Find a formula involving integrals for a particular solution of the differential equation y^(4) – y =g(t). Hint: The functions sint, cost, sinht, and cosht form a fundamental set of solutions of the homogeneous equation.

The coefficient of x7 in the expansion of (3x + 4)10 is 53,248,000.

To find the coefficient of x^7 in the expansion of (3x + 4)^10 using the binomial theorem, we need to identify the term that has x^7.

The binomial theorem states that (a + b)^n = Σ (nCk) * a^(n-k) * b^k, where k goes from 0 to n and nCk denotes the binomial coefficient, which is the combination of choosing k items from n.

In our case, a = 3x, b = 4, and n = 10. We need to find the term with x^7, so the power of a (3x) should be 3 (since 3x raised to the power of 3 is x^7). This means the term will have the form:

10C3 * (3x)^3 * 4^(10-3)

Now we calculate the coefficients:

10C3 = 10! / (3! * (10 - 3)!) = 120
(3x)^3 = 27x^{7}
4^7 = 16384

Now, we multiply the coefficients together:

120 * 27 * 16384 = 53,248,000

Therefore, the coefficient of x^7 in the expansion of (3x + 4)^10 is 53,248,000.

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There are approximately 10.764 square feet in an area of 1.00 square metre. This conversion is a mathematical relation and is applicable in the physical universe.

In order to convert square meters to square feet, you can use the following conversion factor: 1 square meter is equal to 10.764 square feet. So, in an area of 1.00 square meters, there are approximately 10.764 square feet. This conversion is applicable in the physical universe.

The use of a unit depends on the context. For instance, the area of a room is measured in meters, but a pencil's length and thickness are measured in centimetres and millimeters, respectively.

As a result, we must convert from one unit to another. We must comprehend the relationship between units before we can comprehend the idea of unit conversion.

We need to convert between units in order to ensure accuracy and prevent measurement confusion. For example, we do not measure a pencil's length in kilometres. In this scenario, it is necessary to convert from kilometres (km) to centimetres (cm). In most cases, multiplicative conversion factors are used to convert one unit to another of the same quantity.

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Answer:

The first one is the right one

Step-by-step explanation:

The recursive sequence that produces the sequence 4, -14, 58, ... is given by:

a₁ = 4

aₙ = -4aₙ₋₁ - 2, for n ≥ 2

The Mean of the data is 87 secs. The Median is 99 secs.

The Range is 56 secs.

Given the data set for the number of secs that Braden ran in the 200-meter dash as: 56 sec, 99 sec, 112 sec, 56 sec, and 112 sec, first, order the data from lowest to highest.

56, 56, 99, 112, 112

Mean = sum of all data / number of data set = 435/5 = 87 secs.

Median = the middle data value which is 99 secs.

Mode = most appeared data value, thus, there is none that appeared the most. It means there is no mode.

Range = highest data value - lowest data value

= 56 secs.

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The marginal mean and variance of X, nor the covariance of X and Y, we cannot find the correlation coefficient.

To find the correlation coefficient, we first need to find the marginal means and variances of X and Y, as well as their covariance.

Marginal mean of X:

E[tex](X) = ∫∫ xf(x,y) dy dx[/tex]

[tex]= ∫2^∞ ∫0^1 x(16y/x^2) dy dx[/tex]

[tex]= ∫2^∞ [8x] dx[/tex]

= ∞ (diverges)

The integral diverges, so we cannot calculate the marginal mean of X.

Marginal mean of Y:

E [tex](Y) = ∫∫ yf(x,y) dy dx[/tex]

[tex]= ∫2^∞ ∫0^1 y(16y/x^2) dy dx[/tex]

[tex]= ∫2^∞ [8/x^2] dx[/tex]

[tex]= 4[/tex]

Marginal variance of X:

Var[tex](X) = E(X^2) - [E(X)]^2[/tex]

[tex]= ∫∫ x^2f(x,y) dy dx - [E(X)]^2[/tex]

[tex]= ∫2^∞ ∫0^1 x^2(16y/x^2) dy dx - ∞[/tex]

[tex]= ∫2^∞ [8x] dx - ∞[/tex]

= ∞ (diverges)

The integral diverges, so we cannot calculate the marginal variance of X.

