Question 3 dx Find the particular solution of om - 2x + 5y = e-** given that y(0) = 0 and y'(o) day dx? e3= using the method of undetermined coefficients. [15]

a) To find the value of c for which the given vectors are linearly dependent, we need to check if the determinant of the matrix formed by the vectors is zero.

b) To express the first vector as a linear combination of the other two, we need to find the scalars that satisfy the equation: (1 1 c) = α(-10 -1) + β(2 1 2), where α and β are the scalars.

a) For the vectors (1 1 c), (-10 -1), and (2 1 2) to be linearly dependent, the determinant of the matrix formed by these vectors should be zero. Setting up the determinant equation, we have:

| 1 1 c |

|-10 -1 0 |

| 2 1 2 |

Expanding the determinant, we get:

1(-12 - 10) - 1(-102 - 20) + c(-10*1 - (-1)*2) = 0.

Simplifying the equation, we have:

-2 + 20 + 12c = 0,

12c = -18,

c = -18/12,

c = -3/2.

Therefore, the value of c for which the given vectors are linearly dependent is c = -3/2.

b) To express the first vector (1 1 c) as a linear combination of the other two vectors (-10 -1) and (2 1 2), we need to find the scalars α and β that satisfy the equation:

(1 1 c) = α(-10 -1) + β(2 1 2).

Expanding the equation, we have:

1 = -10α + 2β,

1 = -α + β,

c = -α + 2β.

Solving these equations simultaneously, we find:

α = 1/12,

β = 13/12.

Therefore, the first vector (1 1 c) can be expressed as a linear combination of the other two vectors as:

(1 1 c) = (1/12)(-10 -1) + (13/12)(2 1 2).

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The exact value of f(3) is f(3) = e^(15) – e^(5) + 3

To find f(3) in the equation f(x) = e^(5x) – e^(x+2) + log(e^3), we simply substitute x = 3 into the equation.

f(3) = e^(5(3)) – e^(3+2) + log(e^3)

Simplifying the exponents:

f(3) = e^(15) – e^(5) + log(e^3)

Since e^x is the base of the natural logarithm, log(e^3) simplifies to 3.

f(3) = e^(15) – e^(5) + 3

This is the exact value of f(3) in the given equation.

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the simplified expression for tan θ cos θ csc θ is 1.

To express the given expression in terms of sine and cosine and simplify it, we'll start by rewriting the trigonometric functions in terms of sine and cosine:

tan θ = sin θ / cos θ

csc θ = 1 / sin θ

Substituting these expressions into the original expression, we have:

tan θ cos θ csc θ = (sin θ / cos θ) * cos θ * (1 / sin θ)

The cos θ term cancels out with one of the sin θ terms, giving us:

tan θ cos θ csc θ = sin θ * (1 / sin θ)

Simplifying further, we find:

tan θ cos θ csc θ = 1

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The power series representation for f(x) when the index variable of the summation n = 0, is Σ((-1)^(n+2) * (x-3)^(n+2))/(n+2) from n=0 to ∞.

To find the power series representation for f(x), we start by recognizing that f(x) is equal to the sum of terms with coefficients (-1)^(n+2) and powers of (x-3) raised to (n+2). This suggests using a power series of the form Σ(c_n * (x-a)^n), where c_n represents the coefficients and (x-a) represents the power of x.

By substituting a=3, we obtain Σ((-1)^(n+2) * (x-3)^(n+2))/(n+2), where the index variable n starts from 0 and the summation extends to infinity. This power series provides an approximation of f(x) in terms of the given coefficients and powers of (x-3).

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The height of Jane's shadow who is 5.9 feet tall is appoximately 6.3 feet

Given that, a tree 54 feet tall casts a shadow 58 feet long and Jane is 5.9 feet tall.

To find the height of Jane's shadow, we can use proportions and ratios.

