Find the value of x. Area of rectangle = 61

Equation provided is: A(x) = 2x^2 - 5x

Answer:(-3, -6)

Step-by-step explanation:

Since your not moving in the x direction, only your y is going to change.

How far is that point in the y from the y=-1  

(-3,4)  you have to go -5 to get to -1   but keep going another -5 which will bring you to -6 (-1-5=-6)

so your reflected point A'=(-3, -6)

Answer:

2 3/4

Step-by-step explanation:

The longest is 4 1/2 and the shortest is 1 3/4 so we do 4 1/2 - 1 3/4 and you get 2 3/4.

Therefore, the equation that models the number of total downloads y years after launch is: a. [tex]D = 200(1.16)^y:12[/tex].

An equation is a mathematical statement that shows the equality of two expressions. It usually consists of two sides separated by an equal sign (=). The expressions on both sides of the equal sign can include numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division.

Here,

The initial equation D = 200 (1.16) models the number of total downloads, D, for an app Carrie created m months after its launch. We know that there are 12 months in a year. So, we need to convert y years into months to use the given equation.

y years = 12y months

Substituting this value into the equation, we get:

[tex]D = 200(1.16)^{12y/12}:12[/tex]

[tex]D = 200(1.16)^y[/tex]

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The bug crawls a distance of approximately 1.28 meters during the hour.

To estimate the distance using the left-hand Riemann sum, we first divide the time interval [0,1] into subintervals of width Δt=0.2. Then, we evaluate the velocity function at the left endpoint of each subinterval and multiply it by the width of the subinterval. Adding up these products gives us an estimate of the total distance traveled. Using this method, we get an underestimate of 0.8 meters.

To estimate the distance using the right-hand Riemann sum, we evaluate the velocity function at the right endpoint of each subinterval and multiply it by the width of the subinterval. Adding up these products gives us an estimate of the total distance traveled. Using this method, we get an overestimate of 1.6 meters.

To get a new estimate, we average the left-hand and right-hand Riemann sums. So, the new estimate of the total distance traveled by the bug is (0.8+1.6)/2 = 1.2 meters.

Therefore, the bug crawls a distance of approximately 1.28 meters during the hour, with an underestimate of 0.8 meters and an overestimate of 1.6 meters. By taking the average of the two Riemann sums, we get a more accurate estimate of 1.2 meters.

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the value of k that makes f(x, y) = (1/7)x(3 - 2y) a joint probability density function on the given domain D is k = 1/7.

To find the value of k so that the function f(x, y) = kx(3 - 2y) is a joint probability density function on the domain D = {1 ≤ x ≤ 4; 0 ≤ y ≤ 2}, we need to ensure that the total probability over the domain is equal to 1. We can do this by integrating the function over the given domain and setting the result equal to 1:

1 = ∫∫_D f(x, y) dxdy

First, we will integrate the function with respect to x:

1 = ∫[∫_1^4 kx(3 - 2y) dx] dy

1 = ∫[k(3 - 2y)(x^2/2)|_1^4 dy

1 = ∫[k(3 - 2y)(8 - 1/2)] dy

Now, integrate with respect to y:

1 = k(7/2)∫_0^2 (3 - 2y) dy

1 = k(7/2)[(3y - y^2)|_0^2]

1 = k(7/2)(6 - 4)

1 = 7k

To make the total probability equal to 1, we need to find the value of k:
k = 1/7

So, the value of k that makes f(x, y) = (1/7)x(3 - 2y) a joint probability density function on the given domain D is k = 1/7.

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We want to write this in the form given in the question, we can let c_n = e²/n!: \displaystyle \sum_{n=0}\infty c_n(x-2), where c_n = e²/n!

The Taylor series for f(x) = e{x} about x=2 can be written as:

\displaystyle \sum_{n=0}\infty \frac{f{(n)}(2)}{n!}(x-2)n

Since f(x) = e{x}, we can find the derivatives of f(x) and evaluate them at x=2:

f'(x) = e{x}, f''(x) = e{x}, f'''(x) = e{x}, and so on.

So, we have:

f(2) = e²
f'(2) = e²
f''(2) = e²
f'''(2) = e²
and so on.

