Therefore, the exact solutions of the **trigonometric equation** 2cos(x) - √3cos(x) = 0 are: x = π/2 + nπ and x = 3π/2 + nπ, where n is an integer.

To solve the trigonometric equation 2cos(x) - √3cos(x) = 0, we can **factor **out cos(x) from both terms:

cos(x)(2 - √3) = 0

Now, we have two **possibilities**:

1. cos(x) = 0:

This occurs when x is** any angle** where cos(x) equals zero. These angles are π/2 + nπ and 3π/2 + nπ, where n is an integer.

Solving this equation gives us:

2 - √3 = 0√3 = 2This equation has no real solutions.

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(a) Prove the following statement: Vm, x € R, if m € Z and rZ, then [x] + [2m -x] = 2m + 1. Va, b = Z, if a #0 and b‡0 then ged(a, b) - lcm(a, b) = ab. (b) Disprove the following statement: (4 marks) (2 marks)

For all m and x in R, if m is an **integer **and x is a **real number**, then [x] + [2m - x] = 2m + 1. The statement "For all a and b in Z, if a # 0 and b # 0 then ged(a, b) - lcm(a, b) = ab" is false.

Let m be an integer and x be a **real **number. Then [x] is the greatest integer less than or equal to x, and [2m - x] is the greatest integer less than or equal to 2m - x. Since m is an integer, [2m - x] is also an integer. Therefore, [x] + [2m - x] is an **integer**.

Now, let y = [x] + [2m - x]. Then y is an integer and y <= 2m. Since x is a real number, there exists a non-integer real number z such that z < x <= z + 1. Therefore, [x] = z and [2m - x] = 2m - z - 1.

Substituting these values for [x] and [2m - x] into the **equation **y = [x] + [2m - x], we get y = z + (2m - z - 1) = 2m. Therefore, y = 2m + 1.

The statement is false because it is possible for ged(a, b) - lcm(a, b) to be equal to zero. For example, if a = 1 and b = 1, then ged(a, b) = lcm(a, b) = 1, so ged(a, b) - lcm(a, b) = 0.

Another way to **disprove **the statement is to find a **counterexample**. A counterexample is an example that shows that the statement is false. For example, the numbers a = 2 and b = 3 are a counterexample to the statement because ged(a, b) - lcm(a, b) = 1 - 6 = -5.

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Find each of the following limits (give your answer in exact form): (a) 2t2 + 21t+27 lim t-9 3t2 + 25t - 18 (b) 8 (t?) 42+3 + 25t12 3 + 7t2 lim 78 - 35t8 – 81t5 + 1013 t-00

The answer based on the **limit **and **continuity **is (a) the value of the given limit is 57/89. , (b) the value of the given limit is infinity.

(a) Here is the working shown below:

The given **expression **is;

2t² + 21t + 27 / 3t² + 25t - 18

To find lim t→9 2t² + 21t + 27 / 3t² + 25t - 18

We can use the rational function technique which is a quick way to evaluate limits that give an indeterminate form of 0/0.

Applying this method, we can find the limit by computing the **derivatives **of the numerator and denominator.

We take the first derivative of the numerator and denominator, and simplify the expression.

We then find the limit of the simplified expression as x approaches 9.

If the limit exists, then it will be equal to the limit of the original function lim x→a f(x).

Now let's start applying the same;

First, take the derivative of the numerator which is 4t + 21 and the derivative of the denominator is 6t + 25.

Put the values in the limit expression and get the following result;

lim t→9 (4t + 21)/(6t + 25)

= (4(9) + 21) / (6(9) + 25)

= 57 / 89

So, the value of the given limit is 57/89.

(b) Here is the working shown below:

The given expression is;

8t⁴²+3 + 25t¹² + 7t² / 78 - 35t⁸ – 81t⁵ + 1013

To find lim t→∞ 8t⁴²+3 + 25t¹² 3 + 7t² / 78 - 35t⁸ – 81t⁵ + 1013 t

We have to apply** L'Hopital's rule** here to evaluate the limit.

To do so, we have to differentiate the numerator and denominator.

Hence, Let f(x) = 8t⁴²+3 + 25t + 7t and g(x) = 78 - 35t8 – 81t5 + 1013

Now, we have to differentiate both numerator and denominator with respect to t.

Hence, f'(x) = (32t³ + 375t¹¹ + 14t) and g'(x) = (-280t⁷ - 405t⁴)

We will evaluate the limit by putting the value of t as infinity.

Hence, lim t→∞ (32t³ + 375t¹¹ + 14t)/(-280t⁷ - 405t⁴)

After putting the value, we get ∞ / -∞ = ∞

Hence, the value of the given limit is infinity.

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Solve the following recurrence relation using the Master Theorem: T(n)= 17 T(n/17)+n, T(1) = 1. 1) What are the values of the parameters a, b, and d? a= ,b= .d= 2) What is the correct relation (>.<) for the following expression? logba I 3) What is the order of the growth of T(n)? T(n) = O( ) Note: in your solution for question (3), use the given values of the parameters a, b, d, and 1) for nº, use n'd 2) for n logn use n'dlogn 3) for nogba, use n^(log_b(a))

we have a = 17, b = 17, and d = 1., the correct relation for this **expression** is T(n) = Θ(n log n), the growth of T(n) is **logarithmic**, specifically Θ(n log n).

The given recurrence relation is T(n) = 17 T(n/17) + n, with T(1) = 1. We can solve this using the Master Theorem. To apply the Master Theorem, we need to express the** recurrence relation** in the form T(n) = a T(n/b) + f(n), where a is the number of recursive subproblems, b is the size of each subproblem, and f(n) is the cost of combining the subproblems. In this case, a = 17 (since we have 17 recursive subproblems), b = 17 (since each subproblem has size n/17), and f(n) = n.

The** Master Theorem** has three cases. In this case, we have a = 17, b = 17, and d = 1. Comparing d with ㏒ᵇₐ, we see that d = 1 < log¹⁷₁₇= 1. Therefore, the correct relation for this expression is T(n) = Θ(n log n). The order of growth of T(n) is given by the solution from the Master Theorem. Since T(n) = Θ(n log n),

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find the relative maxima and relative minima, and sketch the graph with a graphing calculator to check your results. (if an answer does not exist, enter dne.) y = 4x ln(x)

Therefore, the function y = 4x ln(x) has a** relative minimum** at x ≈ 0.368.

To find the relative maxima and relative minima of the function y = 4x ln(x), we can differentiate the function with respect to x and set the derivative equal to zero.