Hence:

(Height of the tree) : (Length of the tree's shadow) = (Height of Jane) : (Length of Jane's shadow)

Plug in:

Height of the tree = 54

Length of the tree's shadow = 58

Height of Jane = 5.9

Let Length of Jane's shadow = x

54 feet : 58 feet = 5.9 feet : x

54/58 = 5.9/x

Cross multiply:

54 × x = 58 × 5.9

54x = 342.2

x = 342.2/54

x = 6.3 feet

Therefore, the measure of her shadow is approximately 6.3 feet.

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The limit of f(x) as x approaches 3 from the left is undefined. This is because the function f(x) is not defined for values of x less than 3.

In the given function, f(x) takes different forms depending on the value of x. For x values between 0 and 3, f(x) is defined as the square root of (9 - x^2). However, as x approaches 3 from the left, the function is not defined for x values less than 3.

Therefore, we cannot determine the value of f(x) as x approaches 3 from the left.

In summary, the limit of f(x) as x approaches 3 from the left is undefined because the function is not defined for values of x less than 3.

This means that we cannot determine the value of f(x) as x approaches 3 from the left because it is not specified in the given function.

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Answer:

3: 16

Step-by-step explanation:

A ratio has two or more numbers that symbolize relation to each other. Ratios are used to compare numbers, and you can compare them using division.

If 16a = 3b, then:

This means that the ratio a: b is equivalent to the ratio 3: 16.

Therefore, the ratio a: b is 3:16.

Given that Lisa earns a salary of $11.40 per hour at the video rental store and she is paid weekly. Occasionally, she has to work overtime for more than 50 hours but less than 60 hours. For working overtime she is paid at 1.5 times the hourly rate.

When Lisa works overtime, she is paid at 1.5 times her hourly rate for each hour of overtime she works. Since she earns $11.40 per hour, her overtime rate will be:$11.40 x 1.5 = $17.10

Therefore, for each overtime hour, Lisa will be paid $17.10 per hour. Since Lisa works more than 50 hours but less than 60 hours,

we can calculate her overtime pay by using the following formula:

Total overtime pay = (Total overtime hours) x (Overtime pay rate)Total overtime hours = Number of overtime hours worked - 50Total overtime pay = ((Number of overtime hours worked - 50) x $17.10)Let's say Lisa works 55 hours in a week. This means she worked 5 hours of overtime.

Therefore, her overtime pay will be:Total overtime pay = ((55 - 50) x $17.10)Total overtime pay = (5 x $17.10)Total overtime pay = $85.50Hence, Lisa earns $85.50 in overtime pay when she works 55 hours a week.

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After using the limit definition of the derivative, the answer comes as 6x.

The function is f(x) = 3x² - x + 1.

We have to find the derivative of the function using the limit definition of the derivative, f'(x) = limax-0 f( x+ Ax )-f(a).

So, we know that the limit definition of the derivative, f'(x) = limax-0 f(x+ Ax)-f(a) / Ax

By substituting the given values in the above formula, we get; f'(x) = lim Ax-0 {f(x + Ax) - f(x)} / Ax

Now, let us find the derivative of the given function.

Substitute the values in the above formula; f'(x) = lim Ax-0 {f(x + Ax) - f(x)} / Axf'(x) = lim Ax-0 {[3(x + Ax)² - (x + Ax) + 1] - [3x² - x + 1]} / Axf'(x) = lim Ax-0 {[3(x² + 2xAx + A²) - x - Ax + 1] - [3x² - x + 1]} / Axf'(x) = lim Ax-0 {[3x² + 6xAx + 3A² - x - Ax + 1] - [3x² - x + 1]} / Axf'(x) = lim Ax-0 {[6xAx + 3A²] / A}f'(x) = lim Ax-0 {6x + 3Ax}f'(x) = lim Ax-0 {6x} + lim Ax-0 {3Ax}f'(x) = 6x + 0f'(x) = 6xTherefore, the derivative of f(x) = 3x² - x + 1 is f'(x) = 6x.

Answer: f'(x) = 6x.

