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A power series representation for the function is:

x^8 tan^-1(x^3) = ∑[n = 0 to ∞]  (-1)^n (x^(6n + 11)/2n + 1)

In this question we need to find a power series representation for the function.

f(x) = x^8 tan^-1(x^3)

We know that the Maclaurin series:

1/(1 - x) = ∑[n = 0 to ∞] x^n

Replacing x with -x^2 we get,

1/(1 + x^2) = ∑[n = 0 to ∞] (-x^2)^n

               = ∑[n = 0 to ∞] (-1)^n x^2n

Integrating above equation we get,

tan^-1(x) = ∑[n = 0 to ∞]  (-1)^n (x^(2n + 1)/2n + 1) + c

For x = 0 we get c = 0

so, tan^-1(x) = ∑[n = 0 to ∞]  (-1)^n (x^(2n + 1)/2n + 1)

Now we replace x by x^3, we get,

tan^-1(x^3) = ∑[n = 0 to ∞]  (-1)^n (x^3(2n + 1)/2n + 1)

Therefore, the required power series would be,

x^8 tan^-1(x^3) = ∑[n = 0 to ∞]  (-1)^n (x^(6n + 11)/2n + 1)

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Answer:

Please read the assumptions that lead to the proposed answer of options a, b, and e being correct.  Options c and d are easily rejected.  The question is ambiguous about the locations of the angles.

Step-by-step explanation:

Parallel lines have slopes that are identical.  Lines p and q have the same slopes.

It is difficult imagining what angles are represented by x, y, and z.  There are only 2 lines, and they do not intersect.  The locations of the angles are not identified.  The lack of an intersecting point means it is free range as to where we assign the angles.

The attached graph illustrates some possibilities for x, y, and z.  It uses two simple equations as examples, with the same slope to make them parallel.  Assinment of any angles requires the use of an imaginary line that intersects the two parallel lines.  Even then, there aren't many unique locations for the angles.  To illustrate the analysis, lets assign the angles x, y, and z to thelocations shown on the graph.  

Even with this huge step, there is limited information to be able to identify which of the answer options is correct.  We can say that:

x = z

x+y=180

y+z=180

The answer options (below) appear to have similar relationships, so let's agree that the use of the imaginary line, plus the locations of x, y, and z are valid.  Looking at the answer options, we can match choices a, b, and e to the conclusions above.  d makes no sense because it is not an equation.  c says that both lines must be perpendicular to the x axis, a limitation that is relatively immaterial to whether they are parallel.  The condition of being parallel depends on those two angles being the same (e.g., 45 and 45, 22 and 22, etc.) and not require that they both be 90.

Answer Options               Guarantee Parallel?

a. x = z                                     Yes

b. x + y = 180 degree              Yes

c. x + z = 180 degree              No

d. x + y                                     No

e. y + z = 180                           Yes

Answer:

y = -6x - 4

Step-by-step explanation:

Slope intercept form:

y = mx + b

m is the slope

b is the y-intercept

To find the slope, you take two points from the line and plug them into the following equation:

[tex]\frac{y2-y1}{x2-x1}[/tex]

Let's use (0, -4) and (-2, 8):

[tex]\frac{8-(-4)}{-2-0} = \frac{12}{-2}=-6[/tex]

So m = -6

We know that the y-intercept (b) is -4, so we can plug everything into the equation:

y = -6x - 4

The answer is 0.109, or choice B. The probability that only six of 8 Internet users buy their clothes exclusively online is 0.109.

The probability that only 6 out of 8 Internet users buy their clothes exclusively online can be calculated using the binomial distribution. The probability of a single Internet user buying their clothes exclusively online is 0.5, and the probability of an Internet user not buying their clothes exclusively online is 0.5.