Plugging these values into the formula for the Taylor series, we get:

\displaystyle \sum_{n=0}\infty \frac{e²}{n!}(x-2)

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write the taylor series for f(x) = e^{x} about x=2 as \displaystyle \sum_{n=0}^\infty c_n(x-2)^n. Find the first five coefficients.

c0=

c1=

c2=

c3=

c4=

The correct answer is A. The statement is false. R3 is not even a subset of R2. This can be answered by the concept of three-dimensional vector.

The statement is false because R3, which represents a three-dimensional vector space, cannot be a subspace of R2, which represents a two-dimensional vector space. In order for a set to be a subspace, it must satisfy three conditions: (1) it contains the zero vector, (2) it is closed under vector addition, and (3) it is closed under scalar multiplication.

R2 and R3 have different dimensions, and therefore, they do not have the same number of components in their vectors. Consequently, vector addition and scalar multiplication, which are defined component-wise, cannot be applied between vectors from R2 and R3. Therefore, R3 cannot be a subspace of R2.

Therefore, the correct answer is A. The statement is false. R3 is not even a subset of R2

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The directional derivative is (-3/2√10) + (2 ln(2)/√10).

To compute the directional derivative of f(x,y) = y² ln(x) at the point (2,1) in the direction of the vector v = -3i + j, first find the gradient of f and then take the dot product with the unit vector in the direction of v.

The gradient of f(x, y) is given by (∂f/∂x, ∂f/∂y) = (y²/x, 2y ln(x)). At the point (2,1), this becomes (1/2, 2 ln(2)).

Next, find the unit vector of v by dividing v by its magnitude: u = v/||v|| = (-3, 1)/√((-3)² + 1²) = (-3, 1)/√10.

Now, take the dot product of the gradient and the unit vector: ((1/2, 2 ln(2)) · (-3/√10, 1/√10)) = (-3/2√10) + (2 ln(2)/√10).

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Using the depreciation formula we know that the value of the car after 3 years of depreciation will be (C) $18,634.

Depreciation is an annual income tax deduction that enables you to recoup the purchase price or other basis of a specific item over the course of its use.

It is a provision for the property's normal wear and tear, degeneration, or obsolescence.

So, a car's initial cost of acquisition is $14,000. Its worth decreases by 10% per year.

Three years following the time of purchase.

Applying the compound interest formula, we may determine a car's value.

A = P(1 + r)ˣ

Where ˣ be time which is 3 years.

Insert the values in the formula as follows:

A = P(1 + r)ˣ

A = 14000(1+0.1)³

A = 14000(1.1)³

A = 14000 * 1.331

A = $18634

Therefore, using the depreciation formula we know that the value of the car after 3 years of depreciation will be (C) $18,634.

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Answer:

To find out what should be subtracted from -5/4 to get -1, we need to solve the equation if you dont know something in math you can always put it as x first.

-5/4 - x = -1

where x is the number that needs to be subtracted.

To solve for x, we have to simplify the left side of the equation:

-5/4 - x = -1

-5/4 + 4/4 - x = -1  (adding 4/4 to both sides)

-1/4 - x = -1

Now, we can isolate x by adding 1/4 to both sides of the equation:

-1/4 - x = -1

-1/4 + 1/4 - x = -1 + 1/4  (adding 1/4 to both sides)

-x = -3/4

Finally, we can solve for x by multiplying both sides by -1:

-x = -3/4

x = 3/4

Therefore, the number that should be subtracted from -5/4 to get -1 is 3/4.

By strong induction, we have proven that for all n ≥ 30, n cents of postage can be formed using just 4-cent and 11-cent stamps.

To prove that any amount of postage greater than or equal to 30 cents can be formed using just 4-cent and 11-cent stamps, we will use strong induction.

Base Case: For n = 30, we can form 30 cents of postage using three 10-cent stamps.

Inductive Hypothesis: Assume that for all k such that 30 ≤ k ≤ n, we can form k cents of postage using just 4-cent and 11-cent stamps.

Inductive Step: We want to show that we can form (n+1) cents of postage using just 4-cent and 11-cent stamps.

Case 1

We use at least one 11-cent stamp to form (n+1) cents of postage.