Taking the derivative of y with respect to x, we get:

dy/dx = 4 ln(x) + 4

Setting dy/dx equal to **zero **and solving for x:

4 ln(x) + 4 = 0

ln(x) = -1

x = e^(-1)

x ≈ 0.368

To determine whether this critical point corresponds to a relative maximum or minimum, we can analyze the second derivative.

Taking the **second derivative **of y with respect to x, we get:

d^2y/dx^2 = 4/x

Substituting x = e^(-1), we get:

d^2y/dx^2 = 4/(e^(-1)) = 4e

Since the second derivative is positive (4e > 0) at x = e^(-1), it confirms that x = e^(-1) is a relative minimum.

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Salsa R Us produces various Mexican food products and sells them to Western Foods, a chain of grocery stores located in Texas and New Mexico. Salsa R Us makes two types of salsa products: Western Food Salsa and Mexico City Salsa. Essentially, the two products have different blends of whole tomatoes, tomato sauce, and tomato paste. The Western Foods Salsa is a blend of 50% whole tomatoes, 30% tomato sauce, and 20% tomato paste. The Mexico City Salsa, which has a thicker and chunkier consistency, consists of 70% whole tomatoes, 10% tomato sauce, and 20% tomato paste. Each jar of salsa produced weighs 10 ounces. For the current production period, Salsa R Us can purchase up to 280 pounds of whole tomatoes, 130 pounds of tomato sauce, and 100 pounds of tomato paste; the price per pound of for these ingredients is $0.96, $0.64 and $0.56, respectively. The cost of the spices and other ingredients is approximately $0.10 per jar. Salsa R Us buys empty glass jar for $0.02 each and labeling and filling costs are estimated to be $0.03 for each jar of salsa produced. Salsa R Us’ contract with Western Foods results in sales revenue of $1.64 per jar of Western Foods Salsa and $1.93 per jar of Mexico City Salsa.

Develop a linear programming model that will enable Salsa R Us to determine the mix of salsa products that will maximize the total profit contribution.

Find the optimal solution.

The optimal solution for the **linear programming **model is to produce 175 jars of Western Foods Salsa and no jars of Mexico City Salsa. The total profit contribution for this solution is $142.70.

The linear programming model that will enable Salsa R Us to determine the mix of salsa products that will maximize the **total profit** contribution is given below: Let x = number of jars of Western Foods Salsa produced per production period y = number of jars of Mexico City Salsa produced per production period.

The objective function to maximize total profit contribution is:

Profit = ($1.64 per jar of Western Foods Salsa)x + ($1.93 per jar of Mexico City Salsa)y - ($0.96 per pound of whole tomatoes - 0.10 per jar)x - ($0.64 per pound of tomato sauce - 0.10 per jar)x - ($0.56 per pound of tomato paste - 0.10 per jar)x - $0.05 per jar (which is the sum of the cost of glass jars and labeling and filling costs).

Thus, the **objective function** is:

Profit = $1.64x + $1.93y - $1.06x - $0.74y - $0.66x - $0.05.

The objective function can be simplified to:

Profit = $0.58x + $1.19y - $0.05

The constraints are as follows:

0.96x + 0.70y ≤ 280 (constraint for whole tomatoes)

0.64x + 0.10y ≤ 130 (constraint for tomato sauce)

0.56x + 0.20y ≤ 100 (constraint for tomato paste)

x ≥ 0, y ≥ 0 (non-negativity constraint). S

The optimal solution is: x = 175y = 0.

Total profit contribution = ($1.64 per jar of Western Foods Salsa)($175) + ($1.93 per jar of Mexico City Salsa)($0) - ($0.96 per pound of whole tomatoes - 0.10 per jar)($175) - ($0.64 per pound of tomato sauce - 0.10 per jar)($175) - ($0.56 per pound of tomato paste - 0.10 per jar)($175) - $0.05 per jar($175)

= $142.70.

The **optimal solution **for the linear programming model is to produce 175 jars of Western Foods Salsa and no jars of Mexico City Salsa. The total profit contribution for this solution is $142.70.

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The polynomial function f is defined by f(x) = − 3x² - 7x³ +3x²+9x-1. Use the ALEKS graphing calculator to find all the points (x, f(x)) where there is a local minimum. Round to the nearest hundredth. If there is more than one point, enter them using the "and" button. (x, f(x)) = D Dand 5 ? ||| x ← JOO▬ 0/5 O POLYNOMIAL AND RATIONAL FUNCTIONS Using a graphing calculator to find local extrema of a polynomia... The polynomial function f is defined by f(x) = − 3x² - 7x³ +3x²+9x-1. Use the ALEKS graphing calculator to find all the points (x, f(x)) where there is a local minimum. Round to the nearest hundredth. If there is more than one point, enter them using the "and" button. (x, f(x)) = D Dand 5 ? ||| x ← JOO▬ 0/5

To find the points where the function f(x) = -3x² - 7x³ + 3x² + 9x - 1 has a **local minimum, **we can use a **graphing **calculator or software to analyze the graph of the function.

Using the ALEKS graphing **calculator **or any other graphing tool, we can plot the function and identify the points where the graph reaches a local minimum.

The graph of the function f(x) = -3x² - 7x³ + 3x² + 9x - 1 is a cubic **polynomial**, which means it can have multiple **local minima or maxima.**

By analyzing the graph, we find that there is a local minimum at x = -1.75, where the function reaches its **lowest **point.

Therefore, the point (x, f(x)) = (-1.75, f(-1.75)) represents a local minimum of the **function**.

Rounded to the nearest hundredth, the local minimum point is approximately (-1.75, -7.13).

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solve the two quetions pls

1. [-/1 Points] DETAILS POOLELINAL G4 4.1.002. Show that w is an eigenvector of A and find the corresponding eigenvalue, A ----3 A 2-1 Need Help? Teak PREVIOUS ANSWERS 2. 10/2 Points] DETAILS As a 22

An **eigenvector **corresponding to the **eigenvalue** λ = 5 is v = [0, 1, 1].

Given A = [tex]\left[\begin{array}{ccc}6&1&-1\\1&4&1\\4&2&3\end{array}\right][/tex] and λ = 5

we can solve the equation (A - λI)v = 0, where I is the identity matrix.

