Exercise 0.5. Calculate the Fourier series of the function f:(-1,1] →R, f(x) = 1-x^2 Use this series to prove that phi^2/6 = [infinity]Σn=1 1/n^2 (3 + 2 Marks)

Answer:

To find the inverse of the function y = π/2 + sin(x), we need to first swap the positions of x and y:

x = π/2 + sin(y)

Now, we can solve for y:

sin(y) = x - π/2

y = sin⁻¹(x - π/2)

Therefore, the equation for the inverse of the function y = π/2 + sin(x) is y = sin⁻¹(x - π/2).

We are given the function, [tex]\underline{f(x)=x^3-9x^2+20x-12}[/tex], and are asked to find the x and y intercepts of the function.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

What is an intercept?

An intercept is where the graph of a function cross either the x or y axis. The x-intercept(s) crosses the x-axis and the y-intercept(s) crosses the y-axis.

How do find the x-intercept(s)?

To find the x-intercepts let y in your function equal zero, then solve for x.

How do find the y-intercept(s)?

To find the y-intercepts let x in your function equal zero, then solve for y.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Refer to the attached image for the rest.

Answer:

To find the slope of the tangent line to the path of the particle at the point where t = 2, we first need to find the values of x(2) and y(2), as well as their derivatives x'(2) and y'(2).

Using the given parametric functions, we can find:

x(2) = ∫ x'(t) dt = ∫ t sin(t) dt = -t cos(t) + sin(t) + C

where C is the constant of integration.

Since we want x(2), we can evaluate the above expression at t = 2:

x(2) = -2 cos(2) + sin(2) + C

Similarly, we can find:

y(2) = ∫ y'(t) dt = ∫ (5e^(-3t) + 2) dt = (-5/3)e^(-3t) + 2t + C'

where C' is the constant of integration.

Again, since we want y(2), we can evaluate the above expression at t = 2:

y(2) = (-5/3)e^(-6) + 4 + C'

Now we can find the derivatives x'(2) and y'(2) by taking the derivative of x(t) and y(t), respectively, and evaluating them at t = 2:

x'(2) = 2 sin(2) - cos(2)

y'(2) = (5/3)e^(-6)

Therefore, at t = 2, the particle is at the point (x(2), y(2)) = (-2 cos(2) + sin(2) + C, (-5/3)e^(-6) + 4 + C'), and the slope of the tangent line to the path of the particle at this point is given by:

dy/dx = (dy/dt)/(dx/dt) = y'(2)/x'(2)

Substituting the values we found:

dy/dx = [(5/3)e^(-6) + 4 + C']/(2 sin(2) - cos(2))

Since we don't have enough information to find the value of C', we cannot find an exact value for the slope. However, we can simplify the expression by using the trigonometric identities:

sin(2) = 2 sin(1) cos(1)

cos(2) = cos^2(1) - sin^2(1)

where we let t = 1 for simplicity. Then, we can substitute these expressions and simplify:

dy/dx = [(5/3)e^(-6) + 4 + C']/(4 sin(1) cos(1) - cos^2(1) + sin^2(1))

dy/dx = [(5/3)e^(-6) + 4 + C')/(4 sin(1) cos(1) - 1)

Therefore, the slope of the tangent line to the path of the particle at the point where t = 2 is given by the above expression.

Step-by-step explanation:

The slope of the tangent line to the path of the particle at the point where t=2 is approximately 1.55. To find the slope of the tangent line to the path of the particle at the point where t=2,

we need to use the derivatives of x(t) and y(t).

First, we can find the slope of the tangent line by using the formula:

slope = dy/dx = (dy/dt)/(dx/dt)

So, we need to find both dy/dt and dx/dt.

Given that x′(t)=t sin t, we can find dx/dt by taking the derivative of x(t):

dx/dt = x′(t) = t sin t

Given that y′(t)=5e−3t+2, we can find dy/dt by taking the derivative of y(t):

dy/dt = y′(t) = 5e−3t+2

Now, we can find the slope of the tangent line at t=2 by plugging in these values:

slope = (dy/dt)/(dx/dt) = (5e−3t+2)/(t sin t) = (5e−6+2)/(2 sin 2)

Therefore, the slope of the tangent line to the path of the particle at the point where t=2 is approximately 1.55.

