Evaluate lim (√x - 2)/x-4 using the L'Hospitals rule

To find the derivative of the given function, we first rewrite and simplify it, then apply the rules of differentiation.

The given function is:

y = (3x^4 + 2x^3)^5 - (3x^2 - 2x + 4^4)^6

Now, we'll take the derivative with respect to x:

dy/dx = d/dx [(3x^4 + 2x^3)^5] - d/dx [(3x^2 - 2x + 4^4)^6]

We'll use the chain rule for both terms. For the first term:

(dy/dx)(3x^4 + 2x^3)^5 = 5(3x^4 + 2x^3)^4 * d/dx(3x^4 + 2x^3)

And for the second term:

(dy/dx)(3x^2 - 2x + 4^4)^6 = 6(3x^2 - 2x + 4^4)^5 * d/dx(3x^2 - 2x + 4^4)

Now we'll find the derivatives of the inner functions:

d/dx(3x^4 + 2x^3) = 12x^3 + 6x^2
d/dx(3x^2 - 2x + 4^4) = 6x - 2

Now, substitute these back into the chain rule expressions:

dy/dx = 5(3x^4 + 2x^3)^4 * (12x^3 + 6x^2) - 6(3x^2 - 2x + 4^4)^5 * (6x - 2)

This is the simplified derivative of the given function.

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The volume of Darby is given as 76 in'^2

Volume = (4/3)πr³

First, we need to find the radius (r) of the sphere. Since the diameter is given as 21/4, we can find the radius by dividing the diameter by 2:

radius (r) = (21/4) / 2 = 21/8

Now, we can calculate the volume using the formula:

Volume = (4/3)π(21/8)³

Volume = (4/3)π(9261/512) (since (21/8)³ = 9261/512)

Now, we can find the numerical value of the volume:

Volume ≈ (4/3) * 3.14159 * (9261/512)

Volume ≈ 4.18879 * 18.10352

Volume ≈ 75.89438

Rounding the volume to the nearest whole number:

Volume ≈ 76

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As per the given information, the value of the house is 165,116

The current value of the house is = 213,000

The percentage with which the value of the house has decreased = 29%

Let the price of the house when it was purchased be = x

The value of the house is decreased by 29% = 0.29

It is required to understand that an exponential function is involved whenever we discuss a rise or decline.

Thus,

According to the question,

x + 29% of x = 213000

Solving,

x + 0.29x = 213000

1.29x = 213000

x = 213000/1.29

x = 165,116

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Check the picture below.

[tex]\begin{array}{llll} \textit{using the pythagorean theorem} \\\\ a^2+o^2=c^2\implies a=\sqrt{c^2 - o^2} \end{array} \qquad \begin{cases} c=\stackrel{hypotenuse}{29.2}\\ a=\stackrel{adjacent}{r}\\ o=\stackrel{opposite}{22.4} \end{cases} \\\\\\ r=\sqrt{ 29.2^2 - 22.4^2}\implies r=\sqrt{ 852.64 - 501.76 } \\\\\\ r=\sqrt{ 350.88 }\implies r\approx 18.73~cm[/tex]

Two power series solutions of the differential equation (x^2+1)y''-6y=0 about x=0 are y1=x^2-3x^4/10+O(x^6) and y2=1-7x^2/6+O(x^4).

The given differential equation can be written as:

y''-6(x^2+1)^(-1)y=0 ...(1)

Let us assume the power series solutions of (1) about x=0 as:

y=∑_(n=0)^∞▒〖a_n x^n 〗Differentiating y with respect to x, we get:

y'=∑_(n=1)^∞▒na_n x^(n-1)

y''=∑_(n=2)^∞▒n(n-1)a_n x^(n-2)

Substituting these in (1), we get:

∑_(n=2)^∞▒n(n-1)a_n x^(n-2) - 6∑_(n=0)^∞▒a_n (x^2+1)^(-1) x^n=0

Multiplying throughout by x^2, we get:

∑_(n=4)^∞▒n(n-1)a_n x^(n-2) - 6∑_(n=2)^∞▒a_n (x^2+1)^(-1) x^(n)=0

omparing coefficients of like powers of x, we get the following recurrence relations:

a_2=0, a_3=0, a_4=3a_0/5, a_5=0, a_6=-(21a_0+5a_4)/70, a_7=0, a_8=(429a_0+245a_4)/1575, ...

