next renewal occurs within y units of time after time t, which is the same as the probability that the interoccurrence time is less than or equal to y.

To find the conditional mean E[γt|δt = x], we can use the formula for conditional expectation:

E[γt | δt = x] = E[T - δt | δt = x] = E[T | δt = x] - x

where the last step follows from linearity of expectation. To evaluate E[T | δt = x], we can use the survival function S(x) = 1 - F(x), which gives the probability that the next renewal occurs after time x:

E[T | δt = x] = ∫ x to ∞ S(t) dt / S(x)

Differentiating the denominator with respect to x, we get

d/dx S(x) = -d/dx F(x) = -f(x)

where f(x) is the interoccurrence density function. Then,

d/dx (1/S(x)) = f(x) / [tex]S(x)^2[/tex]

and we can use this to evaluate the integral:

E[T | δt = x] = ∫ x to ∞ t f(t) / [tex]S(x)^2[/tex] dt = S(x) / [tex]S(x)^2[/tex] = 1 / S(x)

Therefore, the conditional mean excess life is

E[γt | δt = x] = E[T | δt = x] - x = 1 / S(x) - x

or, equivalently,

E[γt | δt = x] = ∫ x to ∞ y dF(y) / (1 - F(x)) - x

which expresses the conditional mean excess life in terms of the interoccurrence distribution function.

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+) slope = ∆y/∆x = (9-7)/[4-(-2)] = 2/6 = 1/3

Ans: 1/3

Ok done. Thank to me >:333

No, the language L is not a deterministic context-free language (DCFL).

To see if L is deterministic, suppose L is a DCFL. Then there exists a deterministic pushdown automaton (DPDA) that recognizes L.

Consider the string w = a^p b^(p+1) c^(p+1) ∈ L, where p is the pumping length of L. Since w is in L and L is a DCFL, the DPDA for L must accept w.

Assuming that the DPDA for L has only one accepting state. Let q be this accepting state.

By the pigeonhole principle, Let u, v, and x be the three parts of w such that u and v are the substrings of w that correspond to the first two occurrences of q', and x is the remaining suffix of w.

Then we can pump v any number of times and still get a string in L.

We can make the number of b's divisible by the number of c's by choosing an appropriate number of pumps.

However, since v contains at least one b, pumping v will result in a string that contains more b's than c's, which is not in L.

Therefore, we have a contradiction, and L cannot be a deterministic context-free language.

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The maximum speed of the object at x = 6a is 3.12 m/s.

To find the maximum speed, we need to consider the conservation of mechanical energy. At x = 6a, the object's potential energy (PE) is given by the graph. Let's assume the difference in potential energy between 4a and 6a is ΔPE.

1. Calculate ΔPE: 4.91 J is the energy difference between adjacent grid lines, and the object moves two grid lines (from 4a to 6a). So, ΔPE = 4.91 J * 2 = 9.82 J.

2. Determine the object's kinetic energy (KE) at x = 6a: Since the mechanical energy is conserved, the increase in PE will be equal to the decrease in KE. Thus, KE = ΔPE = 9.82 J.

3. Calculate the maximum speed: Using the formula KE = 0.5 * mass * speed^2, we can find the maximum speed: 9.82 J = 0.5 * 0.42 kg * speed^2. Solving for speed, we get 3.12 m/s.

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Answer:

[tex]60 + x = 100[/tex]

[tex]x = 40[/tex]

Diverges; the limit of the terms, a_n, is not 0 as n goes to infinity.

To determine whether the geometric series converges or diverges, we need to first identify the general term a_n and the common ratio r. The series is given as:

(5 + 16(-1)n), where n starts from 1.

The general term for this series is

a_n = 5 + 16(-1)^n

Now, we need to find the common ratio r. Since this series is alternating, the common ratio r can be found by dividing the term a_(n+1) by the term a_n:

r = a_(n+1) / a_n

However, the terms of this series do not have a fixed common ratio, as the (-1)n term causes the series to alternate. This means that the series is not a geometric series, and we cannot determine whether it converges or diverges based on a common ratio.

Instead, let's examine the limit of the terms, a_n, as n goes to infinity:

lim (n→∞) a_n = lim (n) [5 + 16(-1)^n]

As n goes to infinity, the term (-1)n will alternate between -1 and 1, and thus the limit does not exist. Therefore, the series diverges.

Answer: Diverges; the limit of the terms, a_n, is not 0 as n goes to infinity.

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Answer:

Step-by-step explanation:

Answer:

5%

Step-by-step explanation:

he earned 50 so his investment grew from 1000 to 1050

now simply calculate the percentage change

(1050-1000)/1000 *100

=5%

The particular solution of the given differential equation is: y = 2x - 3/2. The correct option is (d).

To solve this, we can use the method of undetermined coefficients, since the right-hand side is a polynomial.

1. First, guess a form for the particular solution: yp(x) = Ax + B, where A and B are constants to be determined.

2. Compute the first and second derivatives:
  yp'(x) = A
  yp''(x) = 0

3. Substitute these derivatives and the guess for yp(x) into the given differential equation:
  0 + 3A + 2(Ax + B) = 4x + 3

4. Equate coefficients of x and the constant terms on both sides:
  2A = 4 (coefficient of x)
  3A + 2B = 3 (constant term)

5. Solve this system of equations:
  A = 2
  B = -3/2

6. Plug A and B back into the guess for the particular solution:
  yp(x) = 2x - 3/2

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The 90% confidence interval for the difference in population proportions of fans who actively disliked the champion in 2000 and 2010 is 0.045, 0.095.

