When constucting the 2^7 design confounded in eight blocks, three independent effects are chosen to generate the blocks, and there are a total if eight interactions confounded with blocks.
a. True
b. False
Explain

The velocity-versus-time graph for a particle moving along the x-axis starts at zero velocity at t = 0 s. It then undergoes uniform acceleration, resulting in a linear increase in velocity over time.

The graph represents a straight line with a positive slope, indicating constant acceleration. The particle's displacement can be determined by calculating the area under the velocity-versus-time graph, which corresponds to the change in position of the particle.

At t = 0 s, the particle is located at x = 0 m and has zero velocity. This is represented by the starting point on the velocity-versus-time graph. As time progresses, the particle undergoes uniform acceleration, which causes its velocity to increase at a constant rate. This results in a straight line on the velocity-versus-time graph.

The slope of the line represents the acceleration of the particle. A positive slope indicates that the particle is accelerating in the positive direction along the x-axis. The steeper the slope, the greater the acceleration.

To determine the displacement of the particle, we can calculate the area under the velocity-versus-time graph. Since the graph is a straight line, the area corresponds to a triangle. The base of the triangle is the time interval, and the height is the average velocity during that interval. The displacement can be calculated using the formula: displacement = average velocity * time.

In summary, the velocity-versus-time graph for a particle moving along the x-axis starts at zero velocity and exhibits a straight line with a positive slope, indicating uniform acceleration. The displacement of the particle can be determined by calculating the area under the graph, which corresponds to the change in position of the particle over time.

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can describe a velocity-versus-time graph for a particle moving along the x-axis with the assumption that at t = 0 s, x = 0 m. However, since I can't directly show you a graph, I'll describe it in words.

Let's assume the velocity-versus-time graph for the particle moving along the x-axis is as follows:

1. From t = 0 s to t = T s:

  - The graph starts at the origin (0,0) indicating that at t = 0 s, the particle's velocity is zero.

  - The graph initially has a positive slope, indicating that the particle is moving in the positive x-direction.

  - As time progresses, the slope remains constant, indicating the particle's velocity is constant.

2. At t = T s:

  - The graph levels off and becomes a horizontal line, indicating that the particle's velocity remains constant.

3. After t = T s:

  - The graph continues as a horizontal line, maintaining the same constant velocity.

Note that without specific values for the slope or the time duration T, I cannot provide exact numerical information. However, I hope this description gives you a general understanding of what the velocity-versus-time graph would look like for a particle moving along the x-axis.

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To solve the given differential equation using Laplace transformation, we'll follow these steps:

Step 1: Apply the Laplace transformation to both sides of the equation.

Taking the Laplace transform of the equation, we have:

L{du(t)/dt} + 2L{dy(t)/dt} + 17L{y(t)} = -10L{dt/dt}

Using the properties of the Laplace transform, we get:

sU(s) - u(0) + 2sY(s) - y(0) + 17Y(s) = -10/s

Step 2: Apply the initial conditions.

Since we have the initial condition dy(0)/dt = 0, and y(0) = 0, we can substitute these values into the equation:

sU(s) - u(0) + 2sY(s) - 0 + 17Y(s) = -10/s

sU(s) + 2sY(s) + 17Y(s) = -10/s

Step 3: Solve for Y(s).

Rearranging the equation to isolate Y(s), we have:

Y(s)(2s + 17) = -10/s - sU(s)

Y(s) = (-10 - sU(s))/s(2s + 17)

Step 4: Take the inverse Laplace transform.

To find the solution y(t), we need to take the inverse Laplace transform of Y(s). However, the given u(t) = e is not in Laplace transform form. Please provide the correct Laplace transform expression for u(t) so that we can proceed with finding the inverse Laplace transform and the solution to the differential equation.

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The angle between the two planes is approximately 63.43°.  Thus, the correct option is 3. neither.

The two planes' equations are

9x+36y−272=1

and

−18x+36y+422=0 respectively.

Now, we have to determine whether they are parallel, perpendicular, or neither.

A plane is represented by an equation of the form

ax+by+cz=d,

where a, b, and c are not all equal to zero.

Two planes are parallel if their normal vectors are parallel to each other.

That is, two planes are parallel if

a1/a2 = b1/b2 = c1/c2,

where (a1,b1,c1) and (a2,b2,c2) are the normal vectors of the two planes.

Let's start by obtaining the normal vectors of the two planes.

9x+36y−272=19x+36y

=273x+12y

=912x+4y

=36

The normal vector of the first plane is (3,1).

−18x+36y+422=0

−18x+36y=-422-9x+18y=-2

11x-2y=-21

The normal vector of the second plane is (1/2,1).

Since neither the direction ratios nor the normal vectors of the two planes are parallel, the two planes are not parallel to each other.

The two planes are perpendicular to each other if the dot product of their normal vectors is zero.

Let's check. (3,1).(1/2,1)

= 3/2+1

= 5/2 ≠ 0

Since the dot product of the normal vectors is not zero, the two planes are not perpendicular to each other.

Therefore, the two planes are neither parallel nor perpendicular.

We must calculate the angle between the two planes.

The angle θ between the two planes is given by the formula

θ = cos⁻¹(|n1.n2|/|n1||n2|),

where n1 and n2 are the normal vectors of the two planes.

θ = cos⁻¹(|(3,1).(1/2,1)|/|(3,1)||(1/2,1)|)

θ = cos⁻¹(5/2/(√10/2√2/2))

θ = cos⁻¹(5/√20)

θ ≈ 63.43°

The correct option is 3. neither.

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Matrices are rectangular arrays of numbers or symbols organized in rows and columns. They are used in mathematics and various fields to represent and manipulate data, perform operations, and solve systems of equations.

The given matrices are;

[tex]A =  \begin{bmatrix} 3 & 1 \\ 3 & -4 \end{bmatrix}[/tex]   and

[tex]B = \begin{bmatrix} -5 & -6 \\ 2 & -4 \end{bmatrix}[/tex]

Now, we are to compute [tex]\(3(2A - 3B)\)[/tex].

Let's begin by calculating 2A and 3B.

