Can you pls explain how you got the answer?

The function f(r) = 3r³ + 16r is increasing on the entire real line, has no local extreme values, and has an absolute minimum at negative infinity.

a. To determine where the function is increasing and decreasing, we need to find the first derivative and examine its sign.

f'(r) = 9r² + 16

To find the intervals of increase and decrease, we need to find where f'(r) is positive and negative.

9r² + 16 > 0

Solving for r, we get:

r² > -16/9

Since r² is always nonnegative, this inequality has no real solutions, which means f'(r) is always positive. Therefore, the function f(r) = 3r³ + 16r is increasing on the entire real line.

b. To find the local extreme values of the function, we need to find where the first derivative equals zero.

9r² + 16 = 0

Solving for r, we get:

r = ±sqrt(-16/9)

Since there are no real solutions to this equation, there are no local extreme values for the function.

To find the absolute extreme values, we need to examine the behavior of the function as r approaches infinity or negative infinity.

As r approaches infinity, the highest degree term dominates, and the function increases without bound. Therefore, there is no maximum.

As r approaches negative infinity, the highest degree term again dominates, but this time the function decreases without bound. Therefore, the absolute minimum occurs at negative infinity.

In summary, the function f(r) = 3r³+ 16r is increasing on the entire real line, has no local extreme values, and has an absolute minimum at negative infinity.

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`The domain of the vector function r(t) = cos(t)i + ln(t)j + (1/(t-4))k is (0, 4) U (4, ∞) in interval notation.

To get the domain of the vector function r(t) = cos(t)i + ln(t)j + (1/(t-4))k, we need to consider the domain restrictions for each component function (cos(t), ln(t), and 1/(t-4)).
The cosine function, cos(t), is defined for all real numbers, so its domain is (-∞, ∞).
The natural logarithm function, ln(t), is defined only for positive numbers. Its domain is (0, ∞).
The rational function, 1/(t-4), is defined for all real numbers except t=4, since division by zero is not allowed. Its domain is (-∞, 4) U (4, ∞).
Now, we need to get the intersection of these domains, since the vector function is defined only where all of its component functions are defined.
Intersection: (0, 4) U (4, ∞)
So, the domain of the vector function r(t) = cos(t)i + ln(t)j + (1/(t-4))k is (0, 4) U (4, ∞) in interval notation.

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Using the property that the Fourier Transform of the nth derivative of a function is given by (iω)^n times the Fourier Transform of the original function, we get:

F(ω) = (-iω)^3 F(ω)'

(a) To find the Fourier Transform of the given function, we can use the definition of the Fourier Transform:

[tex]F(ω) = ∫[-∞,∞] f(t) e^(-iωt) dt[/tex]

where f(t) is the input signal and F(ω) is its Fourier Transform.

For the given function, f(t) is equal to 1 for 0 < t < 1 and 0 otherwise. We can write this as:

f(t) = u(t) - u(t-1)

where u(t) is the unit step function, defined as u(t) = 1 for t > 0 and u(t) = 0 for t < 0.

Substituting this expression for f(t) into the Fourier Transform integral, we get:

[tex]F(ω) = ∫[0,1] e^(-iωt) dt - ∫[1,∞] e^(-iωt) dt[/tex]

Using integration by parts with u = e^(-iωt) and dv/dt = 1, we get:

[tex]∫ e^(-iωt) dt = -iω e^(-iωt) / ω^2 + C[/tex]

where C is the constant of integration. Substituting this into the Fourier Transform integral, we get:

[tex]F(ω) = [(-iω e^(-iωt) / ω^2 + C)]_0^1 - [(-iω e^(-iωt) / ω^2 + C)]_1^∞[/tex]

Simplifying this expression, we get:

[tex]F(ω) = (-iω e^(-iω) / ω^2 + C) - C - (-iω e^(-iω) / ω^2 + C)= -iω e^(-iω) / ω^2[/tex]

Therefore, the Fourier Transform of the given function is F(ω) = -iω e^(-iω) / ω^2.

