f(x) = x², for -L ≤ X ≤ L with f (x + 2 L) = f(x) find fourier series .

Answer:

The correct option is (D) \(y=\frac{4}{7}\ x+\frac{1}{7}\).

Step-by-step explanation:

The two-point form for the equation of straight line is:

\((y-y_{1})=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\ (x-x_{1})\)

The two points provided are:

A = (5, 3)

B = (12, 7)

Compute the equation of the trend line as follows:

\((y-y_{1})=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\ (x-x_{1})\\\\(y-3)=\frac{7-3}{12-5}\ (x-5)\\\\(y-3)=\frac{4}{7}\ (x-5)\\\\y-3=\frac{4}{7}\ x-\frac{20}{7} \\\\y=\frac{4}{7}\ x-\frac{20}{7}+3\\\\y=\frac{4}{7}\ x+\frac{-20+21}{7}\\\\y=\frac{4}{7}\ x+\frac{1}{7}\)

Thus, the equation of the trend line is \(y=\frac{4}{7}\ x+\frac{1}{7}\).

The correct option is (D).

Answer:

D

Step-by-step explanation:

Answer:

the formula is 1/2b×h

now given that the h=4m, and the base is 5

we can use the formula to solve this

I hope this helps and makes sense

Answer:

Step-by-step explanation:

To predict the value of the car at the end of three additional years, we can use exponential decay formula.The formula to calculate exponential decay is given by:A = P (1 - r)^tWhere, A = Final amountP = Initial amountr = Rate of decayt = Time elapsedTherefore, using the formula, we can calculate the value of the car after three years.A = P (1 - r)^tFinal amount, A = $17,480Initial amount, P = $33,940Time elapsed, t = 5 yearsRate of decay, r = (A/P)^(1/t) - 1r = ($17,480/$33,940)^(1/5) - 1r = 0.107 or 10.7%Substituting the values in the formula, we getA = $33,940 (1 - 0.107)^8A = $33,940 (0.893)^8A = $14,836.94Therefore, the predicted value of the car at the end of three additional years is $14,836.94.

Answer:

(x,y)=(-5,5)

Step-by-step explanation:

Double integration in polar coordinates involves integrating a function over a two-dimensional region in the polar coordinate system. This is done by setting up a double integral in terms of r and θ, and integrating first with respect to r and then with respect to θ.

Double integration in polar coordinates involves integrating a function over a two-dimensional region in the polar coordinate system. The steps to double integrate a function in polar coordinates are as follows:

1. Determine the limits of integration for r and θ. These limits define the region over which the function will be integrated. Typically, the limits are determined by the boundaries of the region in the xy plane.

2. Write the function to be integrated in terms of r and θ. The function must be expressed in polar coordinates for integration in polar coordinates.

3. Set up the double integral by writing the function in polar coordinates, multiplying by the appropriate factors of r and integrating with respect to r first and then θ.

4. Integrate the function with respect to r, using the limits of integration for r determined in step 1.

5. Integrate the result from step 4 with respect to θ, using the limits of integration for θ determined in step 1.

The general form of a double integral in polar coordinates is:

∫∫f(r, θ)r dr dθ

where f(r, θ) is the function to be integrated, and r and θ are the polar coordinates.

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y= a(x - 5)^2 - 2.This equation represents a parabola with its vertex at the point (5, -2) and opens upward if a > 0 or downward if a < 0.

To translate the vertex of a quadratic function right 5 units and down 2 units, we can use the vertex form of the quadratic equation:

y = a(x - h)^2 + k

where (h, k) is the vertex of the parabola and a is the coefficient of the quadratic term.

To translate the vertex right 5 units, we need to replace x with (x - 5). This will shift the entire parabola 5 units to the right.

To translate the vertex down 2 units, we need to subtract 2 from the value of k. This will shift the entire parabola 2 units downward.

Therefore, the equation that translates the vertex right 5 units and down 2 units is:

y= a(x - 5)^2 - 2

where a is the coefficient of the quadratic term. This equation represents a parabola with its vertex at the point (5, -2) and opens upward if a > 0 or downward if a < 0.

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Answer:

Step-by-step explanation:

A binomial is a factor of a polynomial has been determined by using the

polynomial division method.

To determine if a binomial is a factor of a polynomial, you can use the polynomial division method. The basic idea is to divide the polynomial by the binomial and check if the remainder is zero. If the remainder is zero, then the binomial is a factor of the polynomial. Here's the step-by-step process:

Write the polynomial and the binomial in standard form, with the terms arranged in descending order of their exponents.

Perform the long division of the polynomial by the binomial, similar to how you would divide numbers. Start by dividing the highest degree term of the polynomial by the highest degree term of the binomial.

Multiply the binomial by the quotient obtained from the division and subtract the result from the polynomial.

