6. Find the exact value of each of the following: a. cos 40° cos 10° + sin 40° sin 10° 5T 5π sin 55 cos75 + cos it sin75 12 12 12 b. sin. C. cos 15°

A. The average cost per unit is given by the formula:

AC(X) = C(X) / X

where C(X) is the total cost of producing X units of the commodity.

B. The average cost per unit is a measure of the cost efficiency of production, and is equal to the total cost divided by the number of units produced. It takes into account both variable costs (such as labor and materials) and fixed costs (such as rent and equipment) and can help businesses make decisions about pricing and production levels.

C. The average cost per unit is typically a U-shaped curve, reflecting the fact that fixed costs are spread out over a larger number of units as production increases, leading to lower average costs per unit. However, as production continues to increase, variable costs may also increase, causing the average cost per unit to rise again.

D. The goal of most businesses is to minimize the average cost per unit, since this will maximize profits. This can be achieved by finding the optimal level of production that minimizes the total cost per unit, taking into account both fixed and variable costs.

E. The average cost per unit is closely related to the concept of economies of scale, which refers to the cost advantages that businesses can achieve by increasing their production levels. As production increases, fixed costs are spread over a larger number of units, leading to lower average costs per unit. This can lead to increased profits and market competitiveness for businesses that can achieve economies of scale.

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Using comparison method, solve for the point of intersection in each, and then graph both lines on the same Cartesian plane:Solution:Comparison Method:

We are given two equations,[tex]y = 5x – 5, y = 7x - 23[/tex]

To find the point of intersection, we set both equations equal to each other:[tex]5x – 5 = 7x - 23[/tex]

Subtract 5x from both sides of the equation:[tex]-5 = 2x - 23Add 23[/tex] to both sides of the equation:[tex]18 = 2x[/tex]

Divide both sides of the equation by [tex]2:9 = x[/tex]

Now that we know that x = 9, we can substitute that value into either of the two original equations.

Let's use the first equation:[tex]y = 5x – 5y = 5(9) - 5y = 45 - 5y = 40[/tex]

Therefore, the point of intersection for the two lines is (9, 40).

Now, let's graph the two lines on the same Cartesian plane:

The graph of lines [tex]y = 5x – 5 and y = 7x - 23[/tex] is shown below:

Graph for y = 5x – 5 and y = 7x - 23

Hence, the graph of the two lines on the same Cartesian plane and the point of intersection for each line is (9, 40).

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(a) The limit is infinity, the series [tex]\Sigma_{n=1}^{\infty} \frac{1}{(4+2n)^{3/2}}[/tex] diverges.

(b) The limit is infinity, the series [tex]\Sigma_{n=1}^{\infty} \frac{1-n}{(n2^{n}}[/tex] diverges.

(a) To determine the convergence of the series [tex]\Sigma_{n=1}^{\infty} \frac{1}{(4+2n)^{3/2}}[/tex], we can use the limit comparison test. Let's compare it to the series [tex]\Sigma_{n=1}^{\infty} \frac{1}{n^{3/2}}[/tex].

Using the limit comparison test, we take the limit as n approaches infinity of the ratio of the terms of the two series:

[tex]lim_{n\rightarrow\infty} [\frac{1/(4+2n)^{3/2}}{(1/n^{3/2}}][/tex]

Simplifying the expression:

[tex]lim_{n\rightarrow\infty} \frac{n^{3/2}}{(4+2n)^{3/2}}[/tex]

Applying the limit comparison test, we compare this expression to 1:

[tex]lim_{n\rightarrow\infty} \frac{[(n^{3/2}) / (4+2n)^{3/2}]}{(1/n)}[/tex]

By simplifying further:

[tex]lim_{n\rightarrow\infty} \frac{[(n^{3/2}) \times (n/4+2n)^{3/2}]}{(1/n)}[/tex]

Taking the limit:

[tex]lim_{n\rightarrow\infty} \frac{[(n^{3/2}) \times (n/4+2n)^{3/2}]}{(1/n)}= lim_{n\rightarrow\infty}\frac{n^{5/2}}{(4+2n)^{3/2}}[/tex]

[tex]lim_{n\rightarrow\infty} \frac{[(n^{3/2}) \times (n/4+2n)^{3/2}]}{(1/n)}[/tex] = ∞

(b) To determine the convergence of the series [tex]\Sigma_{n=1}^{\infty} \frac{1-n}{(n2^{n})}[/tex], we can use the ratio test.

Applying the ratio test, we calculate the limit as n approaches infinity of the absolute value of the ratio of consecutive terms:

[tex]lim_{n\rightarrow\infty}\left|\left[\frac{1-(n+1)}{(n+1)2^{n+1}}\right] \times \left[\frac{(n2^{n})}{ (1-n)}\right]\right|= lim_{n\rightarrow\infty} \left|\left(-\frac{n}{n+1}\right) \times \left(\frac{n2^n}{1-n}\right)\right|[/tex]

[tex]lim_{n\rightarrow\infty}\left|\left[\frac{1-(n+1)}{(n+1)2^{n+1}}\right] \times \left[\frac{(n2^{n})}{ (1-n)}\right]\right|= lim_{n\rightarrow\infty}\left|n \times \frac{2^n}{n+1}\right|[/tex]

Taking the limit:

[tex]lim_{n\rightarrow\infty} \left|n \times \frac{2^n}{n+1}\right|[/tex] = ∞

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The complete question is:

Determine if the following series converge or diverge.

(a) [tex]\Sigma_{n=1}^{\infty} \frac{1}{(4+2n)^{3/2}}[/tex]

(b) [tex]\Sigma_{n=1}^{\infty} \frac{1-n}{(n2^{n})}[/tex]

The series ∑ k=2 to infinity 2 ln(k+1) diverges, and the series ∑ k=1 to infinity k ln(k+1) also diverges.

To determine whether the series ∑ k=2 to infinity 2 ln(k+1) converges conditionally, converges absolutely, or diverges, we need to examine the behavior of the terms.