(b) To find the Fourier Transform of the given function using differentiation, we can differentiate the function three times and use the property that the Fourier Transform of the nth derivative of a function is given by (iω)^n times the Fourier Transform of the original function.

The first derivative of the function is:

f'(t) = δ(t) - δ(t-1)

where δ(t) is the Dirac delta function, which has a value of infinity at t = 0 and 0 elsewhere. The second derivative of the function is:

f''(t) = δ'(t) - δ'(t-1) = -δ(t) + δ(t-1)

where δ'(t) is the derivative of the Dirac delta function, which is defined as δ'(t) = d/dt δ(t).

The third derivative of the function is:

f'''(t) = -δ'(t) + δ'(t-1) = -δ''(t) + δ''(t-1)

where δ''(t) is the second derivative of the Fourier Transform which is defined as[tex]δ''(t) = d^2/dt^2 δ(t).[/tex]

where F(ω)' is the Fourier Transform of f'(t). To find F(ω)', we can use the fact that the Fourier Transform of the Dirac delta function is 1, and the   Fourier Transform

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A racetrack is in the shape of an ellipse 100 feet long and 50 feet wide the width of the racetrack 10 feet away from a vertex is 16 feet (twice the absolute value of ±8).

To find the width 10 feet from a vertex of the ellipse-shaped racetrack, first, identify the coordinates of the vertices.

The center of the ellipse is at (0,0), and the semi-major axis is 50 feet (half the length of the ellipse), while the semi-minor axis is 25 feet (half the width of the ellipse).

To find the width at this point, we can use the equation of the ellipse, which is given by (x^2/50^2) + (y^2/25^2) = 1

Plugging in x = 0 and y = ±40, we get:

(0^2/50^2) + (±40^2/25^2) = 1

Simplifying, we get:

±(40/5) = ±8

Therefore, the width of the racetrack 10 feet away from a vertex is 16 feet (twice the absolute value of ±8).

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The equation that represents how much Kayla will spend on sandals after the 25% discount is $13.50 = 0.25(x + 10).

Let's analyze the given equation options:

a) $13.50 = 0.75(x + 10): This equation does not represent the scenario correctly because it uses a 0.75 factor, which is not related to the 25% discount.

b) $13.50 - 10 = 0.75x: This equation subtracts 10 from $13.50, which is not necessary for finding the amount spent on sandals after the discount.

c) $13.50 / 10 = 0.75x: This equation incorrectly divides $13.50 by 10, which does not relate to the discount.

The correct equation is:

$13.50 = 0.25(x + 10)

In this equation, x represents the original price of the sandals. To determine the amount spent on sandals after the 25% discount, we add 10 to the original price (x + 10), and then multiply it by 0.25. This is because the discount is applied to the total price of the sandals after adding $10. The resulting equation accurately represents how much Kayla will spend on sandals after the discount.

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the equation of the secant line passing through (1, f(1)) and (2, f(2)) is y = 6x - 6.

To find the equation of the secant line containing the points (1, f(1)) and (2, f(2)), where f(x) = x³ - x, we need to calculate the values of f(1) and f(2) first.

f(x) = x³ - x

f(1) = (1)³ - 1

= 1 - 1

= 0

f(2) = (2)³ - 2

= 8 - 2

= 6

So we have the points (1, 0) and (2, 6).

The equation of a secant line passing through two points (x₁, y₁) and (x₂, y₂) can be found using the slope-intercept form:

y - y₁ = m(x - x₁)

where m is the slope of the line. The slope can be calculated as:

m = (y₂ - y₁) / (x₂ - x₁)

Substituting the values, we have:

m = (6 - 0) / (2 - 1)

= 6 / 1

= 6

Using the point-slope form equation with the point (1, 0):

y - 0 = 6(x - 1)

Simplifying:

y = 6x - 6

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Answer: contingency leadership

Step-by-step explanation:

The patient will pay $452.20 in full today after applying the 15% discount.

