Find the particular solution that satisfies the initial condition. (Enter your solution as an equation.)
Differential Equation yy'-9e^x=0 Initial Condition y(0)=7

The correct answer is (a) i.e. the probability that the sum of the numbers on the two dice is greater than 4 is 30/36.

To find the probability that the sum of the numbers on two dice is greater than 4, we need to determine the number of favorable outcomes and divide it by the total number of possible outcomes.

We can start by listing all the possible outcomes when rolling two dice:

1-1, 1-2, 1-3, 1-4, 1-5, 1-6

2-1, 2-2, 2-3, 2-4, 2-5, 2-6

3-1, 3-2, 3-3, 3-4, 3-5, 3-6

4-1, 4-2, 4-3, 4-4, 4-5, 4-6

5-1, 5-2, 5-3, 5-4, 5-5, 5-6

6-1, 6-2, 6-3, 6-4, 6-5, 6-6

Out of these 36 possible outcomes, the outcomes where the sum is greater than 4 are:

1-4, 1-5, 1-6

2-3, 2-4, 2-5, 2-6

3-2, 3-3, 3-4, 3-5, 3-6

4-1, 4-2, 4-3, 4-4, 4-5, 4-6

5-1, 5-2, 5-3, 5-4, 5-5, 5-6

6-1, 6-2, 6-3, 6-4, 6-5, 6-6

There are 30 favorable outcomes. Therefore, the probability that the sum of the numbers on the two dice is greater than 4 is 30/36.

So the correct answer is (a) 30/36.

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To find the number of integers between 400 and 851 inclusive that are divisible by four, we need to determine the number of multiples of four in that range. The first multiple of four in the range is 400, and the last multiple of four is 848. To find how many multiples of four there are, we can subtract the two numbers and divide by four, then add one (because we need to include the first multiple).


- First multiple of four in the range: 400
- Last multiple of four in the range: 848
- Difference between the two: 848 - 400 = 448
- Divide by four: 448 ÷ 4 = 112
- Add one: 112 + 1 = 113

Therefore, there are 113 integers between 400 and 851 inclusive that are divisible by four.

There are 113 integers between 400 and 851 inclusive that are divisible by four.

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The answer is 7/10 given that a book contains animal characters given that it is a graphic Nove. We have 24 books, of which 10 are graphic novels and 11 have animal characters.

Seven of them are graphic novels with animal characters. What we are looking for is the probability of an animal character being present, given that the book is a graphic novel. We can use the Bayes theorem to calculate this. Bayes' Theorem: [tex]P(A|B) = P(B|A)P(A) / P(B)P[/tex](Animal Characters| Graphic Novel) = P(Graphic Novel| Animal Characters)P(Animal Characters) / P(Graphic Novel)By looking at the question, P(Animal Characters) = 11/24,

P(Graphic Novel| Animal Characters) = 7/11, and P(Graphic Novel) = 10/24.P(Animal Characters| Graphic Novel) [tex]= (7/11) (11/24) / (10/24)P[/tex](Animal Characters| Graphic Novel) = 7/10The probability that a book contains animal characters given that it is a graphic novel is 7/10.

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Answer:

700

Step-by-step explanation:

4(175)=700

Answer:

Therefore, Juan will pay $2.22 in sales tax.

Step-by-step explanation:

To find the amount of tax Juan will pay, we can first calculate 8% of the price of the dollhouse, and then round to the nearest cent.

8% of $27.75 = 0.08 × $27.75 = $2.22

Therefore, Juan will pay $2.22 in sales tax.

The growth rate in elevation with respect to horizontal distance as you drive up the road is approximately 0.315 feet of elevation gained for every 1 foot of horizontal distance traveled.

Based on the information given, we know that the elevation at the base of the hill is 0 feet and the elevation at the top is 5000 feet (as the sign reads "Elevation 2500 feet" at the base and "Elevation 7500 feet" at the top). The horizontal distance from the base to the top is 3 miles.

