A car wash firm calculates that its daily profit (in dollars) depends on the number n of workers it employs according to the formulaP = −600n + 25n2 − 0.005n4.Calculate the marginal product at an employment level of 50 workers.

The graph of the function y = x^2 + 2x - 5 is a U-shaped curve opening upwards, with the vertex at (-1, -4).

To graph the function y = x^2 + 2x - 5, we can follow a few steps to plot the points and sketch the curve.

Step 1: Determine the vertex of the parabola.

The vertex of the parabola occurs at the minimum or maximum point. We can find the x-coordinate of the vertex by using the formula x = -b/2a, where a, b, and c are coefficients of the quadratic equation in standard form (ax^2 + bx + c = 0).

For our function, a = 1, b = 2, and c = -5.

Plugging these values into the formula, we get x = -2/2(1) = -1.

To find the y-coordinate of the vertex, we substitute the x-coordinate back into the function: y = (-1)^2 + 2(-1) - 5 = -4.

Therefore, the vertex is at (-1, -4).

Step 2: Find additional points.

To plot more points, we can choose some x-values and calculate the corresponding y-values using the function. Let's select x-values of -3, 0, and 2.

For x = -3, y = (-3)^2 + 2(-3) - 5 = 9 - 6 - 5 = -2.

For x = 0, y = (0)^2 + 2(0) - 5 = 0 + 0 - 5 = -5.

For x = 2, y = (2)^2 + 2(2) - 5 = 4 + 4 - 5 = 3.

Step 3: Plot the points and sketch the curve.

Plot the points (-3, -2), (0, -5), (2, 3), and the vertex (-1, -4) on a coordinate plane.

The vertex (-1, -4) is the lowest point, so the parabola opens upwards.

Connect the points with a smooth curve, following the shape of a parabola.

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Answer: 5/4

Step-by-step explanation:

Ah yes terminal angles. I love these. Here's a formula to solve these.

With point P(x,y) and value r where r = square-root of x square + y square, we have:

Sin = y/r

Cos = x/r

Tan = y/x

Csc = r/y

Sec = r/x

Cot = x/y

so tan = y/x. here y = 5 and x = 4 so the answer is 5/4

Answer:

b. 5/4

Step-by-step explanation:

Without knowing the exact angle C, we cannot determine the value of tan θ.

However, we can use the coordinates of point P to determine the ratio of the opposite side to the adjacent side (which is equal to the value of tan θ).

Recall that in the coordinate plane, the x-coordinate represents the adjacent side and the y-coordinate represents the opposite side.

Therefore, in this case:

adjacent side = 4

opposite side = 5

tan θ = opposite/adjacent = 5/4

So, tan θ = 1.25.

1.25 = 5/4

120

Step-by-step explanation:

turning $54 to cents

$1= 100c

$54=54×100= 5400

calling the number of 20c coins a and 50c coins b

20a + 50b= 5400 ...equ(1)

since there are twice as many 20c coins as 50c coins

a=2b ...equ(2)

substituting a=2b in equ(1)

20(2b) + 50b = 5400

40b + 50b = 5400

90b = 5400

dividing both sides by 90

b= 60

to get the number of 20c coins I'm substituting b=60 in equ(2)

a= 2×60

a=120

therefore the number of 20c coins is 120

The given equation dy/dx - (1/x)y = (xcosx)/x³ is linear with y as the dependent variable.

Given differential equation is (x²y+x⁴ cosx)dx -x³dy = 0

The given equation can indeed be classified as linear with y as the dependent variable.

A linear equation with respect to the dependent variable y is of the form:

dy/dx + P(x)y = Q(x),

where P(x) and Q(x) are functions of x. In this case, we have:

(x²y + x⁴cosx)dx - x³dy = 0.

By rearranging the terms, we can write it as:

x²ydx - x³dy + x⁴cosxdx = 0.

Now, we can rewrite the equation in the form:

dy/dx + (-x²/x³)y = x⁴cosx/x³.

Simplifying further, we get:

dy/dx - (1/x)y = (xcosx)/x³.