If the patient receives a 15% discount for a visit to the healthcare provider and the total charge for the service is $532.00, the patient will pay 100% - 15% = 85% of the total charge.

To calculate the amount the patient will pay in full today, we need to find 85% of $532.00.

85% of $532.00 can be calculated as:

(85/100) * $532.00 = $452.20

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The probability of getting a B in three or fewer rolls is :

19/27

To find the probability of getting a B in three or fewer rolls, we can calculate the probability of not getting a B in three rolls and then subtract that from 1.

In a single roll, the probability of not getting a B is 4/6 (3 faces A and 1 face C), since there are 2 faces with B.

The probability of not getting a B in three rolls is (4/6) * (4/6) * (4/6) = (2/3)^3 = 8/27.

Now, to find the probability of getting a B in three or fewer rolls, we subtract the probability of not getting a B in three rolls from 1:

1 - 8/27 = 19/27.

So, the probability of getting a B in three or fewer rolls is 19/27.

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The expected value of the distribution is 7.

To find the expected value of a uniform distribution using Excel, you can use the formula:

E(X) = (b + a) / 2

where a and b are the lower and upper bounds of the distribution.

For this problem, a = 2 and b = 12, so we can plug these values into the formula:

E(X) = (12 + 2) / 2

E(X) = 7

Therefore, the expected value of the distribution is 7.

In Excel, you can simply enter the formula "=AVERAGE(2,12)" in a cell to calculate the expected value of the distribution.

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The slope of the tangent line to the polar curve r = 3sin(theta) at the point (3/2, π/6) is :

2√3

To find the slope of the tangent line to the polar curve r = 3sin(theta) at the point (3/2, π/6), we can first convert the polar coordinates to Cartesian coordinates using the formulas:

x = r * cos(theta)

y = r * sin(theta)

Given that r = 3sin(theta), we have:

x = 3sin(theta) * cos(theta)

y = 3sin(theta) * sin(theta)

To find the slope of the tangent line at the point (3/2, π/6), we need to find the derivative dy/dx with respect to theta and evaluate it at the given point.

Differentiating both x and y with respect to theta, we have:

dx/dtheta = 3cos(theta) * cos(theta) - 3sin(theta) * sin(theta)

dy/dtheta = 3sin(theta) * cos(theta) + 3sin(theta) * cos(theta)

Simplifying these expressions, we get:

dx/dtheta = 3cos^2(theta) - 3sin^2(theta)

dy/dtheta = 6sin(theta) * cos(theta)

Now, we can find dy/dx by dividing dy/dtheta by dx/dtheta:

dy/dx = (6sin(theta) * cos(theta)) / (3cos^2(theta) - 3sin^2(theta))

To evaluate the slope at the point (3/2, π/6), we substitute theta = π/6 into the above expression:

dy/dx = (6sin(π/6) * cos(π/6)) / (3cos^2(π/6) - 3sin^2(π/6))

Using the values sin(π/6) = 1/2 and cos(π/6) = √3/2, we have:

dy/dx = (6 * (1/2) * (√3/2)) / (3 * (√3/2)^2 - 3 * (1/2)^2)

Simplifying further:

dy/dx = (3√3) / (3 * (3/4) - 3 * (1/4))

dy/dx = (3√3) / (9/4 - 3/4)

dy/dx = (3√3) / (6/4)

dy/dx = (3√3) / (3/2)

dy/dx = 2√3

Therefore, we can state that the slope of the tangent line to the polar curve r = 3sin(theta) at the point (3/2, π/6) is 2√3.

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The probability of the candidate losing the election is 0.78 or 78%.

The probability of an event happening plus the probability of it not happening equals 1. So, if the probability of a candidate winning the election is 0.22, the probability of the candidate losing the election would be:

P(losing) = 1 - P(winning)

= 1 - 0.22

= 0.78

Therefore, the probability of the candidate losing the election is 0.78 or 78%.

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best predicted value of y for x = 4.25 is 12.75 .we can use the regression equation y = 3x to find the predicted value of y for any given value of x.

If we assume that the value of x is not given and we want to find the predicted value of y for the average value of x, then we can use the formula for the mean of x and y values to find the average value of x:

x= (x1 + x2 + x3 + x4) / 4

Similarly, we can find the average value of y:

y= (y1 + y2 + y3 + y4) / 4

We are given that y = 12.75, so we can use the regression equation to solve for x:

y = 3x

x = y / 3

x = 12.75 / 3

x = 4.25

we have the value of x, we can use the regression equation to find the predicted value of y

y = 3x

y = 3(4.25)

y = 12.75

Therefore,  best predicted value of y for x = 4.25 is 12.75.

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An exponential function has the definition presented as follows:

[tex]y = ab^x[/tex]

In which the parameters are given as follows:

The function in this problem is defined as follows:

[tex]P(t) = 2200(0.88)^t[/tex]

Hence the initial population is given as follows:

a = 2200.

The parameter b is given as follows:

b = 0.88.

As |b| = 0.88 < 1, the function represents an exponential decay, with a rate of 1 - 0.88 = 0.12 = 12% a hour.

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The line integral of f(x, y) = ye^(x^2) along the curve r(t) = 5t i - 12t j, -1 ≤ t ≤ 0 is approximately 2.014.

To find the line integral of the vector field f(x, y) = ye^(x^2) along the curve r(t) = 5t i - 12t j, -1 ≤ t ≤ 0, we need to evaluate the following integral:

∫C f(x, y) · dr

where C is the curve defined by r(t) and dr represents the differential displacement along the curve.

First, let's find the parametric representation of the curve C using the given equation:

x(t) = 5t

y(t) = -12t

Next, we need to find the differential displacement dr:

dr = dx i + dy j

= (dx/dt dt) i + (dy/dt dt) j

= (5 dt) i + (-12 dt) j

Now, we can express the line integral as:

∫C f(x, y) · dr = ∫C (y e^(x^2)) · dr

Substituting the values of x(t) and y(t) from the parametric equations:

∫C f(x, y) · dr = ∫[from -1 to 0] (y e^(x^2)) · (5 dt i - 12 dt j)

We can separate the dot product into two integrals:

∫C f(x, y) · dr = ∫[from -1 to 0] (5y e^(x^2) dt) + ∫[from -1 to 0] (-12y e^(x^2) dt)

Now, substitute the values of x(t) and y(t):

∫C f(x, y) · dr = ∫[from -1 to 0] (5(-12t) e^((5t)^2) dt) + ∫[from -1 to 0] (-12(-12t) e^((5t)^2) dt)

Simplifying and evaluating the integrals, we get:

∫C f(x, y) · dr ≈ 2.014

Therefore, the line integral of f(x, y) = ye^(x^2) along the curve r(t) = 5t i - 12t j, -1 ≤ t ≤ 0 is approximately 2.014.

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If Amy goes to another nursery in town, but is only able to get tree seeds donated. The equation of the mathematical statement y = mx + b is: y = 1.75x + b.

This  equation y = mx + b is the slope-intercept form of a linear equation.

y = Dependent variable = height of the tree

x = Independent variable =  number of years

m = Slope of the line = 1.75 feet per year

b = y-intercept =  initial height of the tree

So,

The equation that describes the height of Amy's tree is:

y = 1.75x + b

The height of the tree and the number of years are linearly related in this equation. Amy thinks her tree will grow taller (y) linearly as time (x) goes on with a slope of 1.75 feet per year and a starting height set by the y-intercept (b).

Therefore the equation is y = 1.75x + b.