The series can be written as ∑ k=2 to infinity ln((k+1)^2). Using the logarithmic identity ln(a*b) = ln(a) + ln(b), we can rewrite the series as ∑ k=2 to infinity (ln(k+1) + ln(k+1)).

Now, we can compare this series to known series to determine its convergence. The term ln(k+1) can be thought of as the natural logarithm of k+1, which grows logarithmically. The series ∑ k=1 to infinity ln(k) is known as the natural logarithm series, which diverges.

Since the series ∑ k=2 to infinity (ln(k+1) + ln(k+1)) can be separated into two natural logarithm series, it also diverges.

Therefore, the series ∑ k=2 to infinity 2 ln(k+1) diverges.

Similarly, to determine whether the series ∑ k=1 to infinity k ln(k+1) converges conditionally, converges absolutely, or diverges, we need to examine the behavior of the terms.

The term k ln(k+1) involves both a polynomial term (k) and a logarithmic term (ln(k+1)). As k increases, the logarithmic term grows at a slower rate than the polynomial term. This suggests that the series may converge.

To further analyze the series, we can use the Limit Comparison Test. We compare it to the series ∑ k=1 to infinity k.

By taking the limit as k approaches infinity of the ratio of the terms:

lim k→∞ (k ln(k+1)) / k = lim k→∞ ln(k+1) = ∞

Since the limit is positive and infinite, and the series ∑ k=1 to infinity k is known to diverge, we can conclude that the series ∑ k=1 to infinity k ln(k+1) also diverges.

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4th subject's score can be either 1 point above the mean(4), 1 point below the mean(2), or 3 points above the mean(6), depending on the specific values of the scores.

To determine the score of the 4th subject, we need to consider the overall mean of the sample and the scores of the other three subjects.

Provided that three subjects have scores that are 1 point above the mean each, we can calculate the mean of the sample by adding the scores of the three subjects and dividing by the total number of subjects (n = 4).

Let's denote the mean of the sample as μ.

Since each of the three subjects has a score that is 1 point above the mean, we can express their scores as μ + 1.

To find the mean (μ), we sum up the scores of the three subjects:

μ + 1 + μ + 1 + μ + 1 = 3μ + 3

Since we have four subjects, the mean of the sample (μ) is:

μ = (3μ + 3) / 4

To solve for μ, we can rearrange the equation:

4μ = 3μ + 3

μ = 3

Therefore, the mean of the sample is μ = 3.

Now, let's consider the score of the 4th subject.

We know that the 4th subject's score must be:

a) 1 point above the mean: 3 + 1 = 4 (1 point above the mean)

b) 1 point below the mean: 3 - 1 = 2 (1 point below the mean)

c) 3 points above the mean: 3 + 3 = 6 (3 points above the mean)

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For a circle C with arc ADB has a measure of 258.1 degrees, the measure of ∠ADB is 129.05 degrees.

In a circle, the measure of an inscribed angle is determined by the measure of the intercepted arc. An inscribed angle is an angle formed by two chords or secants within the circle, and its vertex lies on the circle itself.

In this case, given that the arc ADB has a measure of 258.1 degrees. The inscribed angle ∠ADB is formed by the two radii AD and DB, and its vertex is point D.

The key concept to understand here is that the measure of an inscribed angle is equal to half the measure of its intercepted arc. This relationship holds true for any inscribed angle in a circle.

So, to find the measure of ∠ADB, simply divide the measure of arc ADB by 2:

∠ADB = 258.1 degrees / 2 = 129.05 degrees

Therefore, the measure of ∠ADB is 129.05 degrees.

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The rate of change in revenue is -150000.The rate of change in profit is -165000.

Given data:C = 70000 + 30x and R = 200 - x²/40 where the production output in one week is x calculators. Here, the production output in one week is x calculators. And, the production rate is increasing at a rate of 500 calculators when the production output is 6000 calculators.

Now, we need to find the following

:Rate of change in cost = Rate of change in revenue = Rate of change in profit =Solution:

Given,C = 70000 + 30x .......(1)

R = 200 - x²/40 .......

(2)Differentiating equation (1) w.r.t x, we get,dC/dx

= 30 ......(3)

[Since derivative of constant is 0]Differentiating equation (2) w.r.t x, we get,dR/dx

= -x/20 ......

.(4)Now, we have given that the production rate is increasing at a rate of 500 calculators when the production output is 6000 calculators.

So, we can write, dX/dt

= 500 when X = 6000

By using chain rule, we can write,dC/dt

= dC/dx * dx/dt......

..(5)By substituting values from equations (3) and (5), we get,dC/dt

= 30 × 500dC/dt

= 15000

So, the rate of change in cost is 15000.Similarly,dR/dt

= dR/dx * dx/dt By substituting values from equations (4) and (5), we get,dR/dt

= - (6000)/20 * 500dR/dt = -150000

So, the rate of change in revenue is -150000.Now, profit = Revenue - Cost d P/dt

= dR/dt - dC/dt

By substituting values, we get,dP/dt = -150000 - 15000dP/dt

= -165000

So, the rate of change in profit is -165000.Therefore, the rate of change in cost is 15000.The rate of change in revenue is -150000.The rate of change in profit is -165000.

Rate of change in cost = 15000Rate of change in revenue = -150000Rate of change in profit = -165000

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The tangent vector of the position vector [tex]G(s), F(t(s)), is:F(t(s)) = (-(1/sqrt(2)) * sin((s - C) / sqrt(2)), (1/sqrt(2)) * cos((s - C) / sqrt(2)), 1/sqrt(2)).\\[/tex]
To find the distance along curve C, we need to integrate the magnitude of the velocity vector with respect to the parameter t. The velocity vector is defined as the derivative of the position vector with respect to t.

(a) Distance along curve C, s(t):

The velocity vector v(t) is given by:

[tex]v(t) = (x'(t), y'(t), z'(t))[/tex]

where [tex]x'(t), y'(t), and z'(t)[/tex]are the derivatives of x(t), y(t), and z(t), respectively.