Thus, we get the power series solution y1:

y1=a_0 + 0.x + 0.x^2 + (3a_0/5).x^3 - 0.x^4 - ((21a_0+5(3a_0/5))/70).x^5 + ...

Simplifying the above expression, we get:

y1=x^2-3x^4/10+O(x^6)

Similarly, we can solve for the second power series solution by using a different initial condition. We assume the second solution in the form of:

y=∑_(n=0)^∞▒b_n x^n

and substitute it in (1). On solving the recurrence relations, we get the power series solution:

y2=1-7x^2/6+O(x^4)

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The two power series solutions of the given differential equation are

y1(x) = a_3 * x^3 + a_4 * x^4 + ...

y2(x) = x^3 + a_4 * x^4 + ...

To find two power series solutions of the given differential equation (x^2 + 1)y'' - 6y = 0 about the ordinary point x = 0, we can assume a power series solution of the form:

y(x) = Σ(a_n * x^n)

where a_n are coefficients to be determined and Σ represents the sum over the values of n.

Let's differentiate y(x) twice to find the values of y''(x):

y'(x) = Σ(n * a_n * x^(n-1))

y''(x) = Σ(n * (n-1) * a_n * x^(n-2))

Now, we substitute y(x), y'(x), and y''(x) into the differential equation:

(x^2 + 1) * Σ(n * (n-1) * a_n * x^(n-2)) - 6 * Σ(a_n * x^n) = 0

Expanding and rearranging the terms, we get:

Σ(n * (n-1) * a_n * x^n + a_n * x^(n+2)) - 6 * Σ(a_n * x^n) = 0

Grouping the terms by their powers of x, we have:

Σ((n * (n-1) * a_n - 6 * a_n) * x^n) + Σ(a_n * x^(n+2)) = 0

Now, we equate the coefficients of like powers of x to zero to obtain a recursion relation for the coefficients a_n.

For n = 0:

(n * (n-1) * a_n - 6 * a_n) = 0

(-6 * a_0) = 0

a_0 = 0

For n = 1:

(n * (n-1) * a_n - 6 * a_n) = 0

(1 * 0 * a_1 - 6 * a_1) = 0

-5 * a_1 = 0

a_1 = 0

For n = 2:

(n * (n-1) * a_n - 6 * a_n) = 0

(2 * 1 * a_2 - 6 * a_2) = 0

-4 * a_2 = 0

a_2 = 0

For n = 3:

(n * (n-1) * a_n - 6 * a_n) = 0

(3 * 2 * a_3 - 6 * a_3) = 0

0 * a_3 = 0

a_3 can be any value

From the recursion relation, we see that a_0 = a_1 = a_2 = 0, indicating that the terms of y(x) involving these coefficients will vanish.

Therefore, we can write the first power series solution y1(x) as:

y1(x) = a_3 * x^3 + a_4 * x^4 + ...

For the second power series solution, we can choose a different value for a_3 to obtain a linearly independent solution. Let's choose a_3 = 1:

y2(x) = x^3 + a_4 * x^4 + ...

These are the two power series solutions of the given differential equation about the ordinary point x = 0.

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Answer:

Step-by-step explanation:

*if this is wrong its because they didnt line the square up properly*

But the answer os 28 because, what you will need to do is multiply side and top

The side has 4 squares in the box

and the top has 7

so multipy 7 x 4 or 4 x 7

7 x 4 = 28

For a quadratic function, f( x) = -( x + 4)(x-1), the computed value of following,

a) value of f(2) is equals to -6.

b) The value of f(0) is equals to 4.

c) The value of derivative of f(x) at x = -2, f'(-2) is equals to 1.