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The distance between the yacht and the kittiwake. is 160 m

The horizontal distance between Sharon and the yacht

tan 19 = 90 / distance between Sharon and the yacht

distance between Sharon and the yacht = 90 / tan 19

distance between Sharon and the yacht = 261.38 m

The horizontal distance between the Sharon and the kittiwake

tan 15 = distance between the Sharon and the kittiwake / 261.38

distance between the Sharon and the kittiwake = 261.38 x tan 15

distance between the Sharon and the kittiwake = 70.04 m

distance between the yacht and the kittiwake

= 90 + 70.04

= 160.04

= 160 m

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Any value of b within this range would satisfy the original expression "23 > b ≥ -276".

The given expression is "23 > b ≥ -276". This means that b can take any value between -276 and 23, inclusive of -276 but not inclusive of 23.

To understand why this is the case, we can break down the expression into two parts: "23 > b" and "b ≥ -276".

The first part, "23 > b", means that b is less than 23. In other words, b can take any value that is less than 23. For example, b could be 22, 10, or even -100, as long as it is less than 23.

The second part, "b ≥ -276", means that b is greater than or equal to -276. This means that b can take any value that is greater than or equal to -276. For example, b could be -276, -200, or even 0, as long as it is greater than or equal to -276.

Putting these two parts together, we can see that b can take any value that is both less than 23 and greater than or equal to -276. This range of values for b is inclusive of -276 but not inclusive of 23, meaning that b cannot equal 23.

In numerical form, we can write the range of values for b as:

-276 ≤ b < 23

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Answer:

C

Step-by-step explanation:

y=-2x+4

2x+y=4

2x-2x+4=4

4=4

The rule for the table is y = -8x + 88.

The price of the shoes after 8 month is 24 dollars.

The table shows the discount prices for a pair of shoes over several months.

Therefore, the rule for the tables can be represented as follows:

y = mx + b

where

Therefore, using (1,80)(2, 72)

m = 72 - 80 / 2 - 1

m = -8

Hence,

y = -8x + b

using (1, 80)

80 = -8 + b

b = 88

Therefore,

y = -8x + 88

Therefore, let's find the price after 8 months

y = -8(8) + 88

y = -64 + 88

y = 24

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When a=0.02 and n=24, [tex]X_{left}^{2}[/tex] = 9.260 and [tex]X_{right}^{2}[/tex]= 41.638. In order to calculate [tex]X_{left}^{2}[/tex] and [tex]X_{right}^{2}[/tex] when a=0.02 and n=24, we need to use the chi-squared distribution table. This table provides us with the critical values for a given level of significance (alpha) and degrees of freedom (df).

To answer your question, when a=0.02 and n=24, we will find the [tex]X_{left}^{2}[/tex] and [tex]X_{right}^{2}[/tex] values using the Chi-square distribution table.

Step 1: Determine the degrees of freedom. In this case, the degrees of freedom (df) are equal to n-1, so df = 24 - 1 = 23.

Step 2: Determine the significance level (alpha) and divide it by 2. Since a = 0.02, the significance level is [tex]\frac{\alpha}{2} =0.01[/tex] for each tail (left and right) of the distribution.

Step 3: Use the Chi-square distribution table to find the critical values. Look for the values corresponding to the degrees of freedom (23) and significance level (0.01) in each tail.

According to the Chi-square distribution table:
[tex]X_{left}^{2}[/tex]= 9.260
[tex]X_{right}^{2}[/tex]= 41.638

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The probability that a randomly selected point within the circle falls in the red-shaded circle is 0.785

From the question, we have the following parameters that can be used in our computation:

The areas of the above shapes are

Red circle = 3.14 * 11^2 = 379.94

White square = 22^2 = 484

The probability is then calculated as

P = Red circle/White square

So, we have

P = 379.94/484

Evaluate

P = 0.785

Hence, the probability is 0.785

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The linear function for the given statement is y = (-1/2)x + 6.

