On Your Own
1) A sphere has a radius of 7 cm. What is the volume?
round your answer to the
nearest cubic centimeter.
Like example 1
4
2) A sphere has a volume of 72π in³. What is the radius?
round to the nearest tenth
of an inch.
Like example 3

The parabola with equation y=ax^2+bx whose tangent line at (1, 1) has equation y=4x-3; y = 3x^2 - 2x

The tangent line at (1, 1) has equation y = 4x - 3, which means that the slope of the tangent line at that point is 4. We know that the derivative of y = ax^2 + bx is y' = 2ax + b, which gives us the slope of the tangent line at any point on the parabola. So, we can set 2ax + b equal to 4 (the slope of the tangent line) and substitute x = 1 and y = 1 (the point on the tangent line and parabola, respectively).
2a(1) + b = 4
a(1)^2 + b(1) = 1
Simplifying the second equation, we get b = 1 - a. Substituting this into the first equation and simplifying, we get:
2a + 1 - a = 4
a = 3
Therefore, b = 1 - a = -2. The equation of the parabola is y = 3x^2 - 2.
To find the parabola with equation y = ax^2 + bx whose tangent line at (1, 1) has the equation y = 4x - 3, we will first determine the values of a and b.
Since the tangent line touches the parabola at (1, 1), we can substitute these values into both the parabola and tangent line equations:
1 = a(1)^2 + b(1) (Parabola equation)
1 = 4(1) - 3 (Tangent line equation)
From the tangent line equation, we see that it is already satisfied. Now we need to find the derivative of the parabola equation with respect to x to find the slope of the tangent line:
dy/dx = 2ax + b
At the point (1, 1), the slope of the tangent line is equal to the slope of the parabola:
4 = 2a(1) + b
We already know from the parabola equation that:
1 = a + b
Now, we have a system of linear equations:
4 = 2a + b
1 = a + b
Solving the system, we find that a = 3 and b = -2. Therefore, the equation of the parabola is:
y = 3x^2 - 2x

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The value of the triple integral ∭bzex ydv over the box b determined by 0≤x≤4, 0≤y≤3, and 0≤z≤3 is [tex]27e^{12} - 36.[/tex]

What is integration?

Integration is a fundamental concept in calculus that involves finding the integral of a function.

To evaluate the triple integral ∭bzex ydv over the box b determined by 0≤x≤4, 0≤y≤3, and 0≤z≤3, we integrate with respect to z, y, and then x.

∭bzex ydv = [tex]\int\limits^3_0 \int\limits^3_0\int\limits^4_0 bzex y\ dx\ dy\ dz[/tex]

Integrating with respect to x, we get:

[tex]\int\limits^4_0 bzex\ y\ dx\ = bzex\ y\ |^4_0 = bze 4^y - bz[/tex]

Substituting this result into the triple integral, we get:

∭bzex ydv = [tex]\int\limits^3_0 \int\limits^3_0(bze 4^y - bz) dy dz[/tex]

Integrating with respect to y, we get:

[tex]\int\limits^3_0 (bze4^y - bz) dy = (1/4)bze4^y - bzy|_0^3 = (1/4)bz(e^{12} - 1) - 3bz[/tex]

Substituting this result into the triple integral, we get:

∭bzex ydv = [tex](1/4)bz(e^{12} - 1) - 3bz) \int\limits^3_0 dz[/tex]

Integrating with respect to z, we get:

[tex]\int\limits^3_0 (1/4)bz(e^{12} - 1) - 3bz) dz = (9/4)bz(e ^{12} - 1) - 9bz[/tex]

Substituting this result into the triple integral, we get:

∭bzex ydv =[tex](9/4)bz(e^{12} - 1) - 9bz)[/tex]

Now, substituting the limits of integration, we get:

∭bzex ydv = [tex](9/4)(4)(e_{-1} ^{12} - 1) - 9(4) = 27e^{12} - 36[/tex]

Therefore, the value of the triple integral ∭bzex ydv over the box b determined by 0≤x≤4, 0≤y≤3, and 0≤z≤3 is [tex]27e^{12} - 36.[/tex]

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It will take approximately 67 months to have $10,000 to buy a boat with monthly deposits of $140 at a 3% monthly compounded interest rate.