Repeat the division process with the new polynomial obtained from the subtraction step.

Continue dividing until you reach a point where the degree of the polynomial is lower than the degree of the binomial.

If the remainder is zero, then the binomial is a factor of the polynomial. If the remainder is non-zero, then the binomial is not a factor.

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The function f(x) = x^3 - 6x^2 + 12x is not an inverse function.

An inverse function is a function that, when applied to the output of another function, gives the original input. In order for a function to be an inverse, it must pass two tests: (1) the function must be one-to-one, meaning that each input has a unique output, and (2) the composition of the function and its inverse must give the original input.

To check if f(x) is one-to-one, we need to see if different inputs can produce the same output. Differentiating f(x) gives f'(x) = 3x^2 - 12x + 12. Since the discriminant of this quadratic is negative (-108 < 0), it implies that f(x) does not have distinct critical points. Therefore, f(x) is not one-to-one, and it fails the first test to be an inverse function.

Since f(x) fails the first test, there is no need to proceed to the second test. Thus, we can conclude that the function f(x) = x^3 - 6x^2 + 12x is not an inverse function.

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It it will take Charlie a total of 7 months to repay Jayden in full  which could be known through the analytical model calculation .

An analytical model is a mathematical representation of a real-world system or process. It is a simplified, yet rigorous and systematic, way of analysing and understanding complex phenomena. The purpose of an analytical model is to provide insights and predictions about the behavior of a system based on a set of assumptions and inputs.

In this case, the analytical model is a mathematical formula that calculates how long it will take Charlie to repay Jayden based on the payment arrangements they have agreed upon. The model uses the logarithmic function to determine how many times the minimum payment must be made in order to repay the balance in full, and the ceiling function (ceil) to round up to the nearest whole number of months.

Create an analytical model to calculate the number of months it will take Charlie to repay Jayden by using the following formula

number of months = ceil(log2(100 / minimum payment))

Here, minimum payment is half of the remaining balance each month, so it starts at $50 in the first month and is reduced by half each subsequent month.

Using this formula, we can calculate the number of months as follows:

number of months = ceil(log2(100 / 50)) = ceil(log2(2)) = ceil(1) = 1

balance = 100 - 50 = 50

number of months = ceil(log2(50 / 25)) = ceil(log2(2)) = ceil(1) = 1

balance = 50 - 25 = 25

number of months = ceil(log2(25 / 12.5)) = ceil(log2(2)) = ceil(1) = 1

balance = 25 - 12.5 = 12.5...

Continuing this process, we can see that it will take Charlie a total of 7 months to repay Jayden in full.

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Answer:

Hi I love you and I miss you

I don't know the answear sorry

So

EG=y+5=2.5+5=7.5

After 4 hours of discharging, the volume left in the tank would be 500 liters. The correct option is C.

Initially, the tank has a water capacity of 1500 liters.

After three hours of discharging, half of the water is discharged, which means 1500/2 = 750 liters have been removed from the tank.

To find the volume left after 4 hours of discharging, we need to subtract the additional amount discharged in the fourth hour.

Since the discharge rate remains the same, we can calculate the amount discharged in one hour as 750/3 = 250 liters.

Therefore, in the fourth hour, 250 liters would be discharged. Subtracting this from the remaining water after three hours (750 liters), we get 750 - 250 = 500 liters.

Therefore, the correct answer is c. 500 liters.

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Answer:

= 1619.15

Step-by-step explanation:

Hope this is correct :)

Answer:

and independent variable It is a variable that stands alone and isn't changed by the other variables you are trying to measure. For example, someone's age might be an independent variable. and a Just like an independent variable, a dependent variable is exactly what it sounds like. It is something that depends on other factors. For example, a test score could be a dependent variable because it could change depending on several factors such as how much you studied, how much sleep you got the night before you took the test, or even how hungry you were when you took it.

To find the volume of the right prism, we used the Pythagorean theorem to determine the height of the triangular base is 1.197 inches. We then used the formula V = Bh to calculate the volume, which was approximately 2.70 cubic inches.

To find the height of the prism, we need to use the information provided about the triangular base. Since the triangular base is equilateral with a dimension of 1.74 inches, the height of the triangle (and therefore, the height of the prism) can be found by using the Pythagorean theorem.

If we draw a line from the center of the base to the midpoint of one of the sides, we create a right triangle with hypotenuse 1.74 in (which is also the height of the triangle) and one leg equal to half the length of one of the sides of the triangle (since the base of the prism is a square with dimension 1.5 in).

Using the Pythagorean theorem, we can solve for the height of the triangle (and prism)

(1.74/2)² + (1.5/2)² = h²

0.8725 + 0.5625 = h²

h² = 1.435

h ≈ 1.197 inches

Now, we can use the formula V = Bh to find the volume of the prism

V = (1.5 x 1.5) x 1.197 ≈ 2.70 cubic inches

Therefore, the volume of the right prism is approximately 2.70 cubic inches.