To write a linear function for the statement "A candle is 6 inches tall and burns at a rate of 1/2 inch per hour," we will need to use the slope-intercept form of a linear function, which is y = mx + b. In this case, y represents the remaining height of the candle, m represents the rate of burning, x represents time in hours, and b represents the initial height of the candle.

Step 1: Identify the initial height (b). The candle is 6 inches tall, so b = 6.

Step 2: Identify the rate of burning (m). The candle burns at a rate of 1/2 inch per hour, so m = -1/2 (negative because the height decreases as time passes).

Step 3: Write the linear function using the slope-intercept form y = mx + b. Substitute the values of m and b:
y = (-1/2)x + 6

Thus, we can state that the linear function for the given statement is:

y = (-1/2)x + 6.

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The lengths of the diagonals in the isosceles trapezoid are 11.045 units and 7.2 units.

From the given figure, the vertices of the quadrilateral are (1, 6), (3, 0), (-5, 0) and (-1, 6).

From lower left to upper right: (-5, 0) and (1, 6)

Here, length = √(6+5)²+(1-0)²

= √122

= 11.045 units

From lower right to upper left: (3, 0) and (-1, 6)

Here, length = √(-1-3)²+(6-0)²

= √52

= 7.2 units

Therefore, the lengths of the diagonals in the isosceles trapezoid are 11.045 units and 7.2 units.

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Answer:

Vertical angles are congruent, so the two triangles are congruent by AAS (Angle-Angle-Side).

None of the theorems listed can be used to show congruence.

The 95% confidence interval for the mean amount (in milligrams) of nicotine in the sampled brand of cigarettes is C.39.2 to 40.8

We can use a t-table or a calculator to calculate the t-score. The t-score for a 95% confidence interval with 22 degrees of freedom (n-1) is around 2.074.

When we plug in the values, we get:

CI = 40 ± 2.074 * 1.8/√23 = 40 ± 0.763 = (39.237, 40.763)

As a result, the 95% confidence interval for the mean nicotine content of the studied cigarette brand is (39.237, 40.763) mg.

Because 39.2 to 40.8 is the closest response choice, the answer is 39.2 to 40.8.

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We have successfully drawn the given planes when working on a body centered cubic structure.

When working with a body centered cubic structure, it's important to understand that the unit cell consists of a cube with one additional atom at the center of the cube. This gives rise to unique properties and symmetry within the crystal structure.

To draw the planes (100), (010), and (101) within this structure, we can use the Miller indices notation. In this notation, each plane is represented by three integers that correspond to the intercepts of the plane with the three axes of the unit cell.

For example, the (100) plane intersects the x-axis at a point where x=1, and intersects the y- and z-axes at points where y=0 and z=0, respectively. Using the Miller indices notation, we can write this plane as (100).

Similarly, the (010) plane intersects the y-axis at a point where y=1, and intersects the x- and z-axes at points where x=0 and z=0. Therefore, this plane can be written as (010).

Finally, the (101) plane intersects the x-axis at a point where x=1, the y-axis at a point where y=0, and the z-axis at a point where z=1. Using Miller indices notation, we can represent this plane as (101).

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The conditional expectation of X given is E[X|X ≥ E[X]] is µ. The conditional variance is Var[X|X ≥ E[X]] is σ² of the Gaussian random variable X.

Given a Gaussian random variable X with mean µ and variance σ², we need to find the conditional mean and variance given X ≥ E[X].

First, we note that E[X] = µ and Var[X] = σ².

Next, we find the conditional probability P(X ≥ E[X])

P(X ≥ E[X]) = P(X - µ ≥ 0) = P(Z ≥ 0) = 0.5, where Z is the standard normal distribution.

Using Bayes' theorem, we can write the conditional mean and variance as

E[X|X ≥ E[X]] = µ + σφ(Z)/P(X ≥ E[X])

Var[X|X ≥ E[X]] = σ²[1 - φ(Z)²]/P(X ≥ E[X]),

where φ(Z) is the standard normal probability density function.