To determine how long it will take to save $10,000 with monthly deposits of $140 at a 3.0% interest rate compounded monthly, we'll use the future value of a series formula:
FV = P * (((1 + r)^n - 1) / r)
Where:
FV = future value of the series ($10,000)
P = monthly deposit ($140)
r = interest rate per period (0.03 / 12)
n = number of periods (number of months)
Rearrange the formula to solve for n:
n = ln((FV * r / P) + 1) / ln(1 + r)
Plug in the values:
n = ln((10,000 * (0.03 / 12) / 140) + 1) / ln(1 + (0.03 / 12))
n ≈ 62.1
Since we need to round up to the next whole month, it will take approximately 63 months to save $10,000 to buy the boat.

It will take approximately 67 months to have $10,000 to buy a boat. Using the formula for compound interest, we can calculate the future value of monthly deposits of $140 at a rate of 3% compounded monthly:
FV = PMT * ((1 + r)^n - 1) / r
Where:
PMT = $140 (monthly deposit)
r = 0.03/12 (monthly interest rate)
n = number of months
We want to find the value of n that gives us a future value of $10,000:
$10,000 = $140 * ((1 + 0.03/12)^n - 1) / (0.03/12)
Simplifying and solving for n, we get:
n = log(1 + ($10,000 * 0.03/12 / $140)) / log(1 + 0.03/12)
n ≈ 66.8
Since we can't have fractional months, we round up to the next higher month:
n ≈ 67
Therefore, it will take approximately 67 months to have $10,000 to buy a boat with monthly deposits of $140 at a 3% monthly compounded interest rate.

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Using a trigonometric relation we can see that the value of x is 3,549.3 ft

We can see that we have a right triangle, where x is the hypotenuse.

We know one angle of the triangle and the opposite cathetus of said angle.

Then we need to use the trigonometric relation:

sin(a) = (opposite cathetus)/hypotenuse.

Replacing the things that we know we will get.

sin(25°) = 1500ft/x

Solving that for x we will get:

x = 1500ft/sin(25°)

x = 3,549.3 ft

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If h(x) = 6 5f(x) , where f(5) = 6 and f '(5) = 4, then h'(5). h'(5) =20

To find h'(5) given h(x) = 6 + 5f(x), f(5) = 6, and f'(5) = 4, follow these steps:

1. Differentiate h(x) with respect to x: h'(x) = 0 + 5f'(x) (since the derivative of a constant is 0, and we use the chain rule for the second term).

2. Now, h'(x) = 5f'(x).

3. Plug in the given values: h'(5) = 5f'(5).

4. Since f'(5) = 4, substitute this value: h'(5) = 5 * 4.

5. Compute the result: h'(5) = 20.

So, h'(5) = 20.

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The polar form of the equation x = 4 is r = 4 / cos(θ).

To convert the equation x = 4 to polar form:

To convert the equation x = 4 to polar form using variables r and θ (theta),

Follow these steps:

Step 1: Recall the polar to rectangular coordinate conversion formulas:
x = r * cos(θ)
y = r * sin(θ)

Step 2: Replace x in the given equation with the corresponding polar conversion formula:
r * cos(θ) = 4

Step 3: Solve for r:
r = 4 / cos(θ)

So, the polar form of the equation x = 4 is r = 4 / cos(θ).

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The image of triangle EFG after rotation 90 degrees countercloeckwise is shown below.

We know that when we rotate a point P(x, y) 90 degrees counterclockwise about the origin then the coordinates of point after rotation becomes (-y, x)

Here the coordinates of the triangle EFG are:

E(4, -8)

F(4, -1)

G(3, -9)

We need to rotate triangle EFG 90 degrees counterclockwise.

With the help of above statement the coordinates of rotated triangle would be,

E'(8, 4)

F'(1, 4)

G'(9,3)

The transformed triangle is shown below.

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The set of parameters that would give this person the worst grade on the exam relative to others is µ = 70 and σ = 5. This can be found using z-score. The correct option is option c).

To determine which set of parameters would give this person the worst grade on the exam relative to others, we need to find the z-score for the score of 65 under each set of parameters and see which one is the lowest. The z-score is a measure of how many standard deviations a particular value is from the mean.

