find the volume of the solid of revolution generated by revolving about the x-axis the region under the following curve. y=√x from x=0 to x = 10 (the solid generated is called a paraboloid.)

Answers

Answer 1

The volume of the solid of revolution generated by revolving the region under the curve y = √x from x = 0 to x = 10 about the x-axis is approximately 1046.67 cubic units.

To find the volume of the solid of revolution, we can use the method of cylindrical shells. The volume of each cylindrical shell is given by the formula V = 2πrhΔx, where r is the radius of the shell, h is the height of the shell, and Δx is the width of the shell.

In this case, the radius of the shell is given by r = √x, and the height of the shell is h = y = √x. Since we are revolving the region about the x-axis, the width of each shell is Δx.

To find the volume, we integrate the formula V = 2π∫(√x)(√x)dx over the interval [0, 10].

Evaluating the integral, we get V = 2π∫(x)dx from 0 to 10.

Integrating, we have V = 2π[(x^2)/2] from 0 to 10.

Simplifying, V = π(10^2 - 0^2) = 100π.

Approximating π as 3.14159, we find V ≈ 314.159 cubic units.

Therefore, the volume of the solid of revolution generated by revolving the region under the curve y = √x from x = 0 to x = 10 about the x-axis is approximately 1046.67 cubic units.

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Related Questions

Evaluate the derivative of the given function for the given value of x using the product rule. y = (3x - 1)(5-x), x= 6

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We first determine the two elements as "(u = 3x - 1") and "(v = 5 - x") in order to estimate the derivative of the given function, "(y = (3x - 1)(5 - x)" using the product rule.

According to the product rule, if "y = u cdot v," then "y' = u cdot v + u cdot v'" gives the derivative of "y" with regard to "x."

When we use the product rule, we discover:

\(u' = 3\) (v' = -1 is the derivative of (u) with respect to (x)) ((v's) derivative with regard to (x's))

When these values are substituted, we get:

\(y' = (3x - 1)'(5 - x) + (3x - 1)(5 - x)'\)

\(y' = 3(5 - x) + (3x - 1)(-1)\)

Simplifying even more

\(y' = 15 - 3x - 3x + 1\)

\(y' = -6x + 16\)

The derivative at (x = 6) is evaluated by substituting (x = 6) into the

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There are two features we use for entering answers, rest as with a paper exam, you need the opportunity to change an answer if you catch your mistake white checking your work. And the built teature that shows whether or not your answers are correct as you enter them must be disabled. Try answering this question. Perhaps giving a wrong answer first Find a value of A so that 7 and ware parallel. ū - 37 +27 and w - A7 - 107

Answers

The value of A that makes u and w parallel is A = 3/7. To find a value of A such that vectors u = ⟨1, -3, 2⟩ and w = ⟨-A, 7, -10⟩ are parallel, we can set the components of the two vectors proportionally and solve for A.

The first component of u is 1, and the first component of w is -A. Setting them proportional gives -A/1 = -3/7. Solving this equation for A gives A = 3/7. Two vectors are parallel if they have the same direction or are scalar multiples of each other. To determine if two vectors u and w are parallel, we can compare their corresponding components and see if they are proportional. In this case, the first component of u is 1, and the first component of w is -A. To make them proportional, we set -A/1 = -3/7, as the second component of u is -3 and the second component of w is 7. Solving this equation for A gives A = 3/7. Therefore, when A is equal to 3/7, the vectors u and w are parallel.

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Find the antiderivative. Then use the antiderivative to evaluate the definite integral. (A) soux dy 6 Inx ху (B) s 6 In x dy ху .

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(A) To find the antiderivative of the function f(x, y) = 6ln(x)xy with respect to y, we treat x as a constant and integrate: ∫ 6ln(x)xy dy = 6ln(x)(1/2)y^2 + C,

where C is the constant of integration.

(B) Using the antiderivative we found in part (A), we can evaluate the definite integral: ∫[a, b] 6ln(x) dy = [6ln(x)(1/2)y^2]∣[a, b].

Substituting the upper and lower limits of integration into the antiderivative, we have: [6ln(x)(1/2)b^2] - [6ln(x)(1/2)a^2] = 3ln(x)(b^2 - a^2).

Therefore, the value of the definite integral is 3ln(x)(b^2 - a^2).

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16 17
I beg you please write letters and symbols as clearly
as possible or make a key on the side so ik how to properly write
out the problem
16) Elasticity is given by: E(p) = P D'(p) D(p) The demand function for a high-end box of chocolates is given by D(p) = 110-60p+p² -0.04p³ in dollars. If the current price for a box of chocolate is

Answers

The demand for a high-end box of chocolates with a current price of $26 is unit-elastic. To increase revenue, the company should neither raise nor lower prices.

The elasticity of demand can be determined by evaluating the elasticity function E(p) at the given price. In this case, the demand function is [tex]D(p) = 110 - 60p + p^2 - 0.04p^3.[/tex]

To calculate the elasticity, we need to find D'(p) (the derivative of the demand function with respect to price) and substitute it into the elasticity function. Taking the derivative of the demand function, we get:

[tex]D'(p) = -60 + 2p - 0.12p^2[/tex]

Now, we can substitute D'(p) and D(p) into the elasticity function E(p):

[tex]E(p) = -p * D'(p) / D(p)[/tex]

Substituting the values, we have:

[tex]E(26) = -26 * (-60 + 2*26 - 0.12*26^2) / (110 - 60*26 + 26^2 - 0.04*26^3)[/tex]

After evaluating the expression, we find that E(26) ≈ 1.01.

Since the elasticity value is approximately equal to 1, the demand is unit-elastic. This means that a change in price will result in an equal percentage change in quantity demanded.

To increase revenue, the company should consider implementing other strategies instead of changing the price. A price increase may lead to a decrease in quantity demanded by the same percentage, resulting in unchanged revenue.

Therefore, it would be advisable for the company to explore other avenues, such as marketing campaigns, product differentiation, or expanding their customer base, to increase revenue without relying solely on price adjustments.

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The complete question is :

Elasticity is given by: E(p) = - -P.D'(p) D(p) The demand function for a high-end box of chocolates is given by D(p) = 110-60p+p²-0.04p³ in dollars. If the current price for a box of chocolate is $26, state whether the demand is elastic, inelastic, or unit-elastic. Then decide whether the company should raise or lower prices to increase revenue.

he points in the table lie on a line. Find the slope of the line. A table with 2 rows and 5 columns. The first row is x and it has the numbers negative 3, 2, 7, and 12. The second row is y and it has the numbers 0, 2, 4, and 6.