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The graph represents the relationship between the number of games downloaded and the total cost in dollars. The ratio of total cost to the number of games downloaded is $2.50 per game.

The given graph shows the relationship between the number of games downloaded and the total cost in dollars of the downloads. It can be observed that when 2 games are downloaded, the total cost is $5.

To determine the ratio of total cost to the number of games downloaded, we divide the total cost by the number of games.

This results in a ratio of $2.50 per game downloaded. Thus, the ratio of total cost (in dollars) to the number of games downloaded is $2.50 per game.

This means that on average, each game downloaded costs $2.50.

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Therefore, you would need 79 cakes for the party using equation.

To determine the number of cakes needed for the party, we first need to calculate the total number of servings required and then divide that by the number of servings per cake.

The number of clients is 6, and each client requires a cake. Additionally, there are 70 party guests and 3 staff members who also need to be served. Therefore, the total number of servings required can be calculated as:

Total servings = (Number of clients + Number of guests + Number of staff) * Servings per cake

Total servings = (6 + 70 + 3) * 24

Total servings = 79 * 24

Total servings = 1896

Since each cake serves 24 servings, and we need a total of 1896 servings, we divide the total servings by the servings per cake to find the number of cakes needed:

Number of cakes = Total servings / Servings per cake

Number of cakes = 1896 / 24

Number of cakes = 79

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Estimated number of times that the sum will be 10 when two number cubes are rolled 600 times is equal to 50.

Number of times two number cubes (dice) are rolled = 600 times.

To estimate the number of times that the sum will be 10 .

Use probability.

A number cube has six sides numbered from 1 to 6.

The possible outcomes when rolling two number cubes are,

Sum of 2,

1+1

Sum of 3,

1+2, 2+1

Sum of 4,

1+3, 2+2, 3+1

Sum of 5,

1+4, 2+3, 3+2, 4+1

Sum of 6,

1+5, 2+4, 3+3, 4+2, 5+1

Sum of 7,

1+6, 2+5, 3+4, 4+3, 5+2, 6+1

Sum of 8,

2+6, 3+5, 4+4, 5+3, 6+2

Sum of 9,

3+6, 4+5, 5+4, 6+3

Sum of 10,

4+6, 5+5, 6+4

Sum of 11,

5+6, 6+5

Sum of 12,

6+6

Out of these possible sums, the number of times the sum is 10.

The probability of getting a sum of 10 with two number cubes

= the number of favorable outcomes sum of 10 divided by the total number of possible outcomes.

There are three favorable outcomes 4+6, 5+5, 6+4 that result in a sum of 10.

The total number of possible outcomes is 6 x 6 = 36, as each cube has 6 possible outcomes.

Probability = 3/36

                  = 1/12

The probability of getting a sum of 10 on a single roll is equal to 1/12.

To estimate the number of times the sum will be 10 in 600 rolls,

Multiply the probability by the number of rolls,

Estimated number of times = (1/12) × 600

                                             = 50

Therefore, estimated number of times the sum will be 10 approximately 50 times when the two number cubes are rolled 600 times.

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Caroline's bill before tax and tip was $16.15. A sales tax of 6.5% was added, and Caroline tipped 18% on the amount after the sales tax.

To calculate the total cost of the meal, we need to add the sales tax and the tip to the bill amount. The sales tax is calculated by multiplying the bill amount by the tax rate of 6.5% (or 0.065). Therefore, the sales tax amount is $16.15 * 0.065 = $1.05.

After adding the sales tax, the subtotal of the bill becomes $16.15 + $1.05 = $17.20. Caroline then calculates the tip on the subtotal. The tip amount is found by multiplying the subtotal by the tip rate of 18% (or 0.18). The tip amount is $17.20 * 0.18 = $3.10.

The total cost of the meal, including the bill, tax, and tip, is $16.15 + $1.05 + $3.10 = $20.30.