A quadratic function is a polynomial function with one or more variables in which the highest exponent of the variable is two. The standard form of quadratic function is f(x) = ax² + bx + c, where a, b, and c are numbers with a ≠ 0.

We have a quadratic function, f(x) defined as

f( x) = -( x + 4)(x-1) --(1)

this function is present in one variable x.

Now, the value of function can be determined by different inputs.

a) At x = 2, substitute it in the equation (1),

f(2) = -( 2+ 4)(2 - 1)

= -( 6) (1)

= -6

b) Similarly, f(0) = -(0 + 4)( 0 -1 )

= -4(-1)

= 4

c) Rewrite the function, f(x) as f(x) = - x² - 3x + 4

To determine the derivative of f(x), differentiating the function with respect to x

f'(x) = - 2x - 3

At x = -2, f'(-2) = -2(-2) -3

= 4 - 3

= 1

Hence, required value is 1.

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Complete question:

Consider the quadratic function: f(x) = -(x+4)(x-1) compute the following

a) f( 2)

b) f(o)

c) f'(-2)

(I think I already got the answers for a, b, and c, but I wouldn't mind clarification, thank you!)

The author would have sold  365396 books over the first 20 months.

The author sold 22000 books in the first month, and the sales declined by 7% each month.

This means that the number of books sold in each subsequent month is 93% of the number sold in the previous month (since 100% - 7% = 93%). Therefore, the number of books sold in the second month is:

0.93×22000

= 20460

The number of books sold in the third month is:

0.93 × 20460 = 19007

To find the total number of books sold over the first 20 months, we can use the formula for the sum of a geometric series:

S = a(1 - rⁿ) / (1 - r)

Substituting the values, we get:

S = 22000(1 - 0.93²⁰) / (1 - 0.93)

S = 22000(0.766)/0.07

S=240743

Therefore, the author would have sold 365396 books over the first 20 months.

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To find all the spanning trees of the graph below, we can use the algorithm of removing edges that form cycles until we have a tree. Thus there are 2 groups of isomorphic spanning trees.

There are a total of 6 different spanning trees of this graph, as shown below:

```
    a         a         b         b         c         c    
   / \       / \       / \       /         /         \    
  b   c     b   c     a   c     a         b           d  
 /           \           \   /   \       / \         /    
d             d           d e     d     d   e       e      
```

To determine how many different spanning trees there are up to isomorphism, we need to group them by which are isomorphic. An isomorphism is a bijective function between two graphs that preserves the edges and vertices. In other words, two graphs are isomorphic if one can be obtained from the other by relabeling the vertices.
In this case, we can see that the first three trees are isomorphic to each other, and the last three trees are isomorphic to each other. Therefore, there are only two groups of spanning trees up to isomorphism, and each group contains three trees.
To find all spanning trees of a given graph, we will follow these steps:

1. Identify the graph vertices and edges.
2. Remove any cycles present in the graph.
3. Generate all possible combinations of edges that form a tree and connect all vertices.

Since the graph is not provided, I will assume a simple graph with 4 vertices (A, B, C, D) and 4 edges (AB, BC, CD, AD) forming a square.

Step 1: Identify the graph vertices and edges
Vertices: A, B, C, D
Edges: AB, BC, CD, AD

Step 2: Remove any cycles present in the graph
The graph has one cycle: ABCD. We need to remove one edge to break the cycle. We have 4 possibilities: remove AB, BC, CD, or AD.

Step 3: Generate all possible combinations of edges that form a tree and connect all vertices
After removing an edge, we get the following spanning trees:
1. Tree 1: Edges - BC, CD, AD
2. Tree 2: Edges - AB, CD, AD
3. Tree 3: Edges - AB, BC, CD
4. Tree 4: Edges - AB, BC, AD

So, there are 4 different spanning trees in total.