If we use one 11-cent stamp, we need to form (n+1-11) cents of postage using just 4-cent and 11-cent stamps. By our inductive hypothesis, we know that we can form (n+1-11) cents of postage using just 4-cent and 11-cent stamps since 30 ≤ (n+1-11) ≤ n. Thus, we can add one 11-cent stamp to the solution for (n+1-11) cents to get a solution for (n+1) cents.

If we use more than one 11-cent stamp, we can use one less 11-cent stamp and add some combination of 4-cent stamps to get a solution for (n+1) cents. By our inductive hypothesis, we know that we can form the remaining amount using just 4-cent and 11-cent stamps.

Case 2

We use only 4-cent stamps to form (n+1) cents of postage. In this case, we need to form (n+1) cents of postage using only 4-cent stamps, which means we need to use (n+1)/4 stamps. If (n+1) is not divisible by 4, then we can use one 11-cent stamp to make up the difference. Otherwise, we can use (n+1)/4 4-cent stamps to form (n+1) cents of postage.

Since we have shown that we can form (n+1) cents of postage using just 4-cent and 11-cent stamps in both cases, our inductive step is complete.

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The given question is incomplete, the complete question is:

Prove the statement that n cents of postage can be formed with just 4-cent and 11-cent stamps using strong induction, where n ≥ 30.

The 95% confidence interval for the population proportion favoring stricter gun control laws in the district is approximately 0.4612 to 0.5188. This means that 95% are confident that the true proportion of voters in the population who favor stricter gun control laws falls within this range.

The 95% confidence interval for the population proportion favoring stricter gun control laws based on a poll of 1,100 voters in which 49% of them favored stricter laws.

To find the confidence interval, follow these steps:

1. Determine the sample proportion (p-hat): p-hat = favorable votes / total votes = 0.49.

2. Determine the sample size (n): n = 1,100 voters.

3. Calculate the standard error (SE): [tex]SE = \sqrt(p-hat \times (1 - p-hat) / n)[/tex]

[tex]= \sqrt(0.49 \times (1 - 0.49) / 1100) \approx 0.0147.[/tex]

4. Find the critical value (z) for a 95% confidence interval: z = 1.96 (from a standard normal distribution table).

5. Calculate the margin of error (ME): [tex]ME = z \times SE = 1.96 \times 0.0147 \approx 0.0288.[/tex]

6. Find the lower and upper limits of the confidence interval:

Lower limit = p-hat - ME = [tex]0.49 - 0.0288 \approx 0.4612;[/tex]

Upper limit = p-hat + ME = [tex]0.49 + 0.0288 \approx 0.5188.[/tex]

In conclusion, the 95% confidence interval for the population proportion favoring stricter gun control laws in the district is approximately 0.4612 to 0.5188. This means that we are 95% confident that the true proportion of voters in the population who favor stricter gun control laws falls within this range.

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We need a sample size of at least 1536 to be 95% confident that our sample mean will be within 0.0005 inches of the true mean.

To determine how large a sample is needed, we can use the formula for the margin of error:
To determine the required sample size for a 95% confidence interval with a specified margin of error, we'll use the following formula:
n = (Z * σ / E)^2
where:
- n is the sample size
- Z is the Z-score for a given confidence level (1.96 for a 95% confidence interval)
- σ is the population standard deviation
- E is the margin of error (0.0005 inches in this case)

The margin of error = Z-score * (standard deviation / square root of sample size)

Since we want to be 95% confident, the Z-score will be 1.96. We are given that we want the sample mean to be within 0.0005 inches of the true mean, so the margin of error will be 0.0005.

Thus, we can rearrange the formula to solve for the sample size:

Sample size = (Z-score)^2 * (standard deviation)^2 / (margin of error)^2

Since we do not know the population standard deviation, we can use the sample standard deviation as an estimate. Let's assume the sample standard deviation is 0.001 inch.

Plugging in the values, we get:

Sample size = (1.96)^2 * (0.001)^2 / (0.0005)^2

Sample size = 1536

Therefore, we need a sample size of at least 1536 to be 95% confident that our sample mean will be within 0.0005 inches of the true mean.

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20971.52 is the differential  d y for x=4 and dx=0.04.

The differential is the mathematical expression that uses a function of derivative and can be used to approximate to specified function of values. The limit of the quotient y/x, where y is [tex]f(x_0 + x) f(x_0)[/tex] is  derivative of the function at the point x=0, denoted by the symbol [tex]f'({x_0})[/tex].

We can do the following:

[tex]y = (x^2 + 4)^4[/tex]

we know that  derivative of y with respect x,

[tex]dy = f'(x)*dx[/tex]

the differential dy for x=4 and dx=0.4:

where f'(x) is the function's derivative with regard to x.

Using y's derivative with respect to x, we can calculate:

[tex]y' = 4(x^2 + 4)^3 * 2xy' = 8x(x^2 + 4)^3[/tex]

When x = 4, we get:

[tex]y' = 8(4)(4^2 + 4)^3 = 524288[/tex]

When we change x = 4 and d x = 0.4 in the differential d y formula, we obtain:

d y = 524288 * 0.4 = 209715.2

therefore, 209715.2 is the differential dy when x=4 and dx=0.4.

We can  again apply the same derivative  formula to calculate the differential d y for x=4 and d x=0.04:

d y = f'(x)*d x

At x = 4, y' = 524288, as previously discovered.

the substitution of d x = 0.04 and x = 4

At x = 4, y' = 524288, substitute in above

When we substitute  x = 4 and d x = 0.04 in the derivative d y formula, than we get,

d y = 524288 * 0.04 = 20971.52

therefore

20971.52 is the difference dy for x=4 and dx=0.04.

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The area under the standard normal curve to the left of z = -1.76 and to the right of z = 0.07 is 0.5113 square units

To find the area under the standard normal curve to the left of z = -1.76, we can use a standard normal distribution table or a calculator with a normal distribution function. The table or calculator will give us the probability that a standard normal random variable is less than or equal to -1.76.

Using a standard normal distribution table, we can find that the area to the left of z = -1.76 is 0.0392 (rounded to four decimal places).

To find the area under the standard normal curve to the right of z = 0.07, we can subtract the area to the left of z = 0.07 from the total area under the curve, which is 1. Using a standard normal distribution table or calculator, we can find that the area to the left of z = 0.07 is 0.5279. Therefore, the area to the right of z = 0.07 is

1 - 0.5279 = 0.4721

Rounding this to four decimal places, we get 0.4721.

Therefore, the area under the standard normal curve to the left of z = -1.76 and to the right of z = 0.07 is

0.0392 + 0.4721 = 0.5113

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This algorithm segment calculates the geometric mean of the elements in the array c. It does this by iterating over all possible pairs of elements in the array, multiplying the numerator and denominator of the geometric mean calculation by each element in turn, and accumulating the results in the variables p and q.

The final result is then calculated by dividing p by the square root of q. This algorithm has a time complexity of O(n^2) because it contains two nested loops that iterate over the array c.
It appears that wehave an algorithm segment and would like an explanation that includes specific terms. The algorithm segment provided can be described as follows:

1. Initialize two variables, 'p' and 'q', both set to 1.
2. Iterate through the range of 1 to (n-1) using the variable 'i'.
3. For each 'i', iterate through the range of (i+1) to 'n' using the variable 'j'.
4. During the inner loop, update 'p' by multiplying it with the value of 'c[j]' (an element of an array 'c') and update 'q' by multiplying it with the square of 'c[j]'.
5. After completing the inner loop, calculate 'r' by dividing 'p' by 'q'.
6. Proceed to the next iteration of the outer loop with the updated value of 'i'.

This algorithm segment essentially computes the value of 'r' for each 'i' in the range of 1 to (n-1), considering the array 'c' and its elements.

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The particular solution satisfying the initial condition y(0) = 2 is y(x) = 2e^(-8x).

To solve the separable differential equation dy/dx = -8y and find the particular solution satisfying the initial condition y(0) = 2, follow these steps:

Step 1: Identify the given equation and initial condition
The given equation is dy/dx = -8y, and the initial condition is y(0) = 2.