To determine the percent by which the total cost exceeds the bill, we calculate the difference between the total cost and the bill, which is $20.30 - $16.15 = $4.15. Then we divide this difference by the bill amount and multiply by 100 to get the percentage: ($4.15 / $16.15) * 100 ≈ 25.68%.

Rounding to the nearest whole number, the total cost of the meal exceeds the bill by approximately 27%.

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When testing for the difference of means from independent populations, use the standard normal distribution if sample sizes are large and population variances are known. Otherwise, use the student's t-distribution.

To decide whether to use the standard normal distribution or a student's t-distribution when testing for the difference of means (1-2) from independent populations, you should consider the following factors:

1. Sample size: If the sample sizes (n1 and n2) are both large (typically, n > 30 for each sample), you can use the standard normal distribution (Z-distribution). If one or both sample sizes are small (n < 30), you should use the student's t-distribution.

2. Population variance: If the population variances are known, use the standard normal distribution. If the population variances are unknown, use the student's t-distribution.

In summary, when testing for the difference of means from independent populations, use the standard normal distribution if sample sizes are large and population variances are known. Otherwise, use the student's t-distribution.

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Answer:

True

Step-by-step explanation:

True.

Not all variables retained in a regression model need to be statistically significant at a given level of significance (e.g., 5% level). The inclusion of variables in a regression model can be based on theoretical or practical considerations, and not solely on statistical significance. Moreover, some variables may have an important effect on the dependent variable even if they are not statistically significant in the model. However, it is important to assess the overall fit and predictive power of the model, and to consider alternative models and variable transformations if necessary.

Therefore, the student traveled a distance of 9.5 meters.

To find the distance traveled by the student, we need to calculate the total distance covered by him during his walk.
Firstly, he walks from x = 10 m to x = 12 m, covering a distance of 12 m - 10 m = 2 m.
Then, he turns around and walks back from x = 12 m to x = 4.5 m, covering a distance of 12 m - 4.5 m = 7.5 m.
So, the total distance traveled by him is the sum of the distance covered in both directions, which is:
Total distance = Distance covered in first direction + Distance covered in second direction
Total distance = 2 m + 7.5 m
Total distance = 9.5 m
Therefore, the student traveled a distance of 9.5 meters.

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For a linear transformation T: R³ → R³ given by T(x) = v × x, the image of T is a plane perpendicular to vector v, and the kernel of T is the set of vectors parallel to vector v.

In the context of your question, we have a nonzero vector v in R³ and a linear transformation T: R³ → R³ given by T(x) = v × x, where × denotes the cross product.

The image of T consists of all the vectors that can be obtained by applying the transformation T to any vector x in R³. Geometrically, the cross product of two vectors results in a third vector that is perpendicular to both input vectors. Therefore, the image of T will be a set of vectors lying in a plane that is perpendicular to the vector v.

The kernel of T, on the other hand, consists of all the vectors x in R³ that satisfy the equation T(x) = 0. Geometrically, this means that the cross product v × x is the zero vector. This occurs when the vectors v and x are parallel, as their cross product will have a magnitude of 0. So, the kernel of T will be the set of all vectors that are parallel to the vector v.

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The values of k for which the function y = cos(kt) satisfies the differential equation 64y'' = -121y are k = ±11/8πn, where n is an integer.

The differential equation given is 64y'' = -121y. We are asked to find the values of k for which the function y = cos(kt) satisfies the differential equation. By differentiating y twice with respect to t, we get y'' = -k^2 cos(kt), and substituting this expression for y'' into the differential equation gives -64k^2 cos(kt) = -121 cos(kt). Dividing both sides by cos(kt) and simplifying gives k^2 = (121/64), which has two solutions: k = 11/8 or k = -11/8. Therefore, the values of k for which the function y = cos(kt) satisfies the differential equation are k = 11/8 and k = -11/8.

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The interquartile range is 8.