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The correct solution obtained using the method of variation of parameters for the nonhomogeneous differential equation with W = e^(3t), W2 = -e^t, and W = 27 is U = -5e^(3t) + 2e^t.

The method of variation of parameters is a technique used to solve nonhomogeneous linear differential equations. It involves finding a particular solution by assuming it can be expressed as a linear combination of the solutions to the corresponding homogeneous equation, multiplied by unknown functions known as variation parameters.

In this case, we have W = e^(3t) and W2 = -e^t as the solutions to the homogeneous equation. By substituting these solutions into the formula for the particular solution, we can find the values of the variation parameters.

After determining the particular solution, the general solution to the nonhomogeneous differential equation is obtained by adding the particular solution to the general solution of the homogeneous equation

Hence, the correct solution is U = -5e^(3t) + 2e^t.

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The absolute maximum of f(x) = -3 sin(x) over the interval (0, 5) occurs at x = 5, and the absolute minimum occurs at x = 0.

to find the absolute maximum and minimum of the function f(x) = -3 sin(x) over the interval (0, 5), we need to evaluate the function at its critical points and endpoints.

1. critical points:to find the critical points, we take the derivative of f(x) and set it equal to zero:

f'(x) = -3 cos(x) = 0

cos(x) = 0

the solutions to cos(x) = 0 are x = π/2 and x = 3π/2.

2. endpoints:

we also need to evaluate the function at the endpoints of the interval, which are x = 0 and x = 5.

now, we evaluate the function at these points:

f(0) = -3 sin(0) = 0f(5) = -3 sin(5)

to determine the absolute maximum and minimum, we compare the function values at the critical points and endpoints:

-3 sin(0) = 0 (minimum at x = 0)

-3 sin(5) ≈ -2.727 (maximum at x = 5)

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A total of 32 samples will be taken for each possible configuration for the given experiment.

Given that in an experiment to determine the bacterial communities in an aquatic environment, different samples will be taken for each possible configuration of: type of water (saltwater or freshwater), season of the year (winter, spring, summer, autumn), environment (urban or rural).

If two samples are to be taken for each possible configuration, we need to determine the total number of samples required.So, we can get the total number of samples by multiplying the number of options for each factor. For example, there are two types of water, four seasons of the year, and two environments; therefore, there are 2 × 4 × 2 = 16 possible configurations.

Then multiply by two samples for each configuration:16 × 2 = 32

Therefore, a total of 32 samples will be taken for each possible configuration for the experiment.

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The vector perpendicular to the normal vectors of the planes 2x + 4y - z - 10 = 0 and 3x - 2y + 2z = 5 is (5, 2, 1).(option a)

To find a vector perpendicular to the normal vectors of the given planes, we need to determine the normal vectors of the planes first. The normal vector of a plane can be determined by the coefficients of its equation.

For the plane 2x + 4y - z - 10 = 0, the coefficients of x, y, and z are 2, 4, and -1, respectively. So, the normal vector of this plane is (2, 4, -1).

Similarly, for the plane 3x - 2y + 2z = 5, the coefficients of x, y, and z are 3, -2, and 2, respectively. Therefore, the normal vector of this plane is (3, -2, 2).

To find a vector perpendicular to these two normal vectors, we can take their cross product. The cross product of two vectors is a vector that is perpendicular to both of them. Calculating the cross product of (2, 4, -1) and (3, -2, 2) gives us the vector (5, 2, 1).

Hence, the vector (5, 2, 1) is perpendicular to the normal vectors of the given planes.

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Check the picture below.

[tex]\cfrac{2^3}{6^3}=\cfrac{\stackrel{ g }{2}}{V}\implies \cfrac{8}{216}=\cfrac{2}{V}\implies \cfrac{1}{27}=\cfrac{2}{V}\implies V=54~g[/tex]

Statement (b) is equivalent to the statement that a set of vectors (V1, Vp) is linearly independent.

To determine if a set of vectors (V1, Vp) is linearly independent, we need to consider various conditions.