The formula for calculating the z-score is:

z = (x - µ) / σ

where x is the score, µ is the mean, and σ is the standard deviation.

a. µ = 60 and σ = 5

z = (65 - 60) / 5 = 1

b. µ = 70 and σ = 10

z = (65 - 70) / 10 = -0.5

c. µ = 70 and σ = 5

z = (65 - 70) / 5 = -1

d. µ = 60 and σ = 10

z = (65 - 60) / 10 = 0.5

The lowest z-score is -1, which corresponds to option c. This means that most people scored higher than 65, and those who scored lower did so by a smaller margin.

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The statement " An example of continuous data would be the numbers on baseball player jerseys" is false because the numbers on baseball player jerseys represent a finite set of distinct values, which makes them an example of discrete data, not continuous data.

Continuous data is data that can take on any value within a range or interval. This means that the data can be measured and expressed as a decimal or a fraction, and there are an infinite number of possible values within the range. For example, the height of a person can be any value between 5 feet and 6 feet, including all the possible fractions or decimals in between.

On the other hand, discrete data is data that can only take on certain distinct values. These values cannot be measured or expressed as a decimal or a fraction. Examples of discrete data include the number of children in a family, the number of students in a classroom, or the number of books on a shelf.

In the case of baseball player jerseys, the numbers are assigned to players based on a finite set of integers (typically 0 to 99), and there are no fractional or decimal values in between. Therefore, the numbers on baseball player jerseys are an example of discrete data, not continuous data.

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a random variable x, the number of successes, follows a Poisson process, then the probability of success for any two intervals of the same size.

A) is the same

The correct answer is A) is the same.

Probability is a way to gauge how likely something is to happen. Many things are difficult to forecast with absolute confidence. Using it, we can only make predictions about the likelihood of an event happening, or how likely it is. The probability can be between 0 and 1.

In a Poisson process, the probability of success within a certain time interval is determined only by the length of the interval and the rate of success. Therefore, any two intervals of the same size will have the same probability of success, regardless of when the intervals occur.

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f(x) = sqrt(x^2 + x)

Yes, f(x) is O(x^2) because sqrt(x^2 + x) is dominated by x when x is sufficiently large.

f(x) = 17x^(11)

Yes, f(x) is O(x^2) because 17x^11 is dominated by x^2 when x is sufficiently large.

f(x) = x^(2/1000)

Yes, f(x) is O(x^2) because x^(2/1000) is dominated by x^2 when x is sufficiently large.

f(x) = x^42

Yes, f(x) is O(x^2) because x^42 is dominated by x^2 when x is sufficiently large.

f(x) = ⌊x⌋⋅⌈x⌉

Yes, f(x) is O(x^2) because ⌊x⌋⋅⌈x⌉ is bounded above by x^2 when x is sufficiently large.

f(x) = log(2x)

No, f(x) is not O(x^2) because log(2x) grows much more slowly than x^2 when x is sufficiently large.

f(x) = xlog(x)

No, f(x) is not O(x^2) because xlog(x) grows much more slowly than x^2 when x is sufficiently large.

f(x) = sqrt(x^2 + x)

Yes, f(x) is O(x^2) because sqrt(x^2 + x) is dominated by x when x is sufficiently large.

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The correct choice that completes the sentence is: Besides vertical asymptotes, the zeros of the denominator of a rational function give rise to holes oblique so hat , the correct choice is b

Explanation:-

A rational function is a fraction with polynomials in the numerator and denominator. The denominator of a rational function cannot be zero because division by zero is undefined. Therefore, the zeros of the denominator are important in analyzing the behaviour of a rational function.

When a rational function has a zero in the denominator, it creates a vertical asymptote. The function becomes unbounded as it approaches the vertical asymptote from both sides. However, if the numerator has a zero at the same point where the denominator has a zero, the function may have a hole in the graph instead of a vertical asymptote.

In addition to vertical asymptotes and holes, the zeros of the denominator of a rational function can also give rise to oblique asymptotes. An oblique asymptote occurs when the degree of the numerator is exactly one more than the degree of the denominator. In this case, the function approaches a slanted line as x goes to infinity or negative infinity.

Therefore, analyzing the zeros of the denominator of a rational function is crucial in understanding its behaviour and graph. It can help identify vertical asymptotes, holes, and oblique asymptotes, providing valuable insights into the function's behavior.

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The probability that the average amount of paid time lost during a three-month period for the 100 blue-collar workers will exceed 1.5 days is 0.0228.