Answers

The slope of the line passing through the points in the table is 2/5.

Given information,

Rows in Table A = 2

Columns in Table A = 5

Row x has numbers = negative 3, 2, 7, and 12

Row y has numbers = 0, 2, 4, and 6

To find the slope of the line that passes through the points in the table, the formula for slope is used:

Slope (m) = (change in y) / (change in x)

The points (-3, 0) and (12, 6) are from the given table.

Change in x = 12 - (-3) = 12 + 3 = 15

Change in y = 6 - 0 = 6

Slope (m) = (change in y) / (change in x) = 6 / 15 = 2/5

Therefore, the slope of the line passing through the points in the table is 2/5.

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4 7 7 Suppose f(x)dx = 8, f(x)dx = - 7, and s [= Solxjex g(x)dx = 6. Evaluate the following integrals. 2 2 2 2 jaseut-on g(x)dx=0 7 (Simplify your answer.)

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The value of ∫[2 to 7] g(x) dx is -45.

In this problem, we are given: ∫f(x) dx = 8, ∫f(x) dx = -7, and s = ∫[a to b] g(x) dx = 6, and we need to find ∫[2 to 7] g(x) dx. Let’s begin solving this problem one by one. We know that, ∫f(x) dx = 8, therefore, f(x) = 8 dx Similarly, we have ∫f(x) dx = -7, so, f(x) = -7 dx Now, s = ∫[a to b] g(x) dx = 6, so, ∫g(x) dx = s / [b-a] = 6 / [b-a]Now, we need to evaluate ∫[2 to 7] g(x) dx We can write it as follows: ∫[2 to 7] g(x) dx = ∫[2 to 7] 1 dx – ∫[2 to 7] [f(x) + g(x)] dx We can replace the value of f(x) in the above equation:∫[2 to 7] g(x) dx = 5 – ∫[2 to 7] [8 + g(x)] dx Now, we need to evaluate ∫[2 to 7] [8 + g(x)] dx Using the linear property of integrals, we get:∫[2 to 7] [8 + g(x)] dx = ∫[2 to 7] 8 dx + ∫[2 to 7] g(x) dx∫[2 to 7] [8 + g(x)] dx = 8 [7-2] + 6= 50Therefore,∫[2 to 7] g(x) dx = 5 – ∫[2 to 7] [8 + g(x)] dx= 5 – 50= -45Therefore, the value of ∫[2 to 7] g(x) dx is -45.

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Energy problem formulas
Potential Energy = mgh
v = velocity or speed
Kinetic energy = mv²
9 = 9.8 m/s²
m = mass in kg
(Precision of 0.0)
h = height in meters
A baby carriage is sitting at the top of a hill that is 26 m high. The
carriage with the baby has a mass of 2.0 kg.
a) Calculate Potential Energy
(Precision of 0.0)
b) How much work was done to the system to create this potential
energy?

Answers

a. The kinetic energy is 620 J

b. The amount of work done is equal to the kinetic energy. In this case, the work done is 620 J.

Here,

a. The formula for kinetic energy is:

KE = 1/2mv²

where:

KE is the kinetic energy in joules (J)

m is the mass in kilograms (kg)

v is the velocity in meters per second (m/s)

In this case, we have:

m = 3.1 kg

v = 20 m/s

So, the kinetic energy is:

KE = 1/2(3.1 kg)(20 m/s)²

= 620 J

b) How much work is being done to the system to create this kinetic energy?

Work is done to the system to create kinetic energy. The amount of work done is equal to the kinetic energy.

In this case, the work done is 620 J.

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CALCULUS I FINAL FALL 2022 ) 1) Pick two (different) polynomials (1), g(x) of degrec 2 and find lim 2) Find the equation of the tangent line to the curve y + x3 = 1 + at the point (0.1). 3) Pick a

Answers

Post of performing a series of calculations we reach the conclusion that the a) the limit of f(x)/g(x) as x approaches infinity is a/d, b) the equation of the tangent line to the curve [tex]y + x^3 = 1 + 3xy^3[/tex]at the point (0, 1) is y = 3x + 1 and c) the function [tex]f(x) = x^{(-a)}[/tex]is a power function with a negative exponent.

To figure out the limit of [tex]f(x)/g(x)[/tex] as x approaches infinity, we need to apply division for leading the terms of f(x) and g(x) by x².
Let [tex]f(x) = ax^2 + bx + c and g(x) = dx^2 + ex + f[/tex] be two polynomials of degree 2.
Then, the limit of [tex]f(x)/g(x)[/tex] as x reaches infinity is:
[tex]lim f(x)/g(x) = lim (ax^2/x^2) / (dx^2/x^2) = lim (a/d)[/tex]
Then, the limit of f(x)/g(x) as x approaches infinity is a/d.
To calculate the equation of the tangent line to the curve y + x^3 = 1 + 3xy^3 at the point (0, 1),
we need to calculate the derivative of the curve at that point and utilize it to find the slope of the tangent line.
Taking the derivative of the curve with respect to x, we get:
[tex]3x^2 + 3y^3(dy/dx) = 3y^2[/tex]
At the point (0, 1), we have y = 1 and dy/dx = 0. Therefore, the slope of the tangent line is:
[tex]3x^2 + 3y^3(dy/dx) = 3y^2[/tex]
[tex]3(0)^2 + 3(1)^3(0) = 3(1)^2[/tex]
Slope = 3
The point (0, 1) is on the tangent line, so we can apply the point-slope form of the equation of a line to evaluate the equation of the tangent line:
[tex]y - y_1 = m(x - x_1)[/tex]
y - 1 = 3(x - 0)
y = 3x + 1
Henceforth , the equation of the tangent line to the curve [tex]y + x^3 = 1 + 3xy^3[/tex]at the point (0, 1) is y = 3x + 1.
For a positive integer a, the function [tex]f(x) = x^{(-a)}[/tex] is a power function with a negative exponent. The domain of f(x) is the set of all positive real numbers, since x cannot be 0 or negative. .
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The complete question is
1) Pick two (different) polynomials f(x), g(x) of degree 2 and find lim f(x). x→∞ g(x)
2) Find the equation of the tangent line to the curve y + x3 = 1 + 3xy3 at the point (0, 1).
3) Pick a positive integer a and consider the function f(x) = x−a
Need answered ASAP written as clear as possible

A firm manufactures a commodity at two different factories, Factory X and Factory Y. The total cost (in dollars) of manufacturing depends on the quantities, and y produced at each factory, respectively, and is expressed by the joint cost function: C(x, y) = = 1x² + xy + 2y² + 600 A) If the company's objective is to produce 400 units per month while minimizing the total monthly cost of production, how many units should be produced at each factory? (Round your answer to whole units, i.e. no decimal places.) To minimize costs, the company should produce: at Factory X and at Factory Y dollars. (Do not B) For this combination of units, their minimal costs will be enter any commas in your answer.)

Answers

In this case, a = 4 and b = -200, so the y-coordinate of the vertex is:

y = -(-200)/(2*4) = 200/8 = 25

To minimize the total monthly cost of production while producing 400 units per month, we need to determine the optimal quantities to produce at each factory.

Let's solve part A) by finding the critical points of the joint cost function and evaluating them to determine the minimum cost.

The joint cost function is given as:

C(x, y) = x² + xy + 2y² + 600

To find the critical points, we need to take the partial derivatives of C(x, y) with respect to x and y and set them equal to zero:

∂C/∂x = 2x + y = 0   ... (1)

∂C/∂y = x + 4y = 0   ... (2)

Now, let's solve the system of equations (1) and (2) to find the critical points:

From equation (2), we can isolate x:

x = -4y   ... (3)

Substituting equation (3) into equation (1):

2(-4y) + y = 0

-8y + y = 0

-7y = 0

y = 0

Plugging y = 0 back into equation (3), we get:

x = -4(0) = 0

Therefore, the critical point is (0, 0).

To determine if this critical point corresponds to a minimum, maximum, or saddle point, we need to evaluate the second partial derivatives:

∂²C/∂x² = 2

∂²C/∂y² = 4

∂²C/∂x∂y = 1

Calculating the discriminant:

D = (∂²C/∂x²)(∂²C/∂y²) - (∂²C/∂x∂y)²

  = (2)(4) - (1)²

  = 8 - 1

  = 7

Since D > 0 and (∂²C/∂x²) > 0, we conclude that the critical point (0, 0) corresponds to a local minimum.

Now, let's determine the optimal quantities to produce at each factory to minimize costs while producing 400 units per month.

Since we need to produce a total of 400 units per month, we have the constraint:

x + y = 400   ... (4)

Substituting x = 400 - y into the cost function C(x, y), we get the cost function in terms of y:

C(y) = (400 - y)² + (400 - y)y + 2y² + 600

     = 400² - 2(400)y + y² + 400y + 2y² + 600

     = 160000 - 800y + y² + 400y + 2y² + 600

     = 3y² + 600y + y² - 800y + 160000 + 600

     = 4y² - 200y + 160600

To minimize the cost, we need to find the minimum of this cost function.

To find the minimum of the quadratic function C(y), we can use the formula for the x-coordinate of the vertex of a parabola given by x = -b/2a.

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Find (A) the leading term of the polynomial, (B) the limit as x approaches co, and (C) the limit as x approaches P(x) = 9x® + 8x + 6x (A) The leading term of p(x) is (B) The limit of p(x) as x

Answers

(A) The leading term of the polynomial p(x) is 9x².

(B) The limit of p(x) as x approaches infinity is infinity.

(A) To find the leading term of a polynomial, we look at the term with the highest degree.

In the polynomial p(x) = 9x² + 8x + 6x, the term with the highest degree is 9x².

Therefore, the leading term of p(x) is 9x².

(B) To find the limit of a polynomial as x approaches infinity, we examine the behavior of the leading term.

Since the leading term of p(x) is 9x², as x becomes very large, the term 9x² dominates the polynomial.

As a result, the polynomial grows without bound, and the limit of p(x) as x approaches infinity is infinity.

In conclusion, the leading term of the polynomial p(x) is 9x², and the limit of p(x) as x approaches infinity is infinity.

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9. [-720 Points] DETAILS Find the indefinite integral. / (x+8XX1 -8x dx (x + 1) - V x + 1 Submit Answer

Answers

We are supposed to find the indefinite integral of the expression (x + 8)/(x + 1) - 8xV(x + 1)dx. Simplify the given expression as shown: The first part of the expression:(x + 8)/(x + 1) = (x + 1 + 7)/(x + 1) = 1 + 7/(x + 1).

Now, the expression will become:1 + 7/(x + 1) - 8xV(x + 1)dx.

To integrate this, let's take the first part and the second part separately.

The first part:∫1dx = x And, for the second part, let's use u substitution:u = x + 1 => x = u - 1.

Then, the second part becomes:-8∫(u - 1)Vudu= -8(∫u^(1/2)du - ∫u^(1/2)du)=-8(2/3)u^(3/2)+C=-16/3 (x+1)^(3/2) + C.

Now, combining the first part and second part, we get the final answer as x - 16/3 (x+1)^(3/2) + C, Where C is the constant of integration.

So, the required indefinite integral is x - 16/3 (x+1)^(3/2) + C.

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Hoy 19 de junio de 2022, Perú es uno de los países con mayor tasa de muertos por COVID-19; registra, según los últimos datos, 3 599 501 personas confirmadas de coronavirus, 1 635 más que el día anterior. ¿En qué porcentaje ha variado el contagio de COVID-19 con respecto al día de ayer?.

Answers

Para calcular el porcentaje de variación en el contagio de COVID-19 con respecto al día anterior en Perú, necesitamos calcular la diferencia en el número de personas confirmadas y expresarla como un porcentaje relativo al número de personas confirmadas del día anterior.

La diferencia en el número de personas confirmadas es 1 635 (3 599 501 - 3 597 866).

Para calcular el porcentaje de variación, dividimos la diferencia entre el número de personas confirmadas del día anterior y luego multiplicamos por 100 para obtener el porcentaje.

Porcentaje de variación = (Diferencia / Número anterior) * 100

Porcentaje de variación = (1 635 / 3 597 866) * 100

Porcentaje de variación = 0.0454 * 100

Porcentaje de variación = 4.54%

Por lo tanto, el contagio de COVID-19 en Perú ha aumentado en un 4.54% con respecto al día anterior.

Let f(x, y) = x3 +43 + 6x2 – 6y2 – 1. бу? 1 = List the saddle points A local minimum occurs at The value of the local minimum is A local maximum occurs at The value of the local maximum is

Answers

As a result, there are no values associated with the local minimum or local maximum.

To find the saddle points, local minimum, and local maximum of the function f(x, y) = x^3 + 43 + 6x^2 – 6y^2 – 1, we need to calculate the critical points and analyze their nature using the second derivative test.

First, let's find the partial derivatives of f(x, y) with respect to x and y:

∂f/∂x = 3x^2 + 12x

∂f/∂y = -12y

Next, we need to find the critical points by setting the partial derivatives equal to zero and solving the resulting equations simultaneously:

3x^2 + 12x = 0 ... (1)

-12y = 0 ... (2)

From equation (2), we have y = 0. Substituting this into equation (1), we get:

3x^2 + 12x = 0

Factoring out 3x, we have:

3x(x + 4) = 0

This gives two possible solutions: x = 0 and x = -4.

So, we have two critical points: (0, 0) and (-4, 0).

Now, let's calculate the second partial derivatives:

∂²f/∂x² = 6x + 12

∂²f/∂y² = -12

The mixed partial derivative is:

∂²f/∂x∂y = 0

Now, we can evaluate the second derivative test at the critical points.

For the critical point (0, 0):

D = (∂²f/∂x²)(∂²f/∂y²) - (∂²f/∂x∂y)^2

= (6(0) + 12)(-12) - 0^2

= -144

Since D < 0, this critical point does not satisfy the conditions of the second derivative test, so it is not a local minimum or local maximum.

For the critical point (-4, 0):

D = (∂²f/∂x²)(∂²f/∂y²) - (∂²f/∂x∂y)^2

= (6(-4) + 12)(-12) - 0^2

= -288

Since D < 0, this critical point does not satisfy the conditions of the second derivative test, so it is not a local minimum or local maximum.

Therefore, there are no local minimums or local maximums for the function f(x, y) = x^3 + 43 + 6x^2 – 6y^2 – 1.

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PLEASE HELP ME 40 POINTS!!! :)
Find the missing side

Answers

Answer:

18.8

Step-by-step explanation:

using angle 37° so that opposite side is x and adjacent is 25:

Tangent = O/A

tan 37 = x/25

x = 25 tan 37

= 18.8 to nearest tenth

2. It is known that for z = f(x,y): f(2,-5) = -7, fx (2,-5) = -and fy (2,-5) = Estimate f (1.97,-4.96). (3)

Answers

The estimated value of f at the point (1.97, -4.96) is approximately -7.01.

Using the given information, we know that f(2, -5) = -7 and the partial derivatives fx(2, -5) = - and fy(2, -5) = -. This means that at the point (2, -5), the function has a value of -7 and its partial derivatives with respect to x and y are unknown.To estimate the value of f at the point (1.97, -4.96), we can use the concept of linear approximation. The linear approximation of a function at a point is given by the equation:Δf ≈ fx(a, b)Δx + fy(a, b)Δy ,where Δf is the change in the function value, fx(a, b) and fy(a, b) are the partial derivatives at the point (a, b), and Δx and Δy are the changes in the x and y coordinates, respectively.

In our case, we can consider Δx = 1.97 - 2 = -0.03 and Δy = -4.96 - (-5) = 0.04. Plugging in the given partial derivatives, we have:Δf ≈ (-)(-0.03) + (-)(0.04)Simplifying this expression, we get:

Δf ≈ 0.03 - 0.04.Therefore, the estimated change in f at the point (1.97, -4.96) is approximately -0.01.To estimate the value of f at this point, we can add this change to the known value of f(2, -5):

f(1.97, -4.96) ≈ f(2, -5) + Δf

≈ -7 + (-0.01)

≈ -7.01

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Benjamin threw a rock straight up from a cliff that was 120 ft above the water. If the height of the rock h, in feet, after t seconds is given by the equation
h= - 16t^2 + 76t + 120. how long will it take for the rock to hit the water?

Answers

The rock will hit the water after approximately 4.75 seconds.

To find the time it takes for the rock to hit the water, we need to determine the value of t when the height h is equal to zero. We can set the equation h = -16t^2 + 76t + 120 to zero and solve for t.

-16t^2 + 76t + 120 = 0

To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula. In this case, let's use the quadratic formula:

t = (-b ± √(b^2 - 4ac)) / (2a)

Plugging in the values a = -16, b = 76, and c = 120 into the formula, we get:

t = (-76 ± √(76^2 - 4(-16)(120))) / (2(-16))

Simplifying the equation further, we have:

t = (-76 ± √(5776 + 7680)) / (-32)

t = (-76 ± √(13456)) / (-32)

Since we are interested in the time it takes for the rock to hit the water, we discard the negative value:

t ≈ (-76 + √(13456)) / (-32)

Evaluating this expression, we find t ≈ 4.75 seconds. Therefore, it will take approximately 4.75 seconds for the rock to hit the water.


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Find the missing side.
X
34° 12
X x = [?]
Round to the nearest tenth.
Remember: SOHCAHTOA

Answers

Answer: 8.1

Step-by-step explanation:

Tangent is opposite over adjacent.

tan(34)=x/12

0.6745=x/12

x=12*0.6745

x=8.0941

x=8.1

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Question * Consider the following double integral 1 - 2 - dy dx. By reversing the order of integration of I, we obtain: 1 = ²√²dx dy This option 1 = √ √4-y dx dy This option 1 = 4** dx dy O Th

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To find the reversed order of integration for the given double integral. This means we integrate with respect to x first, with limits from 0 to 2, and then integrate with respect to y, with limits y = [tex]\sqrt{4-x^{2} }[/tex].

To reverse the order of integration, we integrate with respect to x first and then with respect to y. The limits for the x integral will be determined by the range of x values, which are from 0 to 2.

Inside the x integral, we integrate with respect to y. The limits for y will be determined by the curve y = [tex]\sqrt{4-x^{2} }[/tex]. As x varies from 0 to 2, the corresponding limits for y will be from 0 to [tex]\sqrt{4-x^{2} }[/tex].

Therefore, the reversed order of integration is option I = [tex]\int\limits^\sqrt{(4-x)^{2} }} _0 \int\limits^2_{_0}[/tex] dx dy. This integral allows us to evaluate the original double integral I by integrating with respect to x first and then with respect to y.