Marginal variance of Y:

Var[tex](Y) = E(Y^2) - [E(Y)]^2[/tex]

[tex]= ∫∫ y^2f(x,y) dy dx - [E(Y)]^2[/tex]

[tex]= ∫2^∞ ∫0^1 y^2(16y/x^2) dy dx - 16[/tex]

[tex]= ∫2^∞ [8/x^2] dx - 16[/tex]

[tex]= 4 - 16/3[/tex]

[tex]= 4/3[/tex]

Covariance of X and Y:

Cov [tex](X,Y) = E(XY) - E(X)E(Y)[/tex]

[tex]= ∫∫ xyf(x,y) dy dx - ∞(4)[/tex]

[tex]= ∫2^∞ ∫0^1 xy(16y/x^2) dy dx - ∞(4)[/tex]

[tex]= ∫2^∞ [8x] dx - ∞(4)[/tex]

[tex]= ∞ - ∞(4)[/tex]

= -∞ (diverges)

The integral diverges, so we cannot calculate the covariance of X and Y.

Since we cannot calculate the marginal mean and variance of X, nor the covariance of X and Y, we cannot find the correlation coefficient.

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The value of surface area of the pyramid is 1/8yd² (option a).

To find the surface area of a square pyramid, we need to add up the area of all its faces.

In this case, we can see from the net that the two equal sides of each triangular face are each 1/2 yard long, and the height of the pyramid is also 1/2 yard. Therefore, the length of the hypotenuse of each triangular face is given by the square root of (1/2)² + (1/2)² = √(2)/2 yards.

The area of each triangular face can be found by multiplying the length of the base (which is also 1/2 yard) by the height (which is 1/2 yard) and then dividing by 2, since the area of a triangle is given by 1/2 times the base times the height.

Therefore, the area of each triangular face is (1/2 x 1/2)/2 = 1/8 square yards.

Since the pyramid has four triangular faces, the total area of all the triangular faces is 4 times 1/8 square yards.

Hence the correct option is (a).

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Answer:

  dw/dt = -cos(t)(2sin(t) +1)

Step-by-step explanation:

You want dw/dt for w = x² -y and x = cos(t), y = sin(t).

  w' = 2xx' -y' . . . . . . derivative with respect to t

  w' = 2cos(t)(-sin(t)) -cos(t) . . . . . substitute given relations

  dw/dt = -cos(t)(2sin(t) +1)

The marginal PDF of x is a function of x^2, and the conditional PDF of y given x=a is a function of y^2.

To find the marginal and conditional PDFs, we need to first determine the value of the constant c.

Since this is a joint PDF, it must satisfy the condition that the integral of the PDF over the entire domain equals 1. Therefore, we have:

integral from -inf to +inf of (integral from -inf to +inf of cx^2 * xy^2 dy)dx = 1

Simplifying this expression, we get:

integral from -inf to +inf of (c/3)x^5 dx = 1

Solving for c, we get:

c = 3/[(2/3)*(pi^2)]

Therefore, the joint PDF is:

f(x,y) = (3/[(2/3)*(pi^2)]) * x^2 * y^2

Now, we can find the marginal PDF of x by integrating f(x,y) over y from negative infinity to positive infinity:

f_x(x) = integral from -inf to +inf of f(x,y) dy = integral from -inf to +inf of (3/[(2/3)*(pi^2)]) * x^2 * y^2 dy

Simplifying this expression, we get:

f_x(x) = (3/[(2/3)*(pi^2)]) * x^2 * integral from -inf to +inf of y^2 dy

The integral of y^2 over the entire domain is equal to infinity, but we can still normalize the marginal PDF by dividing it by its integral over the entire domain. Therefore, we have:

f_x(x) = (3/(pi^2)) * x^2, for -inf < x < +inf

Next, we can find the conditional PDF of y given x = a by dividing the joint PDF by the marginal PDF of x evaluated at x = a:

f(y|x=a) = f(x,y) / f_x(a)

f(y|x=a) = [(2/3)(pi^2)] / (3a^2) * y^2, for 0 < y < +inf

Therefore, the marginal PDF of x is a function of x^2, and the conditional PDF of y given x=a is a function of y^2.

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The trapezoid ABCD have adjacent angles to be supplementary and values of the variable x = 4 while the measure of m∠D = 78°.

The adjacent angles of the the trapezium are supplementary, so their sum is equal to 180°.

m∠A and m∠D are supplementary so;

14x + 46 + 7x + 50 = 180°

21x + 96° = 180°

21x = 180° - 96° {subtract 96° from both sides}

x = 84°/21

x = 4

m∠D = 7(4) + 50

m∠D = 78°

Therefore, the trapezoid ABCD have adjacent angles to be supplementary and values of the variable x = 4 while the measure of m∠D = 78°.