Differentiating x(t), y(t), and z(t) with respect to t, we have:

[tex]x'(t) = -sin(t)y'(t) = cos(t)z'(t) = 1[/tex]

The magnitude of the velocity vector is given by:

[tex]|v(t)| = sqrt((x'(t))^2 + (y'(t))^2 + (z'(t))^2) = sqrt((-sin(t))^2 + (cos(t))^2 + 1^2) = sqrt(sin^2(t) + cos^2(t) + 1) = sqrt(2)\\[/tex]
To find the distance along curve C, we integrate |v(t)| with respect to t:

[tex]s(t) = ∫|v(t)| dt = ∫sqrt(2) dt = sqrt(2)t + C[/tex]

where C is the constant of integration.

(b) Tangent vector of the position vector G(s), F(t(s)):

The position vector G(s) is given by:

G(s) = (x(s), y(s), z(s))

To find the tangent vector of G(s), we need to find the derivative of G(s) with respect to s.

Since s(t) = sqrt(2)t + C, we can solve for t as a function of s:

t(s) = (s - C) / sqrt(2)

Substituting t(s) into the parametric equations for x(t), y(t), and z(t), we have:

[tex]x(s) = cos(t(s)) = cos((s - C) / sqrt(2))y(s) = sin(t(s)) = sin((s - C) / sqrt(2))z(s) = t(s) = (s - C) / sqrt(2)\\[/tex]
The tangent vector F(t(s)) is given by:

[tex]F(t(s)) = (x'(s), y'(s), z'(s))[/tex]

Differentiating x(s), y(s), and z(s) with respect to s, we have:

[tex]x'(s) = -(1/sqrt(2)) * sin((s - C) / sqrt(2))y'(s) = (1/sqrt(2)) * cos((s - C) / sqrt(2))z'(s) = 1/sqrt(2)\\[/tex]
Therefore, the tangent vector of the position vector G(s), F(t(s)), is:

[tex]F(t(s)) = (-(1/sqrt(2)) * sin((s - C) / sqrt(2)), (1/sqrt(2)) * cos((s - C) / sqrt(2)), 1/sqrt(2))\\[/tex]
Note: The constant of integration C affects the starting point along the curve, but it does not affect the direction of the tangent vector.

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The volume of the pyramids obtained using the formula for finding the volumes of pyramids indicates that the volume of the pyramid on the left is larger, and the correct option is the option;

The volume of the pyramid on the left is greater by 8 in³

The volume of a right pyramid is the product of one third and the base area of the pyramid.

Please find attached the possible diagram of the pyramids, created with MS Word;

The volume of a pyramid = (1/3) × Base area × Height

Therefore, we get;

The volume of the pyramid on the left = (1/3) × 8 × 6 × 8 = 128 in.³

The volume of the pyramid on the right = (1/3) × 10 × 9 × 4 = 120 in.³

Therefore, the volume of the pyramid on the left is greater than the volume of the pyramid on the right by 8 in³.

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For a car loan of $8000 to be paid back over 24 months with a 12% annual interest rate, the monthly payment would be approximately $374.17.

To find the monthly payment for a car loan, we can use the formula:

M = (P * r * (1 + r)^n) / ((1 + r)^n - 1)

where M is the monthly payment, P is the amount borrowed, r is the monthly interest rate, and n is the number of months over which the loan is paid back.

In this case, the amount borrowed (P) is $8000 and the loan is paid back over 24 months (n = 24). The annual interest rate is 12%, so we need to convert it to a monthly rate.

First, we divide the annual interest rate by 12 to get the monthly interest rate:

r = 12% / 12 = 0.12 / 12 = 0.01

Now we can substitute the values into the formula:

M = (8000 * 0.01 * (1 + 0.01)^24) / ((1 + 0.01)^24 - 1)

Calculating this expression, we find that the monthly payment (M) for the loan is approximately $374.17.

This means that the borrower would need to pay approximately $374.17 every month for 24 months to fully repay the loan. It's important to note that this calculation assumes a fixed interest rate and does not account for any additional fees or charges that may be associated with the loan.

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The given equation is 2x³+11x²−9x−18=0 and the value of x is to be found out.Factoring is a very useful method of solving cubic equations. One factor can always be found out by putting x=1,2,3, etc. in the equation and finding out whether it is satisfied or not.

If we put x = 1, then the left-hand side is equal to 2 + 11 − 9 − 18 = −14. Thus, x = 1 is not a root of the equation. If we put x = 2, then the left-hand side is equal to 16 + 44 − 18 − 18 = 24. Thus, x = 2 is a root of the equation.

The factor theorem states that if (x − a) is a factor of the polynomial p(x), then p(a) = 0. Using this theorem, we can divide the polynomial 2x³+11x²−9x−18 by (x − 2) and obtain a quadratic equation.

Long Division :

           2x² + 15x + 9
      ________________________
  x - 2 |  2x³ + 11x² - 9x - 18
           2x³ - 4x²
           __________
                 15x² - 9x
                 15x² - 30x
                 ___________
                            21x - 18
                            21x - 42
                            _______
                                     24
The factorization of 2x³+11x²−9x−18 is given by (x−2)(2x²+15x+9). Now, we need to solve the quadratic equation 2x²+15x+9=0.

2x²+15x+9=0
We can use the quadratic formula to solve for x.

x = (-b ± sqrt(b² - 4ac)) / 2a, where a = 2, b = 15, and c = 9.
x = (-15 ± sqrt(15² - 4(2)(9))) / 4
x = (-15 ± sqrt(177)) / 4
x = (-15 + sqrt(177)) / 4, or x = (-15 − sqrt(177)) / 4

Thus, the roots of the cubic equation 2x³+11x²−9x−18=0 are x=2, x=(-15 + sqrt(177)) / 4, and x = (-15 − sqrt(177)) / 4.

The possible rational roots of the cubic function are ±1, ±2, ±3, ±6, ±9 and ±18 and the roots of the equation are x = -1, x = 3/2 and x = -6

To find the roots of the equation 2x³ + 11x² - 9x - 18 = 0, we can use various methods such as factoring, synthetic division, or numerical methods. In this case, let's use the Rational Root Theorem and synthetic division to determine the roots.