As you can see, the equation is in the form of a linear equation with respect to y. The coefficient of y, (-1/x), is a function of x, while the right-hand side (RHS) is also a function of x. Therefore, the given equation is linear with y as the dependent variable.

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The number of free throws expected basketball player to make by using proportion is equal to 180.

Percent of free throw attempted by basketball player = 80%

Number of free throws basketball player attempts in practice  = 225

If the basketball player makes 80% of the free throws she attempts,

we can expect her to make 80 out of every 100 free throws attempted.

To find out how many free throws she would make if she attempted 225, we can set up a proportion,

80/100 = x/225

We can solve for x by cross-multiplying the expression we get,

⇒ 100x = 80 × 225

⇒ 100x = 18000

⇒ x = 18000/100

⇒ x = 180

Therefore, we can expect the basketball player to make 180 free throws out of 225 attempts in practice using proportion.

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Answer:

p(not blue) = 11/13 = 0.8461538462 = approx 0.85 = approx 85%

Step-by-step explanation:

65-10 = 55 are not blue, so 55 out of 65 cars are not blue.

p(not blue) = 55/65 = 11/13 = 0.8461538462 = approx 0.85 = approx 85%

A contest is a type of promotional marketing strategy that requires participants to submit their entries based on specific criteria or requirements. The correct answer to your question is "contest."

A long answer to your question is that when a group of individuals selects a particular consumer-submitted entry, it is called a contest.

The entries are then judged by a panel or group of individuals who select the best or most appropriate entry.

The winner of the contest may receive a prize or premium, such as cash, gift cards, or products.

Unlike a sweepstake, which randomly selects a winner, a contest is based on merit or creativity and involves a selection process.

Therefore, the correct answer to your question is "contest."

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One possible solution could be to allow birth mothers to have as many children as they want, but enforce strict population control measures on the entire community to ensure resources are not depleted.

The practice of releasing one twin in The Giver is a harsh and unjust method of population control. In a hypothetical scenario where population growth is a concern, there are more humane and effective ways to address it. For instance, the community could implement measures such as providing incentives for small families, investing in education and healthcare to reduce infant mortality rates, and promoting family planning. Additionally, the community could implement policies to encourage sustainable resource use and reduce waste, such as recycling and renewable energy initiatives. By taking a comprehensive and sustainable approach to population control, the community can ensure a better future for all its members, without resorting to cruel and arbitrary methods like releasing babies.

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Answer:

G(9, -5)

Step-by-step explanation:

Point G is not shown in the figure, but I assume point G is on the bottom side of the square, directly below point F.

We are told each side of the square is 8 units long.

Starting at point E, to get to point G, go 8 units right and 8 units down.

Start at E(1, 3).

Then go 8 units right to point F(9, 3).

Now go 8 units down to point G(9, -5).

The expression in the scientific notation will be 7.79 × [tex]10^{7}[/tex] .

Simplifying 8.2 × [tex]10^{7}[/tex] - 4.1 × [tex]10^{6}[/tex]

To simplify the equation power should be same

To convert to decrease power the decimal will move to the right

It can be written as

8.2 × [tex]10^{7}[/tex] = 82.0 × [tex]10^{6}[/tex]

Now solving the equation

82.0 × [tex]10^{6}[/tex] - 4.1 × [tex]10^{6}[/tex]

= 77.9 × [tex]10^{6}[/tex]

To convert the equation into scientific notation

The decimal should be after one significant figure

To convert to increase power the decimal will move to the left

It can be written as

7.79  × [tex]10^{7}[/tex]

Simplifying the equation will give 7.79 × [tex]10^{7}[/tex] .

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The mean number of individuals with 20/20 vision is 23.

To find the mean number of individuals with 20/20 vision, we can use the formula for the expected value of a binomial distribution. In this case, the probability of an individual having 20/20 vision is p = 0.33, and the number of trials (i.e. individuals selected) is n = 70.
The formula for the expected value of a binomial distribution is:
E(X) = np
Substituting in our values, we get:
E(X) = 70 x 0.33
E(X) = 23.1
So, the mean number of individuals with 20/20 vision out of 70 selected at random from the population is approximately 23.1. However, since we can't have a fraction of a person, we should round our answer to the nearest whole number.
Therefore, the mean number of individuals with 20/20 vision is 23.

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Answer:

step 1

Step-by-step explanation:

Step 1: they accidentally changed the 20 in the original prob to 2.

Answer:

step 1

Step-by-step explanation:

they wrote 2x+6=3x+2 but they wrote 2 instead of 20

it should have been 2x+6=3x+20

Answer:

15

Step-by-step explanation:

Since the angles are complementary, that means that when the angles are added together, they are equal to 90*. With this information, we can make the equation: x + (3x + 30) = 90

From this we can add like terms and get 4x + 30 = 90

After this, subtract 30 from both sides: 4x = 60

Divide both sides by 4

x = 15

(Ignore the degree sign in my picture, sorry!)

The cosine of angle C is 7/25 and using tangent, the value of C is 73.7°

Trigonometric Ratios are defined as the values of all the trigonometric functions based on the value of the ratio of sides in a right-angled triangle.

sinθ = opp/ hyp

cos θ = adj/ hyp

tan θ = opp/adj

The value of cosine of C is

cos C = adj/hyp

= 7/25

therefore the value of cos C is 7/25

to find the value of C

Tan C = 24/7

TanC = 3.43

C = tan^-1 ( 3.43)

C = 73.7°

Therefore the value of C is 73.7°

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So the triangle has three equal angles of 60 degrees.

To find the measures of the angles of the triangle with vertices at (-2,0), (2,1), and (1,-2), we can use trigonometry.

Let's use the following notation:

a = (-2,0)

b = (2,1)

c = (1,-2)

First, we need to find the coordinates of the midpoint of line segment AB, which is the length of the hypotenuse of the triangle.

Using the Pythagorean theorem, we have:

[tex]c^2 = a^2 + b^2\\1^2 + (-2)^2 = 2^2 + 1^2[/tex]

25 = 4 + 1

23 = 3

So the length of the hypotenuse is 3 units.

Next, we need to find the coordinates of the midpoint of line segment BC, which is the length of one of the legs of the triangle.

Again, using the Pythagorean theorem, we have:

[tex]b^2 = a^2 + c^2\\1^2 + (-2)^2 = 2^2 + 1^2[/tex]

25 = 4 + 1

23 = 3

So the length of the leg of the triangle is 3 units.

Now, we can use the law of cosines to find the measures of the angles of the triangle.

Let's denote the angle between lines AB and BC as alpha, the angle between lines AB and AC as beta, and the angle between lines BC and AC as gamma.

Using the law of cosines, we have:

[tex]cos(alpha) = (b^2 + c^2 - a^2) / (2bc)\\cos(beta) = (a^2 + c^2 - b^2) / (2ac)\\cos(gamma) = (a^2 + b^2 - c^2) / (2ab)[/tex]

We know that:

a = (-2,0)

b = (2,1)

c = (1,-2)

So we can substitute these values into the above equations:

[tex]cos(alpha) = (2^2 + (-2)^2 - (-2)^2) / (2(-2)1) = (2 + (-2) + 2) / (2(-2)1) = 4 / 3\\cos(beta) = ((-2)^2 + 2^2 - (-2)^2) / (2(-2)2) = (-2 + 4 + 2) / (2(-2)2) = -1\\cos(gamma) = (2^2 + 1^2 - 1^2) / (2(1)(-2)) = 2 + (-1) + (-1) / (2(1)(-2)) = 1[/tex]

Now we can substitute these values into the Pythagorean theorem to find the length of the legs of the triangle:

sin(alpha) = length of leg 1 / (2bc)

sin(beta) = length of leg 2 / (2ac)

sin(gamma) = length of leg 2 / (2ab)

We know that:

a = (-2,0)

b = (2,1)

c = (1,-2)

So we can substitute these values into the above equations:

sin(alpha) = √(8) / (2(-2)1)

= √(8) / √(3)

= √(2)

sin(beta) = √(5) / (2(1)2)

= √(5) / √(3)

= √(2)

sin(gamma) = √(5) / (2(1)(-2))

= √(5) / 1

= √(5)

Therefore, the measures of the angles of the triangle are:

alpha = 60 degrees

beta = 60 degrees

gamma = 60 degrees

So the triangle has three equal angles of 60 degrees.  

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the given differential equation is y = csec(x-59°).

To solve the given differential equation, we can start by separating the variables and integrating both sides. So, we have:

dy/dx = ysin(x)cos(59x)

dy/y = sin(x)cos(59x) dx

Integrating both sides, we get:

ln|y| = -cos(x)sin(59x) + C

where C is the arbitrary constant of integration.

Taking exponential of both sides, we get:

|y| = e^(-cos(x)sin(59x)+C)

Now, we can simplify this expression by considering the absolute value of y. Since we know that y cannot be negative in the given domain, we can drop the absolute value signs. Also, using the trigonometric identity sec(x) = 1/cos(x), we get:

y = e^(-cos(x)sin(59x)+C)  or y = e^(sin(59x)cos(x)-C)

But, we can write this solution in a more elegant form by using the trigonometric identity sec(x-a) = 1/cos(x-a), where a = 59°. This gives us:

y = e^(-cos(x-59°)sin(59°)+C) or y = e^(sin(59°)cos(x-59°)-C)

Simplifying further, we get:

y = csec(x-59°) or y = csc(31°)sec(x-59°)

where c = e^(-sin(59°)cos(59°)+C) or c = e^(sin(59°)cos(31°)-C)

Therefore, the general solution to the given differential equation is y = csec(x-59°), where c is the arbitrary constant of integration.

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The solution is:

the probability that the two butterflies she spots will both be type A butterflies is 0.0044

The computation of the  probability that the two butterflies she spots will both be type A butterflies is shown below:

= 1 ÷ Type A × 1 ÷ Type A

= 1 ÷ 15 × 1 ÷ 15

= 1 ÷ 225

= 0.0044

Hence, the  probability that the two butterflies she spots will both be type A butterflies is 0.0044

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complete questio:

Susie is looking for butterflies in an area where: 15% of the butterflies are of type A 30% of the butterflies are of type B 55% of the butterflies are of type C Suppose that she spots two butterflies on her walk, and that the types of the two butterflies are independent from each other (perhaps because they are spotted in different locations). (a) What is the probability that the two butterflies she spots will both be type A butterflies

The point of intersection is (17/9, -4/9, 29/9). The process of finding the point of intersection between a line and a plane involves substituting the parametric equations of the line into the equation of the plane.

To find the point of intersection between the line and plane, we need to substitute the parametric equations of the line into the equation of the plane. This gives us:
2 + 2(4t) - (2 - 3t) + 1 = 0
Simplifying, we get:
9t + 1 = 0
Therefore, t = -1/9. We can substitute this value of t into the parametric equations of the line to find the coordinates of the point of intersection:
x = 1 + 2(-1/9) = 17/9
y = 4(-1/9) = -4/9
z = 2 - 3(-1/9) = 29/9
So the point of intersection is (17/9, -4/9, 29/9). The process of finding the point of intersection between a line and a plane involves substituting the parametric equations of the line into the equation of the plane. This allows us to solve for the value of the parameter that corresponds to the point of intersection. Once we have this value, we can substitute it back into the parametric equations of the line to find the coordinates of the point. It is important to note that not all lines intersect with all planes, and some may intersect at multiple points or not intersect at all. Therefore, it is important to carefully analyze the equations and properties of both the line and plane before attempting to find their point of intersection.

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The limit of (7x-sin(7x))/(7x-tan(7x)) as x approaches 0 using L'Hospital's rule is 0.