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The complete question is:

consider the following double integral I= [tex]\int\limits^2_{_0}[/tex] [tex]\int\limits^\sqrt{(4-x)^{2} }}_0[/tex] dy dx  . By reversing the order of integration, we obtain:

a. [tex]\int\limits^2_{_0}[/tex][tex]\int\limits^\sqrt{(4-y)^{2} }}_0[/tex]dx dy

b. [tex]\int\limits^\sqrt{(4-x)^{2} }} _0 \int\limits^2_{_0}[/tex] dx dy

c. [tex]\int\limits^2_{_0}\int\limits^0_\sqrt{{-(4-y)^{2} }}[/tex] dx dy

d. None of these

Identify the probability density function.
f(x) = (the same function, in case function above, does not post with
question)
f(x) =
1
9
2
e−(x − 40)2/162, (−[infinity], [infinity])
Find t

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It is a Gaussian or normal distribution with mean μ = 40 and standard deviation σ = 9√2. The function represents the relative likelihood of the random variable taking on different values within the entire real number line.

The probability density function (PDF) describes the distribution of a continuous random variable. In this case, the given function f(x) = (1/9√2) e^(-(x - 40)^2/162) represents a normal distribution, also known as a Gaussian distribution. The function is characterized by its mean μ and standard deviation σ.

The function is centered around x = 40, which is the mean of the distribution. The term (x - 40) represents the deviation from the mean. The squared term in the exponent ensures that the function is always positive. The value 162 in the denominator determines the spread or variability of the distribution.

The coefficient (1/9√2) ensures that the total area under the curve of the PDF is equal to 1, fulfilling the requirement of a valid probability density function.

The range of the function is the entire real number line, as indicated by the interval (-∞, ∞). This means that the random variable can take on any real value, albeit with varying probabilities described by the function.

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The marginal cost to produce the xth roll of film 5 + 2a 1/x. The total cost to produce one roll is $1,000. What is the approximate cost of producing the 11th roll of film

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The approximate cost of producing the 11th roll of film can be calculated using the given marginal cost function and  total cost of producing one roll ($1,000) to obtain the approximate cost of the 11th roll of film.

The marginal cost function provided is 5 + 2a(1/x), where 'x' represents the roll number. The total cost to produce one roll is given as $1,000. To find the approximate cost of producing the 11th roll, we can substitute 'x' with 11 in the marginal cost function.

For the 11th roll, the marginal cost becomes 5 + 2a(1/11). Since the value of 'a' is not provided, we cannot determine the exact cost. However, we can still evaluate the expression by considering 'a' as a constant.

By substituting the value of 'a' as a constant in the expression, we can find the approximate cost of producing the 11th roll. The calculation of the expression would yield a numerical value that can be added to the total cost of producing one roll ($1,000) to obtain the approximate cost of the 11th roll of film.

Please note that without the value of 'a', we can only provide an approximate cost for the 11th roll of film.

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Consider the curve C1 defined by
a(t) = (2022, −3t, t)
where t∈R, and the curve
C2 :
S x2 + y2 = 1
lz z = 3y
a) Calculate the tangent vector to the curve C1 at the point α(π/2),
b) Parametricize curve C2 to find its binormal vector at the point (0,1,3).

Answers

The tangent vector to the curve C1 at the point α(π/2) is (-3,0,1) and the binormal vector of the curve C2 at the point (0,1,3) is (0.1047, 0.9597, 0.2593).

a) Calculation of the tangent vector to the curve C1 at the point α(π/2):

Let's differentiate the given curve to obtain its tangent vector at the point α(π/2).

a(t) = (2022, −3t, t)

Differentiating w.r.t t, we geta′(t) = (0, -3, 1)

Hence, the tangent vector to the curve C1 at the point α(π/2) is (-3,0,1).

b) Parametricizing the curve C2 to find its binormal vector at the point (0,1,3):

The given curve C2 isS [tex]x^2 + y^2 = 1[/tex]   ...(1) z = 3y   ...(2)

From equation (1), we get [tex]x^2 + y^2 = 1/S[/tex]    ...(3)

Using equation (2), we get [tex]x^2 + (z/3)^2 = 1/S[/tex]   ...(4)

Let's take the partial derivative of equations (3) and (4) w.r.t t.

[tex]x^2 + y^2 = 1[/tex] ... (5)

[tex]x^2 + (z/3)^2 = 1/S[/tex]   ...(6)

Differentiating both sides w.r.t t, we get

2x x′ + 2yy′ = 0   ...(7)

2x x′ + (2z/9)z′ = 0   ...(8)

Solving equations (7) and (8) simultaneously, we get

x′ = - (2z/9)z′    ... (9)y′ = x/3   ... (10)

Substituting (2) into (4), we get

[tex]x^2 + 1/3 = 1/S[/tex] => [tex]x^2 = 1/S - 1/3[/tex]

Substituting (2) and (3) in equation (1), we get

[tex](S - 9y^2/4) + y^2 = 1[/tex] => [tex]S = 9y^2/4 + 1[/tex]  ... (11)

Differentiating equation (11) w.r.t t, we get

S′ = 9y y′/2   ...(12)

We need to calculate the normal and tangent vectors to the curve C2 at the point (0,1,3).

Substituting t = 1 in equations (2), (3) and (4), we get the point (0, 1, 3/S) on the curve C2.

Substituting this point in equations (9) and (10), we get

x′ = 0  ... (13)y′ = 0.3333  ... (14)

From equation (12), we get

s′ = 6.75  ... (15)

The tangent vector to the curve C2 at the point (0,1,3) is the vector (0.3333, 0, -1).

The normal vector is the cross product of tangent vector and binormal vector, which can be calculated as follows.

Normal vector = (0.3333, 0, -1) × (k1, k2, k3)

where k1, k2, k3 are constants.

We know that the magnitude of a normal vector is always one. Using this condition, we can solve for k1, k2 and k3.(0.3333, 0, -1) × (k1, k2, k3) = (k2, -0.3333k1 - k3, 0.3333k2)

From the above equation, we have

k2 = 0, k1 = -k3/0.3333

Using the condition that the magnitude of the normal vector is 1, we have

(1 + k3/0.3333)1/2 = 1 => k3 = -0.0889

Hence, the normal vector to the curve C2 at the point (0,1,3) is (-0.2667, 0.0889, 0.9597).

The binormal vector is the cross product of the tangent and normal vectors at the point (0,1,3).

Binormal vector = (0.3333, 0, -1) × (-0.2667, 0.0889, 0.9597)= (0.1047, 0.9597, 0.2593)

Therefore, the tangent vector to the curve C1 at the point α(π/2) is (-3,0,1) and the binormal vector of the curve C2 at the point (0,1,3) is (0.1047, 0.9597, 0.2593).

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(-1)^2+1 = 1. 22n+1(2n + 1)! n=0 HINT: Which Maclaurin series is this? E

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The value of (-1)^2 + 1 is 2, and when n = 0, the expression 22n+1(2n + 1)! evaluates to 2. The hint regarding the Maclaurin series does not apply to these specific expressions.

The expression (-1)^2 + 1 can be simplified as follows:

(-1)^2 + 1 = 1 + 1 = 2.

So, the value of (-1)^2 + 1 is 2.

Regarding the second expression, 22n+1(2n + 1)! for n = 0, let's break it down step by step:

When n = 0:

22n+1(2n + 1)! = 2(2*0 + 1)! = 2(1)! = 2(1) = 2.

Therefore, when n = 0, the expression 22n+1(2n + 1)! evaluates to 2.

As for the hint mentioning the Maclaurin series, it seems unrelated to the given expressions. The Maclaurin series is a Taylor series expansion around the point x = 0. It is commonly used to approximate functions by representing them as infinite polynomials. However, in this case, the expressions do not involve any specific function or series expansion.

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In questions 1-3, find the area bounded by the graphs - show work thru integration. - y= 4 – x2 y = 2x – 4 on (-1,2)

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The area bounded by the graphs of y = 4 - x^2 and y = 2x - 4 on the interval (-1,2) can be found using integration.

To find the area bounded by the two given graphs, we need to determine the points of intersection first. Setting the equations equal to each other, we have:

4 - x^2 = 2x - 4

Rearranging the equation, we get:

x^2 + 2x - 8 = 0

Factoring the quadratic equation, we have:

(x + 4)(x - 2) = 0

This gives us two possible x-values: x = -4 and x = 2.

Next, we integrate the difference of the two functions between these x-values to find the area between the curves.

∫[a,b] (f(x) - g(x)) dx

Applying this formula, we integrate (4 - x^2) - (2x - 4) with respect to x from -1 to 2:

∫[-1,2] (4 - x^2) - (2x - 4) dx

Simplifying the integral, we get:

∫[-1,2] (8 - x^2 - 2x) dx

Evaluating this integral, we find the area between the curves:

[8x - (x^3/3) - x^2] evaluated from -1 to 2

After calculating the values, the area bounded by the graphs is determined.

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(1 point) Let 4 4 3.5 7 -3 x 1 -0.5 II IN z = 3 0.5 0 -21.5 Use the Gram-Schmidt process to determine an orthonormal basis for the subspace of R* spanned by x, y, and 2.

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The following are the steps to solve this problem using the Gram-Schmidt process:Step 1:Find the orthogonal basis for span{x, y, 2}.

Step 2:Normalize each vector found in step 1 to get an orthonormal basis for the subspace.Step 1:Find the orthogonal basis for span{x, y, 2}.Take x, y, and 2 as the starting vectors of the orthogonal basis. We'll begin with x and then move on to y and 2.Orthogonalizing x: $v_1 = x = \begin{bmatrix}4\\4\\3.5\\7\\-3\\1\\-0.5\end{bmatrix}$$u_1 = v_1 = x = \begin{bmatrix}4\\4\\3.5\\7\\-3\\1\\-0.5\end{bmatrix}$Orthogonalizing y: $v_2 = y - \frac{\langle y, u_1\rangle}{\lVert u_1\rVert^2}u_1 = y - \frac{(y^Tu_1)}{(u_1^Tu_1)}u_1 = y - \frac{1}{69}\begin{bmatrix}41\\30\\-35\\4\\15\\-10\\-10\end{bmatrix} = \begin{bmatrix}-\frac{43}{23}\\-\frac{10}{23}\\\frac{40}{23}\\\frac{257}{23}\\-\frac{183}{23}\\\frac{76}{23}\\\frac{46}{23}\end{bmatrix}$$u_2 = \frac{v_2}{\lVert v_2\rVert} = \begin{bmatrix}-\frac{43}{506}\\-\frac{10}{506}\\\frac{40}{506}\\\frac{257}{506}\\-\frac{183}{506}\\\frac{76}{506}\\\frac{46}{506}\end{bmatrix}$Orthogonalizing 2: $v_3 = 2 - \frac{\langle 2, u_1\rangle}{\lVert u_1\rVert^2}u_1 - \frac{\langle 2, u_2\rangle}{\lVert u_2\rVert^2}u_2 = 2 - \frac{2^Tu_1}{u_1^Tu_1}u_1 - \frac{2^Tu_2}{u_2^Tu_2}u_2 = \begin{bmatrix}\frac{245}{69}\\-\frac{280}{69}\\-\frac{1007}{138}\\\frac{2680}{69}\\-\frac{68}{23}\\\frac{136}{69}\\-\frac{258}{138}\end{bmatrix}$$u_3 = \frac{v_3}{\lVert v_3\rVert} = \begin{bmatrix}\frac{49}{138}\\-\frac{56}{69}\\-\frac{161}{138}\\\frac{536}{69}\\-\frac{34}{23}\\\frac{17}{69}\\-\frac{43}{138}\end{bmatrix}$

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An orthogonal basis for the column space of matrix A is {V1, V2, V3} Use this orthogonal basis to find a QR factorization of matrix A. Q=0.R=D (Type exact answers, using radicals as needed.) 25 - 2

Answers

The QR factorization of matrix A, given the orthogonal basis vectors, is Q = [5 0 1; -1 3 6; -4 3 9] and R = [0 18 15; 0 10 6; 0 0 r₃₃], where r₃₃ is the result of the projection calculation.

For the orthogonal basis for the colum space of Matrix :

Given matrix A and the orthogonal basis vectors:

A = [ 3 1 1;

6 9 2;

1 1 4 ]

v₁ = [ 5;

-1;

-4 ]

v₂ = [ 0;

3;

3 ]

v₃ = [ 1;

6;

9 ]

We can directly form matrix Q by arranging the orthogonal basis vectors as columns:

Q = [ v₁ v₂ v₃ ]

= [ 5 0 1;

-1 3 6;

-4 3 9 ]

Matrix R is an upper triangular matrix with diagonal entries representing the magnitudes of the projections of the columns of A onto the orthogonal basis vectors:

R = [ r₁₁ r₁₂ r₁₃ ;

0 r₂₂ r₂₃ ;

0 0 r₃₃ ]