The Rational Root Theorem states that if a rational number p/q is a root of the equation, where p is a factor of the constant term and q is a factor of the leading coefficient, then p/q is a possible root.

The constant term of the equation is -18, and its factors are ±1, ±2, ±3, ±6, ±9, and ±18. The leading coefficient is 2, which only has factors of ±1 and ±2. Therefore, the possible rational roots are:

±1, ±2, ±3, ±6, ±9 and ±18

The roots of the original equation 2x³ + 11x² - 9x - 18 = 0 are:

x = -1, x = 3/2 and x = -6

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a) To show that if a sequence a_n is Cauchy, then the sequence b_n = a_n^2 is also Cauchy, we need to prove that for any given epsilon > 0, there exists an integer N such that for all n, m > N, |b_n - b_m| < epsilon.

Since a_n is Cauchy, for any given epsilon > 0, there exists an integer N such that for all n, m > N, |a_n - a_m| < sqrt(epsilon).

Now, let's consider |b_n - b_m| = |a_n^2 - a_m^2| = |(a_n - a_m)(a_n + a_m)|.

By the triangle inequality, |a_n + a_m| ≤ |a_n| + |a_m|.

Therefore, we have |b_n - b_m| ≤ |a_n - a_m| * (|a_n| + |a_m|).

Since |a_n - a_m| < sqrt(epsilon) and |a_n| + |a_m| is a constant, we can choose a larger constant K such that |b_n - b_m| < K * sqrt(epsilon).

This shows that the sequence b_n = a_n^2 is also Cauchy.

b) Let's consider the sequence a_n = (-1)^n. This sequence is not Cauchy because it oscillates between -1 and 1 indefinitely. However, if we consider the sequence b_n = (a_n)^2 = (-1)^n^2 = 1, we have a constant sequence where all terms are equal to 1. This sequence is trivially Cauchy because the difference between any two terms is always 0. Therefore, we have an example where b_n = a_n^2 is Cauchy, but a_n is not Cauchy.

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According to cylindrical coordinates, the mass of the solid Q is 69kπ.

The density function for the solid is given by rho (x,y,z) = k(x^2+y^2).

Where the solid Q is defined by the following inequality:

[tex]Q={(x,y,z):0≤z≤9−x−2y,x^2+y^2≤36}[/tex]

To compute the mass of this solid Q, we first need to calculate the volume of the solid.

The volume V of the solid can be computed using triple integrals in cylindrical coordinates as follows:

[tex]V = ∫∫∫ρ(r, θ, z) r dz dr dθ[/tex]

where [tex]ρ(r, θ, z) = k(r^2)[/tex]

The bounds for the triple integral are as follows:

[tex]r ∈ [0, 6]θ ∈ [0, 2π]z ∈ [0, 9-r cos(θ) - 2r sin(θ)][/tex]

So, the mass of the solid Q is given by:

[tex]M = ∫∫∫ρ(r, θ, z) dV= k∫∫∫(r^3) dz dr dθ= k∫0^{2π}∫0^6∫0^{9-r cos(θ) - 2r sin(θ)}(r^3) dz dr dθM= k∫0^{2π}∫0^6 [(r^3)(9-r cos(θ) - 2r sin(θ))] dr dθ= k∫0^{2π} [2(3/4)(46-6r^2 sin(θ) - 3r^2 cos(θ))] dθ= k(69π)[/tex]

Therefore, the mass of the solid Q is 69kπ.

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(a) The natural domain of the function f(x, y) = xy + 2y − ln(x) − 2ln(y) is (0, ∞) × (0, ∞).

(b) The level curves of the function f(x, y) for the levels z = 2.7, 3, 4, 5, 6, 7, 8, 9, 10, 11 are shown in the Desmos images.(c) The critical points of the function are (1, 2), and the value of the function at these points is 4 − 3ln(2).

(d) The critical point (1, 2) is a saddle point.(e) The range of the function is (-∞, ∞).

(a) The natural domain of the function f(x, y) = xy + 2y − ln(x) − 2ln(y) can be determined by considering the following conditions:xy ∈ R, 2y > 0, ln(x) ∈ R, and ln(y) ∈ R.

Thus, the natural domain of the function is (0, ∞) × (0, ∞).

(b) We need to draw the level curves of the function f(x, y) for the levels z = 2.7, 3, 4, 5, 6, 7, 8, 9, 10, 11 using Desmos. The following images show the required level curves:

(c) To determine the critical points of the function, we need to find the partial derivatives of f(x, y) with respect to x and y and set them to zero.

Then, we can solve the system of equations to find the critical points

                 .f_x(x, y) = y − 1/x = 0f_y(x, y) = x + 2/y = 0

Solving these equations, we getx = 1/√y and y = 2/√x

Substituting y = 2/√x into the first equation, we getx = 1/√(2/√x) ⇒ x = 2y = 2/√x

Thus, the critical points of the function are (1, 2), and the value of the function at these points is:

                          f(1, 2) = 1 × 2 + 2(2) − ln(1) − 2ln(2)

                             = 4 − ln(2) − 2ln(2) = 4 − 3ln(2).

(d) To determine whether the critical points are local extrema or saddle points, we need to use the second derivative test.

The Hessian matrix of the function is given by:H(x, y) = (f_{xy}f_{yx}) = (1 − 1/x^2 1 − 2/y^2)

At the critical point (1, 2), we have:H(1, 2) = (1 − 1 1 − 1/2)

The determinant of this matrix is:d = (1)(-1/2) − (1)(1) = -3/2Since d < 0 and H(1, 2) is symmetric, the critical point (1, 2) is a saddle point.

Using the 3D plot of the graph of this function, we can argue that there is no global minimum or maximum.

(e) The range of the function can be found by considering the maximum and minimum values of the function.

Since the function has no global minimum or maximum, the range of the function is (-∞, ∞).

Hence, the answer to the given question is:

(a) The natural domain of the function f(x, y) = xy + 2y − ln(x) − 2ln(y) is (0, ∞) × (0, ∞).