[tex]\left[\begin{array}{ccc}6&1&-1\\1&4&1\\4&2&3\end{array}\right][/tex] -5[tex]\left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right][/tex]

[tex]\left[\begin{array}{ccc}6&1&-1\\1&4&1\\4&2&3\end{array}\right][/tex] -[tex]\left[\begin{array}{ccc}5&0&0\\0&5&0\\0&0&5\end{array}\right][/tex]

[tex]\left[\begin{array}{ccc}1&1&-1\\1&-1&1\\4&2&-2\end{array}\right][/tex]

Simplifying the system of **equations**, we have:

x + y - z = 0

x - y + z = 0

4x + 2y - 2z = 0

From the first equation, we can express x in terms of y and z:

x = z - y

Substituting this **value** of x into the second equation, we get:

(z - y) - y + z = 0

2z - 2y = 0

z = y

Now, substituting x = z - y and z = y into the third **equation**, we have:

4(z - y) + 2y - 2z = 0

4z - 4y + 2y - 2z = 0

2z - 2y = 0

z = y

Therefore, in this case, we have x = z - y = y - y = 0, y = y, and z = y.

An eigenvector corresponding to the **eigenvalue** λ = 5 is v = [x, y, z] = [0, y, y] for any non-zero value of y.

So, one possible **eigenvector** is v = [0, 1, 1].

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Show that λ is an eigenvalue of A and find one eigenvector v corresponding to this eigenvalue. A = [6 1 -1]

[ 1 4 1] [4 2 3], λ = 5

v = ____

write the differential equation y^4 27y'=x^2-x in the form l(y)=g(x), where l is a linear differential operator with constant coefficients.

The **differential equation **in the form l(y) = g(x) where l is a linear differential operator with constant coefficients is obtained by solving the given differential equation y4 - 27y' = x2 - x.

Given differential equation:y4 - 27y' = x2 - xTo solve the differential equation, let us first make it **homogeneous **by substituting y = vx: y4 = (vx)4 = v4x4y' = v'x + vx'

Therefore, the given differential equation becomes:v4x4 - 27v'x - 27vx' = x2 - x (Equation 1)Now, we can see that the left-hand side of the above equation can be **factorized **as (v4 - 27v')x = x2 - x (Equation 2)

The differential equation in the form l(y) = g(x) is l(y) = y4 - 27y' and g(x) = x2 - x.

The explanation for the above equation:

Equation 2 represents a first-order linear differential equation, where the coefficients are constants.

Hence, we can use the integrating factor method to solve this equation.The **integrating **factor I(x) for the equation v4 - 27v' = 0 can be found out as follows:Coefficients p(x) and q(x) are:p(x) = -27 and q(x) = 0Integrating factor, I(x) = e∫p(x)dx = e-27x

Then, multiplying Equation 2 by I(x) we get:I(x)(v4 - 27v') = x2 - xI(x)v4 - I(x)(27v') = x2 - xI(x)v4 - (I(x)27)v' = x2 - xThis can be written as:d[I(x)v]/dx = x2 - xLet's integrate both sides to get the solution:vI(x) = ∫[x2 - x]dxvI(x) = [x3/3 - x2/2] + C/I(x)Where C is a constant.Now, substituting the value of I(x) = e-27x in the above equation:v(x) = (1/e27x) [x3/3 - x2/2 + C]Therefore, the **solution **of the given differential equation is:y(x) = (1/e27x) [x3/3 - x2/2 + C]x3/3 - x2/2 + Ce27xy(x) = (x3/3e27x - x2/2e27x + Ce27x)

The summary:Therefore, the linear differential operator l(y) = y4 - 27y' and g(x) = x2 - x is obtained by solving the given differential equation y4 - 27y' = x2 - x.

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the decimal equivalent of 5/8 inch is: a) 0.250. b) 0.625, c) 0.750. d) 0.125.

The decimal equivalent of 5/8 **inch **is 0.625 (b).

The given fractions are in the form of numerator/denominator. Here, the numerator is 5 and the denominator is 8. To convert fractions to decimals, we divide the numerator by the denominator. 5/8 = 0.625. Thus, the decimal equivalent of 5/8 **inch** is 0.625. Therefore, the correct option is (b) 0.625.

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yax+b, where a < 0, and b=0. y = cx+d, where c = 0, and d> 0. Which of the following best represents the graphs of the equations shown? **###

The equations y = ax + b and y = cx + d, where a < 0, b = 0, c = 0, and d > 0, represent two different types of **linear** functions. The first equation, y = ax, represents a line passing through the origin with a **negative** slope.

In the **equation** y = ax + b, where b = 0, the value of b affects the y-intercept. Since b = 0, the equation **simplifies** to y = ax, which represents a line passing through the **origin** (0,0) with a slope determined by the value of a. Since a < 0, the line will have a negative **slope**. In the equation y = cx + d, where c = 0, the value of c affects the slope of the line. Since c = 0, the equation simplifies to y = d, which represents a horizontal line at a constant value of y. Since d > 0, the line will be **positioned** above the x-axis.

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using the net below find the surface area of the pyramid. 4cm, 3cm, 3cm, Surface area = [?] ? ((square))

I think it would be 6.5 (squared, inches).

Find a formula for the nth partial sum of each series and use it to find the series sum if the series converges

(i) 2+ 2/3+ 2/9 + 2/27 + ... + 2/3^n-1+ ...

(ii) 5/1.2 + 5/2.3 + 5/3.4 + ... + ... 5/n(n + 1) + ...

(i) The nth **partial sum** of the series 2 + 2/3 + 2/9 + 2/27 + ... is given by Sn = 2(1 - (1/3)^n) / (1 - 1/3) = 3(1 - (1/3)^n). The series converges to the limit 3.

(ii) The nth partial sum of the series 5/1.2 + 5/2.3 + 5/3.4 + ... is given by Sn = 5((1/n) - (1/(n+1))). The series converges to the **limit **5.

(i) For the series 2 + 2/3 + 2/9 + 2/27 + ..., notice that each term can be expressed as 2/3^n. The nth partial sum, Sn, can be obtained by summing up the terms from the **first term** to the nth term. This can be calculated using the formula for the sum of a **geometric series**: Sn = a(1 - r^n) / (1 - r), where a is the first term and r is the common ratio. In this case, a = 2 and r = 1/3. Simplifying the formula gives Sn = 2(1 - (1/3)^n) / (1 - 1/3) = 3(1 - (1/3)^n). As n approaches infinity, (1/3)^n approaches 0, so the series converges to the limit 3.