To find the growth rate in elevation with respect to horizontal distance as you drive up the road, we can use the formula:

growth rate = change in elevation / horizontal distance

In this case, the change in elevation is 5000 feet (from 0 feet at the base to 5000 feet at the top), and the horizontal distance is 3 miles.

We need to convert the units to be consistent, so let's convert 3 miles to feet:

3 miles = 3 x 5280 feet = 15,840 feet

Now we can plug in the values and solve for the growth rate:

growth rate = 5000 feet / 15,840 feet = 0.315

So the growth rate in elevation with respect to horizontal distance as you drive up the road is approximately 0.315 feet of elevation gained for every 1 foot of horizontal distance traveled.

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Answer:

$2.15

Step-by-step explanation:

There are 12 in a dozen.

1.5 dozen = 18.

$38.70/18 = $2.15

It's $2.15 for a single rose.

The value of x in the equation is -3.

The given equation is 4 = (3x+17)/2

We have to find the value of x

Apply cross multiplication

4×2 = 3x+17

8=3x+17

Subtract 17 from both sides

8-17=3x

-9=3x

Divide both sides by 3

x=-3

Hence, the value of x in the equation is -3.

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The estimated probability of getting a green marble from this bag is 9/25.

The probability of getting a green marble from this bag.

We'll use the information provided on each student's results to calculate the probability.
Student 1: Picked a green marble 1 time in 4 experiments.
Student 2: Picked a green marble 5 times in 12 experiments.
Student 3: Picked a green marble 3 times in 9 experiments.
To estimate the probability, we'll add the number of successful green marble picks (1 + 5 + 3 = 9) and divide it by the total number of experiments (4 + 12 + 9 = 25).
Probability of getting a green marble = (Number of successful picks) / (Total number of experiments) = 9 / 25.

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The length of the radius of cone A. is [tex]\frac{\pi}{6}[/tex].

The lateral surface area of cone A is a sector with central angle 6 radians and radius π.

We can use the formula for sector area to find the lateral surface area of the cone.

Area of sector = θ/2π×π²

where θ is the central angle and π is the radius.

Area of cone’s lateral surface area (L) =θ/2π×2πr=rθ.

So, r = L/θ = π/6 (when L=π and θ=6 radians).

The length of the radius of cone A is π/6 which is approximately 0.524.

Therefore, the length of the radius of cone A is [tex]\frac{\pi}{6}[/tex], when unwrapped, given that the lateral surface area of cone A is a sector with central angle 6 radians and radius pi.

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The Laplace transform of a given function, f(t), is denoted by L{f(t)} and can be determined using Table 7.1 and the properties from Table 7.2.

For the given function, f(t) = 12 + [tex]e^t[/tex] × sin(2t), we will use the linearity property and the trigonometric identity.
First, apply the linearity property: L{12 + [tex]e^t[/tex]  sin(2t)} = L{12} + L{[tex]e^t[/tex] × sin(2t)}.
Next, using Table 7.1, find the Laplace transform of each term:
1. L{12} = 12 × L{1} = 12/s
2. L{[tex]e^t[/tex] × sin(2t)} = [tex]e^{(-s)}[/tex]× L{sin(2t)} = (2 /[tex](s^2 + 4)[/tex]) × [tex]e^{(-s)}[/tex]
Now, combine the transforms: L{12 + [tex]e^t[/tex] × sin(2t)} = 12/s + (2 / ([tex]s^2[/tex] + 4)) × [tex]e^{(-s)}[/tex].

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The sequence {hn} defined as h0 = 5/3, h1 = 11/3, hn = 3hn-1 - 4hn-2 + 6n satisfies the given recurrence relation.

To prove that for all n ≥ 0, the sequence {hn} defined as h0 = 5/3, h1 = 11/3, hn = 3hn-1 - 4hn-2 + 6n satisfies the given recurrence relation, we can use mathematical induction.

Base case:

For n = 0, we have h0 = 5/3 which is equal to the given initial value.

For n = 1, we have h1 = 11/3 which is also equal to the given initial value.

Inductive step:

Assume that the recurrence relation holds for some k ≥ 1, i.e., hk = 3hk-1 - 4hk-2 + 6k.

We want to show that it also holds for k+1, i.e., h(k+1) = 3h(k+1)-1 - 4h(k+1)-2 + 6(k+1).

Using the recurrence relation for hk, we have:

hk+1 = 3hk - 4hk-1 + 6k+3 (by substituting k+1 for n in the given recurrence relation)

Similarly, we have:

hk = 3hk-1 - 4hk-2 + 6k (by assumption)

hk-1 = 3hk-2 - 4hk-3 + 6(k-1) (by assumption)

Substituting these values into the expression for hk+1, we get:

hk+1 = 3(3hk-1 - 4hk-2 + 6k) - 4(3hk-2 - 4hk-3 + 6(k-1)) + 6(k+1)

Simplifying the expression, we get:

hk+1 = 9hk-1 - 12hk-2 + 18k - 12hk-2 + 16hk-3 - 24(k-1) + 6(k+1)

hk+1 = 9hk-1 + 4hk-3 - 12hk-2 - 6(k-1) + 6(k+1)

hk+1 = 3(3hk-1 - 4hk-2 + 6k+1) - 4(3hk-2 - 4hk-3 + 6k) + 6(k+1)

hk+1 = 3h(k+1)-1 - 4h(k+1)-2 + 6(k+1)

This shows that the recurrence relation holds for all n ≥ 0 by mathematical induction, and hence the sequence {hn} defined as h0 = 5/3, h1 = 11/3, hn = 3hn-1 - 4hn-2 + 6n satisfies the given recurrence relation.

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When considering two nonnegative numbers p and q such that p+q=6, the difference between the maximum and minimum of the quantity (p^2q^2)/2 is 81 - 0 = 81.

To find the maximum and minimum of the quantity (p^2q^2)/2, we can use the AM-GM inequality.
AM-GM inequality states that for any nonnegative numbers a and b, (a+b)/2 ≥ √(ab).

So, in our case, we can write:
(p^2q^2)/2 = (p*q)^2/2

Let x = p*q, then we have:
(p^2q^2)/2 = x^2/2
Since p and q are nonnegative, we have x = p*q ≥ 0.

Using the AM-GM inequality, we have:
(x + x)/2 ≥ √(x*x)
2x/2 ≥ x
x ≥ 0
So, the minimum value of (p^2q^2)/2 is 0.
To find the maximum value, we need to use the fact that p+q=6.

We can rewrite p+q as:
(p+q)^2 = p^2 + 2pq + q^2
36 = p^2 + 2pq + q^2
p^2q^2 = (36 - p^2 - q^2)^2

Substituting this into the expression for (p^2q^2)/2, we get:
(p^2q^2)/2 = (36 - p^2 - q^2)^2/2
To find the maximum value of this expression, we need to maximize (36 - p^2 - q^2)^2.

Since p and q are nonnegative and p+q=6, we have:
0 ≤ p, q ≤ 6
So, the maximum value of (36 - p^2 - q^2) occurs when p=q=3.

Thus, the maximum value of (p^2q^2)/2 is:
(36 - 3^2 - 3^2)^2/2 = 81

Therefore, the difference between the maximum and minimum of (p^2q^2)/2 is:
81 - 0 = 81.

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To sketch several levels of the function \(f(x, y) = e^x y\), we can plot contour lines corresponding to different function values. Each contour line represents points in the xy-plane where the function takes a constant value.

Here is a sketch showing contour lines for various levels of \(f(x, y) = e^x y\):

```

   |          _______________

   |        _/                |

   |      _/                   |

   |     /                      \

   |    |                        |

   |    |                        |

   |    |                        |

   |     \                      /

   |      \                    /

   |          \______/

   |

   +--------------------------------

```

Each contour line corresponds to a different level of \(f(x, y)\). The lines get closer together as we move away from the origin, indicating an exponential growth pattern.

Please note that the sketch is a rough representation and may not accurately reflect the precise shape and spacing of the contour lines.

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The density difference required for the flow at the exit of the contraction to have uniform velocity is simply -ρ₁.

Assuming steady, incompressible, and inviscid flow, the continuity equation states that the mass flow rate must be conserved, i.e.,

ρ₁A₁U₁ = ρ₂A₂U₂

where ρ₁ and ρ₂ are the densities of the two streams, A₁ and A₂ are the cross-sectional areas of the two streams, U₁ and U₂ are the velocities of the two streams, respectively.

Since the flow at the exit of the contraction has uniform velocity, we can set U₂ = U₁. Also, since the two streams are parallel, we can assume that A₁ = A₂ = A. Therefore, the continuity equation becomes:

ρ₁U₁ = ρ₂U₂ = ρ₂U₁

Now, we can express the density of the second stream in terms of the density of the first stream and the density difference:

ρ₂ = ρ₁ + Ap

Substituting this into the continuity equation, we get:

ρ₁U₁ = (ρ₁ + Ap)U₁

Simplifying this equation, we obtain:

Ap = -ρ₁(U₁/U₁ - 1) = -ρ₁

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The period of the following functions. g ( x ) = cos ( x 4 ) is that it doesn't have any.

To find the period of the function g(x) = cos(x^4), we need to find the smallest positive value of p such that g(x + p) = g(x) for all values of x. That is, we need to find the most minor p such that cos((x + p)^4) = cos(x^4) for all values of x.

Using the identity cos(a + b) = cos(a)cos(b) - sin(a)sin(b), we can expand the left-hand side of the equation:

cos((x + p)^4) = cos(x^4 + 4px^3 + 6p^2x^2 + 4p^3x + p^4)

= cos(x^4)cos(4px^3) - sin(x^4)sin(4px^3)cos(6p^2x^2)

=cos(x^4)sin(4px^3)sin(6p^2x^2) - sin(x^4)cos(4px^3)cos(6p^2x^2) + cos(x^4)cos(4px^3)sin(6p^2x^2)

Let's consider the second condition first. Since x is an integer multiple of π^(1/4), we have: 4px^3 = (4pπ^(3/4))x^3

For this to be an integer multiple of π, we must have p = q/π^(3/4), where q is an integer. Substituting this value of p into the second condition, we get 4qx^3 = rπ

where r is an integer. This implies that x is a multiple of π, which contradicts our assumption that x is an integer multiple of π^(1/4). Therefore, there is no value of p for which g(x + p) = g(x) for all values of x.

In other words, the function g(x) = cos(x^4) does not have a period.

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From central limit theorem, in a sample

a) the sampling distribution of x is normal distribution.

b) The value of P(x>91.3) is equals to the 0.093418.

From the central limit theorem, when the samples of a population are considered then these generate a normal distribution of their own. The sample size must be equal to or higher than 30 in order for the central limit theorem to be true. We have a simple random sample obtained from population with the Sample size, n = 36

Population is skewed right with population mean, µ= 87

Standard deviations, σ = 24

We have to determine the sampling distribution of x.

a) As we see sample size, n = 36 > 30, so the sampling distribution is normal distribution.

b) Using the test statistic value for normal distribution, [tex]z= \frac{ x - \mu }{\frac{\sigma}{\sqrt{n}}} [/tex]. Here, x = 91.3, µ= 87, σ = 24, n = 36. Now, the probability value is, P(x>91.3)

= [tex]P( \frac{ x - \mu }{\frac{\sigma}{\sqrt{n}}} < \frac{ 91.3 - 87 }{\frac{24}{\sqrt{36}}}) [/tex]

= [tex]P(z < \frac{ 4.3}{\frac{24}{6}} )[/tex]

= [tex]P(z < \frac{ 4.3}{4} )[/tex]

= [tex]P(z < 1.32)[/tex]

Using the p-value calculator, the value P(z < 1.32) is equals to the 0.093418. So, P( x < 91.3 ) = 0.093418. Hence, required value is 0.093418.

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Complete question:

A simple random sample of size n=36 is obtained from a population that is skewed right with µ=87 and σ=24.

(a) describe the sampling distribution of x.

b) What is P(x>91.3)?

Answer: No Solution.

Step-by-step explanation:

To solve the system of equations 2x - y = -1 and 4x - 2y = 6 graphically, we can plot the two lines represented by each equation on the same coordinate plane and find the point of intersection, if it exists.

To graph the line 2x - y = -1, we can rearrange it into slope-intercept form:

y = 2x + 1

This equation represents a line with slope 2 and y-intercept 1. We can plot this line by starting at the y-intercept (0, 1) and moving up 2 units and right 1 unit to find another point on the line. Connecting these two points gives us the graph of the line (Look at the first screenshot).

To graph the line 4x - 2y = 6, we can rearrange it into slope-intercept form:

y = 2x - 3

This equation represents a line with slope 2 and y-intercept -3. We can plot this line by starting at the y-intercept (0, -3) and moving up 2 units and right 1 unit to find another point on the line. Connecting these two points gives us the graph of the line (Look at the second screenshot).

We can see from the graphs that the two lines are parallel and do not intersect. Therefore, there is no point of intersection and no solution to the system of equations.

Then, describe the steps involved in calculating taxable income including pension contributions, and provide examples to illustrate how these calculations work.

Finally, you can conclude by emphasizing the importance of proper tax planning and compliance to avoid penalties and other legal issues.

When calculating taxable income, you need to include pension contributions.

Here's how to calculate taxable income including pension contributions:

Step 1: Add up your income for the year, including all sources such as salary, bonuses, rental income, and investment income.

Step 2: Subtract your allowable deductions such as mortgage interest, charitable contributions, and state taxes.

Step 3: Subtract your personal exemptions, which are based on the number of dependents you have.

Step 4: Subtract your pension contributions from your income. These contributions reduce your taxable income, so the higher your contributions, the lower your taxable income.

Step 5: The result of these calculations is your taxable income. You can use this figure to determine how much tax you owe.

Then, describe the steps involved in calculating taxable income including pension contributions, and provide examples to illustrate how these calculations work.

Finally, you can conclude by emphasizing the importance of proper tax planning and compliance to avoid penalties and other legal issues.

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The x-intercepts are approximately 0.54 and 11.46. Since the coefficient of x^2 is positive, the graph opens upwards. Combining all of this information, we can sketch a graph of f(x) that looks like a "U" shape with vertex at (6, -12) and x-intercepts at approximately 0.54 and 11.46.

(a) To express f(x) in standard form, we need to complete the square. First, we can factor out the coefficient of x^2 to get:

f(x) = x^2 - 12x + 24

Next, we add and subtract (12/2)^2 = 36 to the expression inside the parentheses to get:

f(x) = (x^2 - 12x + 36) - 36 + 24

The expression inside the parentheses can be rewritten as (x - 6)^2, so we have:

f(x) = (x - 6)^2 - 12

Therefore, the standard form of the quadratic function f(x) is f(x) = (x - 6)^2 - 12.

(b) To sketch a graph of f, we can first identify the vertex as (6, -12) from the standard form. This is the lowest point on the graph since the coefficient of x^2 is positive. We can also find the x-intercepts by setting f(x) = 0:

(x - 6)^2 - 12 = 0

(x - 6)^2 = 12

x - 6 = ±√12

x = 6 ± 2√3.

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The lengths of the various foam pieces are represented here in inches according to the supplied specs. The following information is provided on a separate sheet of paper, which can be used to answer the questions that are there: 10 in, 20 in, 2 in, 10 in, 5 in, 20 in, 20 in, 8 in, 5 in, 2 in, and 26 in.

The provided measurements suggest that the foam play toy is made up of a number of different foam pieces, each of which has a different length.

One would need to conduct an analysis of the provided measures and give careful consideration to the particular questions that are being asked in order to answer the questions on the separate paper. Because the questions themselves are not included in the information that is provided, it is required to evaluate the prompts that are on the separate page and respond to them in the appropriate manner.

The lengths of the foam pieces can be determined by using the specified measures, which can also be used to answer any queries regarding the arrangement of the foam pieces, the overall length, or any other special inquiries that are mentioned in the https://brainly.com/question/28170201.

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True, This Continuity Editing/Cutting device is used in Classical Hollywood Cinema.

Match on action is a common continuity editing/cutting technique used in Classical Hollywood Cinema, where the editor cuts from one shot to another while maintaining visual continuity between the two shots by showing the continuation of an action or movement from one shot to the next. This helps to create a smooth and seamless flow of action on screen and maintains the illusion of reality for the viewer.

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Answer: -0.3

Step-by-step explanation:

The value of the integral is 640V. we integrate with respect to r:

∫0^4 10Vr^2 r dr = (10/4)(4^4)V = 640V

To evaluate the integral of Vx^2 + y^2 dV over the given region E, we can use cylindrical coordinates since the region lies inside a cylinder.

First, we need to determine the limits of integration for each variable. For z, the limits are -4 to 6, since the region is between the planes z=-4 and z=6. For the cylindrical coordinates, we know that x^2 + y^2 = r^2, so the cylinder can be represented by r = 4. Therefore, the limits for r are 0 to 4, and the limits for theta are 0 to 2π.

Substituting in the cylindrical coordinates into the integral, we get:

∫∫∫E Vr^2 r dz dθ dr

= ∫0^2π ∫0^4 ∫-4^6 Vr^2 r dz dr dθ

Since the integral does not depend on theta or z, we can evaluate them first. The integral with respect to z gives:

∫-4^6 Vr^2 r dz = 10Vr^2 r

Next, we integrate with respect to r:

∫0^4 10Vr^2 r dr = (10/4)(4^4)V

= 640V

Therefore, the value of the integral is 640V.

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To calculate the cost of producing 10 tacos, you will need to consider the cost of all the ingredients and supplies required to make them. This may include tortillas, meat, cheese, lettuce, tomatoes, spices, cooking oil, and any other toppings you plan to use.

Once you have determined the total cost of these items, you can divide it by the number of tacos you are producing to get the cost per taco. To ensure accuracy, it is recommended that you round your final answer to the nearest hundredths place. This means that if your cost per taco is $2.345, you would round it to $2.35. By calculating the cost of producing 10 tacos, you can determine how much you need to charge for each taco to ensure that you make a profit.

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On a Saturday afternoon, a family of five went to see a matinée movie. Everyone paid the same price for their movie tickets. The equation will be 57.50 = 5x + 25 and the value of x is $6.50.

Let the price of one movie ticket be x,

So if there are 5 members and the ticket price is same for all, the total price for 5 movie tickets = Price of one ticket × 5

= x × 5

= 5x

Money spent on drinks = $25

ow, the total money spent is $57.50

Total money spent = money spent on movie tickets + money spent on drinks

So, the equation will be:

57.50 = 5x + 25

Now, on solving the equation:

5x = 57.50 - 25

5x = 32.50

x = 32.50 / 5

x = 6.50

Hence, the price of one ticket is $6.50 and the equation for this question is 57.50 = 5x + 25.

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The power series expansion of f(x) = 4x^2 - 39x + 98 around 5 is f(x) ≈ 3 + (x-5) + 4(x-5)^2.

To expand f(x)=4x^2-39x+98 as a power series around 5, we need to use the formula for a power series:

f(x) = ∑(n=0 to infinity) [f^(n)(a)/n!] * (x-a)^n

where f^(n)(a) represents the nth derivative of f(x) evaluated at x=a. In this case, a=5.

To find the derivatives of f(x), we can use the power rule and the constant multiple rule:

f'(x) = 8x - 39
f''(x) = 8
f'''(x) = 0
f''''(x) = 0
...

Notice that the derivatives beyond the second derivative are all zero. This is because f(x) is a quadratic function, so all higher-order derivatives are zero.

Now we can plug these derivatives into the formula for the power series:

f(x) = f(5) + f'(5)*(x-5) + (f''(5)/2!)*(x-5)^2 + ...

f(5) = 4(5)^2 - 39(5) + 98 = -23

f'(5) = 8(5) - 39 = 1

f''(5) = 8

So the power series expansion of f(x) around x=5 is:

f(x) = -23 + (x-5) + 4/2!*(x-5)^2 + 0*(x-5)^3 + 0*(x-5)^4 + ...

Simplifying this expression, we get:

f(x) = -23 + (x-5) + 2(x-5)^2 + ...

And that's the power series expansion of f(x) around x=5!
Hi! To expand the function f(x) = 4x^2 - 39x + 98 as a power series around 5, we will use the Taylor series expansion formula:

f(x) = f(a) + f'(a)(x-a) + (f''(a)/2!)(x-a)^2 + ...

where a = 5. First, let's find the derivatives of f(x):

f(x) = 4x^2 - 39x + 98
f'(x) = 8x - 39
f''(x) = 8

Now, we'll evaluate the derivatives at a = 5:

f(5) = 4(5)^2 - 39(5) + 98 = 100 - 195 + 98 = 3
f'(5) = 8(5) - 39 = 40 - 39 = 1
f''(5) = 8

Finally, we'll plug these values into the Taylor series expansion formula:

f(x) ≈ 3 + 1(x-5) + (8/2!)(x-5)^2
f(x) ≈ 3 + (x-5) + 4(x-5)^2

So, the power series expansion of f(x) = 4x^2 - 39x + 98 around 5 is f(x) ≈ 3 + (x-5) + 4(x-5)^2.

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Answer:

angle

Step-by-step explanation:

since there is a < sign, that makes it an angle. I'm not sure if that is the whole problem, or if It is missing a picture. Hope this helps!

The correct answer is (c) approximately normal for large samples.

The central limit theorem states that as the sample size increases, the sampling distribution of the sample mean approaches a normal distribution, regardless of the shape of the population distribution. In the case of the difference of two sample means (x1 - x2), the central limit theorem still applies, and the distribution becomes approximately normal as the sample size (n) increases. Therefore, for large sample sizes, the sampling distribution of (x1 - x2) can be approximated by a normal distribution, and the properties of the normal distribution can be used to make statistical inferences about the population mean difference.

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a. distance should not directly affect the performance variable. b. A valid IV can be a challenging task and requires careful consideration of the underlying causal mechanisms and potential confounding factors.

a. In order for distance to be an instrumental variable (IV) for attendance, it must be (i) correlated with attendance, and (ii) uncorrelated with the error term (u) in the attendance equation. Additionally, distance should not directly affect the performance variable.

b. If attendance is correlated with the error term (u) in the attendance equation, then the interaction term between GPA and attendance will also be correlated with u. A possible IV for the interaction term could be a measure of how easily accessible the lecture notes are to the students. If there is a system in place that allows students to access lecture notes online or through a library, then students with lower attendance may still have access to the material covered in the lectures and may perform better if they have good GPA. Thus, this variable may be a good IV for the GPA*attendance term. However, it should be noted that finding a valid IV can be a challenging task and requires careful consideration of the underlying causal mechanisms and potential confounding factors.

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