[tex]2A = $2\begin{bmatrix} 3 & 1 \\ 3 & -4 \end{bmatrix}$= $ \begin{bmatrix} 6 & 2 \\ 6 & -8 \end{bmatrix}$[/tex]

[tex]3B = $3\begin{bmatrix} -5 & -6 \\ 2 & -4 \end{bmatrix}$= $ \begin{bmatrix} -15 & -18 \\ 6 & -12 \end{bmatrix}$[/tex]

[tex]\(2A - 3B = \begin{bmatrix} 6 & 2 \\ 6 & -8 \end{bmatrix} - \begin{bmatrix} -15 & -18 \\ 6 & -12 \end{bmatrix}\)[/tex]

Therefore,

[tex]= $ \begin{bmatrix} 6+15 & 2+18 \\ 6-6 & -8+12 \end{bmatrix}$[/tex]

[tex]= $ \begin{bmatrix} 21 & 20 \\ 0 & 4 \end{bmatrix}$[/tex]

[tex]\(3(2A - 3B) = 3 \begin{bmatrix} 21 & 20 \\ 0 & 4 \end{bmatrix}\)[/tex]

Finally,

[tex]=$ \begin{bmatrix} 63 & 60 \\ 0 & 12 \end{bmatrix}$[/tex].

[tex]\(3(2A - 3B) = \begin{bmatrix} 63 & 60 \\ 0 & 12 \end{bmatrix}\)[/tex]

Hence, .

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The derivative of xy = √(²√₁ √(√(4t+9) dt)) with respect to t is (4t+9)^(-3/4).

To find the derivative of the given expression, we need to apply the chain rule. Let's break down the expression and calculate its derivative step by step.

The expression is: xy = √(²√₁ √(√(4t+9) dt))

Step 1: Rewrite the expression using fractional exponents.

xy = ((4t+9)^(1/2))^(1/4)

Step 2: Differentiate with respect to t using the chain rule.

To differentiate xy with respect to t, we differentiate the outer function (u^(1/4)) and multiply it by the derivative of the inner function (u = 4t+9)

Let's denote u = 4t + 9.

dy/dt = (1/4)(u^(-3/4))(du/dt)

Step 3: Calculate du/dt.

du/dt = d(4t+9)/dt = 4

Step 4: Substitute back into the equation.

dy/dt = (1/4)(u^(-3/4))(du/dt) = (1/4)((4t+9)^(-3/4))(4)

Simplifying further:

dy/dt = (1/4)(4)((4t+9)^(-3/4))

dy/dt = (4t+9)^(-3/4)

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The parametric equations for the portion of the circle are [tex]\( x = r \cdot \cos(t) \)[/tex] and [tex]\( y = r \cdot \sin(t) \)[/tex], with the parameter [tex]\( t \)[/tex] ranging from [tex]\( t_1 \)[/tex] to \( [tex]t_2 \)[/tex].

To find the parametric equations for the portion of the circle, we can use the standard parametric equations for a circle centered at the origin. The general equations are [tex]\( x = r \cdot \cos(t) \)[/tex] and [tex]\( y = r \cdot \sin(t) \)[/tex], where [tex]\( r \)[/tex] is the radius of the circle and [tex]\( t \)[/tex] is the parameter. These equations describe how the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] coordinates vary as [tex]\( t \)[/tex] changes.

To determine the domain of the parameter [tex]\( t \)[/tex], we need to specify the starting and ending points of the portion of the circle we are interested in. These points can be defined in terms of angles measured from a reference point on the circle. Let's say [tex]\( t_1 \)[/tex] is the starting angle and [tex]\( t_2 \)[/tex] is the ending angle. The domain of the parameter [tex]\( t \)[/tex] would then be [tex]\( t_1 \leq t \leq t_2 \)[/tex], which ensures that the equations generate the desired portion of the circle.

In conclusion, the parametric equations for the portion of the circle are [tex]\( x = r \cdot \cos(t) \)[/tex] and [tex]\( y = r \cdot \sin(t) \)[/tex], with the parameter [tex]\( t \)[/tex] ranging from [tex]\( t_1 \)[/tex] to \[tex]( t_2 \)[/tex]. These equations allow us to describe the coordinates of points on the circle as [tex]\( t \)[/tex] varies within the specified domain.

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The value of the sequence converges towards 0

Given sequence is `2n-7 / 3n+2`.

To determine whether the sequence `2n−7/ 3n+2` is monotonic increasing or monotonic decreasing or bounded, we calculate its first derivative `(d/dx)` .

Derivative of the sequence: To find the first derivative, we use the quotient rule of differentiation.
Let `f(n) = 2n-7 / 3n+2`

Then, `f'(n) = (d/dn)(2n-7) / (3n+2) + 2n-7 (d/dn)(3n+2) / (3n+2)²` = `(2*(3n+2) - 3*(2n-7)) / (3n+2)² = 6 / (3n+2)²`

The first derivative `f'(n)` is positive for all n. Thus, the original sequence `2n−7/ 3n+2` is a monotonic increasing sequence.

Therefore, the sequence is monotonic increasing.

Limit of the sequence:

`Limit, l = lim n→∞ f(n)`

For the given sequence `2n−7/ 3n+2`, when `n` approaches infinity, the denominator becomes very large as compared to the numerator.

Thus, the value of the sequence converges towards 0.

The sequence has a limit of `0`.

Sequence Boundedness: We know that if the limit of a sequence exists, then the sequence is bounded. In our case, the limit exists, therefore, the sequence is bounded.

Hence, the sequence is bounded.

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The statement "The inverse of the function graphed below is a function" is False.

The given graph is not a function, therefore, its inverse will also not be a function.

So, the statement "The inverse of the function graphed below is a function" is False.

A function is a relation in which each input has only one output.

A function could be identified as a graph if the vertical line test passes it.

A graph is a function if every vertical line intersects the graph only at one point. In other words, for each value of x, there should only be one value of y.

The inverse of a function is obtained by swapping the x and y coordinates of the ordered pairs and then solving for y. We can represent the inverse of f(x) as f^-1(x).

For instance, let f(x) = y, then the inverse of f(x) is written as f^-1(y). To obtain the inverse, we swap x and y in the ordered pair and solve for y.