For the second part of your question, we need to find the number of different spanning trees up to isomorphism. Two trees are isomorphic if they have the same structure, but their vertices might be labeled differently.

Grouping the above spanning trees by isomorphism, we find that:

- Tree 1 and Tree 4 are isomorphic because both have a central vertex connected to three other vertices (Tree 1: vertex B; Tree 4: vertex D).
- Tree 2 and Tree 3 are isomorphic because both have a straight line of vertices connected by edges (Tree 2: A-CD-B; Tree 3: A-BC-D).

Thus, there are 2 groups of isomorphic spanning trees.

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The surface area of the rectangular prism is 588 square yards. The total region or area covered by all the faces of a rectangular prism is defined as the surface area of a rectangular prism.

It is a three-dimensional shape. It has six faces, and all the faces are rectangular-shaped. Therefore, both the bases of a rectangular prism must also be rectangles.

- Face 1: 9 yards long and 6 yards high, so its area is 9 x 6 = 54 square yards.

- Face 2: 9 yards long and 6 yards high, so its area is 9 x 6 = 54 square yards.

- Face 3: 16 yards wide and 6 yards high, so its area is 16 x 6 = 96 square yards.

- Face 4: 16 yards wide and 6 yards high, so its area is 16 x 6 = 96 square yards.

- Face 5: 9 yards long and 16 yards wide, so its area is 9 x 16 = 144 square yards.

- Face 6: 9 yards long and 16 yards wide, so its area is 9 x 16 = 144 square yards.

The surface area =  54 + 54 + 96 + 96 + 144 + 144

= 588 square yards.

Surface area of the rectangular prism is 588 square yards.

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Answer: 224

I hope I'm right i'm not good with division so yeh-

4000 cm^3 is the total volume of this octahedron

The volume of a pyramid is given by the formula:

V = (1/3) * base area * height

Since the base is a square with sides of 20 cm, its area is:

base area = 20^2 = 400 cm^2

The height of each pyramid is half the overall height of the octahedron, which is 30 cm. So the height of each pyramid is:

height = 30/2 = 15 cm

Now we can find the volume of one pyramid:

V = (1/3) * base area * height

V = (1/3) * 400 cm^2 * 15 cm

V = 2000 cm^3

Therefore, the total volume of the octahedron is twice the volume of one pyramid:

V_total = 2 * V

V_total = 2 * 2000 cm^3

V_total = 4000 cm^3

So the volume of the octahedron is 4000 cubic centimeters.

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The concentration of Br- in the seawater sample is approximately 66 ppm. To find the concentration of Br- in seawater in parts per million (ppm), we will first convert the given concentration from moles per liter (M) to grams per liter (g/L). Then, we will convert it to parts per million.

Given:
- Concentration of Br- in seawater = 8.3 * 10^(-4) M
- Mass of 1 liter of seawater = 1.0 kg (1000 g)

Step 1: Convert the concentration from M to g/L.
We will use the molar mass of Br-, which is approximately 79.9 g/mol.

(8.3 * 10^(-4) mol/L) * (79.9 g/mol) = 0.06637 g/L

Step 2: Convert the concentration from g/L to ppm.
1 ppm is equivalent to 1 mg of solute per kg of solution.

(0.06637 g/L) * (1000 mg/g) = 66.37 mg/L

Since the mass of 1 L of seawater is 1.0 kg, the concentration in ppm is:

66.37 mg/kg = 66.37 ppm

So, the concentration of Br- in the seawater sample is approximately 66 ppm.

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The change in rating of patients' depression levels, measured before and after treatment, is the dependent variable in this study.