The probability of 6 out of 8 Internet users buying their clothes exclusively online can be calculated using the formula:

P(X=6) = (8 choose 6) * (0.5)^6 * (0.5)^2

Plugging in the values, we get:

P(X=6) = (8 choose 6) * (0.5)^6 * (0.5)^2

= 28 * (0.5)^6 * (0.5)^2

= 28 * (0.015625) * (0.25)

= 0.109

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Complete Question:

Fifty percent of the people that use the Internet buy their clothes exclusively online. Find the probability that only six of 8 Internet users buy their clothes exclusively online. Round to three decimal places.  explain answer in 100 words.

A. 0.750

B. 0.109

C. 0.438

D. 0.004

A correct statement about sampling distributions is: Sampling distributions of means get closer to normality as the sample size increases.

The correct answer is an option (A)

In this question we have been given few statements.

We need to select a correct statement about sampling distributions.

We know that the sampling distribution is nothing but a probability distribution that is obtained through repeated sampling of a specific population.

Also according to central limit theorem, sampling distribution of means approaches normal when the sample size increases.

Therefore Sampling distributions of means get closer to normality as the sample size increases.

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1) The inequality used in graph is;

⇒ x > -2

2) The solution for t will be;

⇒ t > - 2

3) The graph of - 3y > 6 is shown in figure.

4) The graph of y/2 < - 3 is shown in figure.

5) The graph of 4y > 3x - 1 is shown in figure.

A relation by which we can compare two or more mathematical expression is called an inequality.

Now,

1) By the graph, we get;

The solution of the graph is,

⇒ x > - 2

Thus, The used inequality is '>'.

2) The inequality is,

⇒ 2t + 7 > - 3 (t + 1)

Solve for t as;

⇒ 2t + 7 > - 3t - 3

⇒ 2t + 3t > - 3 - 7

⇒ 5t > - 10

⇒ t > - 2

3) The inequality is,

⇒ - 3y > 6

⇒ - y > 2

⇒ y < - 2

4) The inequality is,

⇒ y/2 < - 3

⇒ y < - 6

5) The inequality is,

⇒ 4y > 3x - 1.

Thus, The graph of 4y > 3x - 1 is shown in figure.

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Answer:

y = 2x - 7

Step-by-step explanation:

y = mx + b

m is known (m=2):

y = 2x + b

Since one point is known (3,-1), you can solve for b:

-1 = 2(3) + b

b = -7

So the equation is:

y = 2x - 7

Answer: The sum of the first 75 terms for the following sequence is 10,200

Step-by-step explanation: To find the sum of the first 75 terms for the following sequence, we have to first find the 75th term for which the formula used here will be,

[tex]a_{75\\} = a_{1} + (n-1)d\\ = -10 + (74-1)(4)\\ = -10 + 292\\ = 282[/tex]

So, the 75th term of this arithmetic progression is 282

Now we will find the sum of the first 75 terms,

[tex]S_{n} = n(a_{1} + a_{75} )/ 2\\ =75(-10 + 282)/2\\ =75(272)/2\\ =20400/2\\ =10200[/tex]

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bit strings of length 12 contain a) exactly three 1s is 220 b) at most three 1s is 2700 c) atleastthree1s is 220 d) an equal number of 0s and 1s is 66.

a) To calculate the number of bit strings of length 12 containing exactly three 1s, we can use the combination formula. The formula is nCr, where n is the total number of bits (12) and r is the number of 1s (3).The solution is 12C3 = 220.

b) To calculate the number of bit strings of length 12 containing at most three 1s, we can use the summation formula. The formula is sum of nCr, where n is the total number of bits (12) and r is the number of 1s (3). So, the answer is (12C0 + 12C1 + 12C2 + 12C3) = 2700.

c) To calculate the number of bit strings of length 12 containing at least three 1s, we can use the combination formula. The formula is nCr, where n is the total number of bits (12) and r is the number of 1s (3). The solution is 12C3 = 220.

d) To calculate the number of bit strings of length 12 containing an equal number of 0s and 1s, we can use the combination formula. The formula is nCr

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Answer:

The correct answer would be 0.94 per pound.