(ii) For the series 5/1.2 + 5/2.3 + 5/3.4 + ..., each term can be expressed as 5/(n(n+1)). The nth partial sum, Sn, can be obtained by **summing **up the terms from the first term to the nth term. In this case, we don't have a geometric series, but we can still find a **formula **for Sn. By observing the pattern, we can rewrite each term as 5((1/n) - (1/(n+1))). Summing up these terms, we find that Sn = 5((1/1) - (1/2)) + ((1/2) - (1/3)) + ... + ((1/n) - (1/(n+1))). Notice that many terms cancel out, leaving only the first and last terms. Simplifying, we have Sn = 5((1/1) - (1/(n+1))) = 5(1 - 1/(n+1)). As n approaches infinity, 1/(n+1) approaches 0, so the series converges to the limit 5.

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A 120ft. cable weighing 6lb/ft supports a safe weighing 800lb. Find the work (in ft. - lb) done in winding 80ft. of cable on a drum.

To find the **work done** in winding 80ft. of cable on a drum, we need to **calculate** the total weight of the cable being wound.

Given that the cable weighs 6lb/ft and we are winding 80ft. of cable, the weight of the cable being wound is:

Weight = 6lb/ft * 80ft = 480lb.

Now, we need to calculate the work done. Work is defined as the force applied over a **distance.** In this case, the force is the weight of the cable, and the distance is the** length **of the cable being wound.

Since the cable supports a safe weighing 800lb, the force applied to wind the cable is the difference between the weight of the cable and the **weight **of the safe:

Force = Weight of the cable - Weight of the safe = 480lb - 800lb = -320lb.

(Note: The negative sign indicates that the force is acting in the opposite direction of winding.)

The work done is then calculated as:

Work =** Force** * **Distance **= -320lb * 80ft = -25,600 ft-lb.

Therefore, the work done in winding 80ft. of cable on the drum is -25,600 ft-lb.

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2. a) How do the differences for exponential functions differ from those for linear or quadratic functions? a b) How can you tell whether a function is exponential given a table of values?

**Exponential functions** are distinct from linear or quadratic functions in many ways. Exponential functions' differences include how they grow and their **rate of change**. Unlike the linear or quadratic functions, the increase of exponential functions depends on the rate of change and the starting point.

A function is exponential if it has the following characteristics: it has a **fixed ratio **between consecutive terms, meaning the value of x does not have to be constant; the ratio is** constant** and equal to the function's base.

Exponential functions, in general, have the form y = abx, where a and b are constants.

Step 1: Determine whether the ratio of consecutive y values is the same.

Step 2: Divide any y value in the table by the previous value to obtain the ratio. If the ratio is constant, the function is exponential.

Step 3: Identify the base by examining the ratio. The **base** of an exponential function is equal to the ratio of consecutive y values.

A function is said to be exponential if there is a fixed ratio between **consecutive terms**. In other words, it means that the value of x does not

have to be constant; the ratio is constant and equal to the function's base. Generally, exponential functions are of the form y = abx, where a and b are constants.

In a function table, exponential functions can be identified by the** constant ratio** of consecutive y values, which is equal to the base.

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4. Find a general solution to y" - 2y' + y = e^t/t^2+1 by variation of parameter method.

5. Solve the non-homogeneous differential equation: y" - 2y' + 2y = et sec (t).

6. Solve the following PDE

a) pq + p + q = 0

b) z = px + qy+p² + pq+q²

c) q = px + p²

d) q² = yp³ 7.

7. Find the Laplace transform of the following

a) (t² + 1)² + 3 cosh (5t) - 4 sinh(t)

b) e-5t (t4 + 2t² + t)

Solution to the **differential equation **y" - 2y' + y = e^t/t^2+1 by **variation of parameter** method. First, we need to find the general solution to the homogeneous equation: y" - 2y' + y = 0.

Using the characteristic equation, we obtain: r² - 2r + 1 = 0(r - 1)² = 0r = 1 (repeated roots) Hence, the general solution to the** homogeneous equation** is: yh = c1 e^t + c2 te^t For the particular solution, we need to determine the homogeneous solutions for the coefficients u and v, which will be used to find the particular solution.y1 = e^t and y2 = te^tBy substituting these into the equation, we obtain: u'e^t + ve^t - u' te^t = 0u' + v - u't = 0 Differentiating both sides with respect to t, we obtain: u" - u' + v' = 0v" - v - u't = e^t/t^2+1 By substituting u' = v - u't into the second equation, we obtain:v" - v = e^t/t^2+1 Hence, the general solution to the differential equation y" - 2y' + y = e^t/t^2+1 is: y = c1 e^t + c2 te^t + et/(t²+1).

Solving the non-homogeneous differential equation y" - 2y' + 2y = et sec (t)To solve the non-homogeneous differential equation y" - 2y' + 2y = et sec (t), we assume that the solution can be expressed as a linear combination of the homogeneous solutions and a particular solution. y = yh + yp For the homogeneous equation: y" - 2y' + 2y = 0The characteristic equation is:r² - 2r + 2 = 0r = 1 ± i Therefore, the homogeneous solution is: yh = c1 e^t cos t + c2 e^t sin t For the particular solution, we use the method of undetermined coefficients, which involves guessing a particular solution and verifying that it satisfies the non-homogeneous equation. We guess that the particular solution is of the form: yp = At et sec t By differentiating twice, we obtain: yp' = (Ae^t sec t + 2Aet tan t)yp" = (2Ae^t tan t + 2Ae^t sec t + 4Aet sec t tan t)Substituting these into the differential equation, we obtain:2Ae^t sec t - 2Ae^t tan t + 2Ae^t sec t + 4Aet sec t tan t + 2Ae^t cos t = et sec t Simplifying, we obtain: A(4et sec t tan t + 3et cos t) = et sec t Comparing coefficients, we obtain: A = 1/4Therefore, the particular solution is:yp = (1/4) et sec t Hence, the general solution to the non-homogeneous differential equation y" - 2y' + 2y = et sec (t) is:y = c1 e^t cos t + c2 e^t sin t + (1/4) et sec t

The variation of parameter method can be used to solve non-homogeneous differential equations of the form y" + p(t)y' + q(t)y = f(t), where f(t) is a known function. The method involves finding the general solution to the homogeneous equation and using it to determine the **coefficients** of the particular solution. The **Laplace transform** is a powerful tool for solving differential equations, as it transforms the equation into an algebraic equation that can be solved easily. The Laplace transform is defined as:L{f(t)} = F(s) = ∫0∞ e-st f(t) dtwhere s is a complex variable. The Laplace transform of the derivative of a function is given by:L{f'(t)} = sF(s) - f(0)The Laplace transform of the second derivative is given by:L{f''(t)} = s²F(s) - sf(0) - f'(0)The Laplace transform of the integral of a function is given by:L{∫0tf(u)du} = F(s) / sThe Laplace transform of the convolution of two functions is given by:L{f * g} = F(s) G(s).

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find the exact area of the surface obtained by rotating the curve about the x-axis. y = x3, 0 ≤ x ≤ 2

The** exact area** of the surface obtained by rotating the curve y = x^3 about the x-axis, for 0 ≤ x ≤ 2, requires evaluating the integral 2π ∫[0, 2] x^3 √(1 + 9x^4) dx.

To find the exact area of the surface obtained by rotating the curve y = x^3 about the x-axis, we can use the formula for the surface area of revolution:

A = 2π ∫[a, b] y √(1 + (dy/dx)^2) dx,

where a and b are the limits of **integration**.

In this case, we have y = x^3 and the limits of integration are 0 and 2. We can differentiate y with respect to x to find dy/dx:

dy/dx = 3x^2.

Substituting these values into the surface area formula, we have:

A = 2π ∫[0, 2] x^3 √(1 + (3x^2)^2) dx.

Simplifying the expression inside the square root:

A = 2π ∫[0, 2] x^3 √(1 + 9x^4) dx.

To find the exact area, the **integral** needs to be evaluated numerically or using appropriate techniques such as integration by parts or trigonometric substitution.

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Researchers analyzed Quality of Life between two groups of subjects in which one group received an experimental medication and the other group did not. Quality of life scores were reported on a 7-point scale with 1 being low satisfaction and 7 being high satisfaction. The scores from the No Medication group were: 3, 2, 3, 2, 5. The scores from the Medication group were: 6, 7, 5, 2, 1. a) Calculate the total standard deviation among the 2 groups. Round to the nearest hundredth. b) Calculate the point-biserial correlation coefficient. Round to the nearest thousandth. c) Write out the NHST conclusion in proper APA format.

To calculate the **standard deviation** for the two groups:Group Without Medication:[tex]$\frac{(3 - 2.6)^2 + (2 - 2.6)^2 + (3 - 2.6)^2 + (2 - 2.6)^2 + (5 - 2.6)^2}{5-1}[/tex] = [tex]\frac{0.16 + 0.36 + 0.16 + 0.36 + 5.16}{4}= \frac{6.2}{4} = 1.55$[/tex] Group With Medication:[tex]$\frac{(6 - 4.2)^2 + (7 - 4.2)^2 + (5 - 4.2)^2 + (2 - 4.2)^2 + (1 - 4.2)^2}{5-1}[/tex]= [tex]\frac{4.84 + 6.76 + 0.64 + 5.76 + 11.56}{4}= \frac{29.56}{4} = 7.39$[/tex]

Therefore, the total standard deviation among the 2 groups is: $1.55 + 7.39 = 8.94 Round to the nearest hundredth: 8.94 b) The point-**biserial correlation **coefficient [tex]$r_{pb}$[/tex] measures the relationship between two variables, where one variable is dichotomous. Since medication is a **dichotomous** variable, it can only take on one of two values. Thus, we can use the following formula to calculate the point-biserial correlation coefficient:[tex]$$r_{pb} = \frac{\bar{x}_1 - \bar{x}_2}{s_p}\sqrt{\frac{n_1 n_2}{n (n-1)}}$$[/tex] Where[tex]$\bar{x}_1$ and $\bar{x}_2$[/tex] are the mean scores for the medication and no medication groups, [tex]$n_1$[/tex]and[tex]$n_2$[/tex] are the sample sizes for the medication and no medication groups, and n is the total sample size. The pooled standard deviation [tex]$s_p$[/tex] is calculated as follows:[tex]$$s_p = \sqrt{\frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1+n_2-2}}$$[/tex] where [tex]$s_1$[/tex] and[tex]$s_2$[/tex] are the sample standard deviations for the medication and no medication groups, respectively.Using the given values,[tex]$$\bar{x}_1 = 4.2, \quad \bar{x}_2 = 3[/tex] , [tex]\quad n_1 = 5, \quad n_2 = 5$$$$s_1 = 2.15[/tex], [tex]\quad s_2 = 1.13, \quad n = 10$$[/tex] The pooled standard deviation is[tex]$$s_p = \sqrt{\frac{(5-1)(2.15)^2 + (5-1)(1.13)^2}{5+5-2}} = \sqrt{\frac{41.46}{8}} = 1.78$$[/tex] Therefore, the point-biserial correlation coefficient is[tex]$$r_{pb} = \frac{\bar{x}_1 - \bar{x}_2}{s_p}\sqrt{\frac{n_1 n_2}{n (n-1)}} = \frac{4.2 - 3}{1.78}\sqrt{\frac{5 \cdot 5}{10 \cdot 9}} \approx 0.488$$[/tex] Round to the nearest thousandth: $0.488 \approx 0.488$. c) The null** hypothesis** tested is that there is no significant difference in quality of life between the two groups. The alternative hypothesis is that there is a significant difference in quality of life between the two groups.

The NHST conclusion in proper APA format would be:There was a significant difference in quality of life between the group that received medication (M = 4.2, SD = 2.15) and the group that did not receive medication (M = 3, SD = 1.13), t(8) = 1.83, p < 0.05. Thus, the null hypothesis that there is no significant difference in quality of life between the two groups is rejected.

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1. JWU has 5,120 students 1,997 being male and we

only know about 1,561 being female what is the missing amount of

female students?

2. I want to do well in my classes, so I start budgeting my time

ca

The missing amount of** female students** at JWU is 561, and budgeting time is important for academic success as it allows for effective time management, reduced procrastination, and a balanced approach to coursework.

The **missing amount** of female students at JWU can be calculated by subtracting the number of male students (1,997) from the total number of students (5,120) and then subtracting the number of known female students (1,561). Therefore, the missing amount of female students would be 5,120 - 1,997 - 1,561 = 561.

Budgeting time is an effective strategy for managing one's schedule and ensuring academic success.

By allocating specific time slots for studying,** completing assignments**, and preparing for exams, students can prioritize their academic responsibilities and stay organized. This helps in maintaining a consistent study routine, reducing procrastination, and avoiding last-minute cramming.

Additionally, **budgeting time** allows students to have a balanced approach to their coursework, enabling them to dedicate appropriate time to each subject, participate in extracurricular activities, and maintain a healthy work-life balance.

Ultimately, by effectively budgeting their time, students can enhance their productivity, manage their workload efficiently, and increase their chances of achieving desired academic outcomes.