Since the sample size n = 100 is sufficient, we can apply the central limit theorem to roughly approximate the distribution of the sample mean.

Let X represent the total paid time a single blue-collar worker missed during a three-month period. Given that the population mean is 1.3 days and the population standard deviation is 1.0 days, X N(1.3, 1.02) follows.

Let Y be the sample mean of X for a sample of 100 blue-collar workers selected at random. So, according to the central limit theorem, Y = N(1.3, 1.02/100).

We are looking for P(Y > 1.5). By standardized Y, we obtain:

Z is defined as (Y - ) / (n /√(n)) = (1.5 - 1.3) / (1.0 / √(100)). = 2

The probability of the event that average amount of the paid time loss is 0.0288.

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Complete question - In an article in the Journal of Management, Joseph Martocchio studied and estimated the costs of employee absences. Based on a sample of 176 blue-collar workers, Martocchio estimated that the mean amount of paid time lost during a three-month period was 1.3 days per employee with a standard deviation of 1.0 days. Martocchio also estimated that the mean amount of unpaid time lost during a three-monthperiod was 1.4 day per employee with a standard deviation of 1.2 days.

Suppose we randomly select a sample of 100 blue-collar workers. Based on Martocchio's estimates:

(a)What is the probability that the average amount of paid time lost during a three-month period for the 100 blue-collar workers will exceed 1.5 days?

Answer:

answer is (-2,5)

Step-by-step explanation:

gonna make 2 by 5 into Percent

x=0.4y-4

3x-9y=-51

step 2 Add 3 into first column and make it negative

-3x=-1.2y+12

then move y to other side

-3x+1.2y=12

3x-9y=-51

-7.8y=-39

y=5

U got y now

add it into y in first column

x= 2/5(5)-4

then it be 2-4= -2

x=-2

+) Replace x = (2/5)y - 4 into 3x - 9y = -51

[tex] 3 \times ( \frac{2}{5} y - 4) - 9y = - 51 \\ [/tex]

[tex] \frac{6}{5} y - 12 - 9y = - 51[/tex]

[tex]\frac{6}{5} y - 9y = - 51 + 12 = - 39[/tex]

[tex] \frac{ - 39}{5} y = - 39[/tex]

[tex]y = (- 39) \div \frac{( - 39)}{5} = ( - 39) \times \frac{5}{( - 39)} [/tex]

[tex]y = 5[/tex]

[tex]x = \frac{2}{5} y - 4 = \frac{2}{5} \times 5 - 4 = 2 - 4 [/tex]

[tex]x = - 2[/tex]

Ans: (x;y) = (-2;5)

Ok done. Thank to me >:33

It is true that A biased estimate of an odds ratio can exist even if this estimate is very precise.

A biased estimate of an odds ratio can exist even if this estimate is very precise. Bias refers to a systematic error in the estimation process, which can affect the accuracy of the estimate even if it is based on a large sample size or other factors that might increase precision. Therefore, it is important to identify and address sources of bias in the estimation of odds ratios to ensure that the estimates are as accurate as possible.

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Answer:

3400

Step-by-step explanation:

3.4 x 1000 =3400

to go from km a big unit to meters a smaller unit, you multiply by 1000

Doing some changes of units, we can see that the total volume is V = 1.5 gal

We know that the recipe that Janelle follows is the following one:

So we need to do some changes of units, we know that:

1 pint = 0.125 gal

Then:

5 pints = 5*(0.125 gal) = 0.625 gal

Then for the orange juice we have:

1 cup = 0.0625 gal

Then for the 14 cups of apple and grape juice we have:

14*(0.0625 gal) = 0.875 gal

Adding that we have the total volume:

0.625 gal + 0.875 gal = 1.5 gal

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Answer:

[tex] {x}^{2} - 36 = 5x[/tex]

[tex] {x}^{2} - 5x - 36 = 0[/tex]

[tex](x - 9)(x + 4) = 0[/tex]

[tex]x = 9[/tex]

To find the PDF of Z = X + Y when X and Y have the given joint PDF, f(x,y) = 1/900 for 0≤x,y≤3, and 0 otherwise.

Step 1: Identify the range of Z. Since X and Y range from 0 to 3, the minimum value for Z is 0 (when X = 0 and Y = 0) and the maximum value for Z is 6 (when X = 3 and Y = 3).