The formula for the confidence interval for the difference in two population proportions:

(p1 - p2) ± z X sqrt((p1 X (1-p1)/n1) + (p2 X (1-p2)/n2))

where:

p1 and p2 are the sample proportions of fans who actively disliked the champion in 2000 and 2010, respectively.

n1 and n2 are the sample sizes for 2000 and 2010, respectively.

z is the critical value from the standard normal distribution for the desired confidence level. For a 90% confidence level, the critical value is 1.645.

First, let's calculate the sample proportions:

p1 = 0.32

p2 = 0.25

n1 = 1250

n2 = 1300

Substituting these values into the formula, we get:

(0.32 - 0.25) ± 1.645 X sqrt((0.32 X (1-0.32)/1250) + (0.25 X (1-0.25)/1300))

= 0.07 ± 0.025

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the formula for the total number of views (v) the video will have after t days can be expressed as:

[tex]v = a * e^{kt}[/tex]

Assuming the number of views grows according to an exponential model, the formula for the total number of views (v) the video will have after t days can be expressed as:
[tex]v = a * e^{kt}[/tex]

where:
a is the initial number of views
k is the growth rate constant
t is the number of days

This formula is based on the assumption that the rate of growth of views is proportional to the number of views already accumulated. Therefore, as the number of views grows, the rate of growth also increases, resulting in an exponential increase in the total number of views.

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The optimal pair for the problem min 2tx(t)-3t³u(t)dt is u*(t) = -t³/b³, x*(t) = -t⁴/b³.

To find the optimal pair for the problem min 2tx(t)-3t³u(t)dt, we need to use the calculus of variations.

We start by considering the functional [tex]J(u) = \int_{a}^{b} (2tx(t)-3t^3u(t))dt[/tex], where u is the control function that we want to optimize.

We can find the optimal pair (x*, u*) by solving the Euler-Lagrange equation:
d/dt (∂L/∂u') - ∂L/∂u = 0,
where L(t, x(t), u(t), u'(t)) = 2tx(t)-3t³u(t) and u' = du/dt.

After some calculations, we obtain:
-3t² = u''/u',

which is a separable first-order differential equation that we can solve using integration.

We get:
u(t) = c1*t³ + c2,

where c1 and c2 are constants of integration that we can determine using the boundary conditions.

Since we want to minimize J(u), we need to choose the constants that minimize J(u). Using the boundary condition u(a) = u(b) = 0, we get:
c1 = -c2/b³, c2 = 0,

so that:
u(t) = -t³/b³.

Finally, we can compute the corresponding optimal x* using the formula:
[tex]x^*(t) = \int_{a}^{t} (\partial L/ \partial u)du + K[/tex],
where K is a constant of integration that we can determine using the boundary condition x(a) = x(b) = 0.

We obtain:
x*(t) = -t⁴/b³.

Therefore, the optimal pair is given by:
u*(t) = -t³/b³, x*(t) = -t⁴/b³.

Note that we also need to check that this is indeed a minimum by verifying that the second variation of J(u) is positive.

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Answer:

2 3/4

Step-by-step explanation:

The longest is 4 1/2 and the shortest is 1 3/4 so we do 4 1/2 - 1 3/4 and you get 2 3/4.

An eigenvalue for matrix a with eigenvector v, then u(t) eλtv is a solution to the differential du equation = a = au.  This is correct.

If λ is an eigenvalue for matrix A with eigenvector v, then Av = λv. Taking the derivative of u(t)v with respect to t, we get:

du/dt [tex]\times[/tex] v = u(t) [tex]\times[/tex] d/dt(v) = u(t) [tex]\times[/tex] Av = u(t) [tex]\times[/tex]λv

On the other hand, we have:

Au = λu

Multiplying both sides by v, we get:

Avu = λuv

Since v is nonzero (by definition of eigenvector), we can divide both sides by v to get:

Au = λu

So, du/dt [tex]\times[/tex]v = u(t) [tex]\times[/tex]λv = Au(t)v = Au(t)[tex]\times[/tex] (u(t)^(-1)v)

Since u(t)^(-1)v is just a scalar, say c, we have:

du/dt [tex]\times[/tex] v = λc[tex]\times[/tex] u(t)v

Therefore, u(t)v is a solution to the differential equation du/dt = Au, with eigenvalue λ.

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the value of k that makes f(x, y) = (1/7)x(3 - 2y) a joint probability density function on the given domain D is k = 1/7.

To find the value of k so that the function f(x, y) = kx(3 - 2y) is a joint probability density function on the domain D = {1 ≤ x ≤ 4; 0 ≤ y ≤ 2}, we need to ensure that the total probability over the domain is equal to 1. We can do this by integrating the function over the given domain and setting the result equal to 1:

1 = ∫∫_D f(x, y) dxdy

First, we will integrate the function with respect to x:

1 = ∫[∫_1^4 kx(3 - 2y) dx] dy

1 = ∫[k(3 - 2y)(x^2/2)|_1^4 dy

1 = ∫[k(3 - 2y)(8 - 1/2)] dy

Now, integrate with respect to y:

1 = k(7/2)∫_0^2 (3 - 2y) dy

1 = k(7/2)[(3y - y^2)|_0^2]

1 = k(7/2)(6 - 4)

1 = 7k

To make the total probability equal to 1, we need to find the value of k:
k = 1/7

So, the value of k that makes f(x, y) = (1/7)x(3 - 2y) a joint probability density function on the given domain D is k = 1/7.

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