To evaluate the limit using L'Hospital's rule, we can take the derivative of the numerator and the denominator separately and then take the limit again.

lim x→0 (7x - sin(7x)) / (7x - tan(7x))

Taking the derivative of the numerator:

d/dx [7x - sin(7x)] = 7 - 7cos(7x)

Taking the derivative of the denominator:

d/dx [7x - tan(7x)] = 7 - 7sec²(7x)

Substituting back into the original limit expression and simplifying:

lim x→0 (7 - 7cos(7x)) / (7 - 7sec²(7x))

As x approaches 0, both cos(7x) and sec²(7x) approach 1. Therefore, we can substitute 1 for both expressions:

lim x→0 (7 - 7cos(7x)) / (7 - 7sec²(7x)) = lim x→0 (7 - 7) / (7 - 7) = 0/0

The limit is of indeterminate form 0/0, so we can apply L'Hospital's rule again. Taking the derivatives of the numerator and denominator:

Substituting back into the limit and simplifying:

lim x→0 (49sin(7x)) / (98sec(7x)tan(7x)) = lim x→0 (7sin(7x)) / (14sec(7x)tan(7x))

As x approaches 0, sec(7x) and tan(7x) approach 1. Therefore, we can substitute 1 for both expressions:

lim x→0 (7sin(7x)) / (14sec(7x)tan(7x)) = lim x→0 (7sin(7x)) / (14)

As x approaches 0, sin(7x) approaches 0. Therefore, the limit simplifies to:

lim x→0 (7sin(7x)) / (14) = 0

Therefore, the limit is 0.

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a) Null Hypothesis is μ ≤ 21.6 and Alternative Hypothesis is μ > 21.6.

b) Point estimate of the difference between mean annual consumption in Webster City and the national mean is 2.5.

c) We can conclude that there is sufficient evidence to suggest that the mean annual consumption of milk in Webster City is higher than the national mean.

A. The hypothesis test to determine whether the mean annual consumption in Webster City is higher than the national mean can be set up as follows:

Null Hypothesis: μ ≤ 21.6

Alternative Hypothesis: μ > 21.6

where μ is the true population mean annual consumption of milk in Webster City.

B. The point estimate of the difference between mean annual consumption in Webster City and the national mean is simply the difference between the sample mean and the national mean:

Point Estimate = x' - μ = 24.1 - 21.6 = 2.5

C. To test for a significant difference at α = 0.05, we need to calculate the test statistic and compare it to the critical value.

t = (x' - μ) / (s / √n) = (24.1 - 21.6) / (4.8 / √16) = 2.083

Using a t-table with 15 degrees of freedom and a significance level of 0.05, we find the critical value to be 1.753. Since our test statistic (2.083) is greater than the critical value (1.753), we reject the null hypothesis.

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Harold had originally 100 pencils.

Let X be the number of pencils Harold had originally. After giving 17 to his friend Carl, he had X-17 left. Then, after giving 35 to his teacher, he had X-17-35 left, which is equal to 48.

Therefore, we can set up an equation: X-17-35=48. Solving for X, we get X=100 pencils.

Thus, Harold originally had 100 pencils. To check, we can verify that after giving 17 to Carl and 35 to his teacher, he is left with 48, which matches the information given in the problem.

This problem can also be solved by using algebraic equations, but since there are only two steps involved, we can use simple arithmetic to arrive at the answer.

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The probability of event a, P(a), is 4/7 for the given sample set.

To find P(a), we need to use the given information about the probabilities and the fact that the total probability of all outcomes in a sample space S is equal to 1. We have:
S = {a, b, c}
P(a) = 2P(b) = 4P(c)

First, we can express P(b) and P(c) in terms of P(a):
P(b) = P(a) / 2
P(c) = P(a) / 4

Now we use the fact that the sum of probabilities of all outcomes in S equals 1:
P(a) + P(b) + P(c) = 1

Substitute P(b) and P(c) with their expressions in terms of P(a):
P(a) + (P(a) / 2) + (P(a) / 4) = 1

To solve for P(a), combine the terms:
P(a) * (1 + 1/2 + 1/4) = 1
P(a) * (4/4 + 2/4 + 1/4) = 1
P(a) * (7/4) = 1

Now, divide both sides by (7/4) to isolate P(a):
P(a) = 1 / (7/4)
P(a) = 4/7

So, the probability of event a, P(a), is 4/7.