Statement (a) states that a linear combination of V1, Vp is the zero vector if and only if all weights in the combination are zero. This condition is true for linearly independent sets, as no non-trivial linear combination of vectors can result in the zero vector.

Statement (b) asserts that the vector equation x1V1 + x2V2 + ... + x pVp = 0 has only the trivial solution, where x1, x2, ..., xp are the weights. This is precisely the definition of linear independence. If the only solution is the trivial solution (all weights being zero), then the set of vectors is linearly independent.

Statement (c) contradicts the definition of linear independence. If there exist weights, not all zero, that make the linear combination of V1, Vp equal to the zero vector, then the set of vectors is linearly dependent.

Statement (d) and (e) are equivalent and also represent linear independence. If the system with the augmented matrix [A 0] has no free variables or if the matrix equation Ax = 0 has only the trivial solution, then the set of vectors is linearly independent.

Statement (f) is also equivalent to linear independence. If all columns of the matrix A are pivot columns, it means that there are no redundant columns, and hence, the set of vectors is linearly independent.

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The time period is calculated as 1 divided by the frequency. In this case, with a frequency of 290.7247, the time period is approximately 0.0034 seconds.

The time period of a wave or oscillation is the time taken to complete one full cycle. It is inversely proportional to the frequency, which represents the number of cycles per unit time. By dividing 1 by the given frequency of 290.7247, we obtain the time period of approximately 0.0034 seconds. This means that it takes 0.0034 seconds for the wave or oscillation to complete one full cycle.

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The improper integral ∫[-∞, ∞] | sin | + | cos 0 ≥ sin² 0 + cos² 0. [infinity] p sinx+cos x |x| +1 de is divergent.

To determine whether the improper integral | sin | + | cos 0 ≥ sin² 0 + cos² 0. [infinity] p sinx+cos x |x| +1 de converges or diverges, we need to evaluate the integral by breaking it into two separate integrals and then applying the limit test for convergence.

First, we split the integral into two parts:

∫[0, ∞) (|sin x| + |cos x|) dx + ∫[-∞, 0] (|sin x| + |cos x|) dx

Next, we simplify each integral by using the fact that |sin x| ≤ 1 and |cos x| ≤ 1 for all x:

∫[0, ∞) (|sin x| + |cos x|) dx ≤ ∫[0, ∞) (1 + 1) dx = ∞

∫[-∞, 0] (|sin x| + |cos x|) dx ≤ ∫[-∞, 0] (1 + 1) dx = -∞

Since both of these integrals diverge to infinity and negative infinity, respectively, we can conclude that the original improper integral also diverges.

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The line integral of the vector field F along the twisted cubic curve C is 472/3.

To find the line integral of the vector field F(x, y) = xyi + yzj + zxk along the curve C, we need to parameterize the curve C and then evaluate the line integral using the parameterization.

The curve C is given by x = t, y = 12t, and z = 13t + 51.

Let's find the parameterization of C for the given values of x, y, and z.

x = t

y = 12t

z = 13t + 51

We can choose the parameter t to vary from 1 to 2, as given in the problem.

Now, let's calculate the differential of the parameterization:

dr = dx i + dy j + dz k

  = dt i + 12dt j + 13dt k

  = (dt)i + (12dt)j + (13dt)k

Next, substitute the parameterization and the differential dr into the line integral:

∫ F · dr = ∫ (xy)i + (yz)j + (zx)k · (dt)i + (12dt)j + (13dt)k

Simplifying, we have:

∫ F · dr = ∫ (xy + yz + zx) dt

Now, substitute the values of x, y, and z from the parameterization:

∫ F · dr = ∫ (t * 12t + 12t * (13t + 51) + t * (13t + 51)) dt

∫ F · dr = ∫ (12t² + 156t² + 612t + 13t² + 51t) dt

∫ F · dr = ∫ (26t² + 663t) dt

Now, integrate with respect to t:

∫ F · dr = (26/3)t³ + (663/2)t² + C

Evaluate the definite integral from t = 1 to t = 2:

∫ F · dr = [(26/3)(2)³ + (663/2)(2)²] - [(26/3)(1)³ + (663/2)(1)²]

∫ F · dr = (208/3 + 663/2) - (26/3 + 663/2)

∫ F · dr = 472/3

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The maximum height reached by the soccer ball is approximately -67.25 ft. Note that the negative sign indicates that the ball is below the initial height, as it is on its way back down.