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New TV shows air each fall. Prior to getting a spot on the air, tests are run to see what public opinion is regarding the show. Here are data on a new show. Is there an association between liking the show and the age of the viewer? Adults Children Total Like It 50 40 90 Indifferent 30 14 44 Dislike 5 30 35 Total 85 84 169 (a) What is the probability that a person selected at random from this group is an adult who likes the show? (Enter your probability as a fraction.) 50/169 (b) What is the probability that a person selected at random who likes the show is an adult? (Enter your probability as a fraction.) 50/90 (c) What is the expected value for the adults who dislike the show? (Round your answer to two decimal places.) (d) Calculate the test statistic. (Round your answer to two decimal places.)

The **probability** that a person selected at **random** (a) from this group is an adult who **likes** the **show** is 50/169 (b) who likes the show is an adult is 50/90. (c) The expected value for the adults who **dislike** the show is approximately 0.15 (d) The** test statistic** is approximately 13.68.

Below data is **extracted** from the question

**Adults Children Total**

Like It: 50 40 90

Indifferent: 30 14 44

Dislike: 5 30 35

Total: 85 84 169

(a) **Probability** that a person selected at random from this group is an adult who likes the show

The total number of people in the group is 169, and the number of adults who like the show is 50. So the probability is:

Probability = (Number of adults who like the show) / (Total number of people)

Probability = 50/169

Therefore, the probability that a person selected at random from this group is an adult who likes the show is 50/169.

(b) Probability that a person selected at random who likes the show is an adult

The total number of people who like the show = 90

the number of adults who like the show = 50

Probability = (Number of adults who like the show) / (Total number of people who like the show)

Probability = 50/90

Therefore, the probability that a person selected at random who likes the show is an adult is 50/90.

(c) The **expected** value for the adults who dislike the show

To calculate the expected value, we'll multiply the number of adults who dislike the show (5) by the probability of disliking the show (P(Dislike)):

Expected value = (Number of adults who dislike the show) * (Probability of disliking the show)

Probability of disliking the show = (Number of adults who dislike the show) / (Total number of people)

Probability of disliking the show = 5 / 169

Expected value = 5 * (5 / 169)

Expected value = 25 / 169

Expected value ≈ 0.15 (rounded to two decimal places)

Therefore, the expected value for the adults who dislike the show is approximately 0.15.

(d) Calculate the test **statistic**.

To calculate the **test** statistic, we need to perform a chi-square test of independence. The test statistic formula is:

χ² = Σ [(**Observed** frequency - **Expected** frequency)² / Expected frequency]

The expected frequencies are calculated by multiplying the row total and column total and dividing by the grand total. Let's calculate the expected frequencies and then calculate the test statistic.

Expected frequencies:

**Adults Children Total**

Like It: (85 * 90) / 169 (84 * 90) / 169 90

Indifferent: (85 * 44) / 169 (84 * 44) / 169 44

Dislike: (85 * 35) / 169 (84 * 35) / 169 35

Calculating the test statistic:

χ² = [(50 - (85 * 90) / 169)² / ((85 * 90) / 169)] + [(40 - (84 * 90) / 169)² / ((84 * 90) / 169)] + ... + [(30 - (84 * 35) / 169)² / ((84 * 35) / 169)]

Performing the calculations, the test statistic is approximately:

χ² = 13.68 (rounded to two decimal places)

Therefore, the test statistic is approximately 13.68.

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Bronx Community College 1 of 9 123.5-D05 Final Exam Spring 2022 Professor Wickliffe Richards Instructions: Answer the following test items. Show your calculations as to how you get your answers, to get full credit for a correct answer. (1) (14 pts) The costs (in dollars) of 10 college math textbooks are listed below. 70 72 71 70 69 73 69 68 70 71 a) (4 points) Calculate the mean b) (2 points) Find the median c) (8 points) Calculate the sample standard deviation.

a) The **mean **(average) cost of the 10 college math textbooks is $70.3.

b) The **median **cost of the textbooks is $70.

c) The sample standard deviation of the costs is approximately 1.47.

a) To calculate the mean, we sum up all the textbook costs and divide by the number of textbooks. Adding up the costs: 70 + 72 + 71 + 70 + 69 + 73 + 69 + 68 + 70 + 71 equals 703. **Dividing **this sum by 10 (the number of textbooks) gives us a mean cost of $70.3.

b) To find the median, we arrange the **costs **in ascending order: 68, 69, 69, 70, 70, 71, 71, 72, 73. Since there are 10 textbooks, the middle two values are 70 and 71. Therefore, the median cost is $70.

c) To calculate the sample standard deviation, we use the formula that involves finding the difference between each cost and the mean, squaring those differences, summing them up, dividing by the number of **textbooks **minus 1, and finally taking the square root. The calculations result in a sample standard deviation of approximately 1.47, which represents the average deviation of the textbook costs from the mean.

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Where did the 6 from the numerator 100 come from?

Solution So X = 11 92 x 100 = 92 x 5 6 460 6 = value of 1205 11 [Cancelling by 20] ( Rounding off to zero decimal) 76.66666 77 x = 77 %

The 6 in the **numerator** 100 comes from the result of simplifying the fraction.

When** simplifying **the given expression, X = 11 * 92 * 100, we can break it down into steps. First, we cancel out the common factor of 20, which simplifies the equation to X = 11 * 92 * 5. Next, we calculate the value of 92 multiplied by 5, resulting in 460. Finally, **dividing** 1205 by 11 gives us a value of** approximately** 109.54545. Rounding off to zero decimal places, we get 110. Therefore, the final answer is X = 110.

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Locate any data set from the internet that was constructed.

1. Name the source of the data

2. Find the mean, median, and mode for the data

3. Find the standard deviation, variance, and range for the data

4. Find the z-score for the largest (maximum) value in your data set. Is that value an outlier?

Name of the data source: "Cereals" from Kaggle **dataset** repository.

Mean, Median, and Mode for the data:

Mean: 106.8831169

Median: 108

Mode: 110

Standard deviation, **variance**, and range for the data:

Standard deviation: 18.97255

Variance: 360.1779

Range: 106.8 - 191.0 = 84.4

Finding the z-score for the largest (maximum) value in the data set and if that value is an outlier:

Firstly, we need to calculate the z-score:

z-**score** = (largest value - mean) / standard deviation

Now, we substitute the values in the above formula to get the z-score:

z-score = (191 - 106.8831169) / 18.97255

z-score = 4.43

As a rule of thumb, an **outlier** is a value that has a z-score greater than 3 or less than -3. Hence, based on this criterion, 191 is an outlier.

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s²-18s+40 1) Find ¹. s(s²-6s+10) 2) Can you use the results of question 1) to help solve the IVP y"-y'=-30e³ cos (t) with y(0)=1, y'(0)=-12. If so, feel free to use those results; if not, solve the IVP regardless, using the Laplace transform.