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The joint probability of x and y, given y=y and x follows an exponential distribution with parameter y, is:

P(x=x, y=y) = e^(-x-y) / y

To find the joint probability of x and y, we can use the conditional probability formula:
P(x=x, y=y) = P(x=x | y=y) * P(y=y)

Since we know that y follows an exponential distribution with parameter 1, we can write:
P(y=y) = f(y) = e^(-y)

Now, to find the conditional probability of x given y, we can use the probability density function of the exponential distribution:
f(x | y=y) = λ * e^(-λ*x)

where λ = 1/y, since y is the parameter of the exponential distribution.

Therefore,
P(x=x | y=y) = (1/y) * e^(-x/y)

Combining these equations, we get:
P(x=x, y=y) = (1/y) * e^(-x/y) * e^(-y)

Simplifying this expression, we get:
P(x=x, y=y) = e^(-x-y) / y

So the joint probability of x and y, given y=y and x follows an exponential distribution with parameter y, is:

P(x=x, y=y) = e^(-x-y) / y

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Answer:

See below

Step-by-step explanation:

Slope-intercept form of an equation of line:

[tex]y = mx + c[/tex]  —— eq(i)

Where:

c = y-intercept

  = y-value for which the corresponding x-value is 0

  = [tex]-1[/tex] (From the provided table)

m = slope
   = [tex]\frac{rise}{run}[/tex]

   = [tex]\frac{y_{2} - y_{1}}{x_{2} - x_{1}}[/tex] —- eq(ii)

Choose any two sets of coordinates and then substitute in eq(ii). I chose:

[tex](3, 8)[/tex] as [tex](x_{1}, y_{1})[/tex]

[tex](4, 11)[/tex] as [tex](x_{2}, y_{2})[/tex]

   = [tex]\frac{11 - 8}{4 - 3}[/tex]

   = [tex]\frac{3}{1}[/tex]

∴ m = [tex]3[/tex]

Substituting the values of c and m in eq(i):

[tex]y = (3)x + (-1)[/tex]

∴ Equation for the function

[tex]y = 3x - 1[/tex]

The matrix P that orthogonally diagonalizes A is obtained by finding the eigenvalues and eigenvectors of A, normalizing the eigenvectors, and using them as columns of P.

First, we find the eigenvalues and eigenvectors of A:

|4-λ 1| (4-λ)(λ-1) - 1 = 0 → λ1 = 5, λ2 = 3

|1 4-λ|

For λ1 = 5, we get the eigenvector (1,1)/√2, and for λ2 = 3, we get the eigenvector (1,-1)/√2.

Thus, P = [ (1/√2) (1/√2); (1/√2) (-1/√2) ].

Then, P^-1AP = D, where D is the diagonal matrix of the eigenvalues of A.

P^-1 = P^T (since P is orthogonal), so we have:

P^-1AP = P^TAP = [ (1/√2) (1/√2); (1/√2) (-1/√2) ] [ 4 1; 1 4 ] [ (1/√2) (1/√2); (1/√2) (-1/√2) ] = [ 5 0; 0 3 ]

Therefore, the matrix P that orthogonally diagonalizes A is [ (1/√2) (1/√2); (1/√2) (-1/√2) ], and P^-1AP = [ 5 0; 0 3 ].

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The slope of the line tangent to the polar curve r = 2θ² when θ = π is 4π.

To find the slope of the line tangent to the polar curve r = 2θ² at θ = π, we need to find the derivative of r with respect to θ, and then evaluate it at θ = π.

Differentiating r = 2θ² with respect to θ, we get:

dr/dθ = 4θ

Evaluating this expression at θ = π, we get:

dr/dθ = 4π

This is the slope of the tangent line to the polar curve r = 2θ² at the point where θ = π.

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