To find the maximum height reached by the soccer ball, we can use the kinematic equation for vertical motion under constant acceleration due to gravity:

h = h₀ + v₀t - (1/2)gt²

Where:

h is the final height (maximum height)

h₀ is the initial height (5 ft)

v₀ is the initial velocity (48 ft/s)

g is the acceleration due to gravity (-32 ft/s²)

t is the time it takes to reach the maximum height (unknown)

At the maximum height, the velocity will be 0, so we can set v = 0 and solve for t:

0 = v₀ - gt

Rearranging the equation, we have:

gt = v₀

Solving for t:

t = v₀ / g

Now we can substitute this value of t into the equation for height to find the maximum height:

h = h₀ + v₀t - (1/2)gt²

h = 5 + 48(v₀ / g) - (1/2)g(v₀ / g)²

h = 5 + 48(v₀ / g) - (1/2)(v₀ / g)²

h = 5 + 48(48 / -32) - (1/2)(48 / -32)²

h = 5 - 72 - (1/2)(3/2)

h = 5 - 72 - 9/4

h = -67 - 9/4

h ≈ -67.25 ft

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Answer:

The difference between two matrices of the same size is calculated by subtracting the corresponding elements of the two matrices.

Let’s apply this to matrices A and B:

A - B = [3 -1; 2 0; -3 3] - [3 3; -5 4; -4 2] = [0 -4; 7 -4; 1 1]

So the correct answer is B) Matrix consisting of 3 rows and 2 columns. Row 1 shows 0 and negative 4, row 2 shows 7 and negative 4, and row 3 shows 1 and 1.

The given series, 3n^2, converges from x = 1 (including the left endpoint) to x = 9 (including the right endpoint).

To determine the convergence of the series 3n^2, we need to find the values of x for which the series converges. In this case, the series is defined as the sum of 3 times n squared, where n starts from 1.

The series 3n^2 is a polynomial series of the form an^2, where a = 3. For polynomial series, the series converges for all real values of x. Therefore, the series converges for all values of x in the given range from 1 to 9.

In conclusion, the series 3n^2 converges from x = 1 to x = 9. This means that the sum of the series exists and is finite within this range.

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The value of f(1) is 7 cos(1) - 8 sin(1). Given the function f(x) = 7 sin(x) + 8 cos(x), we want to find the value of f(1).

To do so, we substitute x = 1 into the function. Plugging in x = 1, we have f(1) = 7 sin(1) + 8 cos(1). This simplifies to f(1) = 7 cos(1) - 8 sin(1) using the trigonometric identity sin(a) = cos(a - π/2). Thus, the value of f(1) is 7 cos(1) - 8 sin(1). It is important to note that the given expression f'(1) - 7 cos(x) - 8 sin(2) is unrelated to finding the value of f(1) and appears to be a separate expression or equation.

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The shooting method is used to solve the second-order ordinary differential equation 7d^2y/dx^2 - 2dy/dx - yx = 0 with the boundary conditions y(0) = 5 and y(20) = 8.

To solve the differential equation using the shooting method, we convert it into a system of two first-order equations. Let y = y0 and z = dy/dx, where z represents the derivative of y with respect to x. Then, we have the following system:

dy/dx = z

dz/dx = (2z + yx) / 7

By specifying the initial condition y(0) = 5, we have y0 = 5. To find the appropriate initial condition for z, we use the shooting method. We start by assuming an initial condition for z, say z0, and solve the above system of equations from x = 0 to x = 20. We compare the value of y at x = 20 with the desired boundary condition y(20) = 8.