Step 2: Separate the variables
To separate the variables, divide both sides by y and multiply by dx:
(dy/y) = -8 dx

Step 3: Integrate both sides
Integrate both sides with respect to their respective variables:
∫(1/y) dy = ∫-8 dx

The result is:
ln|y| = -8x + C₁

Step 4: Solve for y
To solve for y, use the exponential function:
y = e^(-8x + C₁) = e^(-8x)e^(C₁)

Let e^(C₁) = C₂ (since C₁ and C₂ are both constants):
y = C₂e^(-8x)

Step 5: Apply the initial condition
Now, apply the initial condition y(0) = 2:
2 = C₂e^(-8 * 0)
2 = C₂

Step 6: Write the particular solution
Finally, substitute the value of C₂ back into the equation:
y(x) = 2e^(-8x)

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The number line and graph of the inequality −3 ≤ x ≤ 0 represents -3 and 0 both are included points.

The inequality is written as,

−3 ≤ x ≤ 0

Plot the given inequality  -3 ≤ x ≤ 0 on the number line.

On the number line, we can represent this as ,

Value of x is in between -3 and 0.

Number line is attached.

The interval between -3 and 0, including both endpoints, represents the region that satisfies the inequality.

On the coordinate plane, we can represent this inequality on the x-axis as a shaded region between -3 and 0, including both endpoints:

Graph of the inequality is also attached here.  

The shaded region between -3 and 0, including both endpoints, represents the region that satisfies the inequality.

Therefore, the inequality region include both the endpoints -3 and 0 on number line and coordinate plane.

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The given question is incomplete, I answer the question in general according to my knowledge:

Find the region which satisfies the inequality −3≤ x ≤0 on the number line or coordinate plane.

The statements that are true are "f(x) may have a global maximum at more than one x-value." and "f(x) may or may not have global extrema." Therefore, options A. and B. are true.

Consider a continuous function f(x) defined on the interval -∞ to ∞. Let's consider the given statements:

A. f(x) may have a global maximum at more than one x-value:

This statement is true. A function can have multiple x-values where the global maximum occurs.

B. f(x) may or may not have global extrema:

This statement is true. Depending on the function, it may have a global minimum, a global maximum, both, or neither.

C. f(x) may have a global minimum or a global maximum, but cannot have both:

This statement is false. A continuous function defined on an unbounded domain can have both a global minimum and a global maximum, such as a parabolic function.

D. f(x) must have both a global maximum and a global minimum:

This statement is false. There's no guarantee that a continuous function defined on an unbounded domain must have both a global maximum and a global minimum.

E. f(x) cannot have any global extrema:

This statement is false. A continuous function defined on an unbounded domain can have global extrema.

Therefore, options A. and B. are true.

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Suppose that the distribution of body temperature of healthy people is approximately normal with a mean (µ) of 98.6°F and a standard deviation (σ) of 0.5°F. This means that the majority of healthy individuals have body temperatures close to 98.6°F, and the temperatures typically vary within a range of 0.5°F above or below the mean.

Based on the information you provided, we know that the distribution of body temperature of healthy people is approximately normal with a mean of 98.6° and a standard deviation of 0.5°. This means that most healthy people have a body temperature that falls within a range of about 98.1° to 99.1°, since that range is within one standard deviation of the mean. However, there will still be some healthy people who fall outside of that range, since the normal distribution is a continuous distribution and there is always some variability in any population. It's also worth noting that while 98.6° is often cited as the "normal" body temperature, this is actually just an average and many healthy people will have slightly higher or lower body temperatures depending on a variety of factors.

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Every day, the mass of the sample shrinks by a factor of 0.04.

An exponential function has the definition presented as follows:

y = ab^x.

In which the parameters are given as follows:

The growth or decay of an exponential function depends on the parameter b as follows:

The decay factor k of the exponential function, when |b| < 1, is obtained as follows:

b = 1 - k

k = 1 - b.

The parameter b for this problem is given as follows:

b = 0.96.

Hence it represents decay, and the factor is obtained as follows:

k = 1 - 0.96

k = 0.04.

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[tex]\tan(A )=\cfrac{\stackrel{opposite}{12}}{\underset{adjacent}{18}} \implies \tan(A)=\cfrac{2}{3}\implies A =\tan^{-1}\left( \cfrac{2}{3} \right)\implies A \approx 33.7^o[/tex]

Make sure your calculator is in Degree mode.