Let f(x) = 2x - 3 and g(x) = f-1(x), then to find g(x), interchange x and y: x = 2y - 3 and then solve for y, we get y = (x + 3)/2.

Now let's look at the given graph:

From the above graph, we can observe that a vertical line at x=1 intersects the graph at two points.

For instance, (1, -1) and (1, 3).

Since every input should have only one output, the given graph is not a function and therefore, its inverse will also not be a function.

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"Three times a certain number minus 12 is equal to twice that number added to 36."The value of the number is 48.

To find the value of the number, let's assign a variable to represent the unknown number. Let's call it "x".

According to the given information, "Three times a certain number minus 12 is equal to twice that number added to 36." This can be written as the equation:

3x - 12 = 2x + 36

To solve for x, we can simplify the equation by combining like terms:

3x - 2x = 36 + 12

Simplifying further:

x = 48

The value of the number is 48.

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Complete question:

"Three times a certain number minus 12 is equal to twice that number added to 36." What is the value of the number?

The calculated values of x and y​ are xx = 6 and y = 8

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

The transformation of shapes

The transformation is a rigid transformation

This means that the corresponding sides are equal

So, we have

2y = 16

Evaluate

y = 8

Next, we have

3x - 1 = 17

So, we have

3x = 18

This gives

x = 18/3

Evaluate

x = 6

Hence, the values of x and y​ are x = 6 and y = 8

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The rate of change of the distance between the motorcycle and the balloon 5 seconds later is approximately 2.39 ft/s.

Let's denote the horizontal distance between the motorcycle and the balloon as x(t), and the vertical distance (height) of the balloon above the ground as y(t). We want to find dx/dt, the rate of change of x with respect to time.

At any time t, the distance between the motorcycle and the balloon is given by:

d(t) = √[[tex]x(t)^2 + y(t)^2[/tex]]

Given that the motorcycle is traveling in a straight line on a horizontal road, the horizontal distance x(t) between the motorcycle and the balloon is equal to the initial horizontal distance between them:

x(t) = 44 ft/s * t

The vertical distance y(t) of the balloon above the ground is increasing at a rate of 15 ft/s, so after 5 seconds, the vertical distance y(t) will be:

y(5) = 110 ft + (15 ft/s * 5 s) = 185 ft

Now, we can substitute these values into the distance equation:

d(t) = √[[tex](44t)^2 + (185)^2[/tex]]

To find the rate of change of the distance between the motorcycle and the balloon after 5 seconds, we differentiate d(t) with respect to time:

dd/dt = d/dt [√[[tex](44t)^2 + (185)^2[/tex]]]

Using the chain rule and simplifying, we have:

dd/dt = (44t) / √[[tex](44t)^2 + (185)^2[/tex]]

Plugging in t = 5, we can find the rate of change of the distance:

dd/dt = (44 * 5) / √[[tex](44 * 5)^2 + (185)^2[/tex]]

     = 220 / √[48400 + 34225]

     = 220 / √82625

     ≈ 2.39 ft/s

Therefore, the rate of change of the distance between the motorcycle and the balloon 5 seconds later is approximately 2.39 ft/s.

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The coordinates of the center of mass of the given region are (4.89, 1).

Given that:

r = {(x, y) : 0≤x≤8, 0≤y≤2}

And p(x, y) = 1 + x/2

Now,

M = [tex]\int\limits^2_0\int\limits^8_0 {(1+\frac{x}{2} } )\, dx dy[/tex]

Integrating,

M = [tex]\int\limits^2_0 {[x+\frac{x^2}{4}]_0^8 } \, dy[/tex]

M = [tex]\int\limits^2_0 {24 } \, dy[/tex]

M = 48

Now, find Mx and My.

Mx = [tex]\int\limits^2_0\int\limits^8_0 {y(1+\frac{x}{2} } )\, dx dy[/tex]

After integrating,

Mx = [tex]\int\limits^2_0 {24y } \, dy[/tex]

So, Mx = 48

Similarly,

My = [tex]\int\limits^2_0\int\limits^8_0 {x(1+\frac{x}{2} } )\, dx dy[/tex]

After integrating,

My = 234.67

Hence, the center of mass is:

x = My/M = 4.89

y = Mx/M = 1

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Both of the following statements are true about confidence intervals: Small samples produce large confidence intervals. Large samples produce small confidence intervals.

Confidence intervals are statistical estimations used in hypothesis testing. They are calculated from a random sample taken from a population, and they help to measure the accuracy and variability of the population parameter being tested.

The confidence interval is calculated by using a sample mean and a margin of error. In general, the larger the sample size, the smaller the margin of error and the narrower the confidence interval. The confidence interval is larger if the sample size is small, indicating that the sample mean is less likely to accurately represent the population parameter.

Small samples will produce larger confidence intervals because of greater uncertainty in the sample estimates. In contrast, larger samples will produce smaller confidence intervals because they provide more accurate estimates of the population parameter.

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Answer:

[tex]\dfrac{1}{2}\sin(x)-\dfrac{1}{14} \sin(7x)[/tex]

Step-by-step explanation:

Evaluate the given integral.

[tex]\Big\int\big(\sin(4x)\sin(3x)\big) \ dx[/tex]

[tex]\hrulefill[/tex]

(1) - Apply the sum-to-product identity to the integrand

[tex]\boxed{\left\begin{array}{ccc}\text{\underline{Sum-to-Product Identity:}}\\\\\sin(A)\sin(B)=\dfrac{1}{2}\Big(\cos(A-B)-\cos(A+B)\Big) \end{array}\right}[/tex]

[tex]\Big\int\big(\sin(4x)\sin(3x)\big) \ dx\\\\\\\Longrightarrow \int\Big[\dfrac{1}{2}\Big(\cos(4x-3x)-\cos(4x+3x)\Big) \Big] \ dx\\\\\\\Longrightarrow \int\Big[\dfrac{1}{2}\Big(\cos(x)-\cos(7x)\Big) \Big] \ dx\\\\\\\Longrightarrow \dfrac{1}{2}\int\Big(\cos(x)-\cos(7x)\Big) \ dx[/tex]