Substituting the values, we get

E[X|X ≥ E[X]] = µ + σφ(0)/0.5 = µ

Var[X|X ≥ E[X]] = σ²[1 - φ(0)²]/0.5 = σ²

Therefore, the conditional mean and variance given X ≥ E[X] are both equal to the original mean µ and variance σ² of the Gaussian random variable X.

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Answer:

A. 38, 49, 60

Step-by-step explanation:

1. Find the difference between each number in the sequence

To go from -6 to 5, you add 11

To go from 5 to 16, you add 11

To go from 16 to 27, you add 11.

Therefore, the difference in all of the numbers is 11, so the pattern should continue and you should add 11 to the last number (27) making the next numbers 38, 49, 60

74% of the trials are completed within 20 days (rounded to the nearest whole number).

b. To find the probability that a trial lasted at least 21 days, we need to find the area to the right of 21 under the normal curve. Using a standard normal table or calculator, we can find:

z = (21 - 22) / 6 = -0.1667

P(X ≥ 21) = P(Z ≥ -0.1667) = 0.5675

So the probability that a trial lasted at least 21 days is 0.5675.

c. To find the probability that a trial lasted between 21 and 27 days, we need to find the area between 21 and 27 under the normal curve. Again using a standard normal table or calculator, we can find:

z1 = (21 - 22) / 6 = -0.1667

z2 = (27 - 22) / 6 = 0.8333

P(21 ≤ X ≤ 27) = P(-0.1667 ≤ Z ≤ 0.8333) = 0.3454

So the probability that a trial lasted between 21 and 27 days is 0.3454.

d. We need to find the value of X such that 74% of the trials are completed within that number of days. Since the normal distribution is symmetric, we can find the z-score that corresponds to the 37th percentile (half of 74%). Using a standard normal table or calculator, we can find:

P(Z ≤ z) = 0.37

z = -0.3528

Now we can use the z-score formula to find X:

z = (X - μ) / σ

-0.3528 = (X - 22) / 6

X - 22 = -2.1168

X = 19.8832

So 74% of the trials are completed within 20 days (rounded to the nearest whole number).

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Answer:

11668

2+2 = 4

+2*6*6*6*3*3*3

= 11668

Step-by-step explanation:

Answer:

Please mark me the brainliest.

Step-by-step explanation:

11668, trust me i used the calculator

The percent of area under a normal curve between the mean and −1.18 standard deviations is 38.10% and between z=−1.6 and z = 0.7 is 70.32% and the​ z-score that best satisfies the condition is z=−0.4.

To find the percent of area under a normal curve between the mean and −1.18 standard deviations from the mean, we need to use a standard normal distribution table or calculator.

The area to the left of −1.18 standard deviations is 0.1190, and the area to the left of the mean is 0.5000. To find the area between them, we subtract the smaller area from the larger area:

0.5000 - 0.1190 = 0.3810

Therefore, the percent of area under a normal curve between the mean and −1.18 standard deviations is 38.10%.

To find the percent of the total area under the standard normal curve between z=−1.6 and z = 0.7, we again need to use a standard normal distribution table or calculator.

The area to the left of −1.6 is 0.0548, and the area to the left of 0.7 is 0.7580. To find the area between them, we subtract the smaller area from the larger area:

0.7580 - 0.0548 = 0.7032

Therefore, the percent of the total area between z=−1.6 and z = 0.7 is 70.32%.

To find the​ z-score that best satisfies the condition that 36% of the total area is to the left of z, we need to use a standard normal distribution table or calculator.

We look for the z-score that corresponds to a cumulative probability of 0.36. This is approximately −0.4, which means that 36% of the total area under the standard normal curve is to the left of z=−0.4. Therefore, the​ z-score that best satisfies the condition is z=−0.4.

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The basic eigenvector corresponding to λ₁ = 4 is v₁ = [3.75, 4], and the basic eigenvector corresponding to λ₂ = -3 is v₂ = [5, 6].