If the value of y at x = 20 is greater than 8, we adjust the initial condition z0 and repeat the process. If the value is less than 8, we increase z0 and repeat. By iteratively adjusting the initial condition for z, we find the appropriate value that satisfies y(20) = 8.

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The general solution to the given system is x(t) = c₁e^(2t)[-1, 2] + c₂te^(2t)[-1, 2], where c₁ and c₂ can be any constants.

The given system is represented by the matrix equation x'(t) = AX(t), where A is the coefficient matrix. In order to find the eigenvectors, we need to solve the characteristic equation det(A - λI) = 0, where λ is the eigenvalue and I is the identity matrix.

In this case, the characteristic equation becomes:

det(A - λI) = det([[8-λ, -6], [20, 4-λ]]) = (8-λ)(4-λ) - (-6)(20) = (λ-2)(λ-10) = 0

The eigenvalues are λ₁ = 2 and λ₂ = 10. Since there is a repeated eigenvalue, we need to find the corresponding eigenvector(s) using the eigenvector equation (A - λI)v = 0.

For λ₁ = 2:

(A - 2I)v₁ = [[8-2, -6], [20, 4-2]]v₁ = [[6, -6], [20, 2]]v₁ = 0

Solving this system of equations yields the eigenvector v₁ = [-1, 2].

Now, we can construct the general solution using the formula x(t) = c₁e^(λ₁t)v₁ + c₂te^(λ₁t)v₁, where c₁ and c₂ are constants.

Therefore, the general solution to the given system is x(t) = c₁e^(2t)[-1, 2] + c₂te^(2t)[-1, 2], where c₁ and c₂ can be any constants.

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The estimated value for σe is approximately 0.79.

To estimate the parameters θ and σe for the MA(1) process using the method of least squares, set up the system of equations based on the observed data and solve for the parameters.

In a MA(1) process, the observed data Yt can be expressed as:

Yt = θet-1 + et

where Yt is the observed value at time t, et is the error term at time t, and θ is the parameter we want to estimate.

Given the observed data Y1 = 0, Y2 = -1, and Y3 = 1/2, we can substitute these values into the equation to obtain three equations:

Y1 = θe0 + e1   (equation 1)

Y2 = θe1 + e2   (equation 2)

Y3 = θe2 + e3   (equation 3)

Since the process mean is known to be zero, we can assume the mean of the error term et is zero.

From equation 1, we have:

0 = θe0 + e1

e1 = -θe0

From equation 2, we have:

-1 = θe1 + e2

Substituting e1 = -θe0 from equation 1, we get:

-1 = -θ^2e0 + e2

From equation 3, we have:

1/2 = θe2 + e3

Substituting e2 = -θ^2e0 - 1 from equation 2, we get:

1/2 = -θ^3e0 + e3

now have a system of equations in terms of θ and e0. By substituting e0 = 1, we can solve for θ:

-1 = -θ^2 - 1

θ^2 = 0

θ = 0

Therefore, the estimated value for θ is 0.

To estimate σe, we can substitute θ = 0 into any of the original equations. Let's use equation 1:

0 = 0 * e0 + e1

e1 = 0

From equation 2:

-1 = 0 * e1 + e2

e2 = -1

From equation 3:

1/2 = 0 * e2 + e3

e3 = 1/2

The error terms are e1 = 0, e2 = -1, and e3 = 1/2. To estimate σe, we can calculate the sample standard deviation of these error terms:

σe = √[ (e1^2 + e2^2 + e3^2) / (n - 1) ]

   = √[ (0^2 + (-1)^2 + (1/2)^2) / (3 - 1) ]

   = √[ (1 + 1/4) / 2 ]

   = √[5/8]

   ≈ 0.79

Therefore, the estimated value for σe is approximately 0.79.

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The power series representation for f(x) is Σ(n=0 to ∞) [6(x-3)^n/(5^n)], with f(3) = 4 and the convergence radius |x-3| < 5.