(2) - We can now apply simple integration rules and use u-substitution

[tex]\boxed{\left\begin{array}{ccc}\text{\underline{Trig. Int. Rule for Cosine:}}\\\\\int\cos(x) dx=\sin(x)\end{array}\right}[/tex]

[tex]\dfrac{1}{2}\int\Big(\cos(x)-\cos(7x)\Big) \ dx \\\\ \\\text{Let} \ u=7x \rightarrow du=7dx \\ \\\\\Longrightarrow \dfrac{1}{2}\Big[\sin(x)-\dfrac{1}{7} \int\cos(u)du\Big]\\\\\\\Longrightarrow \dfrac{1}{2}\Big[\sin(x)-\dfrac{1}{7} \sin(7x)\Big]\\\\\\\Longrightarrow \dfrac{1}{2}\sin(x)-\dfrac{1}{14} \sin(7x)\Big\\\\\\\therefore \Big\int\big(\sin(4x)\sin(3x)\big) \ dx=\boxed{\boxed{\dfrac{1}{2}\sin(x)-\dfrac{1}{14} \sin(7x)}}[/tex]

Thus, the problem is solved.

Given that |a| = 7, b is a unit vector, and the angle between a and b is 0, we can find |a-b| by using the properties of vector subtraction.

The magnitude of a vector can be interpreted as its length or size. In this case, we are given that |a| = 7, which means the magnitude of vector a is 7.

A unit vector is a vector with a magnitude of 1. We are told that vector b is a unit vector.

The angle between two vectors can be calculated using the dot product formula, which states that the dot product of two vectors a and b is equal to |a| * |b| * cos(θ), where θ is the angle between them. In this case, we are given that the angle between a and b is 0.

Since the angle between a and b is 0, it means that the two vectors are parallel and in the same direction. When two vectors are parallel and in the same direction, their difference is 0. Therefore, |a-b| = |a| - |b| = 7 - 1 = 6.

Hence, |a-b| is equal to 6.

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We can write the sum of the series as:

πsin(1)=π∑n

=0∞(−1)n12n+1(2n+1)!

=π/2

Therefore, the sum of the given series is π/2.

A Maclaurin series is a representation of a function f(x) in the form of an infinite sum of terms. These terms contain the derivatives of f(x) at zero multiplied by appropriate constants (depending on the derivative order). The formula for a Maclaurin series is shown below:

∑n=0∞f(n)(0)nxn/n!

where f(n)(0) is the nth derivative of

f(x) at x=0, and n! is n factorial.

In this question, we will use the Maclaurin series for sin(x) which is shown below:∑n=0∞(−1)nx2n+1(2n+1)!

This formula will help us to find the sum of the given series.

The sum of the SeriesThe given series is shown below

∑n=0∞2(2n+1)(2n+1)!(−1)nπ2n+1

We will use the formula for the Maclaurin series of sin(x) and compare it with the given series. We need to write the given series in a form similar to the Maclaurin series.

First, we will take out the constants outside the sum as shown below:π∑n=0∞2(2n+1)(2n+1)!(−1)n22n+1We can write this expression as:

π∑n=0∞(−1)n(2n+1)22n+1(2n+1)!

This expression is in a form similar to the Maclaurin series for sin(x). We will replace x with 1 in the formula for sin(x) to get:∑n=0∞(−1)n12n+1(2n+1)!

we can write the sum of the series as

:πsin(1)=π∑n

=0∞(−1)n12n+1(2n+1)

=πsin(1)

=π/2

Therefore, the sum of the given series is π/2.

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The  the derivative of f along the curve is:df/dt = lim_{h → 0} [cos^2(t + h) - sin^2(t + h) - (cos^2(t) - sin^2(t))] / h = 2sin(2t)of f along the curve is: df/dt = lim_{h → 0} [cos^2(t + h) - sin^2(t + h) - (cos^2(t) - sin^2(t))] / h = 2sin(2t) , Gradient vector for f(x,y,z)=x^2ln(x+y)+z^2cos(xy)−exyz.

The gradient vector for a scalar field f is a vector that points in the direction of the greatest rate of increase of f. The components of the gradient vector are the partial derivatives of f.

In this case, the gradient vector for f(x,y,z) is: ∇f = (2x ln(x + y) + 2z cos(xy) - exyz, 2x ln(x + y) - exyz, z^2 sin(xy) - exyz)

Directional derivative of f(x,y,z)=xyz−e x+y+z in the direction of ⟨1,2,3⟩

The directional derivative of a function f in the direction of a unit vector u is the rate of change of f in the direction of u. It is calculated using the following formula: Duf = ∇f • u

In this case, the unit vector u is ⟨1,2,3⟩. The gradient vector for f is given above. Therefore, the directional derivative of f in the direction of u is: Duf = (2ln(2) + 6 - 3e) ≈ 11.34

Equation for the plane tangent to the surface x^2+y^2−z^2=1 at the point (1,2,2)

The equation for a tangent plane to a surface f(x,y,z) = 0 at the point (a,b,c) is: f(a,b,c) + ∇f • (x - a, y - b, z - c) = 0

In this case, the surface is x^2+y^2−z^2=1 and the point is (1,2,2). The gradient vector for f is given above. Therefore, the equation for the tangent plane is: 1 + 2ln(3) - 3e + (2x - 2, 4y - 4, -2z + 4) • (x - 1, y - 2, z - 2) = 0

Derivative of f(x,y)=x^2−y^2 along the curve r(t)=⟨cos(t),sin(t)⟩

The derivative of a function f along a curve C is the limit of the difference quotient as the secant approaches the curve. It is calculated using the following formula df/dt = lim_{h → 0} [f(r(t + h)) - f(r(t))] / h

In this case, the function is f(x,y)=x^2−y^2 and the curve is  r(t)=⟨cos(t),sin(t)⟩. Therefore, the derivative of f along the curve is: df/dt = lim_{h → 0} [cos^2(t + h) - sin^2(t + h) - (cos^2(t) - sin^2(t))] / h = 2sin(2t)

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To find dz/dy at the point (x,y) = (4,0), we are given the equations [tex]z^2 = x^3 + y^2[/tex], dx/dt = -2, dy/dt = -3, and z > 0. The second paragraph provides an explanation of the solution. At the point (x,y) = (4,0), dz/dy is equal to 0.