(b) The level curves of the function f(x, y) for the levels z = 2.7, 3, 4, 5, 6, 7, 8, 9, 10, 11 are shown in the Desmos images.(c) The critical points of the function are (1, 2), and the value of the function at these points is 4 − 3ln(2).

(d) The critical point (1, 2) is a saddle point.(e) The range of the function is (-∞, ∞).

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The correct answers are as follows:

(a) The relative rate of change at t = 27 is [tex]0.041 * a * e^{1.107}[/tex].

(b) The elasticity of demand for [tex]D(p) = p^{5500[/tex] is [tex]E(p) = 5500 / p[/tex].

The function h(t) is given as [tex]h(t) = 2 + 1 * a * e^{0.041 * t}[/tex], where a and t are variables.

(a) To find the relative rate of change at each 27, we need to calculate the derivative of h(t) with respect to t and evaluate it at t = 27.

Taking the derivative of h(t) with respect to t, gives

[tex]h'(t) = 0.041 * 1 * a * e^{0.041 * t[/tex]

Substituting t = 27 into the derivative, gives

[tex]h'(27) = 0.041 * 1 * a * e^{0.041 * 27[/tex]

Simplifying further,

[tex]h'(27) = 0.041 * a * e^{1.107[/tex]

Therefore, the relative rate of change at t = 27 is[tex]0.041 * a * e^{1.107}[/tex].

(b) To determine whether the demand function [tex]D(p) = p^{5500[/tex] is elastic, inelastic, or unit-elastic at the price of p, we need to calculate the elasticity of demand.

The elasticity of demand (E) is given by the formula E(p) = (p * D'(p)) / D(p), where D'(p) is the derivative of D(p) with respect to p.

Differentiating D(p) = p^5500 with respect to p, we obtain D'(p) = 5500 * p^5499.

Substituting these values into the elasticity formula, we have

[tex]E(p) = (p * 5500 * p^{5499}) / (p^{5500})[/tex].

Simplifying further,

[tex]E(p) = 5500 / p.[/tex]

Therefore, the elasticity of demand is [tex]E(p) = 5500 / p[/tex].

Thus, the correct answers are as follows:

(a) The relative rate of change at t = 27 is [tex]0.041 * a * e^{1.107}[/tex].

(b) The elasticity of demand for [tex]D(p) = p^{5500[/tex] is [tex]E(p) = 5500 / p[/tex].

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The shop's busiest hour is at 11 am where it processes about 125 transactions.

To find the busiest hour for the coffee shop, we need to determine the hour during the opening hours (from 6 am to 8 am) when the number of transactions, N(t), is the highest.

The function [tex]N(t) = -t^3 + 5t^2 + 25t[/tex] represents the number of transactions processed t hours after the shop opens at 6 am.

To find the busiest hour, we can analyze the function and identify the hour that yields the maximum value of N(t).

We can start by taking the derivative of N(t) with respect to t to find the critical points where the function reaches its maximum or minimum values:

[tex]N'(t) = -3t^2 + 10t + 25.[/tex]

Setting N'(t) = 0 and solving for t, we can find the critical points. In this case, the equation becomes:

[tex]-3t^2 + 10t + 25 = 0.[/tex]

By solving this quadratic equation, we find two critical points:

t = -1.67 and t = 5.

Since the time cannot be negative in this context, we discard the negative value and focus on the positive critical point t = 5.

Therefore, the busiest hour for the coffee shop is 5 hours after it opens at 6 am, which corresponds to 11 am.

By substituting the value t = 5 into the N(t) function, we can find the number of transactions during the busiest hour:

[tex]N(5) = -(5)^3 + 5(5)^2 + 25(5)[/tex] = -125 + 125 + 125 = 125.

Hence, during the busiest hour at 11 am, the coffee shop processes 125 transactions.

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To test the claim about the population mean \( \mu \) at the given level of significance \( \alpha \) using the provided sample statistics, a t-test can be employed.

Assuming the population is normally distributed, the t-test will help determine whether the sample mean is significantly different from the claimed population mean.

A t-test is used to assess whether the difference between the sample mean and the population mean is statistically significant. The formula for the t-test statistic is given by:

\[ t = \frac{\bar{x} - \mu}{s/\sqrt{n}} \]

where:

- \( \bar{x} \) is the sample mean,

- \( \mu \) is the population mean,

- \( s \) is the sample standard deviation, and

- \( n \) is the sample size.

To conduct the t-test, we compare the calculated t-value with the critical t-value obtained from the t-distribution table or statistical software. The critical t-value is determined based on the desired level of significance \( \alpha \) and the degrees of freedom (df = n - 1).

If the calculated t-value is greater than the critical t-value (t_calc > t_crit) or falls in the rejection region, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

Therefore, by conducting the t-test, we can determine whether the sample mean provides enough evidence to support or refute the claim about the population mean.

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There are no groups in a one-sample t-test.

The null hypothesis is that there is no significant difference between the sample mean and the population mean.

The significance level is not given in the question.

A two-tailed test is appropriate for this study.

The alternative hypothesis is that there is a significant difference between the sample mean and the population mean.

The degrees of freedom is equal to n - 1, therefore, 20 - 1 = 19.

The t-observed value is calculated as follows:

t = (X - μ) / (s / √n)t = (13.7 - 14.1) / (0.8 / √20)t = -1 / 0.178t = -5.62

The t-critical value(s) can be obtained from a t-table or calculator. Assuming a two-tailed test and a 95% confidence level, the t-critical value is ±2.093.

Since the t-observed value (-5.62) is outside the t-critical values (-2.093 and 2.093), the researcher should reject the null hypothesis.

The p-value can be obtained from a t-table or calculator. The p-value is less than 0.001.

Cohen’s d = (X - μ) / sCohen’s d = (13.7 - 14.1) / 0.8Cohen’s d = -0.5

The decision to reject or retain the null hypothesis would depend on the calculated p-value.

If the calculated p-value is less than 0.01, the researcher would reject the null hypothesis.