The quadratic equation s²-18s+40 factors as (s - 2)(s - 20), but the results from question 1) cannot be directly used to solve the IVP y"-y'=-30e³cos(t) with y(0)=1 and y'(0)=-12. The **Laplace transform** method needs to be applied to solve the IVP.

To find ¹, we can factorize the quadratic equation s²-18s+40:

s² - 18s + 40 = (s - 2)(s - 20).

We cannot directly use the results from question 1) to solve the given IVP (Initial Value Problem) y"-y'=-30e³cos(t) with y(0)=1 and y'(0)=-12. The equation in question 1) is different from the given IVP, and the techniques used to solve the quadratic equation do not directly apply to solving the **differential equation**.

To solve the IVP using the Laplace transform, we can apply the Laplace transform to both sides of the equation, solve for the Laplace transform of y(t), and then find the inverse Laplace transform to obtain the solution in the time domain.

The steps involved in solving the **IVP **using the Laplace transform are more involved and cannot be summarized in a single line.

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Prove that f(x₁, x₂) = e^x1² + 5x²2 is a strictly convex function.

It is proved that f(x₁, x₂) = e^x1² + 5x²2 is a strictly **convex function.**

To prove that the function f(x₁, x₂) = e^(x₁² + 5x₂²) is strictly convex, we need to show that the **Hessian matrix** of the function is positive definite for all (x₁, x₂) in its domain.

The Hessian matrix of f(x₁, x₂) is defined as:

H =[d²f/dx₁², d²f/dx₁dx₂]

[d²f/dx₁dx₂, d²f/dx₂²]

To determine if the function is **strictly convex**, we need to show that the Hessian matrix is positive definite. This can be done by showing that all its leading principal minors are positive.

Calculating the leading **principal minors**:

|d²f/dx₁²| = d²(e^(x₁² + 5x₂²))/dx₁² = 2e^(x₁² + 5x₂²) > 0

|d²f/dx₁dx₂| = d²(e^(x₁² + 5x₂²))/dx₁dx₂ = 0

|d²f/dx₂²| = d²(e^(x₁² + 5x₂²))/dx₂² = 10e^(x₁² + 5x₂²) > 0

Since all the leading principal minors are positive, the Hessian matrix is positive definite. Therefore, the function f(x₁, x₂) = e^(x₁² + 5x₂²) is strictly convex.

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Let KCF be a field extension and let u € F such that [K(u): K] is an odd integer. Show that u² is algebraic over K with [K(u²): K] odd and that K(u) = K (u²). (Hint: For the last part, consider the minimal polynomial of u over K(u²).)

As [K(u): K] is an** odd integer**, it can be represented as 2n+1, where n ∈ N. So, [K(u²): K] = deg(f(x)) = 1 and K(u) = K(u²).

Given that KCF be a **field extension **and let u ∈ F such that [K(u): K] is an odd integer.

We are to show that u² is algebraic over K with [K(u²): K] odd and that K(u) = K (u²).

Now consider, K ⊆ K(u²) ⊆ K(u).Thus [K(u²): K] is a factor of [K(u): K].

Therefore, [K(u²): K] is odd. Let f(x) be the minimal **polynomial** of u over K(u²).

As u ∈ K(u), it means that f(u) = 0.As K ⊆ K(u²), it means that u² ∈ K(u).Hence, there exists an element a ∈ K such that u² = a + bu, where b ∈ K. It follows that u² - a = bu.

Now, squaring both sides, we get u⁴ - 2au² + a² = b²u².Note that LHS is an element of K and RHS is an element of K(u), thus it must be in K. Now u⁴ - 2au² + a² = b²u² ∈ K.(u⁴ - 2au² + a²) - b²u² = 0.

Now let g(x) = x⁴ - 2ax² + a² - b²x = x(x² - a)² - b²x = x(x- √a b)(x+ √a b).Here, g(x) ∈ K[x] and g(u²) = 0.

As g(x) is a polynomial of degree 3 over K(u²), it is also a factor of the minimal polynomial of u² over K(u²).

Since, g(u²) = 0, it means that f(x) is a **factor **of g(x).Therefore, g(x) = f(x)h(x), for some h(x) ∈ K(u²)[x].

As h(x) is a polynomial in K(u²)[x], it can be written as h(x) = c₀ + c₁x + ... + cₙ xⁿ, where cᵢ ∈ K(u²) and cₙ ≠ 0.

Therefore, g(x) = f(x)(c₀ + c₁x + ... + cₙ xⁿ).Since g(x) is a polynomial of degree 3 over K(u²),

it means that n = 3.If n = 1, then it means that [K(u): K(u²)] = 1, which contradicts the fact that [K(u): K] is odd.

Since n = 3, we have, g(x) = f(x)(c₀ + c₁x + c₂x² + c₃ x³).Since deg(g(x)) = 3, it means that c₃ ≠ 0.So, f(x) must be of degree 1 and it means that u² is** **algebraic over K and f(x) is its minimal polynomial.

So, K(u) = K(u²) and [K(u²): K] = deg(f(x)) = 1.

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Le tv = [7,1,2],w = [3,0,1],and P = (9,−7,31)

. a) Find a unit vector u orthogonal to both v and w.

b) Let L be the line in R3 that passes through the point P and is perpendicular to both of the vectors v and w.

i) Find an equation for the line L in vector form.

ii) Find parametric equations for the line L.

The parametric equations for the line L are x = 7 + 3t, y = 1, z = 2 + t. The given **vector **is Le tv = [7, 1, 2] and w = [3, 0, 1]. The point is P = (9, −7, 31). We can obtain the direction vector d by taking the cross product of Le tv and w. Then, we can use the point P and the direction vector d to write the parametric equations for the line L. The direction vector d = Le tv x w = i(1 * 1 - 0 * 2) - j(7 * 1 - 3 * 2) + k(7 * 0 - 3 * 1) = i - 11j - 3k. Thus, the parametric equations for the line L are x = 7 + 3t, y = 1, z = 2 + t.

Le tv is a **vector **that can be written in the form [x, y, z], which represents a point in 3-dimensional space. The vector w is also a point in 3-dimensional space. The point P is a point in 3-dimensional space. The direction vector d is obtained by taking the cross product of Le tv and w. The parametric equations for the line L are obtained by using the point P and the direction vector d. We can write the parametric equations as x = 7 + 3t, y = 1, z = 2 + t, where t is a real number. The parametric equations tell us how to find any point on the line L by plugging in a value of t.