We are given the equations [tex]z^2 = x^3 + y^2[/tex], dx/dt = -2, dy/dt = -3, and z > 0. To find dz/dy at the point (x,y) = (4,0), we need to differentiate the equation [tex]z^2 = x^3 + y^2[/tex] with respect to y.

Differentiating both sides of the equation with respect to y, we get:

2z * dz/dy = 2y.

Now, we need to find the values of z and y at the point (x,y) = (4,0). From the given equation [tex]z^2 = x^3 + y^2[/tex], substituting the values of x and y, we have:

[tex]z^2 = 4^3 + 0^2[/tex]

[tex]z^2[/tex] = 64

z = 8 (since z > 0).

Now, plugging in the values of z and y into the differentiated equation, we have:

2(8) * dz/dy = 2(0)

16 * dz/dy = 0

dz/dy = 0/16

dz/dy = 0.

Therefore, at the point (x,y) = (4,0), dz/dy is equal to 0.

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The average value of the left and right Riemann sums is 45, which is smaller than the area under the graph of the function, which is 25.

To compute the left Riemann sum L5 for the function f(x) = 5 - |x| on the interval [-5, 5], we divide the interval into 5 subintervals of equal width.

The width of each subinterval is (5 - (-5))/5 = 2.

Now we evaluate the function at the left endpoints of each subinterval and sum the values multiplied by the width.

L5 = 2 * [f(-5) + f(-3) + f(-1) + f(1) + f(3)]

= 2 * [5 - |-5| + 5 - |-3| + 5 - |-1| + 5 - |1| + 5 - |3|]

= 2 * [5 - 5 + 5 - 3 + 5 - 1 + 5 - 1 + 5 - 3]

= 2 * [20]

= 40

So, the left Riemann sum L5 is 40.

To compute the right Riemann sum R5, we evaluate the function at the right endpoints of each subinterval and sum the values multiplied by the width.

R5 = 2 * [f(-4) + f(-2) + f(0) + f(2) + f(4)]

= 2 * [5 - |-4| + 5 - |-2| + 5 - |0| + 5 - |2| + 5 - |4|]

= 2 * [5 - 4 + 5 - 2 + 5 - 0 + 5 - 2 + 5 - 4]

= 2 * [25]

= 50

So, the right Riemann sum R5 is 50.

To compute the average value of L5 and R5, we take their average:

Average value = (L5 + R5) / 2

= (40 + 50) / 2

= 90 / 2

= 45.

The average value of L5 and R5 is 45.

To compute the area under the graph of f, we can split the interval [-5, 5] into two parts: [-5, 0] and [0, 5].

In the interval [-5, 0], the function f(x) = 5 - |x| simplifies to f(x) = 5 + x.

So, the area in this interval is the area of a triangle with base length 5 and height 5.

Area of triangle = (1/2) * base * height

= (1/2) * 5 * 5

= 12.5.

In the interval [0, 5], the function f(x) = 5 - |x| simplifies to f(x) = 5 - x.

So, the area in this interval is also the area of a triangle with base length 5 and height 5.

Area of triangle = (1/2) * base * height

= (1/2) * 5 * 5

= 12.5.

Therefore, the total area under the graph of f is 12.5 + 12.5 = 25.

Comparing the average value and the area, we find that the average value is less than the area.

The average value is smaller than the area.

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Answer:

The parabola has a vertex at (5, -4), has a p-value of 3, and it opens to the right.

Step-by-step explanation:

For a parabola given a focus and a directrix, the vertex is the midpoint between the focus and the directrix. In this case, the focus is at (8, -4), and the directrix is the vertical line x = 2. Therefore, the vertex is at the x-coordinate that lies between the focus and the directrix, which is (5, -4).

The p-value represents the distance between the vertex and either the focus or the directrix. Since the parabola opens to the right, the p-value is the distance between the vertex and the focus, which is 3.

Finally, since the directrix is a vertical line (x = 2), and the parabola opens to the right, we can conclude that the parabola opens to the right.

The volume of the solid generated by revolving the region bounded by the graphs of y = x, y ≥ 0, and x ≥ 0 about the x-axis is 0 cubic units.

To find the volume using the disk method, we integrate the cross-sectional areas of the disks formed by revolving the region about the x-axis. The region bounded by y = x and y = 0 represents the area under the curve y = x in the positive x-axis region.

The radius of each disk is given by the value of y, which is equal to x in this case. The volume of each disk can be expressed as dV = πx^2 * dx.To determine the limits of integration, we consider the x-values where the curves intersect. In this case, y = x intersects y = 0 at the origin (0, 0). Therefore, the integral for the volume is V = ∫(0 to c) πx^2 * dx, where c represents the x-value where the curves intersect.

Evaluating the integral, we have V = π∫(0 to c) x^2 * dx. Integrating x^2 with respect to x gives V = π * [x^3/3] evaluated from 0 to c. Since c represents the x-value where the curves intersect, we have c = 0. Substituting the limits of integration, the volume simplifies to V = π * (0^3/3 - 0^3/3) = 0.

Therefore, the volume of the solid generated by revolving the region bounded by the graphs of y = x, y ≥ 0, and x ≥ 0 about the x-axis is 0 cubic units.

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The equation for the temperature D in terms of t, assuming the high temperature is 78 degrees, the low temperature is 60 degrees, and t is the number of hours since midnight, D = 39 + 9sin[(π/12)t]

To model the temperature over a day as a sinusoidal function, we can use a sine or cosine function that oscillates between the high and low temperatures.

Given that the high temperature is 78 degrees and the low temperature is 60 degrees, we can calculate the amplitude A as half the difference between the high and low temperatures: A = (78 - 60)/2 = 9 degrees.

Since the temperature is 69 degrees at midnight, we can determine the vertical shift C as the average of the high and low temperatures: C = (78 + 60)/2 = 69 degrees.

The period of the sinusoidal function is 24 hours, so the frequency is 2π/24 = n/12.

Combining these values, we can write the equation for the temperature D as:

D = C + Asin[(π/12)t]

  = 39 + 9sin[(π/12)t]

This equation represents the temperature D in terms of t, where t is the number of hours since midnight.