Step-by-step explanation:

Answer:

60

Step-by-step explanation:

Let x = number of people at the party.

0.9x wore red dresses.

2/25x wore green dresses.

rest of people wore white dresses

The rest of the people is

x - 0.9x - 2/25 x

x - 0.9x - 0.08x = 15

0.02x = 15

x = 750

0.9x = 675 (red)

2/25 x = 60 (green)

0.02x = 15 (white)

Answer: 60

The surface area of the cube is increasing at a rate of 49 cm²/min.

If the length of an edge of a cube is l cm.,

its volume V is l3 and surface area A is 6l2.

Differentiating V=l3 w.r.t. time,

we get,

dV/dt = 3l2dldt

dV/dt=10 cm3min,

when l=90

dl/dt=103×902

=12430

As A=6l2

dA/dt =12l×dldt

=12×90×12430

=49 cm2min

The total number of cubic units that a cube occupies in three dimensions is its volume. Six faces, twelve edges, and eight vertices make up the three-dimensional shape of a cube. Therefore, the area encircled by a cube's six faces is considered to equal its volume.

It differs from 2D shapes in that it adds a third dimension, known as height or thickness, in addition to length and breadth.

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15 quarts are required to plan ice cream party for 60 students which is

[tex]\frac{1}{4}[/tex] th fraction of quart.

1 quarts = 4 cups

so 1 cup = [tex]\frac{1}{4}[/tex] th fraction of quart

so in  order to plan party for 60 students we need 60 cups as it is mentioned that each student gets 1 cup of ice cream.

and 1 quarts = 4 cups

so let we need x quarts for the planning of ice cream party

so total cups = 4 *x

so 4*x =60  , x= 60/4 , x=15

so 15 quarts are required for the ice cream party of 60 students  .

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Answer: The first word in the alternative statistical hypothesis for an experiment with three levels of a single independent variable (IV) is "differences." This is because the alternative hypothesis typically involves the existence of differences between the levels of the IV.

Step-by-step explanation:

Here is how to formulate the alternative hypothesis for an experiment with three levels of a single IV:

Identify the independent variable (IV) and the levels of the IV. In this case, there is a single IV with three levels.

Determine the relationship between the IV and the dependent variable (DV). In other words, consider how the levels of the IV may affect the DV.

Write the alternative hypothesis in terms of the relationship between the IV and the DV. The alternative hypothesis should state that there are differences between the levels of the IV in their effect on the DV.

For example, if the IV is "type of exercise" and the levels are "running," "cycling," and "swimming," and the DV is "improvement in cardiovascular fitness," the alternative hypothesis might be "There are differences in the improvement in cardiovascular fitness between the three types of exercise."

What is the null hypothesis in an experiment?

The null hypothesis in an experiment is a statement that there is no relationship between the IV and the DV, or that there are no differences between the levels of the IV in their effect on the DV. The null hypothesis is typically denoted as H0. In the example given above, the null hypothesis might be "There is no difference in the improvement in cardiovascular fitness between the three types of exercise."

This means that the hypothesis is testing whether there are significant differences between the means of the three levels of the IV. Therefore, the first word in the alternative statistical hypothesis for any experiment with three levels of a single IV is "differences."

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The first word in the alternative statistical hypothesis for any experiment with three levels of a single IV is "difference," which refers to the difference between the means of the three levels of the IV.

What is hypothesis?

Hypothesis is an educated guess or prediction made in response to a question or problem. It is based on prior knowledge and experience, as well as available evidence. The purpose of a hypothesis is to suggest a possible explanation for an observed phenomenon and then to test the validity of the hypothesis through further research or experimentation. If the hypothesis is proven correct, it can become a theory. Hypotheses can also be disproven or modified through further testing.

This is a reference to the null hypothesis, which states that there is no difference between the means of the different levels of the IV. The alternative hypothesis is that there is a difference between the means of the different levels of the IV.