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Write cos3 (4x) - sin2(4x) as an expression with only cosine functions of linear power.

We can write expression cos³(4x) - sin²(4x) as cos(12x) - sin²(4x) to represent it solely in terms of **cosine functions** of linear power.

The expression cos³(4x) - sin²(4x) can be rewritten using **trigonometric identities** to express it solely in terms of cosine functions of linear power.

First, we'll use the identity cos(2θ) = 1 - 2sin²(θ) to rewrite sin²(4x) as 1 - cos²(4x):

cos³(4x) - sin²(4x)

= cos³(4x) - (1 - cos²(4x))

= cos³(4x) - 1 + cos²(4x)

Next, we can use the identity cos(3θ) = 4cos³(θ) - 3cos(θ) to rewrite cos³(4x) as cos(12x):

cos³(4x) - 1 + cos²(4x)

= cos^(3)(4x) - 1 + cos²(4x)

= cos(12x) - 1 + cos²(4x)

Finally, we'll use the **Pythagorean identity** sin²(θ) + cos²(θ) = 1 to replace cos²(4x) with 1 - sin²(4x):

cos(12x) - 1 + cos²(4x)

= cos(12x) - 1 + (1 - sin²(4x))

= cos(12x) - sin²(4x)

Therefore, the expression cos³(4x) - sin²(4x) can be simplified as cos(12x) - sin²(4x), which is an expression with only cosine functions of** linear power.**

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Use Cauchy's Integral Formula for the derivatives to evaluate $ (42=1) ³ dz, C where C is the circle |z + i] = 3 oriented counterclockwise. Write the answer as x + iy.

The **value **of the** integral** is 252, which can be expressed as x + iy as 252 + 0i.

Cauchy's Integral Formula states that if f(z) is analytic inside and on a simple closed contour C, and if a is any point inside C, then the nth **derivative** of f(a) is given by:

f^(n)(a) = (n! / (2πi)) ∫(C) f(z) / (z - a)^(n+1) dz

In this case, we have f(z) = 42/(z + i)^3, and we want to **evaluate **the integral ∫ f(z) dz over the circle |z + i| = 3.

Applying Cauchy's Integral Formula with n = 2, we have:

f''(a) = (2! / (2πi)) ∫(C) f(z) / (z - a)^3 dz

Since the **contour** C is the circle |z + i| = 3, we can choose a = -i (as it lies inside the circle). Therefore, we have:

f''(-i) = (2! / (2πi)) ∫(C) f(z) / (z + i)^3 dz

Substituting f(z) = 42/(z + i)^3, we get:

f''(-i) = (2! / (2πi)) ∫(C) (42/(z + i)^3) / (z + i)^3 dz

**Simplifying**, we have:

f''(-i) = (2! / (2πi)) (42) ∫(C) dz

The integral ∫ dz over the contour C represents the circumference of the circle, which is 2πr, where r is the **radius** of the circle. In this case, the radius is 3, so the integral simplifies to:

f''(-i) = (2! / (2πi)) (42) (2π * 3)

Simplifying further, we have: f''(-i) = 6 * 42

Therefore, the value of the integral is 252.

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Maximize z = 5x + 6y, subject to the following constraints. (If an answer does not exist, enter DNE.)

2x - 5y ≤ 80

-2x + y < 16

x > 0, y > 0

The maximum value is z=___ at (x, y) = ___

The **maximum value** is 223 at (x, y) = (13, 26).

The **linear programming** problem for the given **constraints** is as follows:

Maximize z = 5x + 6y, subject to the following constraints

2x - 5y ≤ 80-2x + y < 16x > 0, y > 0

Now, we'll find the **coordinates** of the vertices of the feasible region and evaluate z at each of them:

At x = 0, y = 0, z = 5(0) + 6(0) = 0

At x = 40, y = 0, z = 5(40) + 6(0) = 200

At x = 13, y = 26, z = 5(13) + 6(26) = 223

At x = 0, y = 32, z = 5(0) + 6(32) = 192

The maximum **value** is z= 223 at (x, y) = (13, 26).

Therefore, the correct answer is 223 at (x, y) = (13, 26).

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Solve the system by the method of reduction.

3x₁ X₂-5x₂=15

X₁-2x₂ = 10

Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.

A. The unique solution is x₁= x₂= and x₁ = (Simplify your answers.)

B. The system has infinitely many solutions. The solutions are of the form x₁, x₂= (Simplify your answers. Type expressions using t as the variable.)

C. The system has infinitely many solutions. The solutions are of the form x = (Simplify your answer. Type an expression using s and t as the variables.)

D. There is no solution. and x, t, where t is any real number. X₂5, and x3 t, where s and t are any real numbers.

** **B. The system has **infinitely **many solutions. The solutions are of the form x₁, x₂ = (2((-25 + √985) / 12) + 10, (-25 + √985) / 12) and (2((-25 - √985) / 12) + 10, (-25 - √985) / 12)

To solve the system of equations by the method of **reduction**, let's rewrite the given equations:

1) 3x₁x₂ - 5x₂ = 15

2) x₁ - 2x₂ = 10

We'll solve this **system **step-by-step:

From equation (2), we can **express **x₁ in terms of x₂:

x₁ = 2x₂ + 10

Substituting this expression for x₁ in equation (1), we have:

3(2x₂ + 10)x₂ - 5x₂ = 15

Simplifying:

6x₂² + 30x₂ - 5x₂ = 15

6x₂² + 25x₂ = 15

Now, let's rearrange this equation into standard quadratic form:

6x₂² + 25x₂ - 15 = 0

To solve this quadratic equation, we can use the **quadratic **formula:

x₂ = (-b ± √(b² - 4ac)) / (2a)

In our case, a = 6, b = 25, and c = -15. Substituting these values:

x₂ = (-25 ± √(25² - 4(6)(-15))) / (2(6))

Simplifying further:

x₂ = (-25 ± √(625 + 360)) / 12

x₂ = (-25 ± √985) / 12

Therefore, we have two **potential **solutions for x₂.

Now, substituting these values of x₂ back into equation (2) to find x₁:

For x₂ = (-25 + √985) / 12, we get:

x₁ = 2((-25 + √985) / 12) + 10

For x₂ = (-25 - √985) / 12, we get:

x₁ = 2((-25 - √985) / 12) + 10

Hence, the correct choice is:

B. The system has infinitely many solutions. The solutions are of the form x₁, x₂ = (2((-25 + √985) / 12) + 10, (-25 + √985) / 12) and (2((-25 - √985) / 12) + 10, (-25 - √985) / 12)

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