Step 2: Find the marginal PDFs of X and Y. Since X and Y are uniformly distributed, we have f_X(x) = 1/3 for 0≤x≤3 and f_Y(y) = 1/3 for 0≤y≤3.

Step 3: Compute the convolution of the marginal PDFs.

To find the PDF of Z = X + Y, we need to compute the convolution of f_X(x) and f_Y(y): f_Z(z) = ∫ f_X(x) * f_Y(z-x) dx

Now, let's compute the convolution for different ranges of Z:

a) 0≤z≤3: f_Z(z) = ∫(1/3)(1/3) dx from x=0 to x=z f_Z(z) = (1/9)[x] from 0 to z f_Z(z) = z/9

b) 3

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Answer:

2

Step-by-step explanation:

(½ × 8 + 6) ÷ 5

Order of operations: BODMAS - (Brackets, Orders, Division, Multiplication, Addition, Subtraction)

So Brackets first, and within the bracket we do mulplication first, then addition.

(4 + 6) ÷ 5

10 ÷ 5

Division is left so obviously

Ans : 2

Answer:

vertex = (- 4, - 2 )

Step-by-step explanation:

given a parabola in standard form

f(x) = ax² + bx + c ( a ≠ 0 )

then the x- coordinate of the vertex is

[tex]x_{vertex}[/tex] = - [tex]\frac{b}{2a}[/tex]

f(x) = 2x² + 16x + 30 ← is in standard form

with a = 2 , b = 16 , then

[tex]x_{vertex}[/tex] = - [tex]\frac{16}{2(2)}[/tex] = - [tex]\frac{16}{4}[/tex] = - 4

substitute x = - 4 into f(x) for corresponding y- coordinate

f(- 4) = 2(- 4)² + 16(- 4) + 30

      = 2(16) - 64 + 30

      = 32 - 34

     = - 2

vertex = (- 4, - 2 ) or x = - 4 , y = - 2

The largest possible value of f(8) is 14.

Since f is continuous on [4,8] and differentiable on (4,8), we can apply the Mean Value Theorem (MVT) on the interval [4,8]. The MVT states that there exists a c in (4,8) such that

f(8) - f(4) = f'(c)(8-4)

or equivalently,

f(8) = f(4) + f'(c)(8-4).

Since f(4) = -6 and f'(x) ≤ 10 for all x in (4,8), we have

f(8) = -6 + f'(c)(8-4) ≤ -6 + 10(8-4) = 14.

Therefore, the largest possible value of f(8) is 14. This maximum value can be achieved by a function that is increasing at the maximum rate of 10 on the interval (4,8) and passes through the point (4,-6).

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Answer:

simplifying

Step-by-step explanation:

To find the equation of the tangent plane to the given surface at the specified point.

Given surface: z = 6(x - 1)^2 + 6(y + 3)^2

Specified point: (2, -2, 14)

Step 1: Find the partial derivatives with respect to x and y. ∂z/∂x = 12(x - 1) ∂z/∂y = 12(y + 3)

Step 2: Evaluate the partial derivatives at the specified point (2, -2, 14). ∂z/∂x|_(2,-2,14) = 12(2 - 1) = 12 ∂z/∂y|_(2,-2,14) = 12(-2 + 3) = 12

Step 3: Use the tangent plane equation: z - z₀ = ∂z/∂x(x - x₀) + ∂z/∂y(y - y₀), where (x₀, y₀, z₀) is the specified point. z - 14 = 12(x - 2) + 12(y + 2)

Step 4: Simplify the equation. z - 14 = 12x - 24 + 12y + 24 z = 12x + 12y + 14

So, the equation of the tangent plane to the given surface at the specified point (2, -2, 14) is z = 12x + 12y + 14.

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Answer:

Step-by-step explanation:

The y-intercept of the line of fit is 41.16, which means that when the machine is first filled, it starts with approximately 41.16 fluid ounces of diet soda.

In the context of the data, the y-intercept represents the initial amount of diet soda in the machine before any soda is purchased. This information can be useful for determining how much soda is being purchased by customers over time, as it provides a baseline for comparison.

Therefore, the correct answer is: The y-intercept is 41.16. The machine starts with 41.16 ounces of diet soda.

Answer:

y-intercept is 41.16. The machine starts with 41.16 ounces of diet soda.