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There can be 8 possible outcomes present in the sample space

The probability that one switch is not working is 3% or 0.03

We need to find the number of possible outcomes in the sample space that can be possible

The first switch can be working or not

The second switch might be working or not

so does the third switch, it might be working or not working

And each switch is independent to one another

The number of outcomes can be

2 x 2 x 2 = 8

Therefore, the number of outcomes are there in the sample space is 8

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Answer:

  45

Step-by-step explanation:

You want to know the number of line segments that can be drawn between pairs of points in a group of 10 distinct points.

From the first point, 9 segments can be drawn.

From the second point 8 more segments can be drawn. (The segment to the first point has already been counted.)

From the third point, 7 more segments can be drawn. The total number of segments will be the sum of integers from 1 through 9:

  9(10)/2 = 45

45 line segments can be drawn.

__

Additional comment

This is identical to the "handshake" problem: the number of handshakes in a group of 10 people if each shakes hands with all the others.

The approximation of [tex]f_x(3,5)[/tex] using the contour diagram of f(x, y) is [tex]-\frac{2}{3}[/tex]

As per the given data we have determine the value of approximation of [tex]f_x(3,5)[/tex] using the contour diagram of f(x, y).

A contour or a contour line may be defined as the line of intersection of a level surface with the surface of ground.

This means every point on a contour line has the same altitude as that of the assumed intersecting surface.

The set of the curves f(x, y)=c

From the given contour plot the curves intersects with the line y = 5 are f( x,  y ) = 12 and f (x, y) = 10

f(x, y) = 12 intersects the line y = 5 at the point x = 3

So, f(3,5) = 12

f(x, y) = 10 intersects the line y = 5 at the point x = 6

so, f(6,5) = 10

[tex]$f_x(3,5)= \lim _{h \rightarrow 0} \frac{f(3+h, 5)-f(3,5)}{h}$[/tex]

Where approximate h = 3

[tex]f_x(3,5) & =\frac{f(3+3,5)-f(3,5)}{3}[/tex]

[tex]=\frac{f(6,5)-f(3,5)}{3}[/tex]

[tex]=\frac{10-12}{3}[/tex]

[tex]f_x(3,5) & =\frac{-2}{3}[/tex]

Therefore the answer is [tex]-\frac{2}{3}[/tex].

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Approximate  [tex]f_x(3,5)[/tex] using the contour diagram of f(x, y) shown below.

Diagram attached at the end of solution

Answer:

the answer will be 4.75 units

Step-by-step explanation:

explained

The midsegment of a trapezoid is a line segment that connects the midpoints of the nonparallel sides of the trapezoid. To find the length of the midsegment, we need to first find the midpoints of the sides IJ and KL.

The midpoint of a line segment is the point that is exactly halfway between the two endpoints of the segment. We can find the midpoint of a line segment by taking the average of the x-coordinates and the average of the y-coordinates of the endpoints.

The coordinates of the endpoints of the side IJ are (-3, -5) and (8, -5), so the midpoint of this side is:

((-3 + 8)/2, (-5 + (-5))/2) = (2.5, -5)

The coordinates of the endpoints of the side KL are (-3, -9) and (3, -9), so the midpoint of this side is:

((-3 + 3)/2, (-9 + (-9))/2) = (0, -9)

To find the length of the midsegment, we can use the distance formula to find the distance between the two midpoints:

sqrt((2.5 - 0)^2 + (-5 - (-9))^2) = sqrt(2.5^2 + 4^2) = sqrt(6.25 + 16) = sqrt(22.25) = 4.75

Therefore, the length of the midsegment of the trapezoid is approximately 4.75 units.

Answer:

18

Step-by-step explanation:

[tex]6/ 1/3[/tex] =

6 x 3

18

Answer:

Step-by-step explanation:

for n=1 it is 7(1+2)=21

for n=2 it is 7(2+2)=28

for n=3 it is 7(3+2)=35

The number of miles the whole trip is 15 miles

For instance, the equation 3x + 5 = 14 consists of the two equations 3x + 5 and 14, which are separated by the 'equal' sign.