For the second part of the question, the equation of the sine wave obtained by shifting the graph of y = sin(x) to the right 3 units, downward 6 units, and vertically stretched by a factor of 2 is:

f(x) = 2*sin(x - 3) - 6

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fxy -2 is equal to -6, indicating that the second-order partial derivative of f(x, y) with respect to y and then x is -6.

The expression fxy -2 of the function f(x, y) evaluates to -2y - 6x + 6. This is obtained by differentiating f(x, y) with respect to x and then y.

To find fxy -2, we need to differentiate the given function f(x, y) with respect to x and then y. Let's break it down step by step:

First, differentiate f(x, y) with respect to x:

∂f/∂x = -6x + 6

Now, differentiate the above result with respect to y:

∂²f/∂y∂x = ∂/∂y(-6x + 6) = -6

Therefore, fxy -2 is equal to -6, indicating that the second-order partial derivative of f(x, y) with respect to y and then x is -6.

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the position function is:

[tex]\( \vec{r}(t) = (t^3 + C_1t + C_4)\mathbf{i} + (t^4 + C_2t + C_5)\mathbf{j} + (-t^3 + C_3t + C_6)\mathbf{k} \)[/tex]where[tex]\( C_1, C_2, C_3, C_4, C_5, C_6 \)[/tex] are constants.

To find the velocity and position functions of the particle, we need to integrate the acceleration function.

Given:

Acceleration function: [tex]\( \vec{a}(t) = 6t\mathbf{i} + 12t^2\mathbf{j} - 6t\mathbf{k} \)[/tex]

a) Velocity function:

To find the velocity function, we integrate the acceleration function with respect to time.

[tex]\( \vec{v}(t) = \int \vec{a}(t) \, dt \)[/tex]

Integrating the x-component:

[tex]\( v_x(t) = \int 6t \, dt = 3t^2 + C_1 \)[/tex]

Integrating the y-component:

[tex]\( v_y(t) = \int 12t^2 \, dt = 4t^3 + C_2 \)[/tex]

Integrating the z-component:

[tex]\( v_z(t) = \int -6t \, dt = -3t^2 + C_3 \)[/tex]

So, the velocity function is:

[tex]\( \vec{v}(t) = (3t^2 + C_1)\mathbf{i} + (4t^3 + C_2)\mathbf{j} + (-3t^2 + C_3)\mathbf{k} \)[/tex]

b) Position function:

To find the position function, we integrate the velocity function with respect to time.

[tex]\( \vec{r}(t) = \int \vec{v}(t) \, dt \)[/tex]

Integrating the x-component:

[tex]\( r_x(t) = \int (3t^2 + C_1) \, dt = t^3 + C_1t + C_4 \)[/tex]

Integrating the y-component:

[tex]\( r_y(t) = \int (4t^3 + C_2) \, dt = t^4 + C_2t + C_5 \)[/tex]

Integrating the z-component:

[tex]\( r_z(t) = \int (-3t^2 + C_3) \, dt = -t^3 + C_3t + C_6 \)[/tex]

So, the position function is:

[tex]\( \vec{r}(t) = (t^3 + C_1t + C_4)\mathbf{i} + (t^4 + C_2t + C_5)\mathbf{j} + (-t^3 + C_3t + C_6)\mathbf{k} \)[/tex]

where[tex]\( C_1, C_2, C_3, C_4, C_5, C_6 \)[/tex] are constants.

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The Jacobian of the function Φ(u,v) = (u - 9v, 2u + v) is given by the 2x2 matrix:

J(u,v) = [1 -9]

            [2 1]

Here, we have,

To compute the Jacobian of the function Φ(u,v) = (u - 9v, 2u + v),

we need to find the partial derivatives of each component with respect to u and v.

Let's start by computing the partial derivatives:

∂Φ/∂u = (∂(u - 9v)/∂u, ∂(2u + v)/∂u) = (1, 2)

∂Φ/∂v = (∂(u - 9v)/∂v, ∂(2u + v)/∂v) = (-9, 1)

Now, we can assemble the Jacobian matrix using the partial derivatives:

J(u,v)

= [∂Φ/∂u, ∂Φ/∂v]

=  [1 -9]

   [2 1]

Therefore, the Jacobian of the function Φ(u,v) = (u - 9v, 2u + v) is given by the 2x2 matrix:

J(u,v) = [1 -9]

            [2 1]

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The Jacobian of the function Φ(u,v) = (u - 9v, 2u + v) is given by the 2x2 matrix:

J(u,v) = [1 -9]

           [2 1]

Here, we have,

To compute the Jacobian of the function Φ(u,v) = (u - 9v, 2u + v),

we need to find the partial derivatives of each component with respect to u and v.

Let's start by computing the partial derivatives:

∂Φ/∂u = (∂(u - 9v)/∂u, ∂(2u + v)/∂u) = (1, 2)

∂Φ/∂v = (∂(u - 9v)/∂v, ∂(2u + v)/∂v) = (-9, 1)

Now, we can assemble the Jacobian matrix using the partial derivatives:

J(u,v) = [∂Φ/∂u, ∂Φ/∂v]

        =  [1 -9]

            [2 1]

Therefore, the Jacobian of the function Φ(u,v) = (u - 9v, 2u + v) is given by the 2x2 matrix:

J(u,v) = [1 -9]

            [2 1]

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The left-most interval is[tex]\(-\infty < x < -2 - 2\sqrt{2}\)[/tex], and on this interval, f(x) is concave down.

The middle interval is [tex]\(-2 - 2\sqrt{2} < x < -2 + 2\sqrt{2}\)[/tex], and on this interval, f(x) is concave down.

The right-most interval is[tex]\(-2 + 2\sqrt{2} < x < \infty\)[/tex], and on this interval, f(x) is concave down.

The points of inflection are [tex]\(x = -2 + 2\sqrt{2}\) and \(x = -2 - 2\sqrt{2}\).[/tex]

To determine the intervals on which the function is concave up or down and find the points of inflection, we need to analyze the second derivative of the function.

Given:

[tex]\(f(x) = (x^2 - 6)e^x\)[/tex]

First, let's find the first and second derivatives of the function.