Step-by-step explanation:

The area of the composite shape is 40.57 square m and the perimeter is 34.57 meters

The surface area of composite shapes can be found by breaking the composite shape down into simpler shapes and then finding the surface area of each individual shape.

Here, we have

Area = Area of rectangle + circle

So, we have

Area = 4 * 7 + 22/7 * (4/2)^2

Area = 40.57 square m

So, the area is 40.57 square m

For the perimeter, we have

Perimeter = 2 * (4 + 7) + 2 * 22/7 * (4/2)

Perimeter = 34.57

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The value of the population standard deviation necessary to produce a standard error of 3 points is 18 points. The value of the population standard deviation necessary to produce a standard error of 12 points is 72 points. The value of the population standard deviation necessary to produce a standard error of 2 points is 12 points.

To calculate the value of the population standard deviation (σ) necessary to produce each of the following standard error values for a sample of n = 36 scores, we can use the formula:

σM = σ / √n
where σM is the standard error of the mean, σ is the population standard deviation, and n is the sample size.

1. If σM = 12 points, then:
12 = σ / √36
12 = σ / 6
σ = 12 x 6
σ = 72 points

Therefore, the value of the population standard deviation necessary to produce a standard error of 12 points is 72 points.

2. If σM = 3 points, then:
3 = σ / √36
3 = σ / 6
σ = 3 x 6
σ = 18 points

Therefore, the value of the population standard deviation necessary to produce a standard error of 3 points is 18 points.

3. If σM = 2 points, then:
2 = σ / √36
2 = σ / 6
σ = 2 x 6
σ = 12 points

Therefore, the value of the population standard deviation necessary to produce a standard error of 2 points is 12 points.

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[tex]6 + 36x + 72x^2 + 96x^3[/tex] / 3! + ... this is the Maclaurin series for f(x). Note that since e^x has a well-known Maclaurin series, we were able to simplify the original expression before finding the series.

The problem asks us to find the Maclaurin series for the function:

[tex]f(x) = 6e^x e^4x[/tex] sigma^infinity_n=0 (1^n / n!)

To do this, we first need to recognize that the expression inside the sigma notation is actually the Maclaurin series for e^x:

sigma^infinity_n=0 (1^n / n!) = e^x

We can substitute this expression into the original function to get:

[tex]f(x) = 6e^x e^4x e^x[/tex]

Now we can simplify this expression using the laws of exponents:

[tex]f(x) = 6e^x * e^(4x) * e^x[/tex]

f(x) = 6e^(6x)

Now we need to express this function as a Maclaurin series. We can start by writing out the first few terms of the series:

[tex]f(x) = 6e^(6x)[/tex]

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

[tex]= 6 + 36x + 72x^2 + 96x^3 / 3! + ...[/tex]

This is the Maclaurin series for f(x). Note that since e^x has a well-known Maclaurin series, we were able to simplify the original expression before finding the series.

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The closed-form expression for the given summation is (3n^2 + 7n) / (2n^2). Using this formula, the sums for n = 10, 100, 1000, and 10,000 are 37/20, 307/200, 3007/2000, and 30007/20000, respectively.

The given expression can be rewritten using the summation formulas as

∑nj=1 3j+2/n2 = (3(1)+2)/n2 + (3(2)+2)/n2 + ... + (3(n)+2)/n2

Let's simplify this expression by factoring out the common term of 1/n2

= (3/n2)(1 + 2 + ... + n) + (2/n2)(1 + 1 + ... + 1)

= (3/n2)(n(n+1)/2) + (2/n2)(n)

= (3n(n+1) + 4n) / (2n2)

= (3n^2 + 7n) / (2n^2)

Therefore, we have the closed-form expression for S(n) as

S(n) = (3n^2 + 7n) / (2n^2)

Using this formula, we can find the sums for n = 10, 100, 1000, and 10,000

S(10) = (3(10^2) + 7(10)) / (2(10^2)) = 37/20

S(100) = (3(100^2) + 7(100)) / (2(100^2)) = 307/200

S(1000) = (3(1000^2) + 7(1000)) / (2(1000^2)) = 3007/2000

S(10000) = (3(10000^2) + 7(10000)) / (2(10000^2)) = 30007/20000

Therefore, the sums for n = 10, 100, 1000, and 10,000 are 37/20, 307/200, 3007/2000, and 30007/20000, respectively.

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