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The area under the standard normal curve between z=-2.95 and z=2.61 is approximately 0.9942.

What is curve?

A curve is a geometrical object that is made up of points that are continuous and connected to form a line or a path. It can be defined mathematically by an equation or parametrically by a set of equations that describe the x and y coordinates of points on the curve as a function of a parameter such as time or distance along the curve.

Using a standard normal distribution table, we can find the area under the curve between z=-2.95 and z=2.61 as follows:

Area = Phi(2.61) - Phi(-2.95)

Where Phi(z) represents the cumulative distribution function of the standard normal distribution.

From the standard normal distribution table, we find:

Phi(2.61) = 0.9959

Phi(-2.95) = 0.0017

Therefore, the area under the curve between z=-2.95 and z=2.61 is:

Area = 0.9959 - 0.0017 = 0.9942

Rounding to four decimal places, we get:

Area ≈ 0.9942

Therefore, the area under the standard normal curve between z=-2.95 and z=2.61 is approximately 0.9942.

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To approximate the sum of the series with an error less than 0.0001, we need to find the partial sum up to the value of n found in part a. From part a, we know that n = 9.

So, we need to find the sum of the first 9 terms of the series. Using the formula for the nth term of the series, we can write:
an = 1/(n*(n+1))

So, the first few terms of the series are:

a1 = 1/2
a2 = 1/6
a3 = 1/12
a4 = 1/20
a5 = 1/30
a6 = 1/42
a7 = 1/56
a8 = 1/72
a9 = 1/90

To find the sum of the first 9 terms, we can simply add these terms:

s9 = a1 + a2 + a3 + a4 + a5 + a6 + a7 + a8 + a9
s9 = 0.5 + 0.1667 + 0.0833 + 0.05 + 0.0333 + 0.0238 + 0.0179 + 0.0125 + 0.0111
s9 = 0.8893

To ensure that our approximation has an error less than 0.0001, we need to check the error term. We know that the error term for the nth partial sum is bounded by the (n+1)th term of the series. So, in this case, the error term is bounded by a10:

a10 = 1/110

We want the error to be less than 0.0001, so we need:

a10 < 0.0001
1/110 < 0.0001

Therefore, we know that s9 is an approximation of the actual sum of the series with an error of less than 0.0001.

Rounding s9 to 4 decimal places, we get:
s9 = 0.8893

So, the sum of the series with an error less than 0.0001 is approximately 0.8893.

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True. This can be determined using the formula for calculating the z-score: Z = (X - M) / (S / sqrt(n)), where X is the score, M is the mean, S is the sample standard deviation, and n is the sample size. Substituting the given values, we get:

Z = (72 - 76) / (8 / sqrt(1)) = -0.5

Therefore, a score of X = 72 corresponds to Z = -0.50, given that the sample has a mean of M = 76 and a sample standard deviation of S = 8.
True. Given a sample with a mean (M) of 76 and a sample standard deviation (S) of 8, you can calculate the Z-score for a score of X = 72 using the formula:

Z = (X - M) / S

Z = (72 - 76) / 8

Z = (-4) / 8

Z = -0.50

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The probability approximation for P(p≥0.14) is 0.0495

(a) The mean of the sample proportion p is equal to the population proportion, which is 0.12:

μp = 0.12

(b) The standard deviation of the sample proportion p can be calculated as:

σp = sqrt[(0.12(1-0.12))/1650]

  = 0.0121

Therefore, the standard deviation of the sample proportion p is 0.0121.

(c) To compute an approximation for P(p≥0.14), we can use the central limit theorem and assume that the distribution of the sample proportion p is approximately normal.

The mean and standard deviation of the sample proportion have already been calculated in parts (a) and (b).

z = (0.14 - 0.12) / 0.0121

 = 1.65

Using a standard normal distribution table or calculator, the probability that a standard normal random variable is greater than or equal to 1.65 is approximately 0.0495.

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Answer:

VU and TU

Step-by-step explanation:

the marked angle between the lines VU and TU is ∠ VUT or ∠ TUV

that is the 2 lines forming the angle between them

Answer:

VU and TU

Step-by-step explanation:

i did this and the rest of it to

Answer:

2(2.5) + 2(4.5) + 2(1.75) + 3.25

= 5 + 9 + 3.5 + 3.25 = 14 + 6.75 = 20.75

= 20 3/4 feet

The length of the wall is 20 3/4 feet.