To find the eigenvalues of matrix A, we need to solve the characteristic equation:

|A - λI| = 0

where I is the identity matrix of the same size as A, and λ is the eigenvalue we are trying to find.

For matrix A given as:

a=[8 -15]

[6 -10]

we have:

|A - λI| =

|8 - λ -15 |

|6 -10- λ |

Expanding the determinant, we get:

(8 - λ)(-10 - λ) - (-15)(6) = 0

Simplifying the expression, we get:

λ² - 2λ - 12 = 0

Using the quadratic formula, we get:

λ₁ = 4

λ₂ = -3

Therefore, the distinct eigenvalues of A are λ₁ = 4 and λ₂ = -3.

Next, we find the eigenvectors corresponding to each eigenvalue. We do this by solving the system of equations:

(A - λI)x = 0

For λ₁ = 4:

A - λ₁I =

|8 - 4 -15 |

|6 -10 - 4 |

=

|4 -15 |

|6 -14|

RREF:

|1 -3.75|

|0 0 |

Thus, we have a free variable x₂. Setting x₂ = 4, we get the basic eigenvector:

v₁ = [3.75, 4]

Therefore, there is one basic eigenvector corresponding to eigenvalue λ₁ = 4.

For λ₂ = -3:

A - λ₂I =

|8 15 |

|6 7 |

RREF:

|1 -5/6|

|0 0 |

Thus, we have a free variable x₂. Setting x₂ = 6, we get the basic eigenvector:

v₂ = [5, 6]

Therefore, there is one basic eigenvector corresponding to eigenvalue λ₂ = -3.

In summary, the distinct eigenvalues of matrix A are λ₁ = 4 and λ₂ = -3. There is one basic eigenvector corresponding to each eigenvalue. The basic eigenvector corresponding to λ₁ = 4 is v₁ = [3.75, 4], and the basic eigenvector corresponding to λ₂ = -3 is v₂ = [5, 6].

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Answer:

The sectarian violence is creating the situations leading to the fracturing of states. For example, in Syria, sectarian violence between Sunni and Shia Muslims has fueled a civil war that has led to the fracturing of the country. Similarly, in Iraq, sectarian violence between Sunni and Shia Muslims has contributed to the fracturing of the country, with ISIS taking advantage of the situation to establish a caliphate. The article also notes that sectarian violence has contributed to the instability of Lebanon and Bahrain.

Step-by-step explanation:

Answer:

The correct answer is d. As the value of x increases, the value of the error term also increases.

Step-by-step explanation:

The simple regression model has several major assumptions, including:

Linearity: The relationship between the dependent variable and the independent variable(s) is linear.

Independence: The error terms are independent of each other.

Homoscedasticity: The variance of the error terms is constant for all levels of the independent variable(s).

Normality: The error terms are normally distributed.

No perfect multicollinearity: There is no perfect linear relationship between the independent variables.

Option d violates the assumption of independence, which states that the error terms are independent of each other and are not affected by the value of the independent variable(s). If the value of the error term increases as the value of x increases, then the error terms are not independent of the independent variable(s). This can lead to biased and unreliable estimates of the regression coefficients, and the resulting model may not accurately predict the dependent variable.

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We apply the Gram-Schmidt process to these vectors to find an orthonormal basis:

v1 = x1 = [3, -4, 1

To find a basis of the null space N(A), we need to find all vectors x such that Ax = 0, where 0 is the zero vector.