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To show that `(n + 3)7 ∈ Θ(n7)`, we need to prove that `(n + 3)7 = Θ(n7)`.This can be done by showing that `(n + 3)7 = O(n7)` and `(n + 3)7 = Ω(n7)` .Now, let's prove the two parts separately:

Proof for `(n + 3)7 = O(n7)`.

We want to prove that there exists a positive constant c and a non-negative constant k such that `(n + 3)7 ≤ cn7` for all `n ≥ k`.Using the Binomial theorem, we can expand `(n + 3)7` as:```
(n + 3)7
= n7 + 7n6(3) + 21n5(3)2 + 35n4(3)3 + 35n3(3)4 + 21n2(3)5 + 7n(3)6 + 37
≤ n7 + 21n6(3) + 21n5(3)2 + 35n4(3)3 + 35n3(3)4 + 21n2(3)5 + 7n(3)6 + n7
≤ 2n7 + 21n6(3) + 21n5(3)2 + 35n4(3)3 + 35n3(3)4 + 21n2(3)5 + 7n(3)6
≤ 2n7 + 84n6 + 441n5 + 2205n4 + 10395n3 + 45045n2 + 153609n + 729
```Thus, we can take `c = 153610` and `k = 1` to satisfy the definition of big-Oh notation. Hence, `(n + 3)7 = O(n7)`.Proof for `(n + 3)7 = Ω(n7)`We want to prove that there exists a positive constant c and a non-negative constant k such that `(n + 3)7 ≥ cn7` for all `n ≥ k`.Using the Binomial theorem, we can expand `(n + 3)7` as:```
(n + 3)7
= n7 + 7n6(3) + 21n5(3)2 + 35n4(3)3 + 35n3(3)4 + 21n2(3)5 + 7n(3)6 + 37
≥ n7
```Thus, we can take `c = 1` and `k = 1` to satisfy the definition of big-Omega notation. Hence, `(n + 3)7 = Ω(n7)`.

As we have proved that `(n + 3)7 = O(n7)` and `(n + 3)7 = Ω(n7)`, therefore `(n + 3)7 = Θ(n7)`.Thus, we have shown that `(n + 3)7 ∈ Θ(n7)`.From the proof, we can see that we used the Binomial theorem to expand `(n + 3)7` and used algebraic manipulation to bound it from above and below with suitable constants. This technique can be used to prove the time complexity of various algorithms, where we have to find the tightest possible upper and lower bounds on the number of operations performed by the algorithm.

Hence, we have shown that `(n + 3)7 ∈ Θ(n7)` for non-negative integer n.

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Step-by-step explanation:

4 points appear-

12 lines appear..?

1 plane appears

4 collinear points.

There's 2 non-coplanar points??

4 intersections??

I hope this helped. I could be wrong, but that's how I solved it mentally.

Answer:

343

Step-by-step explanation:

Not sure how to explain it

The standard parameterization for the tangent line  is x = 12t + 18 , y = 3t + 13 , z = 54t + 54

Equation of this type is known as a parametric equation; it uses an independent variable known as a parameter (commonly represented by t) and dependent variables that are defined as continuous functions of the parameter and independent of other variables.

Given that,

Given that, t = t0 = 3, I + (3t + 4) j + (2t 3) k

In order to get the tangent line to the curve at t=3, (x, y, z) = 18, 13, and 54,

r'(t) = 4t, 3, > and r'(t=3) = 12, 3, 54>. (18, 13, 54)

finally,

x = 12t + 18

y = 3t + 13

z = 54t + 54

The tangent line's usual parameterization is as follows:

x=12t+18, y=3t+13, and z=54t+54

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Answer:

Step-by-step explanation:

C

the equation would be y=2x+4

The solution is Option C.

The linear equation is given as y = 2x + 4

The equation of a line is expressed as y = mx + b where m is the slope and b is the y-intercept

And y - y₁ = m ( x - x₁ )

y = y-coordinate of second point

y₁ = y-coordinate of point one

m = slope

x = x-coordinate of second point

x₁ = x-coordinate of point one

The slope m = ( y₂ - y₁ ) / ( x₂ - x₁ )

Given data ,

Let the equation be represented as A

Now , the value of A is

Let the x values be = { -4 , -2 , 0 , 2 }

Let the y values be = { -4 , 0 , 4 , 8 }

The slope of the line is m = ( y₂ - y₁ ) / ( x₂ - x₁ )

Substituting the values in the equation , we get

Slope m = ( 0 - ( -4 ) / ( -2 - ( - 4 ) )

On simplifying the equation , we get

Slope m = 4/2 = 2

Now , the equation of line is

y - y₁ = m ( x - x₁ )

Substituting the values in the equation , we get

y - 0 = 2 ( x - ( -2 ) )

On simplifying the equation , we get

y = 2x + 4

Hence , the linear equation is y = 2x + 4

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After solving the given inequality, the obtained value will be x ≤ -45.