Sure, here are the steps to solve this problem:

1. Since the scale of the blueprint is 1:40, it means that any 1 unit on the blueprint represents 40 units on the actual building.

2. The room on the building measures 3.4 m by 4.8 m.

3. So for the dimensions of the room on the blueprint, we divide the measurements by the scale ratio.

4. 1:40 scale means 1 unit = 40 units.

5. So,

3.4 m / 40 units = 0.085 units = 0.08 units (round to 0.08 units)

4.8 m / 40 units = 0.12 units

6. Therefore, the room on the blueprint measures 0.08 units by 0.12 units.

Let me know if this explanation helps or if you have any other questions! I'm happy to help further.

step-by-step:

Room dimensions on building: 3.4 m by 4.8 m

Scale of blueprint: 1 : 40

Step 1) 1 unit on blueprint = 40 units on building

Step 2) 3.4 m / 40 units = 0.085 units (round to 0.08 units)

Step 3) 4.8 m / 40 units = 0.12 units

Step 4) Room dimensions on blueprint = 0.08 units by 0.12 units

Does this help explain the steps? Let me know if any part is still confusing!

(a) The closest z-value to 0.2119 is -0.81, so c = -0.81.

(b) The closest z-value to 0.9515 is 1.43, so c = 1.43 or -1.43.

(c) The closest z-value to 0.9948 is 2.62, so c = 2.62.

(d) The closest z-value to 0.2546 is -0.53, so c = 0.53 or -0.53.

(a) For a standard normal distribution, we can find the value of c such that P(z < c) = 0.2119 using a standard normal distribution table or calculator. From the table, we can see that the closest probability value to 0.2119 is 0.2119 = 0.5893 - 0.3771.

This corresponds to z = -0.81 (the closest z-value to 0.2119 is -0.81), so c = -0.81.

(b) For a standard normal distribution, we can find the value of c such that P(-c < z < c) = 0.9030 using symmetry.

Since the distribution is symmetric about the mean, P(-c < z < c) = 2P(z < c) - 1 = 0.9030. Solving for P(z < c), we get P(z < c) = (1 + 0.9030)/2 = 0.9515.

From the standard normal distribution table or calculator, we find that the closest probability value to 0.9515 is 0.9515 = 0.3450 + 0.6064.

This corresponds to z = 1.43 (the closest z-value to 0.9515 is 1.43), so c = 1.43 or -1.43.

(c) Similarly, for P(z < c) = 0.9948, we find the closest probability value in the standard normal distribution table or calculator to be 0.9948 = 0.4999 + 0.4948.

This corresponds to z = 2.62 (the closest z-value to 0.9948 is 2.62), so c = 2.62.

(d) For P(z > c) = 0.6915, we can use symmetry to find the value of c. Since the distribution is symmetric about the mean, P(z > c) = P(z < -c) = 0.6915.

From the standard normal distribution table or calculator, we find that the closest probability value to 0.6915 is 0.6915 = 0.2546 + 0.4364.

This corresponds to z = -0.53 (the closest z-value to 0.2546 is -0.53), so c = 0.53 or -0.53.

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Answer: Option B: 67.0

Step-by-step explanation: To find Q3, we need to first find the median (Q2) of the dataset.

Arranging the data in order, we get:

49, 52, 52, 55, 55, 67, 74

The median (Q2) is the middle value of the dataset, which is 55.

Next, we need to find the median of the upper half of the dataset, which consists of the values:

55, 67, 74

The median of this upper half is 67.

Therefore, Q3 (the third quartile) is 67.0, option B.

Okay, let's break this down step-by-step:

* The shipping crate holds up to 24 cubic feet of space.

* The box measures:

Width: 2ft 1 1/2in = 2.75ft

Length: 1 1/2ft = 1.5ft

Height: 0.8ft

* To convert to cubic feet:

Width x Length x Height = (2.75ft) x (1.5ft) x (0.8ft) = 4.2 cubic feet

* So each box takes up 4.2 cubic feet of space.

* To fill the 24 cubic feet in the crate:

24 cubic feet / 4.2 cubic feet per box = 5 boxes

Therefore, the shipping crate can hold up to 5 of those rectangular boxes.