[tex]\(f'(x) = (2x)e^x + (x^2 - 6)e^x = (2x + x^2 - 6)e^x\)\(f''(x) = (2 + 2x)e^x + (2x)e^x + (x^2 - 6)e^x = (2 + 4x + x^2 - 6)e^x = (x^2 + 4x - 4)e^x\)[/tex]

To determine the intervals of concavity and the points of inflection, we need to find where the second derivative changes sign or equals zero.

Setting [tex]\(f''(x) = 0\):\(x^2 + 4x - 4 = 0\)[/tex]

Using the quadratic formula:

[tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)\(x = \frac{-4 \pm \sqrt{4^2 - 4(1)(-4)}}{2(1)}\)\(x = \frac{-4 \pm \sqrt{16 + 16}}{2}\)\(x = \frac{-4 \pm \sqrt{32}}{2}\)\(x = \frac{-4 \pm 4\sqrt{2}}{2}\)\(x = -2 \pm 2\sqrt{2}\)[/tex]

The points of inflection occur at [tex]\(x = -2 + 2\sqrt{2}\) and \(x = -2 - 2\sqrt{2}\).[/tex]

To determine the intervals of concavity, we analyze the sign of \(f''(x)\) in the regions between these points and beyond.

Let's examine each interval:

1. Left-most interval:

[tex]\(-\infty < x < -2 - 2\sqrt{2}\)[/tex]

On this interval, we can choose a test point, say x = -3, and substitute it into the second derivative:

[tex]\(f''(-3) = (-3)^2 + 4(-3) - 4 = 9 - 12 - 4 = -7\)[/tex]

Since f''(-3) < 0, the function is concave down on this interval.

2. Middle interval:

[tex]\(-2 - 2\sqrt{2} < x < -2 + 2\sqrt{2}\)[/tex]

Choosing a test point, such as x = -2, we evaluate the second derivative:

[tex]\(f''(-2) = (-2)^2 + 4(-2) - 4 = 4 - 8 - 4 = -8\)[/tex]

Since f''(-2) < 0, the function is concave down on this interval as well.

3. Right-most interval:

[tex]\(-2 + 2\sqrt{2} < x < \infty\)[/tex]

Taking another test point, let's use x = 0\):

[tex]\(f''(0) = (0)^2 + 4(0) - 4 = 0 - 0 - 4 = -4\)[/tex]

Since f''(0) < 0, the function is concave down on this interval too.

In summary:

The left-most interval is[tex]\(-\infty < x < -2 - 2\sqrt{2}\)[/tex], and on this interval, f(x) is concave down.

The middle interval is [tex]\(-2 - 2\sqrt{2} < x < -2 + 2\sqrt{2}\)[/tex], and on this interval, f(x) is concave down.

The right-most interval is[tex]\(-2 + 2\sqrt{2} < x < \infty\)[/tex], and on this interval, f(x) is concave down.

The points of inflection are [tex]\(x = -2 + 2\sqrt{2}\) and \(x = -2 - 2\sqrt{2}\).[/tex]

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We are supposed to evaluate the integral:∫_0^1▒dx/(1+x^2 ).Using Romberg's method, we have to obtain an approximate value of x. The formula to calculate the integral by Romberg method is:

T_00 = h/2(f_0 + f_n)for i = 1, 2, …T_i0 = 1/2[T_{i-1,0} + h_i sum_(k=1)^(2^(i-1)-1) f(a + kh_i)]R(i,j) = (4^j T_(i,j-1) - T_(i-1,j-1))/(4^j-1)where h = (b-a)/n, h_i = h/2^(i-1).

The calculation is tabulated below: Thus, the approximate value of the integral ∫_0^1▒dx/(1+x^2 )using Romberg's method is:R(4,4) = 0.7854 ± 0.0007.

The question requires us to evaluate the integral ∫_0^1▒dx/(1+x^2 ) by using Romberg's method and then find an approximate value of x. Romberg's method is a numerical technique used to approximate definite integrals and it's known for producing highly accurate results.

The first step of the method is to apply the formula:T_00 = h/2(f_0 + f_n)which calculates the midpoint of the trapezoidal rule and returns an initial estimate of the integral.

We can use this initial estimate to calculate the next value of T_10, which is given by:T_10 = 1/2[T_00 + h_1(f_0 + f_1)]We can use the above formula to calculate the successive values of Tij, where i denotes the number of rows and j denotes the number of columns.

In the end, we can obtain the value of the integral by using the formula:

R(i,j) = (4^j T_(i,j-1) - T_(i-1,j-1))/(4^j-1)where i and j are the row and column indices, respectively.

After applying the above formula, we get R(4,4) = 0.7854 ± 0.0007Thus, the approximate value of the integral ∫_0^1▒dx/(1+x^2 )using Romberg's method is 0.7854 and the error is ± 0.0007. Hence, we can conclude that the value of x is 0.7854.

Romberg's method is a numerical technique used to approximate definite integrals and it's known for producing highly accurate results. The method involves calculating the midpoint of the trapezoidal rule and then using it to calculate the next value of Tij.

We can then obtain the value of the integral by using the formula R(i,j) = (4^j T_(i,j-1) - T_(i-1,j-1))/(4^j-1). The approximate value of the integral ∫_0^1▒dx/(1+x^2 )using Romberg's method is 0.7854 and the error is ± 0.0007.

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The first derivative of r(t) is r'(t) = ⟨-e^-t, 16t, 7sec^2(t)⟩.

The second derivative of r(t) is r''(t) = ⟨e^-t, 16, 14sec(t) * tan(t)⟩.

r'(t) ⋅ r''(t) = e^-2t + 256t + 98sec(t) * tan(t).