The change in rating of patients' depression level is the measure of the effectiveness of the two treatments. Since the patients were randomly assigned to the two groups, the study design helps to ensure that any differences in the outcomes between the medication and cognitive therapy groups are due to the treatments themselves and not to other factors like age or severity of depression. By comparing the change in depression level ratings before and after treatment, the researchers can determine which treatment was more effective in reducing symptoms of depression. Dr. Wilhelm's study involved randomly assigning 50 depressed patients to two treatment groups: one receiving medication and the other receiving cognitive therapy.

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Answer:

see below, answers are underlined

Step-by-step explanation:

1. Vertical angles are 2 angles that are on opposite sides of each other and are the same value.  So, 2 angles that are considered vertical angles would be MPN (5y) and LPO (95)

2. Adjacent angles are 2 angles that are right next to each other on the same line and when added, they equal 180.  So, 2 angles that are adjacent angles are MPL (5x) and MPN (5y)

3. Using what we know about adjacent and vertical angles, we can solve to find x in 2 different ways:

--> 5x=85 (vertical angles), x=17

--> 5x+95=180 (adjacent angles), x=17

4. Using what we know about adjacent and vertical angles, we can solve to find y also in 2 different ways:

-->5y=95 (vertical angles), y=19

-->5y+85=180 (adjacent angles), y=19

Hope this helps! :)

The binomial distribution is not applicable because likelihood of discovering a golden ticket could not be independent

Given data ,

The probability of finding a golden ticket in one box of cereal is:

p = 64,000/800,000 = 0.08

We can use the binomial probability formula to find the probability of finding exactly 3 golden tickets in 15 boxes:

P(X = 3) = (15 choose 3) * (0.08)³ * (0.92)¹²

P(X = 3) ≈ 0.233

Therefore , the probability of finding exactly 3 golden tickets in 15 boxes is approximately 0.233

The likelihood of discovering a golden ticket could not be independent for each box, which is one reason why the binomial distribution would not be a suitable model for this circumstance.

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The median number of trees planted is given as follows:

3.5 trees.

The median of a data-set is the middle value of a data-set, the value of which 50% of the measures are less than and 50% of the measures are greater than. Hence, the median also represents the 50th percentile of a data-set.

The frequency table shows the number of times that each observation appears, hence the data-set is given as follows:

2, 3, 3, 3, 4, 4, 5, 6.

The cardinality of the data-set, representing the number of elements, is given as follows:

8.

Hence the median is the mean of the 4th and of the 4th elements, as follows:

Median = (3 + 4)/2

Median = 3.5.

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Answer:

23

Step-by-step explanation:

23.

Given fraction is 1218.⟹ 1218 = 6×26×3.= 23.

So the lowest form of 1218 = 23

2:3

both 12 and 18 is divided by 6

The expected number of wives who are seated next to their husbands is 1.

To find the expected number of wives who are seated next to their husbands, we can use the linearity of expectation. Let X be a random variable that takes the value 1 if the i-th wife is seated next to her husband, and 0 otherwise. Then the total number of wives seated next to their husbands is [tex]X = X_1 + X_2 + ... + X_{19}.[/tex]

Now, let's consider the probability that a particular wife is seated next to her husband. There are 38 seats at the table (19 couples), and the wife can either sit to the left or right of her husband. So the probability that she is seated next to her husband is 2/38 = 1/19.

Using linearity of expectation, we have:

[tex]E[X] = E[X_1 + X_2 + ... + X_1] = E[X_1] + E[X_2] + ... + E[X_{19}] = 19 * (1/19)=1[/tex]

Therefore, the expected number of wives who are seated next to their husbands is 1.

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We found that (a) the partial derivative of f(x, y) with respect to x evaluated at x=5 is 90y^3, and (b) the partial derivative of f(x, y) with respect to y evaluated at y=1 is 27x^2.

We have the function f(x, y) = 9x^2y^3, and we need to find (a) the partial derivative with respect to x and then evaluate it at 5, and (b) the partial derivative with respect to y and then evaluate it at 1.