The 68% confidence interval is given by:X ± t * (s / √n) = 13.7 ± 1.042 * (0.8 / √20)= 13.7 ± 0.468The 68% confidence interval is (13.232, 14.168)

The 90% confidence interval is given by:X ± t * (s / √n) = 13.7 ± 1.725 * (0.8 / √20)= 13.7 ± 0.776The 90% confidence interval is (12.924, 14.476).

The 98% confidence interval is given by:X ± t * (s / √n) = 13.7 ± 2.878 * (0.8 / √20)= 13.7 ± 1.295The 98% confidence interval is (12.405, 14.995)

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General solution of the given differential equations are as follows:

18. xy' = 2y + x³ cos x

The given differential equation is of the form xy′ − 2y = x³ cos x

This is a linear differential equation of first order, so the general solution can be written as

y(x) = e^(∫P(x)dx)(∫Q(x)e^(-∫P(x)dx)dx + C)

Where P(x) = -2/x and Q(x) = x³ cos x

Now, we have to solve the equation by using integrating factor= e^(∫-2/xdx)

= e^(-2lnx) = 1/x

²Multiplying throughout by the integrating factor gives(x^{-2}y)' = x cos x

Integrating with respect to x,

we gety(x) = (-1/3)x^{-3} cos x + (1/9) x^3 sin x + C/x219. y' + y cotx

= cos x

The given differential equation is of the form y′ + P(x)y = Q(x),

where P(x) = cot x

Now, the integrating factor can be found by using the formula= e^(∫P(x)dx)

= e^ln(sin x)

= sin x

Multiplying throughout by the integrating factor gives(sin x y)' = cos x Integrating with respect to x,

we gety(x) = sin x + Ccos x20.

y'= 1 + x + y + xy,

y(0) = 0The given differential equation is of the form y′ + P(x)y = Q(x),

where P(x) = 1 + x

Now, the integrating factor can be found by using the formula= e^(∫P(x)dx)

= e^(∫(1 + x)dx)

= e^(x + x²/2)

Multiplying throughout by the integrating factor gives(e^{x + x²/2} y)'

= e^{x + x²/2} (x + 1)

Integrating with respect to x, we gety(x) = e^{-x-x^2/2} ∫e^{x+x^2/2} (x + 1)dx + Ce^{-x-x^2/2}21.

xy' = 3y + x⁴ cos x, y(2л) = 0

The given differential equation is of the form xy′ − 3y = x⁴ cos x

This is a linear differential equation of first order, so the general solution can be written as y(x)

= e^(∫P(x)dx)(∫Q(x)e^(-∫P(x)dx)dx + C)

Where P(x) = -3/x and

Q(x) = x⁴ cos x

Now, we have to solve the equation by using integrating factor= e^(∫-3/xdx)

= e^(-3lnx) = 1/x³

Multiplying throughout by the integrating factor gives(x³y)' = x cos x Integrating with respect to x,

we get y(x) = (-1/3)x^{-3} cos x + (1/9) x^3 sin x + C/x³22.

y' = 2xy + 3x² exp(x²),

y(0) = 5

The given differential equation is a first-order linear differential equation of the form y′ + P(x)y = Q(x),

where P(x) = 2x and Q(x)

= 3x² exp(x²)

Now, the integrating factor can be found by using the formula= e^(∫P(x)dx) = e^(∫2xdx) = e^(x²)

Multiplying throughout by the integrating factor gives(e^{x²} y)'

= 3x² e^(2x²)Integrating with respect to x,

we gety(x) = ∫3x² e^(2x²) e^{-x²} dx + Ce^{-x²}y(x)

= (3/2) ∫2x e^{x²} dx + Ce^{-x²}y(x)

= (3/4) e^{x²} + Ce^{-x²}23.

xy' + (2x − 3) y = 4x⁴

The given differential equation is a first-order linear differential equation of the form y′ + P(x)y = Q(x),

where P(x) = (2x − 3)/x and Q(x) = 4x³

Multiplying throughout by the integrating factor, e^(∫(2x-3)/xdx), gives(xy)'e^(-3lnx) + y(x)e^(-3lnx)

= 4x³e^(-3lnx)Multiplying by x^{-3} throughout both sides of the equation, we have(x^{-2}y)' - 3x^{-2}y = 4x

Integrating both sides,

we get y(x) = x^3 - (16/5) x^{-2} + C/x^3

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The average rate of change of the function \( g(x) = x^2 - 5 \) from -4 to 1 is 6. The average rate of change represents the average slope of the function over the given interval.

To find the average rate of change of a function, we need to calculate the difference in the function values divided by the difference in the input values over the given interval. In this case, the interval is from -4 to 1.

First, let's calculate the function values at the endpoints of the interval:

[tex]\( g(-4) = (-4)^2 - 5 = 16 - 5 = 11 \)[/tex]

[tex]\( g(1) = (1)^2 - 5 = 1 - 5 = -4 \)[/tex]

Next, we calculate the difference in function values: -4 - 11 = -15.

Then, we calculate the difference in input values: 1 - (-4) = 5.

Finally, we divide the difference in function values by the difference in input values to obtain the average rate of change:

[tex]\( \text{Average rate of change} = \frac{{-15}}{{5}} = -3 \).[/tex]

Therefore, the average rate of change of [tex]\( g(x) = x^2 - 5 \)[/tex] from -4 to 1 is -3. This means that, on average, the function decreases by 3 units for every 1 unit increase in the input within the given interval.

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(b) dT/dx | x=0 = 0; The patient's temperature is not changing at the start of the illness.

(c) dT/dx | x=1 ≈ 0.00899; The patient's temperature is changing at a rate of approximately 0.00899 degrees per hour 1 hour after the start of the illness.

(d) dT/dx | x=3 ≈ -0.00616; The patient's temperature is changing at a rate of approximately -0.00616 degrees per hour 3 hours after the start of the illness.

(e) dT/dx | x=8 ≈ -0.00041; The patient's temperature is changing at a rate of approximately -0.00041 degrees per hour 8 hours after the start of the illness.

To find dT/dx, we need to differentiate the temperature function T(x) = 9x /[tex](x^2 + 98.6)[/tex] with respect to x.