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The work done in dragging the air compressor up the incline is 4,168.24 J.

To solve this problem, we need to determine the work done in dragging the air compressor up the incline.

First, we need to determine the change in height of the compressor:

Δy = y2 - y1

Δy = 4.4 m - 1.3 m

Δy = 3.1 m

Next, we need to determine the work done against gravity in lifting the compressor:

W_gravity = mgh

W_gravity = (25 kg)(9.81 m/s^2)(3.1 m)

W_gravity = 765.98 J

Finally, we need to determine the work done against friction in dragging the compressor:

W_friction = μmgd

where μ is the coefficient of kinetic friction, g is the acceleration due to gravity, and d is the distance moved.

We can assume that the compressor is moved at a constant speed, so the work done against friction is equal to the work done by the applied force.

To find the applied force, we can use the fact that the net force in the x-direction is zero:

F_applied,x = F_friction,x

F_applied,x = μmgcosθ

where θ is the angle of the incline (measured from the horizontal) and cosθ = (r2 - r1)/d.

d = |r2 - r1| = √[(8.3 m - 1.3 m)² + (4.4 m - 1.3 m)²]

d = 8.24 m

cosθ = (r2 - r1)/d

cosθ = [(8.3 m - 1.3 m)/8.24 m]

cosθ = 0.888

μ = F_friction,x / (mgcosθ)

μ = F_applied,x / (mgcosθ)

μ = (F_net,x - F_gravity,x) / (mgcosθ)

μ = (0 - mg(sinθ)) / (mgcosθ)

μ = -tanθ

where sinθ = (Δy / d) = (3.1 m / 8.24 m) = 0.376.

μ = -tanθ = -(-0.376) = 0.376

F_applied = F_net = F_gravity + F_friction

F_applied = F_gravity + μmg

F_applied = mg(sinθ + μcosθ)

F_applied = (25 kg)(9.81 m/s^2)(0.376 + 0.376(0.888))

F_applied = 412.58 N

W_friction = F_appliedd

W_friction = (412.58 N)(8.24 m)

W_friction = 3,402.26 J

Therefore, the total work done in dragging the compressor up the incline is:

W_total = W_gravity + W_friction

W_total = 765.98 J + 3,402.26 J

W_total = 4,168.24 J

So the work done in dragging the air compressor up the incline is 4,168.24 J.

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To find the elasticity of the demand function 2p + 3q = 90 at the price p = 15, we need to first solve for q at that price level.

2(15) + 3q = 90

30 + 3q = 90

3q = 60

q = 20

So, at a price level of p = 15, the quantity demanded is q = 20.

Next, we need to find the derivative of the demand function with respect to price:

dQ/dp = -2/3

Then, we can use the formula for elasticity:

Elasticity = (dQ/dp) * (p/Q)

Elasticity = (-2/3) * (15/20)

Elasticity = -0.5

Therefore, the elasticity of the demand function 2p + 3q = 90 at the price p = 15 is -0.5.
To find the elasticity of the demand function 2p 3q = 90 at the price p = 15, we need to first find the corresponding quantity (q) and then calculate the price elasticity of demand.

Step 1: Solve for q in terms of p
2p 3q = 90
3q = 90 - 2p
q = (90 - 2p) / 3

Step 2: Substitute p = 15 into the equation
q = (90 - 2(15)) / 3
q = (90 - 30) / 3
q = 60 / 3
q = 20

Now we have the point (p, q) = (15, 20) on the demand curve.

Step 3: Differentiate the demand function with respect to p
dq/dp = -2/3

Step 4: Calculate the price elasticity of demand (E)
E = (dq/dp) * (p/q)
E = (-2/3) * (15/20)

E = -0.5

The elasticity of the demand function 2p 3q = 90 at the price p = 15 is -0.5.

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If X is a continuous random variable having the probability density:

f(x)={2(x−1) 1 0 elsewhere then the variance of X is 1/6.

To find the variance of X, we need to use the formula:

Var(X) = [tex]E(X^2)[/tex] - [tex][E(X)]^2[/tex]

To find the expectation E(X) and E(X^2) for the continuous random variable X with the given probability density function (pdf), we integrate the respective expressions over the entire support of the random variable.