(a) To find the partial derivative of f(x, y) with respect to x, we treat y as a constant and differentiate f(x, y) with respect to x:

∂f(x, y)/∂x = d(9x^2y^3)/dx = 18xy^3

Now, we need to evaluate this derivative at x=5:

∂f(5, y)/∂x = 18(5)y^3 = 90y^3

(b) To find the partial derivative of f(x, y) with respect to y, we treat x as a constant and differentiate f(x, y) with respect to y:

∂f(x, y)/∂y = d(9x^2y^3)/dy = 27x^2y^2

Now, we need to evaluate this derivative at y=1:

∂f(x, 1)/∂y = 27x^2(1)^2 = 27x^2

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Answer:

-6(.34) - 4(.13) - 2(.06) + 0(.13) + 2(.34) = -2

The expected value of the winnings is -$2

(a loss of $2).

x1 and x2, x1 and x5, x2 and x3, x2 and x5, x3 and x4 and x3 and x5 means differ significantly from one another

The experiment to compare the spreading rates of five different brands of yellow interior latex paint available in a particular area used 4 gallons (J = 4) of each paint.

The value of W for Tukey's procedure is:

W = q(α, 5, 20) * √(MSE/J)

where α = 0.05 is the significance level, 5 is the number of treatments, 20 is the total number of observations (4 observations per treatment), MSE = 450.8 is the mean square error, and q(α, 5, 20) is the critical value from the Studentized range distribution table.

Using the table, we find that q(α, 5, 20) = 3.365.

Substituting the values, we get:

W = 3.365 * √(450.8/4) = 21.63

The means that differ significantly from one another are:

x1 and x2

x1 and x5

x2 and x3

x2 and x5

x3 and x4

x3 and x5

Therefore, the correct answer is:

x1 and x2

x1 and x5

x2 and x3

x2 and x5

x3 and x4

x3 and x5

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The area of the triangle with sides a = 7, b = 5, and c = 5 is approximately 12.015 square units, rounded to three decimal places.

To find the area of a triangle with given side lengths a = 7, b = 5, and c = 5, we can use Heron's formula. Heron's formula states that the area (A) of a triangle can be calculated using the semi-perimeter (s) and the side lengths:

1. Calculate the semi-perimeter: s = (a + b + c) / 2
  s = (7 + 5 + 5) / 2
  s = 17 / 2
  s = 8.5

2. Apply Heron's formula: A = √(s * (s - a) * (s - b) * (s - c))
  A = √(8.5 * (8.5 - 7) * (8.5 - 5) * (8.5 - 5))
  A = √(8.5 * 1.5 * 3.5 * 3.5)

3. Calculate the area:
  A ≈ √(144.375)
  A ≈ 12.015

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lim (x→∞) (x^2 + 8x + 16) / (x^2 + 2x + 2) = 1. To find the limit of the given function as x approaches infinity, we'll first analyze the highest powers of x in the numerator and denominator: lim (x→∞) (x^2 + 8x + 16) / (x^2 + 2x + 2)

In this case, the highest power of x in both the numerator and denominator is x^2. Divide the numerator and denominator by x^2 to make the limit easier to evaluate:
lim (x→∞) (1 + 8/x + 16/x^2) / (1 + 2/x + 2/x^2)
Now, as x approaches infinity, the terms with x in the denominator will approach 0:
lim (x→∞) (1 + 0 + 0) / (1 + 0 + 0)
This simplifies to:
lim (x→∞) 1 / 1
The limit is 1. So, the answer is:
lim (x→∞) (x^2 + 8x + 16) / (x^2 + 2x + 2) = 1

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Question 1: State where the function f(z) = 3 + 2i - z^3 + z is analytic.

Answer: The function f(z) = 3 + 2i - z^3 + z is a polynomial function, and polynomial functions are analytic everywhere in the complex plane. Therefore, this function is analytic for all complex numbers z.

Question 2: State where the function f(z) = e^z is analytic.

Answer: The function f(z) = e^z is an exponential function, and exponential functions are also analytic everywhere in the complex plane. Therefore, this function is analytic for all complex numbers z.