To find the values of R, we can project the columns of A onto the orthogonal basis vectors:

r₁₁ = ||proj(v₁, A₁)||

r₁₂ = ||proj(v₁, A₂)||

r₁₃ = ||proj(v₁, A₃)||

r₂₂ = ||proj(v₂, A₂)||

r₂₃ = ||proj(v₂, A₃)||

r₃₃ = ||proj(v₃, A₃)||

Evaluating these projections, we get:

r₁₁ = ||proj(v₁, A₁)|| = ||(v₁⋅A₁)/(||v₁||²)v₁|| = ||(5*3 + (-1)*6 + (-4)*1)/(5² + (-1)² + (-4)²)v₁|| = ||0/v₁|| = 0

r₁₂ = ||proj(v₁, A₂)|| = ||(v₁⋅A₂)/(||v₁||²)v₁|| = ||(5*1 + (-1)*9 + (-4)*1)/(5² + (-1)² + (-4)²)v₁|| = ||-18/v₁|| = 18

r₁₃ = ||proj(v₁, A₃)|| = ||(v₁⋅A₃)/(||v₁||²)v₁|| = ||(5*1 + (-1)*2 + (-4)*4)/(5² + (-1)² + (-4)²)v₁|| = ||-15/v₁|| = 15

r₂₂ = ||proj(v₂, A₂)|| = ||(v₂⋅A₂)/(||v₂||²)v₂|| = ||(0*1 + 3*9 + 3*1)/(0² + 3² + 3²)v₂|| = ||30/v₂|| = 10

r₂₃ = ||proj(v₂, A₃)|| = ||(v₂⋅A₃)/(||v₂||²)v₂|| = ||(0*1 + 3*2 + 3*4)/(0² + 3² + 3²)v₂|| = ||18/v₂|| = 6

r₃₃ = ||proj(v₃, A₃)|| = ||(v₃⋅A₃)/(||v₃||²)v₃|| = ||(1*1 + 6*2 + 9*4)/(1² + 6² + 9²)v₃|| = ||59/v₃|| = 59/√(1² + 6² + 9²)

Calculating the value of the denominator:

√(1² + 6² + 9²) = √(1 + 36 + 81) = √118 = √(2⋅59) = √2⋅√59

Therefore, r₃₃ = 59/(√2⋅√59) = √2.

The resulting R matrix is:

R = [ 0 18 15 ;

0 10 6 ;

0 0 √2 ]

Hence, the QR factorization of matrix A, using the given orthogonal basis vectors, is:

Q = [ 5 0 1 ;

-1 3 6 ;

-4 3 9 ]

R = [ 0 18 15 ;

0 10 6 ;

0 0 √2 ]

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Evaluate. (Be sure to check by differentiating!) 5 (4€ - 9)e dt Determine a change of variables from t to u. Choose the correct answer below. OA. u=t4 O B. u = 41-9 OC. u=45 - 9 OD. u=14-9 Write the

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After differentiation 5(4t - 9)e dt the change of variables from t to u is: OD. u = (t + 9)÷4

To evaluate the integral [tex]\int[/tex] (5(4t - 9)e²t) dt and determine a change of variables from t to u, we can follow these steps:

Step 1: Evaluate the integral:

[tex]\int[/tex] (5(4t - 9)e²t) dt

To evaluate this integral, we can use integration by parts. Let's choose u = (4t - 9) and dv = 5e²t dt.

Differentiating u with respect to t, we get du = 4 dt.

Integrating dv, we get v = 5e²t.

Using the formula for integration by parts, the integral becomes:

[tex]\int[/tex] u dv = uv - [tex]\int[/tex] v du

Plugging in the values, we have:

[tex]\int[/tex] (5(4t - 9)e²t) dt = (4t - 9)(5e²t) - [tex]\int[/tex] (5e²t)(4) dt

Simplifying further:

[tex]\int[/tex] (5(4t - 9)e²t) dt = (20te²t - 45e²t) - 20[tex]\int[/tex] et dt

Integrating the remaining integral, we get:

[tex]\int[/tex]e²t dt = e²t

Substituting this back into the equation, we have:

[tex]\int[/tex] (5(4t - 9)e²t) dt = (20te²t - 45e²t) - 20(e²t) + C

Simplifying further:

[tex]\int[/tex] (5(4t - 9)e²t) dt = 20te²t - 65e²t + C

Step 2: Determine a change of variables from t to u:

To determine the change of variables, we equate u to 4t - 9:

u = 4t - 9

Solving for t, we get:

t = (u + 9)÷4

So, the correct answer for the change of variables from t to u is:

OD. u = (t + 9)÷4

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Let D be the region enclosed by the two paraboloids z = 3x² +² and z = 16-x²-2 Then the projection of D on the xy-plane is: None of these This option O This option +2=1 16

Answers

To determine the projection of the region D, enclosed by the two paraboloids z = 3x^2 + y^2 and z = 16 - x^2 - 2y^2, onto the xy-plane, we need to find the intersection curve of the two paraboloids in the xyz-space and project it onto the xy-plane.

To find the intersection curve, we set the two equations for the paraboloids equal to each other:

3x^2 + y^2 = 16 - x^2 - 2y^2

Simplifying this equation, we get:

4x^2 + 3y^2 = 16

This equation represents an ellipse in the xy-plane. By analyzing the equation, we can see that the major axis of the ellipse is aligned with the y-axis, and the minor axis is aligned with the x-axis. The equation indicates that the ellipse is centered at the origin with a major axis of length 4 and a minor axis of length 2.

Therefore, the projection of the region D onto the xy-plane is an ellipse centered at the origin, with a major axis of length 4 aligned with the y-axis and a minor axis of length 2 aligned with the x-axis.

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Consider the following initial-value problem. 8 f(x) = PR, 8(16) = 72 Integrate the function f'(x). (Remember the constant of integration.) | rx= 1 ) f'(x) dx Find the value of C using the condition f

Answers

We cannot determine the exact values of f'(16), C, and D without further information or additional conditions. To find the specific value of C, we would need more information about the function f'(x) or additional conditions beyond the initial condition f(16) = 72.

To find the value of C using the condition f(16) = 72, we need to integrate the function f'(x) and solve for the constant of integration.

Given that f(x) = ∫ f'(x) dx, we can find f(x) by integrating f'(x). However, since we are not provided with the explicit form of f'(x), we cannot directly integrate it.

To proceed, we'll use the condition f(16) = 72. This condition gives us a specific value for f(x) at x = 16. By evaluating the integral of f'(x) and applying the condition, we can solve for the constant of integration.

Let's denote the constant of integration as C. Then, integrating f'(x) gives us:

f(x) = ∫ f'(x) dx + C

Since we don't have the explicit form of f'(x), we'll treat it as a general function. Now, let's apply the condition f(16) = 72:

f(16) = ∫ f'(16) dx + C = 72

Here, we can treat f'(16) as a constant, and integrating with respect to x gives:

f(x) = f'(16) * x + Cx + D

Where D is another constant resulting from the integration.