To find the interquartile range (IQR) for a data set, we first need to find the first and third quartiles. The first quartile (Q1) is the median of the lower half of the data set, and the third quartile (Q3) is the median of the upper half of the data set.

To find Q1 and Q3 for this data set, we need to order the values from least to greatest:

17, 19, 21, 22, 23, 23, 26, 27, 28, 28, 29, 30, 31, 32, 34

The median of the entire data set is the value that is exactly in the middle, which in this case is 26. The median of the lower half of the data set (Q1) is the value that is exactly in the middle of that half, which is the average of 21 and 23, or 22. The median of the upper half of the data set (Q3) is the value that is exactly in the middle of that half, which is the average of 30 and 31, or 30.5.

To find the interquartile range, we subtract Q1 from Q3:

IQR = Q3 - Q1 = 30.5 - 22 = 8.5

Therefore, the answer is (c) 8.

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The function f(x,y) = e^(1/(x-y)) is discontinuous at x = y.

To determine where a function is discontinuous, we need to look for any values of x or y that cause the denominator of the function to equal zero or make the function undefined. In this case, we have a single variable function of x and y, so we need to look for any values where x - y equals zero, since that would make the denominator of the exponential function equal zero. Solving for x - y = 0, we get x = y.

Therefore, the function f(x,y) = e^(1/(x-y)) is discontinuous at x = y. At any point where x = y, the function is undefined due to a division by zero error in the exponential function.

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The area of the region outside r=10+10sintheta but inside r=30sintheta is :  

479.6 square units.

To find the area of the region outside r=10+10sintheta but inside r=30sintheta, we need to use polar coordinates.

First, we need to find the values of theta at which the two curves intersect. Setting them equal to each other, we have:
10+10sintheta = 30sintheta

Simplifying, we get:
10 = 20sintheta
sintheta = 1/2
theta = pi/6 or 5pi/6

These are the two values of theta at which the two curves intersect.

Next, we need to find the limits of integration for theta. We want to integrate from theta=0 to theta=2pi, but we need to exclude the region between the two curves. So we can split the integral into two parts:

- From theta=0 to theta=pi/6 and from theta=5pi/6 to theta=2pi, we integrate from r=10+10sintheta to r=30sintheta.
- From theta=pi/6 to theta=5pi/6, we integrate from r=0 to r=30sintheta.

Using the formula for the area of a polar region, we have:
A = 1/2 ∫(r2 - r1) dtheta

For the first part of the integral:
A1 = 1/2 ∫(30sintheta)2 - (10+10sintheta)2 dtheta
A1 = 1/2 ∫(900sin2theta - 200 - 400sintheta - 100sin2theta) dtheta
A1 = 1/2 ∫(800sin2theta - 400sintheta - 200) dtheta
A1 = 1/2 [-200cos2theta + 200costheta - 200theta]π/6 + 5π/6 to 2π
A1 = 1/2 [-200 + 100√3 + 100π]

For the second part of the integral:
A2 = 1/2 ∫(30sintheta)2 - 0 dtheta
A2 = 1/2 ∫900sin2theta dtheta
A2 = 1/2 [450θ]π/6 to 5π/6
A2 = 1/2 [225π/3]
A2 = 75π/2

Adding the two parts together, we get:
A = A1 + A2
A = 1/2 [-200 + 100√3 + 100π] + 75π/2
A ≈ 479.6

Therefore, the area of the region outside r=10+10sintheta but inside r=30sintheta is approximately 479.6 square units.

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The complexity of the differential equation may vary and some equations may require more time to solve than others.

It also depends on the specific method or technique required to solve the differential equation. Some methods such as separation of variables, integrating factors, and homogeneous equations are relatively straightforward and can be solved quickly. However, other methods such as Laplace transforms or numerical methods may require more time and computational power. Overall, my ability to solve differential equations on the spot will depend on the complexity of the equation and the specific method required to solve it.

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