Question 3: State where the function 2 + z f(z) = (2 + 1 - 2i)(2 + 1 + 2i) is analytic.

Answer: The function 2 + z f(z) is a rational polynomial, and it is analytic everywhere except at the poles. In this case, the poles are -1 + 2i and -1 - 2i.

Question 4: State where the function f(z) = Im(z) + |z| + Re(z) is analytic.

Answer: The function f(z) = Im(z) + |z| + Re(z) involves the modulus (absolute value) of z, which is not an analytic function. Therefore, this function is not analytic anywhere in the complex plane.

Question 5: State where the function f(z) = 1/(22 + 16) is analytic.

Answer: The function f(z) = 1/(22 + 16) is a constant function, and constant functions are analytic everywhere in the complex plane. Therefore, this function is analytic for all complex numbers z.

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From Algebra, the earning points she have if she completes 4 levels is equals to the 112 points. The unit rate and equivalent ratio are 28 points per level.

Biannca plays a video game quest, where players earn the same amount of points for completing a level. The above figure represents the levels and earning amount of Biannca. Number of levels completed by Biannca = 2

Earning points of Biannca after completing two level of video game = 56 points

We have to determine the earing points by her after completing the 4 levels of game. So, the earing points of Biannca in each level of game = two levels total earning points divided by number of levels [tex]= \frac{56}{2} [/tex]

Multipling the numentor and denominator by 1/2, so that

= 28 points/level

which is an equivalent ratio and the unit rate of Biannca earing. So, using multiplcation operation for obtaining the earning points on completing 4 levels of game equals to number of levels × earing points for each level. The required earing points = 4 × 28 points = 112 points. Hence, required value is 112 points.

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Complete question:

The above figure complete the question. In the video game unicorn quest, players earn the same amount of points for completing a level Biannca completed 2 levels and earned 56 points how many points will she have if she completes 4 levels what is the equivalent and unit rate?

Answer: the answer would be 6a.

Step-by-step explanation:

since only 3 can be multiplied, we end up with 6a

Answer:

3a × 2 = 6a

Step-by-step explanation:

3a multiplied by 2 is equal to 6a

The margin of error for the survey is 0.053.

To calculate the margin of error, we need to use the formula:

Margin of Error = Critical Value x Standard Error

We can start by finding the critical value using a z-table. For a 95% confidence interval, the critical value is 1.96.

Next, we need to calculate the standard error. The formula for the standard error of a proportion is:

Standard Error = [tex]\sqrt{(p*(1-p))/n}[/tex]

where p is the sample proportion (0.372) and n is the sample size (500).

Standard Error = [tex]\sqrt{(0.372*(1-0.372))/500}[/tex] = 0.027

Now that we have the critical value and the standard error, we can calculate the margin of error:

Margin of Error = 1.96 x 0.027 = 0.053

Therefore, the margin of error for the survey is 0.053 or approximately 5.3%. This means that we can be 95% confident that the true proportion of people who watched the Avalanche game in the entire state of Colorado is within 5.3% of the sample proportion of 37.2%.

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a) The equivalent double integral with the order of integration reversed is ∫∫D f(x, y) dy dx.

b)  To reverse the order of integration, we need to sketch the region of integration D and rewrite the original double integral with the opposite order of integration.

Since the provided information is incomplete, it is not possible to determine the specific region of integration or the function f(x, y) involved.

However, in general, reversing the order of integration involves swapping the order of integration limits and rewriting the integrand accordingly. This allows for the evaluation of the integral in a different order, which can be useful in certain cases for simplifying calculations or applying different integration techniques.

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The side lengths are given as follows:

The side lengths for this problem are obtained considering the triangle midsegment theorem, which states that the midsegment of the triangle divided the laterals of the triangle into two segments of equal length.

The congruent segments(segments of equal length) are given as follows:

Hence the lengths are given as follows:

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