To find the power series representation for f(x), we start with the general form of a power series: Σ(n=0 to ∞) [a_n(x - c)^n]. In this case, we have f(3) = 5 - 1, which implies that f(3) is the constant term of the series, equal to 4.

The coefficient a_n can be calculated by taking the n-th derivative of f(x) and evaluating it at x = 3. By finding the derivatives and evaluating them at x = 3, we get a_n = 6/5^n. Thus, the power series representation for f(x) is Σ(n=0 to ∞) [6(x-3)^n/(5^n)], where |x-3| < 5, indicating the convergence radius of the series.

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The critical numbers of f(x) = 2x³ - 9x² are x = 0 and x = 3. f'(x) is positive on the interval (4/9, ∞), implying that the function is increasing again on this interval.

1. To find the critical numbers of f(x) = 2x³ - 9x², we need to find the values of x where the derivative of the function is equal to zero or undefined.

First, let's find the derivative of f(x):

f'(x) = 6x² - 18x

Next, we set the derivative equal to zero and solve for x:

6x² - 18x = 0

Factoring out 6x, we have:

6x(x - 3) = 0

Setting each factor equal to zero, we get two critical numbers:

6x = 0  =>  x = 0

x - 3 = 0  =>  x = 3

Therefore, the critical numbers of f(x) = 2x³ - 9x² are x = 0 and x = 3.

2. To determine the open intervals on which the function is increasing or decreasing, we can analyze the sign of the derivative f'(x) on different intervals.

Using the critical numbers found in the previous step, we can create a sign chart:

Interval | f'(x)

-----------------

(-∞, 0)  |  -

(0, 3)   |  +

(3, ∞)   |  -

From the sign chart, we can see that f'(x) is negative on the interval (-∞, 0), which means the function is decreasing on this interval. It is positive on the interval (0, 3), indicating that the function is increasing there. Finally, f'(x) is negative on the interval (3, ∞), implying that the function is decreasing again on this interval.

3. For the function f(x) = x³ - (2/3)x², we can find the open intervals on which the function is increasing or decreasing by following similar steps as in the previous question.

First, let's find the derivative of f(x):

f'(x) = 3x² - (4/3)x

Setting the derivative equal to zero and solving for x:

3x² - (4/3)x = 0

Factoring out x, we have:

x(3x - 4/3) = 0

Setting each factor equal to zero, we get two critical numbers:

x = 0

3x - 4/3 = 0  =>  3x = 4/3  =>  x = 4/9

The critical numbers are x = 0 and x = 4/9.

Using these critical numbers, we can create a sign chart:

Interval | f'(x)

-----------------

(-∞, 0)  |  +

(0, 4/9) |  -

(4/9, ∞) |  +

From the sign chart, we can determine that f'(x) is positive on the interval (-∞, 0), indicating that the function is increasing on this interval. It is negative on the interval (0, 4/9), indicating that the function is decreasing there. Finally, f'(x) is positive on the interval (4/9, ∞), implying that the function is increasing again on this interval.

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Answer:

2x (2x + 1) + 4(5x + 6)

2(x + 2) (2x + 1)

Step-by-step explanation:

The solution to the inequality is (-21, ∞) ∩ [3/2, ∞).

To solve the inequality 50 < 8 - 2x ≤ 5, we need to solve each part separately.

First, let's solve the left side of the inequality:

50 < 8 - 2x

Subtract 8 from both sides:

42 < -2x

Divide both sides by -2 (note that the inequality flips when dividing by a negative number):

-21 > x

So we have x > -21 for the left side of the inequality.

Next, let's solve the right side of the inequality:

8 - 2x ≤ 5

Subtract 8 from both sides:

-2x ≤ -3

Divide both sides by -2 (note that the inequality flips when dividing by a negative number):

x ≥ 3/2

So we have x ≥ 3/2 for the right side of the inequality.

Combining both parts, we have:

x > -21 and x ≥ 3/2

In interval notation, this can be written as:

(-21, ∞) ∩ [3/2, ∞)

So the solution to the inequality is (-21, ∞) ∩ [3/2, ∞).

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