If f and g are differentiable, then d dx[f(x) g(x)] = f 0 (x) g 0 (x).- TRUE

This statement is true. The product rule of differentiation states that

d/dx[f(x)g(x)] = f'(x)g(x) + f(x)g'(x).

Therefore, if f(x) and g(x) are differentiable, then,

d/dx[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)

= f0(x)g(x) + f(x)g0(x),

which is equivalent to:

d/dx[f(x)g(x)] = f0(x)g(x) + f(x)g0(x).

Therefore, the statement is true.
The statement is TRUE (T). If f and g are differentiable, then the product rule applies when differentiating the product f(x)g(x). The product rule states that the derivative of a product of two functions is:
d/dx [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
This is not the same as f'(x)g'(x), which is stated in the question.

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The solution for the equation log₃(x⁸) * log₃(x) = 2 is [tex]x = 9^{(1/9)}[/tex].

In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number x to the base b is the exponent to which b must be raised, to produce x.

We have to solve the equation log₃(x⁸) * log₃(x) = 2.

Rewrite the given equation using the properties of logarithms.
log₃(x⁸) * log₃(x) = log₃(x⁸) + log₃(x¹)

(using the property of logarithms that [tex]log_a(b) \times log_a(c) = log_a(b) + log_a(c)[/tex])

Simplify the expression.
log₃(x⁸) + log₃(x¹) = log₃(x⁸ × x¹)

(using the property of logarithms that [tex]log_a(b) + log_a(c) = log_a(b c)[/tex])

Rewrite the equation.
log₃(x⁸ * x¹) = 2

Eliminate the logarithm using the property of logarithms that if [tex]log_a(b) = c[/tex], then [tex]a^c = b[/tex].
3² = x⁸ × x¹

Simplify the equation.
9 = x⁹

Solve for x.
[tex]x = 9^{(1/9)}[/tex]
This is the required solution.

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The value of the double integral ∬d(xy - y²) dA over the region D is 1/36.

In mathematics, a double integral is a type of integral that extends the concept of a single integral to two dimensions. It is used to calculate the signed area or volume of a two-dimensional or three-dimensional region, respectively.

To evaluate the double integral ∬d(xy - y²) dA over the region bounded by the x-axis, the lines x = −1, y = 1, and y = x, we need to set up the limits of integration for both x and y.

First, let's consider the boundaries of the region.

The x-axis forms the lower boundary, and the line y = x forms the upper boundary.

The line x = −1 is the left boundary, and the line y = 1 is the right boundary.

To determine the limits of integration, we can express the region D as follows:

D: −1 ≤ x ≤ y, 0 ≤ y ≤ 1.

Now, we can set up the double integral:

∬d(xy - y²) dA = ∫[y=0 to y=1] ∫[x=-1 to x=y] (xy - y²) dx dy.

Let's evaluate this integral step by step.

First, we integrate with respect to x:

∫(xy - y²) dx = (1/2)x²y - y²x.

Next, we integrate the result with respect to y:

∫[(1/2)x²y - y²x] dy = (1/2)x²(1/2)y² - (1/3)y³x.

Now, we can evaluate the double integral:

∬d(xy - y²) dA = ∫[y=0 to y=1] [(1/2)x²(1/2)y² - (1/3)y³x] dy.

Plugging in the limits and evaluating the integral, we get:

∬d(xy - y²) dA = ∫[0 to 1] [(1/2)x²(1/2)y² - (1/3)y³x] dy

= [(1/2)x²(1/2)(1/3)y³ - (1/4)(1/3)y⁴x] evaluated from y = 0 to y = 1

= [(1/2)x²(1/6) - (1/12)x] - [0]

= (1/12)x² - (1/12)x.

Finally, we integrate the remaining expression with respect to x:

∫[(1/12)x² - (1/12)x] dx = (1/36)x³ - (1/24)x².

Therefore, the value of the double integral ∬d(xy - y²) dA over the given region is:

∬d(xy - y²) dA = ∫[x=-1 to x=1] [(1/36)x³ - (1/24)x²] dx

= [(1/36)(1)³ - (1/24)(1)²] - [(1/36)(-1)³ - (1/24)(-1)²]

= (1/36 - 1/24) - (-1/36 - 1/24)

= 1/72 + 1/72

= 1/36.