To do this, we set up the augmented matrix [A | 0] and row reduce:

[ 1 2 1 3 2 | 0 ]

[ 4 1 0 6 1 | 0 ]

[ 1 1 2 4 5 | 0 ]

R2 - 4R1 -> R2:

[ 1 2 1 3 2 | 0 ]

[ 0 -7 -4 6 -7 | 0 ]

[ 1 1 2 4 5 | 0 ]

R3 - R1 -> R3:

[ 1 2 1 3 2 | 0 ]

[ 0 -7 -4 6 -7 | 0 ]

[ 0 -1 1 1 3 | 0 ]

R2 / -7 -> R2:

[ 1 2 1 3 2 | 0 ]

[ 0 1 4/7 -6/7 1 | 0 ]

[ 0 -1 1 1 3 | 0 ]

R1 - 2R2 - R3 -> R1:

[ 0 0 0 0 0 | 0 ]

[ 0 1 4/7 -6/7 1 | 0 ]

[ 0 0 11/7 -1/7 1 | 0 ]

We can write the system of equations corresponding to this row echelon form as:

x2 + (4/7)x3 - (6/7)x4 + x5 = 0

(11/7)x3 - (1/7)x4 + x5 = 0

Solving for the variables in terms of the free variables x3, x4, and x5, we get:

x1 = -[(4/7)x3 - (6/7)x4 - x5]/2

x2 = -(4/7)x3 + (6/7)x4 - x5

x3 = x3 (free variable)

x4 = x4 (free variable)

x5 = x5 (free variable)

So the null space N(A) is the set of all vectors of the form:

x = [ -[(4/7)x3 - (6/7)x4 - x5]/2, -(4/7)x3 + (6/7)x4 - x5, x3, x4, x5 ]

To find an orthogonal basis for N(A), we can use the Gram-Schmidt process. Let's call the columns of A a1, a2, a3, a4, and a5.

First, we need to find a basis for N(A) by setting the free variables to 1 and the others to 0:

x1 = [3, -4, 1, 0, 0]

x2 = [-2, 3, 0, 1, 0]

x3 = [-2, 1, 0, 0, 1]

Next, we apply the Gram-Schmidt process to these vectors to find an orthonormal basis:

v1 = x1 = [3, -4, 1

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Using the quadratic formula to find the solutions, we get x = (-10 ± √40)/6

From the question, we have the following parameters that can be used in our computation:

3x² + 10x + 5=0

Using the quadratic formula, we have

x = (-10 ± √(10² - 4 * 3 * 5))/(2 * 3)

Evaluate the products and the exponents

So, we have

x = (-10 ± √40)/6

Hence, the solution is x = (-10 ± √40)/6

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At the end of three years, Daniel would have saved $6749.

When interest is calculated on a principal amount of money using the compound interest method, the interest gained is periodically added back to the principal and interest is then calculated on the new principal amount.

We know that;

A = P(1 + r/n)^nt

A =amount

r = rate

P = principal

n = Number of times compounded

t = time

Thus;

A = 6000(1 + 0.04)^3

A = $6749

Thus the compound amount saved is  $6749

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The probability of the arrow landing on an odd number is the number of odd numbers divided by the total number of possible outcomes. Therefore, the probability of the arrow landing on an odd number is  0.5 or 50%.


To find the probability that the arrow will point at an odd number on a circular board with 8 equal parts, we'll first determine the total number of odd numbers present and then divide that by the total number of possible outcomes.

Step 1: Identify the odd numbers on the board. They are 1, 3, 5, and 7. The game consists of spinning the arrow on a circular board with 8 equal parts, which means there are 8 possible outcomes or numbers. Since we want to know the probability of landing on an odd number, we need to count how many odd numbers are on the board. In this case, there are four odd numbers: 1, 3, 5, and 7.

Step 2: Count the total number of odd numbers. There are 4 odd numbers.

Step 3: Count the total number of possible outcomes. Since the board is divided into 8 equal parts, there are 8 possible outcomes.

Step 4: Calculate the probability. The probability of the arrow pointing at an odd number is the number of odd numbers divided by the total number of possible outcomes.

Probability = (Number of odd numbers) / (Total number of possible outcomes)
Probability of landing on an odd number = Number of odd numbers / Total number of possible outcomes
Probability of landing on an odd number = 4 / 8

Step 5: Simplify the fraction. The probability of the arrow pointing at an odd number is 1/2 or 50%.

So, the probability that the arrow will point at an odd number is 1/2 or 50%.

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At the .005 significance level with a one-tailed test, the critical z-value is 2.33. Since our calculated z-value (2.58) is greater than the critical value, we reject the null hypothesis and conclude that there is evidence to support the claim that the proportion of men who own cats is larger than 90% at the .005 significance level.