Using the quotient rule, we have:

dT/dx = [ [tex](x^2 + 98.6)(9) - (9x)(2x) ] / (x^2 + 98.6)^2[/tex]

Simplifying this expression gives:

dT/dx = [[tex]9(x^2 + 98.6) - 18x^2 ] / (x^2 + 98.6)^2[/tex]

Now, let's evaluate dT/dx at the given times:

(b) x = 0:

Substituting x = 0 into the derivative expression, we have:

dT/dx | x=0

= [ [tex]9(0^2 + 98.6) - 18(0)^2 ] / (0^2 + 98.6)^2[/tex]

= 0 /[tex](98.6)^2[/tex]

= 0

Interpretation: At the start of the illness (0 hours), the patient's temperature is not changing, indicating a stable condition.

(c) x = 1:

Substituting x = 1 into the derivative expression, we have:

dT/dx | x=1

= [tex][ 9(1^2 + 98.6) - 18(1)^2 ] / (1^2 + 98.6)^2[/tex]

[tex][ 9(1^2 + 98.6) - 18(1)^2 ] / (1^2 + 98.6)^2[/tex]

= 891 / 99203.36

≈ 0.00899

Interpretation: 1 hour after the start of the illness, the patient's temperature is changing at a rate of approximately 0.00899 degrees per hour.

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The relation that is a function in the option are C and D.

Function relates input and output. A function is an expression, rule, or law that defines a relationship between one variable (the independent variable or the domain) and another variable (the dependent variable or the range).

Therefore, a function relates each element of a set with exactly one

element of another set (possibly the same set).

Therefore, the relation that is a function in the option are C and D.

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a. Cc = (0.23 mm)^2 / (0.42 mm * 0.15 mm) = 0.354.

b. Cc = (0.39 mm)^2 / (0.79 mm * 0.28 mm) = 0.256.

a. The uniformity coefficient (Cu) is calculated by dividing the D60 (effective size) by the D10 (coarsest size) of the gravel. In this case, the D60 is 0.42 mm and the D10 is 0.15 mm. Therefore, Cu = 0.42 mm / 0.15 mm = 2.8.

The coefficient of gradation (Cc) is calculated by dividing the square of the D30 (median size) by the product of the D60 and D10. In this case, the D30 is 0.23 mm. Therefore, Cc = (0.23 mm)^2 / (0.42 mm * 0.15 mm) = 0.354.

Based on the calculated values, we can determine the grading of the soil. A well-graded soil has a Cu value greater than 4 and a Cc value between 1 and 3. In this case, the gravel has a Cu value of 2.8, indicating that it is poorly graded.

b. To determine the Cu and Cc for the given sand, we can use the provided grain size distribution data.

Cu is calculated by dividing the D60 by the D10. In this case, the D60 is 0.79 mm and the D10 is 0.28 mm. Therefore, Cu = 0.79 mm / 0.28 mm = 2.82.

Cc is calculated by dividing the square of the D30 by the product of the D60 and D10. In this case, the D30 is 0.39 mm. Therefore, Cc = (0.39 mm)^2 / (0.79 mm * 0.28 mm) = 0.256.

Based on the calculated values, we can determine the grading of the soil. A well-graded soil has a Cu value greater than 4 and a Cc value between 1 and 3. In this case, the sand has a Cu value of 2.82, indicating that it is poorly graded.

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The equation of the line is y = 4x. Option B

First, we need to know that the general equation of a line is expressed as;

y = mx + c

Such that the parameters of the formula are expressed as;

From the information given, we have that the points in the line are;

(1,-4)

(2,-8)

Now, let us determine the slope of the line, we have to take the change in the value of the points

Slope, m = -8 -(-4)/2-1

expand the bracket

Slope, m = 4

Then, the intercept is;

-8 = 4(2) + c

c = 0

Equation of the line would be;

y = 4x

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The rate at which the distance between the cyclist and the balloon is increasing 5 seconds later is 135 m/s.

Let's assume the distance between the cyclist and the balloon at time t is given by d(t). We are interested in finding the rate of change of d(t) with respect to time t, which is denoted as d'(t) or simply the derivative of d(t).

Given:

Vertical velocity of the balloon (b) = 0.4 m/s

Horizontal velocity of the cyclist (c) = 5 m/s

The distance between the cyclist and the balloon (d) can be found using the Pythagorean theorem:

d² = (23 + b * t)² + (c * t)²

Differentiating both sides of the equation with respect to t:

2d * d' = 2(23 + b * t) * (b) + 2(c * t) * (c)

Simplifying the equation:

d * d' = (23 + 0.4t) * 0.4 + (5t) * 5

At t = 5 seconds, we can substitute the value to find the rate of change of the distance between the cyclist and the balloon:

d(5) * d'(5) = (23 + 0.4 * 5) * 0.4 + (5 * 5) * 5

Solving the equation:

d(5) * d'(5) = (23 + 2) * 0.4 + 25 * 5

= (25) * 0.4 + 125

= 10 + 125  

= 135

Therefore, the rate at which the distance between the cyclist and the balloon is increasing 5 seconds later is 135 m/s.

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To create a Pareto Chart for the local bank's phone calls, we will rank the reasons for customer calls in decreasing order of occurrence. The cumulative percentage line will also be included.

Based on the given check sheet tally, we have the following data:

Reason for Calls:

1. Account Balance Inquiries: 40

2. Card Issues: 30

3. Loan Inquiries: 25

4. Transaction Disputes: 15

5. Other: 50

Step 1: Calculate the total number of calls.

Total Calls = Sum of all tallies = 40 + 30 + 25 + 15 + 50 = 160

Step 2: Calculate the percentage of each reason.

Percentage = (Tally / Total Calls) * 100

Reason for Calls:

1. Account Balance Inquiries: (40 / 160) * 100 = 25%

2. Card Issues: (30 / 160) * 100 = 18.75%

3. Loan Inquiries: (25 / 160) * 100 = 15.625%

4. Transaction Disputes: (15 / 160) * 100 = 9.375%

5. Other: (50 / 160) * 100 = 31.25%

Step 3: Calculate the cumulative percentage.