Therefore, the value of the double integral is 1/36.

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To calculate the iterated integral of the function (6x^2y - 2x) with respect to y from y=0 to y=3 and with respect to x from x=1 to x=2, first integrate the function with respect to y. Then evaluate the integral at the given limits for y. Next, integrate the resulting expression with respect to x and evaluate the integral at the given limits for x. The final result will be the value of the iterated integral.

1. First, integrate the function with respect to y:

∫(6x^2y - 2x) dy = 3x^2y^2 - 2xy + C(y)

2. Now, evaluate the integral at the given limits for y:

[3x^2(3)^2 - 2x(3)] - [3x^2(0)^2 - 2x(0)] = 27x^2 - 6x

3. Next, integrate this result with respect to x:

∫(27x^2 - 6x) dx = 9x^3 - 3x^2 + C(x)

4. Finally, evaluate the integral at the given limits for x:

[9(2)^3 - 3(2)^2] - [9(1)^3 - 3(1)^2] = (72 - 12) - (9 - 3) = 60 - 6 = 54

So, the iterated integral of the given function is 54.

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The product of zeros of cubic polynomial x³ - 3x² - x + 3 is 3

The zeroes of a cubic polynomial are the values of x at which the polynomial equals zero.

Given the cubic polynomial x³ - 3x² - x + 3, we desire to find the product of the zeroes of the polynomial. We proceed as follows.

For a cubic polynomial ax³ + bx² + cx + d with factors (x - l)(x - m)(x - n), and zeroes, l, m and n respectively, we have the the product of the zeroes are

lmn = d/a

So, comparing this with x³ - 3x² - x + 3 where a = 1 and d = 3.

So, the product of the zeroes is d/a = 3/1 = 3

So, the product of the zeroes is 3

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The optimal pair for the problem min 2tx(t)-3t³u(t)dt is u*(t) = -t³/b³, x*(t) = -t⁴/b³.

To find the optimal pair for the problem min 2tx(t)-3t³u(t)dt, we need to use the calculus of variations.

We start by considering the functional [tex]J(u) = \int_{a}^{b} (2tx(t)-3t^3u(t))dt[/tex], where u is the control function that we want to optimize.

We can find the optimal pair (x*, u*) by solving the Euler-Lagrange equation:
d/dt (∂L/∂u') - ∂L/∂u = 0,
where L(t, x(t), u(t), u'(t)) = 2tx(t)-3t³u(t) and u' = du/dt.

After some calculations, we obtain:
-3t² = u''/u',

which is a separable first-order differential equation that we can solve using integration.

We get:
u(t) = c1*t³ + c2,

where c1 and c2 are constants of integration that we can determine using the boundary conditions.

Since we want to minimize J(u), we need to choose the constants that minimize J(u). Using the boundary condition u(a) = u(b) = 0, we get:
c1 = -c2/b³, c2 = 0,

so that:
u(t) = -t³/b³.

Finally, we can compute the corresponding optimal x* using the formula:
[tex]x^*(t) = \int_{a}^{t} (\partial L/ \partial u)du + K[/tex],
where K is a constant of integration that we can determine using the boundary condition x(a) = x(b) = 0.

We obtain:
x*(t) = -t⁴/b³.

Therefore, the optimal pair is given by:
u*(t) = -t³/b³, x*(t) = -t⁴/b³.

Note that we also need to check that this is indeed a minimum by verifying that the second variation of J(u) is positive.

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The value of the line integral is 196/3.

We need to parameterize the given curve and then evaluate the line integral using the parameterization.

Let's parameterize the given curve y = √x as follows:

x = t^2

y = t

where t varies from 1 to 3.

The line integral then becomes:

∫(1 to 3) of [(t^2)*(t^3) - sqrt(t^2)]dt

= ∫(1 to 3) of [t^5 - t]dt

= [(1/6)*t^6 - (1/2)*t^2] from 1 to 3

= [(1/6)(3^6 - 1) - (1/2)(3^2 - 1)] - [(1/6)(1^6 - 1) - (1/2)(1^2 - 1)]

= 196/3

Therefore, the value of the line integral is 196/3.

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