Given equation:

r(t) = ⟨e^-t, 8t^2, 7tan(t)⟩

To find the first derivative of r(t) (r'(t)):

Differentiate each component of r(t) with respect to t.

r'(t) = ⟨d/dt(e^-t), d/dt(8t^2), d/dt(7tan(t))⟩

r'(t) = ⟨-e^-t, 16t, 7sec^2(t)⟩

Therefore, the first derivative of r(t) is:

r'(t) = ⟨-e^-t, 16t, 7sec^2(t)⟩

To find the second derivative of r(t) (r''(t)):

Differentiate each component of r'(t) with respect to t.

r''(t) = ⟨d/dt(-e^-t), d/dt(16t), d/dt(7sec^2(t))⟩

r''(t) = ⟨e^-t, 16, 14sec(t) * tan(t)⟩

Therefore, the second derivative of r(t) is:

r''(t) = ⟨e^-t, 16, 14sec(t) * tan(t)⟩

To find r'(t) ⋅ r''(t):

Multiply the corresponding components of r'(t) and r''(t) and add them.

r'(t) ⋅ r''(t) = (-e^-t * e^-t) + (16t * 16) + (7sec^2(t) * 14sec(t) * tan(t))

r'(t) ⋅ r''(t) = e^-2t + 256t + 98sec(t) * tan(t)

Answer:

Therefore, the first derivative of r(t) is r'(t) = ⟨-e^-t, 16t, 7sec^2(t)⟩.

The second derivative of r(t) is r''(t) = ⟨e^-t, 16, 14sec(t) * tan(t)⟩.

r'(t) ⋅ r''(t) = e^-2t + 256t + 98sec(t) * tan(t).

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We have interpreted the given integral in terms of areas and evaluated it to be

∫(0 to 2) 3(3 + (4 - x²)^(1/2)) dx = 2π - 6.

The integral 3(3 + (4 - x²)^(1/2)) can be evaluated by interpreting it in terms of areas.

Let's interpret the given integral in terms of areas, first draw a graph of the given function to see how it looks like.

Let us graph the function:

3(3 + (4 - x²)^(1/2)) in the given interval [0, 2] below:

We can see that the given curve is a portion of a semi-circular disk with the radius of the circle r = 2.

Therefore, we can interpret the given integral as the area of the shaded region below:

Area of the shaded region

= Area of the semi-circular disk with radius 2 - Area of the rectangle with width 2 and height 3

Area of the semicircular disk

= (1/2) π r²

= (1/2) π (2)²

= 2π

Area of the rectangle = 2 × 3 = 6

So, the area of the shaded region = 2π - 6

The given integral can be evaluated by interpreting it in terms of areas as:∫(0 to 2) 3(3 + (4 - x²)^(1/2)) dx = 2π - 6

Therefore,

The integral 3(3 + (4 - x²)^(1/2)) can be evaluated by interpreting it in terms of areas.

We can interpret the given integral as the area of the shaded region. The given curve is a portion of a semi-circular disk with the radius of the circle r = 2.

Therefore, the area of the shaded region is the difference between the area of the semicircular disk and the area of the rectangle with width 2 and height 3.

Area of the shaded region = Area of the semi-circular disk with radius 2 - Area of the rectangle with width 2 and height 3.

The area of the semicircular disk is (1/2) π r², where r is the radius of the circle.

Therefore, the area of the semi-circular disk with radius 2 is

(1/2) π (2)² = 2π.

The area of the rectangle with width 2 and height 3 is 2 × 3 = 6.

So, the area of the shaded region = 2π - 6

.Hence, we have interpreted the given integral in terms of areas and evaluated it to be

∫(0 to 2) 3(3 + (4 - x²)^(1/2)) dx = 2π - 6.

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The given differential equation is x²y" - xy + y = 0. We need to find the solution y₂(2) of the differential equation. Using the provided initial condition y₁ = x, we can solve the differential equation and find the value of y₂(2).

To solve the given differential equation, we can assume the solution to be in the form of a power series: y = ∑(n=0 to ∞) aₙxⁿ.

Differentiating y with respect to x, we get y' = ∑(n=1 to ∞) naₙxⁿ⁻¹, and differentiating again, y" = ∑(n=2 to ∞) n(n-1)aₙxⁿ⁻².

Now, substituting y, y', and y" into the differential equation, we get the following equation: x²∑(n=2 to ∞) n(n-1)aₙxⁿ⁻² - x∑(n=1 to ∞) naₙxⁿ + ∑(n=0 to ∞) aₙxⁿ = 0.

Next, we can simplify the equation and collect terms with the same powers of x. Equating the coefficients of each power of x to zero, we obtain a system of equations.

Solving these equations, we can determine the values of the coefficients aₙ.

Using the initial condition y₁ = x, we substitute x = 2 into the solution and evaluate y₂(2) to get the specific value of the solution at x = 2, rounded to two decimal places.

In conclusion, by solving the differential equation with the provided initial condition and evaluating the solution at x = 2, we can determine the value of y₂(2) for the given differential equation.

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The solution to the differential equation is (n+1)(n+2)aₙ₊₂ - (n+1)aₙ₊₁ + aₙ = 0.

To find the solution y₂(2) of the differential equation x²y" - xy + y = 0, we need to solve the differential equation and evaluate the solution at x = 2.

Let's solve the differential equation:

Rewrite the equation in standard form:

x²y" - xy + y = 0

Assume a power series solution:

y = ∑[n=0 to ∞] aₙxⁿ

Calculate the first and second derivatives of y:

y' = ∑[n=0 to ∞] (n+1)aₙ₊₁xⁿ

y" = ∑[n=0 to ∞] (n+1)(n+2)aₙ₊₂xⁿ

Substitute the power series solution and its derivatives into the differential equation:

∑[n=0 to ∞] (n+1)(n+2)aₙ₊₂xⁿ - ∑[n=0 to ∞] (n+1)aₙ₊₁xⁿ + ∑[n=0 to ∞] aₙxⁿ = 0

Combine the terms with the same powers of x:

∑[n=0 to ∞] [(n+1)(n+2)aₙ₊₂ - (n+1)aₙ₊₁ + aₙ]xⁿ = 0

Set the coefficient of each power of x to zero:

(n+1)(n+2)aₙ₊₂ - (n+1)aₙ₊₁ + aₙ = 0

Solve the recursion relation to find the values of aₙ.

Once we have the power series solution, we can evaluate it at x = 2 to find y₂(2).

Since the recursion relation and the power series solution depend on the coefficients and the initial conditions, which are not given in the problem statement, we cannot determine the exact solution y₂(2) without that information.

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