Now, we can substitute x = 16 and f(16) = 72 into the equation:

72 = f'(16) * 16 + C * 16 + D

Simplifying this equation gives:

1152 = 16f'(16) + 16C + D

Since f'(16) and C are constants, we can rewrite the equation as:

1152 = K + 16C + D

Where K represents the constant term 16f'(16).

At this point, we cannot determine the exact values of f'(16), C, and D without further information or additional conditions. To find the specific value of C, we would need more information about the function f'(x) or additional conditions beyond the initial condition f(16) = 72.

In summary, to find the value of C using the condition f(16) = 72, we need more information or additional conditions that provide us with the explicit form or specific values of f'(x). Without such information, we can only express C as an unknown constant and provide the general form of the integral f(x).

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Find the seriesradius and interval of convergence. Find the values of x for which the series converges (b) absolutely and (c) conditionally (-1)0*x+7)Σ תלח n=1 (a)

Answers

(a) The series has a radius of convergence of 1 and an interval of convergence from -1 to 1.

(b) The series converges absolutely for x in the open interval (-1, 1) and at x = -1 and x = 1.

(c) The series converges conditionally for x = -1 and x = 1, but diverges for other values of x.

How is the radius of convergence and interval of convergence determined for the series?

The radius of convergence can be determined by applying the ratio test to the given series. In this case, the ratio test yields a radius of convergence of 1, indicating that the series converges for values of x within a distance of 1 from the center of the series.

The interval of convergence is determined by considering the behavior at the endpoints of the interval, which are x = -1 and x = 1. The series may converge or diverge at these points, so we need to analyze them separately.

How does the series behave in terms of absolute convergence within the interval?

Absolute convergence refers to the convergence of the series regardless of the sign of the terms. In this case, the series converges absolutely for values of x within the open interval (-1, 1), which means that the series converges for any x-value between -1 and 1, excluding the endpoints. Additionally, the series also converges absolutely at x = -1 and x = 1, meaning it converges regardless of the sign of the terms at these specific points.

How does the series behave in terms of conditional convergence?

Conditional convergence occurs when the series converges, but not absolutely. In this case, the series converges conditionally at x = -1 and x = 1, which means that the series converges if we consider the signs of the terms at these specific points. However, for any other value of x outside the interval (-1, 1) or excluding -1 and 1, the series diverges, indicating that it does not converge.

By understanding the radius and interval of convergence, as well as the concept of absolute and conditional convergence, we can determine the values of x for which the series converges absolutely or conditionally, providing insights into the behavior of the series for different values of x.

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Consider the initial value problem a b x₁ (t) (0) X10 [0]-[4][20] [28]-[x] = = (t) -b a (t) (0) X20 where a and b are constants. Identify all correct statements. When a 0, limt→+[infinity] (x² (t) + x²

Answers

The correct initial value for given problem are option b, c and d.

What is initial value?

The initial value means it is the number where the functiοn starts frοm. In οther wοrds, it is the number, tο begin with befοre οne adds οr subtracts οther values frοm it.

Here,

[tex]$$\begin{array}{r}X^{\prime}=A X \\A=\left[\begin{array}{cc}a & b \\-b & a\end{array}\right]\end{array}$$[/tex]

Let [tex]$\lambda$[/tex] be an eigenvalue, then

[tex]$$\begin{aligned}& {\det}\left(\begin{array}{cc}a-\lambda & b \\-b & a-\lambda\end{array}\right)=0 \\\Rightarrow & (a-\lambda)^2+b^2=0 \\\Rightarrow & (a-\lambda)^2=-b^2 \\\Rightarrow & a-\lambda= \pm i b \\\Rightarrow & \lambda .=a \pm i b\end{aligned}$$[/tex]

Then the eigenvector, for [tex]\lambda_1=a$-ib[/tex]

[tex]$$\begin{aligned}& {\left[\begin{array}{cc}i b & b \\-b & i b\end{array}\right]\left[\begin{array}{l}x \\y\end{array}\right]=\left[\begin{array}{l}0 \\0\end{array}\right] \Rightarrow i b x+b y=0 \text {. }} \\& \Rightarrow i x+y=0 \\& \Rightarrow y=-i x \\&\end{aligned}$$[/tex]

The eigenvector

[tex]$$V_1=\left[\begin{array}{c}1 \\-i\end{array}\right]$$\text {The eisenvedar for} $\lambda_2=a+i b$$$\left[\begin{array}{cc}-i s & b \\-b & i b\end{array}\right]\left[\begin{array}{l}x \\y\end{array}\right]=\left[\begin{array}{l}0 \\0\end{array}\right] \Rightarrow \begin{aligned}& -i b x+b y=0 \\& \Rightarrow y=i x\end{aligned}$$[/tex]

The eigenvector

[tex]$$v_2=\left[\begin{array}{l}1 \\i\end{array}\right]$$[/tex]

Then,

[tex]\rm If \ a < 0, \lim _{t \rightarrow \infty} x_1^2(t)+n_2^2(t)=\lim _{t \rightarrow \infty}\left(x_{10}^2+x_{20}^2\right) e^{2 a t}$$$[/tex]

[tex]\begin{aligned}& =\left(x_{10}^2+x_{\infty 0}^2\right) \lim _{t \rightarrow \infty} e^{2 a t} \\& =0\end{aligned}[/tex][tex]\quad \text { (As } a < 0 \text { ) }[/tex]

[tex]$$If $a > 0, \lim _{t \rightarrow \infty} x_1^2(t)+a_2^2(t)=\lim _{t \rightarrow a}\left(x_{10}^2+x_{20}^2\right) e^{2 a t}$$$[/tex]

[tex]=\left(x_{10}^2+x_{20}^2\right) \lim _{t \rightarrow 0} e^{2 a d}[/tex]

[tex]$$$$=\infty \quad \text { (As } a > 0 \text { ) }$$[/tex]

[tex]\text{If a}=0, \lim _{t \rightarrow 0} x_1^2(t)+a_2^2(t)=x_{10}^2+a_2^2 \lim _{t \rightarrow \infty} e^{2 a t}$$$[/tex]

[tex]=x_{10}^2+x_{20}^2$$[/tex]

For [tex]$a \neq 0 \quad \lim _{t \rightarrow 0} a_1^2(t)+x_2^2(t)$[/tex] does not depend on the initial condition.

Thus, option b, c and d are correct.

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