Inequalities are mathematical expressions where neither side is equal. In inequality, as opposed to equations, we compare the two values. Less than (or less than and equal to), larger than (or greater or equal to), or not similar to signs are used in place of the equal sign in between.

As per the given inequality in the question,

-x/3 ≥ 15

-x ≥ 45

Multiply the equation with a minus sign, due to which the sign of inequality will change.

So, the inequality will be,

x ≤ -45

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Step-by-step explanation:

the correct answer is 1/7

Answer:

Step-by-step explanation:

?

In standard nomenclature, a normally distributed dataset is represented as \(N(µ, σ^2)\), where µ is the mean and \(σ^2\)is the variance (square of the standard deviation).

A normally distributed phenomenon using standard nomenclature can be described as follows:

A dataset is said to be normally distributed if it follows a bell-shaped curve, which is symmetrical around the mean (µ) and characterized by its standard deviation (σ). In standard nomenclature, a normally distributed dataset is represented as \(N(µ, σ^2)\), where µ is the mean and \(σ^2\)is the variance (square of the standard deviation).

For example, if we consider the heights of adult males in a large population, we may observe that the distribution is normally distributed with a mean height (µ) of 175 cm and a standard deviation (σ) of 10 cm. In this case, the nomenclature for this normally distributed phenomenon would be N(175, 100), as the variance is \(10^2 = 100\).

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The correct option for the matrix will be:

a) If an n x n matrix A has fewer than n distinct eigenvalues, then A is not diagonalizable.

It could have repeated eigenvalues as long as the basis of each eigenspace is equal to the multiplicity of that eigenvalue.

b) If A is diagonalizable the A2 is diagonalizable

If A is diagonalizable then there exists an invertible matrix

c) If Rn has a basis of eigenvectors of A, then A is diagonalizable.

d) A is diagonalizable if and only if A has n eigenvalues, counting multiplicity.

e) If A is diagonalizable, then A is invertible.

It’s invertible if it doesn’t have a zero as eigenvalue but this doesn’t affect diagonalizable.

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Answer:

113.1 cm³

Step-by-step explanation:

diameter = 2 X radius

Volume of sphere = (4/3) X π X r ³

= (4/3) π (3)³

= 36π

= 113.1 cm³ to nearest tenth

At the hot wings restaurant, a fraction of 5/9 of the patrons ordered the inferno hot wings, and 1/8 of those patrons passed out from the intensity of the sauce. The fraction of patrons who passed out are 5/72.

Given that 5/9 of the patrons ordered the inferno hot wings, this fraction represents the portion of patrons who were exposed to the intense sauce. Out of this group, 1/8 passed out due to the sauce's intensity.

To find the fraction of patrons who passed out, we multiply the fractions 5/9 and 1/8:

(5/9) * (1/8) = 5/72.

Therefore, the fraction 5/72 represents the proportion of patrons who passed out from the intensity of the inferno hot wing sauce.

This information is important for understanding the effects of the spicy sauce and can be used by the restaurant to gauge the intensity of the dish and potentially make adjustments to cater to different preferences. Additionally, it provides insights into the customer experience and can influence future menu decisions or considerations regarding the heat levels of their offerings.

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Answer:

200 miles

Step-by-step explanation:

1 3/4 can be turned into a decimal as 1.75

Now let’s create fractions to prepare for cross multiplication

1.75/50=7/x

1.75x=7(50)

1.75x=350

/1.75.  /1.75

x=200

200 miles

hopes this helps please mark brainliest

Answer:

The function rule is f(x)=2x+3. It cut the three out.

Step-by-step explanation: cut the three out

The correlation is a significant non-zero value.

The number of samples is n.

n = 25

The correlation of the coefficient is r.

r = -0.40

When correlation is significant,

\(H _{0} : p = 0\)

When correlation is non-zero,

\(H _{ \alpha } : p ≠0\)

The test statistic is,

\(TS = \frac{r \times \sqrt{n - 2} }{ \sqrt{1 - r {}^{2} } } \)

\( = \frac{0.4 \times \sqrt{25 - 2} }{ \sqrt{1 - ( - 0.4) ^{2} } } \)

\( = \frac{ - 1.918 }{ \sqrt{0.84} } \)

= -2.093

The test statistic is -2.093.

The correlation is,

\(H _{ \alpha } : - 2.93 ≠0\)

The correlation is not equal to zero and is significant.

Therefore, the correlation is a significant non-zero value.

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