Given the pdf:

f(x) = { 2(x - 1), 1, 0 elsewhere }

We can calculate E(X) as follows:

E(X) = ∫x*f(x) dx

      = ∫x*2(x - 1) dx

      = 2∫([tex]x^2[/tex] - x) dx

      = 2[([tex]x^3[/tex]/3) - ([tex]x^2[/tex]/2)] evaluated from 0 to 1

      = 2[(1/3) - (1/2) - (0 - 0)]

      = 2[(1/3) - (1/2)]

      = 2[-1/6]

      = -1/3

Similarly, we can calculate E(X^2) as follows:

E(X^2) = ∫[tex]x^2[/tex]*f(x) dx

         = ∫[tex]x^2[/tex]*2(x - 1) dx

         = 2∫([tex]x^3[/tex] - [tex]x^2[/tex]) dx

         = 2[([tex]x^4[/tex]/4) - ([tex]x^3[/tex]/3)] evaluated from 0 to 1

         = 2[(1/4) - (1/3) - (0 - 0)]

         = 2[(1/4) - (1/3)]

         = 2[1/12]

         = 1/6

Therefore, the expectation E(X) is -1/3 and E(X^2) is 1/6 for the given continuous random variable X with the specified pdf.

Therefore, the variance of X is 1/6.

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1. The inverse function, f^(-1)(x) = (x + 2)/5.

2. The derivative of the inverse function, (f^(-1))'(x) = 1/5.

3. (f^(-1))'(3) = 1/5.

We know that a function is invertible if and only if it is one-to-one and onto. In this case, we can easily see that f(x) is a one-to-one function because different inputs always give different outputs, and it is also onto because any real number can be obtained as an output. Therefore, f(x) is invertible.

To find (f^-1)'(3), we need to use the formula for the derivative of the inverse function:

(f^-1)'(3) = 1 / f'(f^-1(3))

First, we need to find f^-1(x). We can do this by solving the equation y = 5x - 2 for x in terms of y:

y = 5x - 2

y + 2 = 5x

x = (y + 2) / 5

Therefore, f^-1(x) = (x + 2) / 5.

Now we can find f'(x):

f(x) = 5x - 2

f'(x) = 5

Next, we need to find f^-1(3):

f^-1(3) = (3 + 2) / 5 = 1

Finally, we can use the formula to find (f^-1)'(3):

(f^-1)'(3) = 1 / f'(f^-1(3)) = 1 / f'(1) = 1 / 5

Therefore, the answer is b) 1/15.

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Answer:

[tex](-8, 15)[/tex]

Step-by-step explanation:

First, solve this system. Since [tex]y=y[/tex],

[tex]2x-1 = 2(x-1)+1\\2x-1=2x-2+1\\2x-1=2x-1[/tex]

Thus, they are the same equation.

Now, plug in to find:

[tex]-1 = 2(0)-1 (correct)\\3 = 2(2) -1 (correct)\\-5 = 2(-2) - 1 (correct)\\15 \neq 2(-8) - 1 (incorrect)\\15 = 2(8) - 1 (correct)[/tex]

Thus [tex](-8, 15)[/tex] is the ordered pair that doesn't work.

The final expression of integral ∫(x²+1)/[(x-5)(x-4)²] dx  is

= -1/9 ln|x-4| - 1/9(x-4)⁻¹ + C

To evaluate the integral ∫(x²+1)/[(x-5)(x-4)²] dx, we can use partial fraction decomposition and then integrate each term separately:

First, we decompose the rational function into partial fractions:

(x²+1)/[(x-5)(x-4)²] = A/(x-5) + B/(x-4) + C/(x-4)²

Multiplying both sides by the denominator and simplifying, we get:

x² + 1 = A(x-4)²+ B(x-5)(x-4) + C(x-5)

Expanding the right-hand side and equating coefficients, we get:

Therefore, the partial fraction decomposition of the rational function is:

(x²+1)/[(x-5)(x-4)²] = -1/9/(x-4) + 1/9/(x-4)²

The integral now becomes:

∫(x²+1)/[(x-5)(x-4)²] dx = -1/9∫1/(x-4) dx + 1/9∫1/(x-4)² dx

Integrating each term separately, we get:

where C1 and C2 are constants of integration.

Substituting these values back into the original integral, we get:

∫(x²+1)/[(x-5)(x-4)²] dx = -1/9ln|x-4| + 1/9(-1/(x-4)) + C

Simplifying further, we get:

∫(x²+1)/[(x-5)(x-4)²] dx = -1/9 ln|x-4| - 1/9(x-4)⁻¹ + C

where C is a constant of integration.