The number of miles the whole trip is 15 miles.Given that, Liam's family has completed 40% of a trip.

In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign =.

Let the number of miles be x.

so, 40% of x=30

⇒ 40/100 ×x=30

⇒ 0.4x=30

⇒ x=30/0.4

⇒ x=12 miles

Therefore, the number of miles the whole trip is 15 miles.

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Answer:

40

Step-by-step explanation:

A total of 112 combinations are possible out of all possible combinations  when a coin is tossed  10 times and having total of 4 tails and have different start and end faces.

For the given question we have 2 conditions:

condition 1:

when the start turn have head face then the end turn must have tails face as it is given that start and end have different faces.

so in this case 1st and last position is fixed so we have exactly 8 positions.

and given that out of 10 times there must be total of 4 tails and we discussed that last face already have tail face so now out of 8 available positons we have exactly 3 tails in it and rest 5 must be heads.

so we have 8 available positions and we can select any 5 out of them and the number of ways of selection of this is = 8 combination 5  = [tex]8_C__5[/tex] =56 ways

conditon 2:

 when the start turn have tail face then the end turn must have head face as it is given that start and end have different faces.

so in this case 1st and last position is fixed so we have exactly 8 positions

and given that out of 10 times there must be total of 4 tails and we discussed that first face already have tail face so now out of 8 available positons we have exactly 3 tails in it and rest 5 must be heads.

so we have 8 available positions and we can select any 5 out of them and the number of ways of selection of this is =8 combination 5 =  [tex]8_C__5[/tex] =56 ways

so total number of ways is = no. of ways condition 1+ no. of ways condition 2 = 56 +56 = 112 ways

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Answer: 20 days

Step-by-step explanation:

First, we find the food provisioned for 1 man will last for how many days

Take 80 times 30 = 2400 days

Now we know that 40 more joined

80 + 40 = 120 serve

So, we take 2400 divided by 120 = 20 days

Answer:

D

Step-by-step explanation:

If 48 times 1/4 is 12, 12 divided by 1/4, the inverse operation, is 48.

Answer: Solution: 0.625

Step-by-step explanation: First, you have to set up the fraction as a division problem, so we have to remember that when making a fraction to a decimal, the numerator becomes the dividend, and the denominator becomes the divisor, which the dividend goes inside the bracket, and the divisor goes outside the bracket. Now, you would divide as usual, and get 0.625. I hoped this helped!  NOTE: I put in an attachment to show what I'm talking about.

Answer:

Step-by-step explanation:  do 5÷8 and u get 0.625

Answer:

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

Add up all the numbers to get the end of month figure.

Convert mixed fractions to sum of whole and fractions, then combine whole numbers and fractions, bringing all to common denominator of 16.

If the given function is f '(t) = 16/(1 + t²), the value of the function f will be f = 16 tan (t) - 24.8.

It is described as a particular kind of relationship with a set domain and range. Every value in the domain has a specific relationship with one value in the range, according to the function.

Given that,

f '(t) = 16/(1 + t²)

f(1) = 0

In order to find f integrate the function,

f = ∫(16/(1 + t²)) dt

f = 16 ∫1/(1 + t²) dt

f = 16 tan t + C

As, f(1) = 0

0 = 16 tan 1 + C

C = - 16(1.55) = -24.8

Thus, the value of function is f = 16 tan (t) - 24.8.

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The number of homes in a community of 12, 000 which would be heated by electricity, given the rate of houses heated, is c. 3,360 homes

To find the number of homes in the community which would be heated by electricity, first find the percentage of homes that are heated according to the U.S. Census :

= Number of houses heated / Total number

= 7 / 25

= 28 %

When given a community of 12, 000 homes therefore, the number of homes which would be heated are:

= Percentage of houses heated x Community size

= 28 % x 12, 000

= 3,360 homes

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Answer:

c. 3,360 homes

Step-by-step explanation: