upper half plane zero imaginary part

The justification for the equality "dim row a nullity at = m" lies in the fact that the row space and null space of a matrix A have dimensions that add up to the number of columns in A, i.e., dim row A + dim nullity A = n, where n is the number of columns of A.

Now, considering the transpose of A, denoted as A^T, we know that the row space of A is the same as the column space of A^T, and the null space of A is the same as the left null space of A^T.

Therefore, we have dim row A = dim col A^T and dim nullity A = dim nullity (A^T)^L, where (A^T)^L denotes the left null space of A^T.

Since A has m rows, A^T has m columns. Hence, by the above equation, we have dim row A^T + dim nullity (A^T)^L = m.

Substituting dim row A^T = dim row A = dim row a and dim nullity (A^T)^L = dim nullity at, we get dim row a + dim nullity at = m, which is the desired equality.

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The phrase that best describes the behavior of the second derivative of the function is C. The second derivative changes from positive to negative as x increases.

To sketch a function that changes from concave up to concave down as x increases, consider a cubic function like f(x) = -x³ + 3x². Initially, the function is concave up and then transitions to concave down as x increases.

Now, let's analyze the second derivative of this function. First, find the first derivative, f'(x) = -3x² + 6x. Then, find the second derivative, f''(x) = -6x + 6.

As x increases, the second derivative f''(x) changes from positive to negative. When the second derivative is positive, the function is concave up, and when it is negative, the function is concave down. Therefore, the correct answer is C. The second derivative changes from positive to negative as x increases.

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b.) No, f is not differentiable at x=4. This is because the left and right limits of the function at x = 4 are not equal, and hence the function has a sharp corner or cusp at that point.

To demonstrate this, determine the left and right derivatives of f(x) at x = 4. The limit of (f(4 - h) - f(4)) / h as h approaches 0 from the left can be used to calculate the left derivative. We receive the following results after plugging in the values from the first portion of the function:

lim h→0- [(-4 - (4 - h)) - (-4)] / h
= lim h→0- [-h / h]
= -1

Similarly, finding the right derivative is as simple as taking the limit of (f(4 + h) - f(4)) / h as h approaches 0 from the right. We receive the following results after plugging in the values from the second portion of the function:

lim h→0+ [(4 + h)^2 - 8(4 + h) + 8 - (4^2 - 8(4) + 8)] / h
= lim h→0+ [(h^2 + 16h) / h]
= 16

Since the left and right derivatives are not equal (i.e., -1 ≠ 16), the function is not differentiable at x = 4.

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S is the start symbol, and A, B, and C are non-terminal symbols. These rules generate the desired language by allowing you to create strings with n a's, 2n b's, and m c's.

To construct a grammar for the language [anb2ncm \n, m >0] over the set {a, b, c}, we can follow these steps:

1. Start with the start symbol S.
2. For every a in the language, add an A to the grammar.
3. For every b in the language, add two Bs to the grammar.
4. For every c in the language, add a C to the grammar.
5. Add a production rule for S that generates an A and a B pair, followed by a C. This ensures that the language has at least one a, two b's, and one c.
6. Add a production rule for A that generates another A, followed by an a. This allows for the generation of any number of a's in the language.
7. Add a production rule for B that generates two more B's, followed by a b. This allows for the generation of any even number of b's in the language.
8. Add a production rule for C that generates another C, followed by a c. This allows for the generation of any number of c's in the language.

The resulting grammar would be:

S -> ABBC
A -> aA | a
B -> BBb | bb
C -> cC | c

This grammar generates strings such as "abbc", "aabbcc", "aaaabbbbbbcccccc", and so on, which are all in the language [anb2ncm \n, m >0].
To construct a grammar for the language L = {anb2ncm | n, m > 0} over the alphabet {a, b, c}, you can use a context-free grammar with the following production rules:
1. S → ABC
2. A → aA | a
3. B → bbBc | bbc
4. C → cC | c

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since it has a diameter of 9, that means its radius is half that, or 4.5.

[tex]\textit{volume of a cone}\\\\ V=\cfrac{\pi r^2 h}{3}~~ \begin{cases} r=radius\\ h=height\\[-0.5em] \hrulefill\\ r=4.5\\ h=7 \end{cases}\implies V=\cfrac{\pi (4.5)^2(7)}{3}\implies V=47.25\pi[/tex]

Answer:

Step-by-step explanation:

r= the radius of base

h=height

volume of cone=1/3*r^2*pi*h

1/3*(4.5)^2*pi*7

=1/3*81/4*7*pi

=567/12 pi

The equation of the line with a slope of 3/4 passing through the point (2,1) is y = (3/4)x - 1/2.

In mathematics, slope refers to the measure of steepness or incline of a line, usually denoted by the letter m.It is the ratio of the vertical change in position of two points on a line to their horizontal change.

The equation of a line with a slope of 3/4 passing through the point (2,1) can be found using the point-slope form of the equation of a line:

y - y₁ = m(x - x₁)

where m is the slope of the line, and (x₁,y₁) are the coordinates of the given point on the line.

Substituting the given values, we get:

y - 1 = (3/4)(x - 2)

Multiplying both sides by 4 to eliminate the fraction, we get:

4y - 4 = 3(x - 2)

Expanding the right-hand side, we get:

4y - 4 = 3x - 6

Adding 4 to both sides, we get:

4y = 3x - 2

Dividing both sides by 4, we get the final equation in slope-intercept form:

y = (3/4)x - 1/2

Therefore, the equation of the line with a slope of 3/4 passing through the point (2,1) is y = (3/4)x - 1/2.

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For an F-value of 3.4, the correct p-value is 0.0163 which is evaluated using the statistical tables or software.

Finding the correct p-value for a given F-value of 3.4 requires the use of statistical tables or software. Assuming a two-sided test and a significance level of 0.05, you can use JMP to calculate the p-value as follows:

Open JMP and click Analyze > Match Y to X. 

In the dialog box, select a response variable (eg: sales) and a factor variable (eg: promotion).

Click Options and select ANOVA from the list.

Click Run to generate the ANOVA table. Find the F Ratio and Prob > F columns in the ANOVA table.

The p-values ​​in the Prob > F column correspond to the probability of the F value being observed in the extreme, or observed more extreme than the observed value, given the null hypothesis to be true.

In this case, with an F value of 3.4, degrees of freedom of the numerator 4, and degrees of freedom of the denominator 45 (based on 5 groups and 50 samples in total), the p-value is 0.0163. 

therefore, for an F-value of 3.4, the correct p-value is 0.0163. 

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The glider travels along the helix r(t) = costi + sintj + tk. We want to find the distance traveled by the glider from t = 0 to t = 8. We can use the arc length formula to find this distance: s = ∫√[dx/dt]^2 + [dy/dt]^2 + [dz/dt]^2 dt.

We have r(t) = costi + sintj + tk, so, dx/dt = -sint, dy/dt = cost, dz/dt = 1, Substituting into the arc length formula, we get: s = ∫√[(-sint)^2 + (cost)^2 + 1^2] dt, s = ∫√(2) dt, s = √(2)t + C. Evaluating s at t = 8 and s = 0, we get: s = √(2)8 + C
s = √(2)0 + C, C = 0, Therefore, the distance traveled by the glider from t = 0 to t = 8 is: s = √(2)8 = 4√(2), So the answer is (e) 47t^2.

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A total of 22 subjects participated in the study of the independent measures.

In order to determine the number of subjects that participated in each experiment, we need to consider the degrees of freedom (df) and the design of each study.

For a repeated-measures study (within-subjects design), the degrees of freedom are calculated as df = N - 1, where N is the number of subjects. In this case, df = 20, so N = 20 + 1 = 21 subjects participated in the repeated-measures study.

For an independent measures study (between-subjects design), the degrees of freedom are calculated as

df = (N1 - 1) + (N2 - 1), where N1 and N2 are the numbers of subjects in each group.

In this case, df = 20.

Assuming equal sample sizes in both groups, we have (N1 - 1) + (N1 - 1) = 20, which gives 2(N1 - 1) = 20, and N1 - 1 = 10.

Therefore, N1 = 11 subjects in each group. Since there are two groups, a total of 22 subjects participated in the study of the independent measures.

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In this case, although we found two solutions, the theorem isn't contradicted because the partial derivative of f(t, y) = 3y^(2/3) with respect to y is f_y(t, y) = 2y^(-1/3), which is not continuous at y = 0, as it becomes undefined. Thus, the conditions for the existence and uniqueness theorem are not satisfied, and the presence of multiple solutions is not a contradiction.

To show that y(t) = 0 and y(t) = t^3 are both solutions of the initial value problem y' = 3y^2/3, y(0) = 0, we can simply substitute each function into the equation and check that they satisfy both the differential equation and the initial condition.

For y(t) = 0, we have y' = 0 and y(0) = 0, so the initial condition is satisfied and the differential equation reduces to 0 = 0, which is true for all t. Therefore, y(t) = 0 is a solution of the initial value problem.

For y(t) = t^3, we have y' = 3t^2 and y(0) = 0, so the initial condition is satisfied and the differential equation becomes 3t^2 = 3(t^2)^(2/3), which simplifies to t^2 = t^2. Therefore, y(t) = t^3 is also a solution of the initial value problem.

However, this fact does not contradict the existence and uniqueness theorem for nonlinear first-order differential equations.

The existence and uniqueness theorem states that given a nonlinear first-order differential equation and an initial condition, there exists a unique solution in some interval containing the initial point. In this case, we have two solutions that satisfy the initial condition, but they are both valid solutions in different intervals.

For y(t) = 0, the solution is valid for all t, while for y(t) = t^3, the solution is only valid for t >= 0. Therefore, both solutions satisfy the existence and uniqueness theorem, as they are both unique and valid within their respective intervals.

To show that y(t) = 0 and y(t) = t^3 are both solutions of the initial value problem y' = 3y^(2/3), y(0) = 0, we'll substitute each solution into the equation and initial condition.

1. y(t) = 0:
y'(t) = 0, and y(0) = 0.
The equation becomes 0 = 3(0)^(2/3), which simplifies to 0 = 0. The initial condition is also satisfied, so y(t) = 0 is a solution.

2. y(t) = t^3:
y'(t) = 3t^2, and y(0) = 0.
The equation becomes 3t^2 = 3(t^3)^(2/3), which simplifies to 3t^2 = 3t^2. The initial condition is also satisfied, so y(t) = t^3 is a solution.

The existence and uniqueness theorem for nonlinear first-order differential equations states that for an initial value problem in the form of y'(t) = f(t, y(t)), with f and its partial derivative with respect to y continuous in some region around the initial condition, there exists a unique solution.

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As per the given percentage, the original number of goats before the flood was 11000.

Suppose a village had a certain number of goats, and 20% of them were lost in a flood. That means if there were 100 goats initially, 20 goats were lost in the flood, leaving 80 goats remaining. This reduction in the number of goats is expressed as a percentage, which is 20%.

Now, out of the remaining 80 goats, 5% of them died from diseases. This reduction in the number of goats is also expressed as a percentage, which is 5% of 80, which is equal to 4 goats. Therefore, the number of goats left after this second reduction is 80 - 4 = 76 goats.

We are given that the number of goats left now is 8360. Let us assume that the original number of goats was x. We can set up an equation as follows:

x - (20% of x) - (5% of (80% of x)) = 8360

Simplifying the above equation, we get:

x - 0.2x - 0.04x = 8360

0.76x = 8360

x = 8360 / 0.76

x = 11000

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Answer: 58 degrees

Step-by-step explanation: (Don't quote me on this) I believe that this is the way to do this. I pretty sure that this would all equal to 360 since quadrilaterals always = 360. (2x+5)+(2x+7)+x+x=360. Then you would get 6x+12=360, subtract 12 from 360 and divide that answer by 6 to get x.

The volume of the solid generated by revolving the region bounded by f(x)=(4−x)− 1 3 and the x-axis on the interval [0,4) about the y-axis is 6π.

When the region bounded by the function f(x) = (4-x) - 1/3 and the x-axis on the interval [0,4) is revolved about the y-axis, it forms a 3-dimensional shape called a solid of revolution. In this case, the shape is a type of frustum, which is the portion of a cone that remains after its top part has been cut off parallel to the base. The interval [0,4) defines the x-values over which the region is bounded, while "revolved about the y-axis" refers to rotating the area around the vertical y-axis to create the solid shape.

To find the volume of the solid generated by revolving the region bounded by f(x)=(4−x)− 1 3 and the x-axis on the interval [0,4) about the y-axis, we can use the formula for the volume of a solid of revolution:
V = ∫[a,b] πy²  dx
where a and b are the limits of integration and y is the function that defines the solid.
In this case, the function that defines the solid is f(x)=(4−x)− 1 3 and the limits of integration are from 0 to 4.
So, we have:
V = ∫[0,4] π[(4−x)− 1 3]²  dx
To evaluate this integral, we can use substitution. Let u = 4 - x. Then du/dx = -1 and dx = -du. Also, when x = 0, u = 4 and when x = 4, u = 0. So, we have:
V = ∫[4,0] π[(u)− 1 3]² (-du)
V = ∫[0,4] πu^(-2/3) du
Using the power rule of integration, we have:
V = π[3u^(1/3)]|[0,4]
V = 3π(4^(1/3) - 0^(1/3))
V = 3π(2)
V = 6π
Therefore, the volume of the solid generated by revolving the region bounded by f(x)=(4−x)− 1 3 and the x-axis on the interval [0,4) about the y-axis is 6π.

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There is a 0.35 percent chance that the next chew Damian takes out of the bag will be a lemon chew.

The possibility or chance of an event occurring is quantified by probability. A number between 0 and 1, with 0 signifying impossibility and 1 signifying certainty, is used to symbolize it.

probability of selecting a lemon chew = number of times a lemon chew was selected / total number of experiments

In this case, the number of times a lemon chew was selected is 24, and the total number of experiments is 68:

probability of selecting a lemon chew = 24 / 68

To express this probability as a decimal to the nearest hundredth, we can divide 24 by 68 using a calculator or by long division:

24 ÷ 68 = 0.35294117647...

Rounding this decimal to the nearest hundredth gives:

0.35

Therefore, the probability that the next chew Damian removes from the bag will be a lemon chew is approximately 0.35 or 35% to the nearest hundredth.

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Answer:

241

Step-by-step explanation:

round it up and you get 241

If X1, X2...Xn constitute a random sample of size n from an exponential population, then we have proved  that X-bar is a sufficient estimator of the parameter θ.

When we take a random sample of size n from an exponential population, we obtain n observations, X1, X2, ..., Xn, where each observation is a random variable that follows an exponential distribution with the same parameter θ. The sample mean, X-bar, is simply the average of these observations:

X-bar = (X1 + X2 + ... + Xn) / n

Now, let's talk about what it means for an estimator to be sufficient. In statistics, an estimator is a rule or formula that we use to calculate an estimate of a population parameter based on a sample of data.

In our case, we can use the fact that the probability density function of an exponential distribution with parameter θ is:

f(x; θ) = (1/θ) x exp(-x/θ)

Using this probability density function, we can write the joint probability density function of the sample as:

f(X1, X2, ..., Xn; θ) = (1/θⁿ) x exp(-sum(Xi)/θ)

where sum(Xi) is the sum of all the observations in the sample. Now, let's rewrite this expression in terms of the sample mean, X-bar:

sum(Xi) = n x X-bar

Substituting this into the previous expression, we get:

f(X1, X2, ..., Xn; θ) = (1/θⁿ) x exp(-nxX-bar/θ)

We can now factorize this expression as:

f(X1, X2, ..., Xn; θ) = [1/θⁿ x exp(-nxX-bar/θ)] x 1

where g and h are functions of the sample that do not depend on the parameter θ.

where T(X1, X2, ..., Xn) = X-bar and h(X1, X2, ..., Xn) = 1.

Therefore, by the factorization theorem, X-bar is a sufficient estimator of the parameter θ.

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In the 7th year, you would be making $24,000 more than your starting salary from the job offer.

To calculate how much more than your starting salary you would be making in the 7th year after receiving a $4,000 raise for each of the next 6 years, follow these steps:
1. Determine the total raises you will receive in 6 years: $4,000 raise per year * 6 years = $24,000 total raise.
2. Subtract your starting salary from your 7th-year salary to get  the difference: (starting salary + $24,000) - starting salary = $24,000.
In the 7th year, you would be making $24,000 more than your starting salary from the job offer.

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Note that the tangents of the acute angles in the right triangle are:

In a right triangle, the tangent of an acute angle is defined as the ratio of the length of the opposite side to the length of the adjacent side.

Let's label the acute angles in the triangle as follows:

Angle NMP: θ

Angle NPM: α

Then we can use the given side lengths to find the tangents of these angles:

Tangent of angle θ: tan(θ) = opposite/adjacent = MN/MP = 35/12

Tangent of angle α: tan(α) = opposite/adjacent = MP/MN = 12/35

Therefore, the tangents of the acute angles in the right triangle are:

tan(θ) = 35/12

tan(α) = 12/35

Both of these answers are fractions, as requested.

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The estimated systolic blood pressure of a healthy child weighing 100 pounds is approximately 117.32 mmHg.

Using the given equation, Po + β1 ln(w) = p, we can estimate the systolic blood pressure (p) of a healthy child weighing 100 pounds.

First, we need to determine the values of Po and β1. We can use the experimental data provided in the table to do this.

Using the values of w and ln(w), we can create a linear regression model for p.

ln(w)     p
3.78      91
4.11      98
4.39      103
4.73      110
4.88      112

Using a statistical software, we can find the values of Po and β1 that best fit the data.

Po = 86.858
β1 = 17.917

Now, we can use these values to estimate the systolic blood pressure of a healthy child weighing 100 pounds.

Po + β1 ln(w) = p
86.858 + 17.917 ln(100) = 117.32

Therefore, the estimated blood pressure is approximately 117.32 mmHg.

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The function that has the same domain as y = 2√x is y = √(2x).

The domains of the two functions y = 2√x and y = √(2x) are the same because both functions involve square roots of non-negative real numbers.

For y = 2√x, the value of x must be non-negative (i.e., x ≥ 0) since the square root of a negative number is undefined in the real number system. Therefore, the domain of y = 2√x is the set of non-negative real numbers or [0, ∞).

For y = √(2x), the expression inside the square root must also be non-negative. Therefore, 2x ≥ 0, which implies that x ≥ 0. Therefore, the domain of y = √(2x) is also [0, ∞).

Since the domains of both functions are the same, we can say that they have the same domain.

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The demand function for good y is y* = -(p2/p1) * (x*/2)

To find the demand for good y, we can set the MRS (marginal rate of substitution) from part 1 equal to the price ratio at an interior optimal bundle.

Let's call the optimal bundle (x*, y*) and the prices of goods x and y as px and py, respectively. Then the condition for optimal consumption is given by:

MRS = -px / py = p1 / p2 (assuming a two-good model)

where p1 and p2 are the prices of goods x and y, respectively, and px/py is the price ratio.

Solving this equation for y, we get:

y* = -(p2/p1) * (x*/2)

This gives us the demand function for good y in terms of its price (p2) and the price of good x (p1) and the optimal quantity of good x (x*).

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Answer:

FALSE: It could just have a saddle point in between the maxima (imagine a mountain with two peaks: it doesn't have a local minimum elevation).

We can find the compositions f o g, g o g, and f o g by plugging the function expressions into each other and simplifying.

f o g(n) = f(g(n)) = f(3n - 1) = 2(3n - 1) + 1 = 6n - 1

g o g(n) = g(g(n)) = g(3n - 1) = 3(3n - 1) - 1 = 8n - 4

f o g(n) = f(g(n)) = f(3n - 1) = 2(3n - 1) + 1 = 6n - 1

g o f(n) = g(f(n)) = g(2n + 1) = 3(2n + 1) - 1 = 6n + 2

Therefore, the compositions are:

f o g(n) = 6n - 1

g o g(n) = 8n - 4

f o g(n) = 6n - 1

g o f(n) = 6n + 2

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Answer:

There are 1320 ways in which 15 runners can be awarded for a 200-meter sprint race three different medals.

Step-by-step explanation: Whenever we are supposed to find the ways in which certain things have to be arranged, we use the concept of Permutation.

The number of possible ways to award gold, silver, and bronze medals for a 200-meter sprint with 15 runners is 2,730.

1. Select the gold medal winner: There are 15 runners, so there are 15 choices for the gold medal.

2. Select the silver medal winner: Since the gold medalist is already chosen, there are 14 remaining runners to choose from for the silver medal.

3. Select the bronze medal winner: With gold and silver medalists chosen, there are now 13 remaining runners to choose from for the bronze medal.

Multiply the number of choices together: 15 (gold) x 14 (silver) x 13 (bronze) = 2,730 possible ways to award the medals.

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The minimum score required to get an A is 83 in a given case.

Using the z-score formula: [tex]z = (x - μ) / σ[/tex], we have:

[tex]z1 = (20 - 22) / 5 = -0.4\\\z2 = (26 - 22) / 5 = 0.8[/tex]

Using a z-table or calculator, the probability of a randomly selected turkey weighing between 20 and 26 pounds is:

[tex]P(-0.4 < z < 0.8) = 0.564[/tex]

b. Using the z-score formula:[tex]z = (x - μ) / σ,[/tex] we have:

[tex]z = (12 - 22) / 5 = -2[/tex]

Using a z-table or calculator, the probability of a randomly selected turkey weighing below 12 pounds is:

[tex]P(z < -2) = 0.023[/tex]

We need to find the z-score that corresponds to the top 15% of the distribution, and then convert it back to the raw score (exam score) using the formula:[tex]z = (x - μ) / σ.[/tex]

Using a z-table or calculator, we find that the z-score corresponding to the top 15% is approximately 1.04.

So, 1.04 = (x - 60) / 20

Solving for x, we get:

[tex]x = 60 + 20(1.04)\\x = 82.8[/tex]

Rounding up to the nearest integer, t

The minimum score required to get an A is 83.

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The given function f'(x) is:
f'(x) = 5 - 20x^3 + 21x^6
Integrating each term, we get:
f(x) = 5x - (20/4)x^4 + (21/7)x^7 + C
f(x) = 5x - 5x^4 + 3x^7 + C
Now, we'll use the initial condition f(1)=0:
0 = 5(1) - 5(1)^4 + 3(1)^7 + C
C = -3
So, the antiderivative f(x) is:
f(x) = 5x - 5x^4 + 3x^7 - 3

To find an antiderivative f(x) with function f′(x) = f(x) = 5 20 x3 21x6 and f(1)=0, we need to integrate f'(x) which will give us f(x).

First, we need to separate the terms in f'(x) since they are not combined. We can write f'(x) as:

f'(x) = 5 + 20x^3 + 21x^6

To integrate this, we need to use the power rule for integration, which states:

∫xn dx = (1/(n+1)) x^(n+1) + C

where C is the constant of integration.

Using this rule, we can integrate each term in f'(x) separately:

∫5 dx = 5x + C1

∫20x^3 dx = (20/4) x^4 + C2 = 5x^4 + C2

∫21x^6 dx = (21/7) x^7 + C3 = 3x^7 + C3

where C1, C2, and C3 are constants of integration.

Now we can combine these integrals to find f(x):

f(x) = 5x + 5x^4 + 3x^7 + C

where C is the constant of integration.

To find the value of C, we use the fact that f(1) = 0.

Substituting x = 1 into the equation for f(x), we get:

f(1) = 5(1) + 5(1)^4 + 3(1)^7 + C = 5 + 5 + 3 + C = 13 + C

Since f(1) = 0, we can solve for C:

13 + C = 0

C = -13

Therefore, the antiderivative f(x) with f′(x)=f(x)=5 20x3 21x6 and f(1)=0 is:

f(x) = 5x + 5x^4 + 3x^7 - 13.

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Answer:

415

Step-by-step explanation:

26. AH = w, BF = x, FC = z, DH = y 27. The contrary sides of the quadrilateral are harmonious, and they're resemblant because the quadrilateral is symmetric with respect to the center of the circle.

excursions to circles are lines that cross the circle at a single point. Point of tangency refers to the position where a digression and a circle meet. The circle's compass, where the digression intersects it, is vertical to the digression. Any twisted form can be considered a digression. Tangent has an equation since it's a line.

26. We know that,

From a point out side of the circle the two excursions to the circle are equal.

therefore, AH = AE = w

BF = BE = x

FC = CG = z

DH = DG = y

27. The contrary sides of the quadrilateral are harmonious, and they're resemblant because the quadrilateral is symmetric with respect to the center of the circle.

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The correct option  is:

She cannot reject the null hypothesis at α = 0.05 because 7.5 is not contained in the 95% confidence interval.

Option B is correct

A confidence interval is described as a range of estimates for an unknown parameter. A confidence interval is also computed at a designated confidence level; the 95% confidence level is most common, but other levels, such as 90% or 99%, are sometimes used.

For the true population mean, a confidence interval provides a range of likely values, and in this instance, the 95% confidence interval is (5.8, 6.4).

Thus, if we were to conduct this study repeatedly, we could anticipate that the genuine population mean would, 95% of the time, fall between 5.8 and 6.4.

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The value of the limit  [tex]\lim _{t \rightarrow \infty} \frac{\sqrt{t} t^2 6-t^2}{t t^2 6-t^2}[/tex] is 0.

We are given the expression:

[tex]\lim _{t \rightarrow \infty} \frac{\sqrt{t} t^2 6 -t^2}{t t^2 6 -t^2}[/tex]

Factor out the highest power of t in the numerator and the denominator:
[tex]\lim _{t \rightarrow \infty} \frac{t^2(\sqrt{t} 6-1)}{t^2(t 6 -1)}[/tex]

Cancel out the t² terms:
[tex]\lim _{t \rightarrow \infty} \frac{(\sqrt{t} 6-1)}{(t 6 -1)}[/tex]

Divide each term by t:
[tex]\lim _{t \rightarrow \infty} \frac{(\sqrt{t} 6/t-1/t)}{(t 6/t -1/t)}[/tex]

As t approaches infinity, the terms with 1/t go to zero:
[tex]\lim _{t \rightarrow \infty} \frac{(\sqrt{t} 6/t-0)}{(t 6/t -0)}\\=lim _{t \rightarrow \infty} \frac{(\sqrt{t} /t)}{(t /t)}[/tex]

Simplify the expression:
[tex]\lim _{t \rightarrow \infty} {(\sqrt{t} /t)}= \lim _{t \rightarrow \infty}(\frac{1}{\sqrt{t} } )[/tex]

Step 6: As t approaches infinity, the expression goes to zero:
[tex]\lim _{t \rightarrow \infty}(\frac{1}{\sqrt{t} } )=0[/tex]

So, the limit of the given expression as t approaches infinity is 0.

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