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The value of the integral is π/16 - (π/4sqrt(2)).

To convert the integral to polar coordinates, we need to express x and y in terms of r and θ. The region of integration is the area under the curve √(4-x^2), which is a semicircle with radius 2 centered at the origin, and above the x-axis. This region can be described as:

0 ≤ θ ≤ π (since we are integrating over the upper semicircle)

0 ≤ r ≤ 2cos(θ) (since r ranges from 0 to 2 and x = rcos(θ))

So, the integral in polar coordinates becomes:

∫(from θ=0 to π) ∫(from r=0 to 2cos(θ)) √(4-r^2cos^2(θ)) e^(-r^2) r dr dθ

To evaluate this integral, we first integrate with respect to r:

∫(from θ=0 to π) [- e^(-r^2)/2 √(4-r^2cos^2(θ))] (from r=0 to 2cos(θ)) dθ

= ∫(from θ=0 to π) [- (1/2) e^(-4cos^2(θ)) + (1/2) e^(-r^2)cos^2(θ)] dθ

We can now integrate with respect to θ:

= [- (1/2) ∫(from θ=0 to π) e^(-4cos^2(θ)) dθ] + [(1/2) ∫(from θ=0 to π) e^(-r^2)cos^2(θ) dθ]

The first integral is a bit tricky, but can be evaluated using a well-known result from calculus called the Gaussian integral:

∫(from θ=0 to π) e^(-4cos^2(θ)) dθ = π/2sqrt(2)

For the second integral, we use the fact that cos^2(θ) = (1/2)(1+cos(2θ)):

(1/2) ∫(from θ=0 to π) e^(-r^2)cos^2(θ) dθ = (1/4) ∫(from θ=0 to π) e^(-r^2)(1+cos(2θ)) dθ

= (1/4) [∫(from θ=0 to π) e^(-r^2) dθ + ∫(from θ=0 to π) e^(-r^2)cos(2θ) dθ]

The first integral evaluates to π/2, while the second integral evaluates to 0 (since the integrand is an odd function of θ). Therefore:

(1/2) ∫(from θ=0 to π) e^(-r^2)cos^2(θ) dθ = (1/8) π

Substituting these results back into the original integral, we get:

integral^2_-2 integral √(4-x^2) e-x^2-y^2 dy dx = [- (1/2) (π/2sqrt(2))] + [(1/2) (1/8) π]

= - (π/4sqrt(2)) + (π/16)

= π/16 - (π/4sqrt(2))

So the value of the integral is π/16 - (π/4sqrt(2)).

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we need to add at least 12 terms to find the sum of the series to within 0.01.

To find the sum of the series [infinity] 1 [n(1 ln n)3] n = 1 within 0.01, we need to use the Cauchy condensation test.

First, we need to check the convergence of the series. We can use the integral test:

[tex]\int_1^{oo}{x(lynx)^3}dx[/tex]
[tex]=\int u^3du\\\\= (\frac{1}{4}) u^4 + C\\\\= (\frac{1}{4}) [1 ln x]^4 + C[/tex]
As x approaches infinity, the integral converges, and therefore, the series also converges.

Now, using the Cauchy condensation test, we have:

[tex]2^n [2^n (1 ln 2^n)3]\\\\= 2^{4n} [(n ln 2)3]\\\\= (8 ln 2)3 (n ln 2)3\\\\= (8 ln 2)^3 [\frac{1}{2}^{3n}}] [(n ln 2)^3]\\\\[/tex]

The series [infinity][tex](8 ln 2)^3 [\frac{1}{2}^{3n}] [(n ln 2)^3] n = 1[/tex]converges, and its sum is equal to[tex]\frac{ [(8 ln 2)^3]}{[2^3 - 1]}.[/tex]

We can use the error formula for alternating series to estimate how many terms we need to add to find the sum to within 0.01:

[tex]error \leq a_{(n+1)}[/tex]

where [tex]a_n = (8 ln 2)^3 [{1/2}^{3n}] [(n ln 2)3][/tex]

Let's solve for n:

[tex]0.01 \leq a_{(n+1)}\\0.01 \leq (8 ln 2)^3 [1/2^{(3(n+1))}] [(n+1) ln 2]3[/tex]

n ≥ 11.24

Therefore, we need to add at least 12 terms to find the sum of the series to within 0.01.

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