Cumulative Percentage = Sum of Percentages

Reason for Calls:

1. Account Balance Inquiries: 25%

2. Card Issues: 25% + 18.75% = 43.75%

3. Loan Inquiries: 43.75% + 15.625% = 59.375%

4. Transaction Disputes: 59.375% + 9.375% = 68.75%

5. Other: 68.75% + 31.25% = 100%

Step 4: Create the Pareto Chart.

Reason for Calls:

1. Other (50)

2. Account Balance Inquiries (40)

3. Card Issues (30)

4. Loan Inquiries (25)

5. Transaction Disputes (15)

(Note: The reasons are listed in decreasing order of occurrence based on the tallies.)

In the Pareto Chart, we can see that the "Other" category has the highest number of calls. To address the high number of calls in the "Other" column, further analysis and categorization can be done to identify the specific sub-reasons contributing to this category. By understanding the underlying causes, the bank can develop targeted strategies to address the most common reasons within the "Other" category and potentially reduce the overall number of calls in the future.

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The final three-dimensional structure of XeO4 will have a trigonal bipyramidal shape, with the Xenon atom in the center and the four oxygen atoms arranged in a plane around it.

To draw the three-dimensional structure of XeO4, we need to consider the valence electrons of each atom and their arrangement around the central atom (Xe).

1. Determine the total number of valence electrons:
- Xenon (Xe) is in group 8, so it has 8 valence electrons.
- Oxygen (O) is in group 6, so each oxygen atom contributes 6 valence electrons.
- Since we have four oxygen atoms, the total number of valence electrons is 8 + 4(6) = 32.

2. Place the central atom:
- The central atom is Xenon (Xe). Draw Xe in the center.

3. Connect the outer atoms:
- Each oxygen atom will be connected to the central Xenon atom by a single bond. Place the oxygen atoms around the Xenon atom.

4. Distribute the remaining electrons:
- After connecting the oxygen atoms, we have used 4 electrons (1 from each oxygen) and 4 single bonds. So we have 32 - 4 = 28 electrons remaining.

5. Add lone pairs and complete the octets:
- Start by adding lone pairs to each oxygen atom until they have a complete octet (8 electrons).
- Distribute the remaining electrons as lone pairs on the central Xenon atom.
- If there are still remaining electrons, place them as lone pairs on the oxygen atoms.

The final three-dimensional structure of XeO4 will have a trigonal bipyramidal shape, with the Xenon atom in the center and the four oxygen atoms arranged in a plane around it. Each oxygen atom will have a lone pair, and the Xenon atom will have two lone pairs.

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The enthalpy change when 1,2-dibromoethane is mixed into solution A to form solution B can be determined by calculating the heat of solution.

To calculate the heat of solution, we need to know the enthalpy of mixing and the molar enthalpies of the pure substances involved. The enthalpy of mixing is the difference between the enthalpy of the solution and the sum of the enthalpies of the individual components.

In this case, we need the enthalpy of mixing between 1,2-dibromoethane and cyclohexane. The enthalpy of mixing can be positive or negative, indicating an endothermic or exothermic process, respectively.

To find the enthalpy change, we can use the equation:

ΔH(solution) = n(CH2BrCH2Br) * ΔH(CH2BrCH2Br) + n(C6H12) * ΔH(C6H12) + ΔH(mixing)

where ΔH(solution) is the enthalpy change, n is the number of moles, ΔH(CH2BrCH2Br) is the molar enthalpy of 1,2-dibromoethane, ΔH(C6H12) is the molar enthalpy of cyclohexane, and ΔH(mixing) is the enthalpy of mixing.

By substituting the given values and calculating the enthalpy change, we can determine the enthalpy change when 1,2-dibromoethane is mixed into solution A to form solution B.

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The range of the function on the graph is all the real numbers greater than or equal to 0. Option B is the correct answer.

The graph of the function is a parabola that opens upwards, which means that the range of the function is all the real numbers greater than or equal to 0. The function can never take on a value less than 0, because the parabola never touches or crosses the x-axis.

The other answer choices are incorrect because they do not include all the possible values of the function. For example, the answer choice O. all the real numbers is incorrect because the function can never take on a negative value.

The answer choice O. all the real numbers greater than or equal to 2 is incorrect because the function can take on values greater than 2, such as 3, 4, and so on.

The answer choice O. all the real numbers greater than or equal to -3 is incorrect because the function can take on values greater than -3, such as 0, 1, and so on.

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The general solution for the given second-order linear homogeneous differential equation, y" + 4y' + 13y = e^x - cosx, is

y = c1e^((-2+3i)x) + c2e^((-2-3i)x) + (1/12)*e^x - (1/169)cosx + Csinx.


To find the general solution for the given second-order linear homogeneous differential equation, y" + 4y' + 13y = e^x - cosx, we need to solve the associated homogeneous equation and then find a particular solution for the non-homogeneous part.

The associated homogeneous equation is y" + 4y' + 13y = 0. To solve this equation, we assume a solution of the form y = e^(rx), where r is a constant.

Plugging this into the equation, we get the characteristic equation r^2 + 4r + 13 = 0. Solving this quadratic equation yields the roots r1 = -2 + 3i and r2 = -2 - 3i.

The general solution for the homogeneous equation is given by y_h = c1*e^((-2+3i)x) + c2*e^((-2-3i)x), where c1 and c2 are arbitrary constants.

To find a particular solution for the non-homogeneous part, we can use the method of undetermined coefficients. Since the non-homogeneous part includes terms e^x and cosx, we assume a particular solution of the form y_p = A*e^x + (B*cosx + C*sinx), where A, B, and C are constants.

Plugging this particular solution into the differential equation, we find that A = 1/12 and B = -1/169, while C can take any value.

Therefore, a particular solution is y_p = (1/12)*e^x - (1/169)*cosx + C*sinx.

The general solution for the given differential equation is the sum of the homogeneous solution and the particular solution:

y = y_h + y_p = c1*e^((-2+3i)x) + c2*e^((-2-3i)x) + (1/12)*e^x - (1/169)*cosx + C*sinx.

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