To test the claim that the proportion of men who own cats is larger than 90% at the .005 significance level, we can conduct a one-tailed hypothesis test. Our null hypothesis (H0) is that the proportion of men who own cats is less than or equal to 90%, while our alternative hypothesis (Ha) is that the proportion is greater than 90%.

We can use a z-test for proportions to calculate the test statistic and p-value. Let's assume we sample 200 men and find that 186 own cats. This gives us a sample proportion of 0.93.

Using the formula for the z-test for proportions, we get:

z = (0.93 - 0.9) / sqrt(0.9 * 0.1 / 200) = 2.58

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The expected number of heads = 1.5

The correct answer is an option (B)

We know that the formula for the expected value is:

E (x) = ∑ x P ( x )

where P(x) represents the probability of outcome X

and E(x) is the expected value of x

We need to find the expected number of heads.

From the probability distribution table of a 3-coin toss, the expected number of heads would be,

E(H) = 0(1/8) + 1(3/8) + 2(3/8) + 3(1/8)

E(H) = 0 + 3/8 + 6/8 + 3/8

E(H) = 12/8

E(H) = 1.5

The correct answer is an option (B)

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Answer:

Step-by-step explanation:

X(t) = Xo/[1 + (1 - Xo)/Xo exp(-[r - o^2/2]t - oW(t))]

E[X(t)] = Xo/(1 + (1 - Xo)/Xo exp(-r t)),

V[X(t)] = Xo^2 exp(rt)/(1 + (1 - Xo)/Xo exp(rt))^2 - Xo^2/(1 + (1 - Xo)/Xo exp(-r t))^2.

The given equation is a stochastic differential equation (SDE) of the form dX(t) = a(X(t))dt + b(X(t))dW(t), where W(t) is a Wiener process (Brownian motion), a(X(t)) = rX(t)(1 - X(t)), b(X(t)) = oX(t), and Xo is the initial condition.

To solve this SDE, we use Itô's lemma, which states that for a function f(X(t)) of a stochastic process X(t), the SDE for f(X(t)) is given by df(X(t)) = (∂f/∂t)dt + (∂f/∂X)dX(t) + 1/2(∂^2f/∂X^2)(dX(t))^2.

Applying Itô's lemma to the function f(X(t)) = ln(X(t)/(1 - X(t))), we get df(X(t)) = [1/X(t) + 1/(1 - X(t))]dX(t) - 1/2[X(t)^(-2) + (1 - X(t))^(-2)](dX(t))^2.

Substituting a(X(t)) and b(X(t)) in the above expression, we get d[f(X(t))] = [r(1 - 2X(t))dt + o(1 - 2X(t))dW(t)] - 1/2[r^2X(t)(1 - X(t))^2 + o^2X(t)^2]dt.

Integrating both sides of the above expression from time 0 to t and using the initial condition X(0) = Xo, we get ln[X(t)/(1 - X(t))] = ln[Xo/(1 - Xo)] + [r - o^2/2]t + oW(t).

Solving for X(t), we get X(t) = Xo/[1 + (1 - Xo)/Xo exp(-[r - o^2/2]t - oW(t))].

Taking the expectation and variance of X(t), we get:

E[X(t)] = Xo/(1 + (1 - Xo)/Xo exp(-r t)),

V[X(t)] = Xo^2 exp(rt)/(1 + (1 - Xo)/Xo exp(rt))^2 - Xo^2/(1 + (1 - Xo)/Xo exp(-r t))^2.

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The slices that will always result in a square plane section include;

The intersection of a three-dimensional object with a plane creates a two-dimensional shape known as a "plane section." When this section is square in shape, it is referred to as a "square plane section."

A cube possesses six faces that are all squares. Slicing the cube parallel to one of its sides produces a square plane section every time, given that the side being sliced also happens to be a square. Meanwhile, cutting the cube perpendicular to any of its other faces generates a comparable result: a square plane section.

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