The total profit P(x) (in thousands of dollars) from the sale of x hundred thousand automobile tires is approximated by P(x) = - x2 +9x2 + 165x - 400, X2 5. Find the number of hundred thousands of tires that must be sold to maximize profit. Find the maximum profit The maximum profit is $ when hundred thousand tires are sold.

The solution to the initial value problem 64y'' - y = 0, with y(-8) = 1 and y'(-8) = -1, is approximately:

y(t) ≈ -4.038e^(t/8) + 5.038e^(-t/8)

To solve the initial value problem 64y'' - y = 0, with initial conditions y(-8) = 1 and y'(-8) = -1, use the method of solving second-order linear homogeneous differential equations.

First, let's find the characteristic equation:

64r^2 - 1 = 0

Solving the characteristic equation, we have:

r^2 = 1/64

r = ±1/8

The general solution of the homogeneous equation is given by:

y(t) = c1e^(t/8) + c2e^(-t/8)

Now, let's apply the initial conditions to find the particular solution.

1. Using the condition y(-8) = 1:

y(-8) = c1e^(-1) + c2e = 1

2. Using the condition y'(-8) = -1:

y'(-8) = (c1/8)e^(-1) - (c2/8)e = -1

system of two equations:

c1e^(-1) + c2e = 1

(c1/8)e^(-1) - (c2/8)e = -1

Solving this system of equations, we find:

c1 ≈ -4.038

c2 ≈ 5.038

Therefore, the particular solution is:

y(t) ≈ -4.038e^(t/8) + 5.038e^(-t/8)

Hence, the solution to the initial value problem 64y'' - y = 0, with y(-8) = 1 and y'(-8) = -1, is approximately:

y(t) ≈ -4.038e^(t/8) + 5.038e^(-t/8)

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The result will be in the form a + bi, where a and b are real numbers, representing the real and imaginary parts of the root, respectively.

To determine the indicated roots of a complex number, we need to consider the form of the complex number and the root we are trying to find. The indicated roots can be found using the nth root formula in rectangular form.

For a complex number in rectangular form a + bi, the nth roots can be found using the formula: z^(1/n) = (r^(1/n))(cos(θ/n) + i sin(θ/n))

Here, r represents the magnitude of the complex number and θ represents the argument (angle) of the complex number.To find the indicated roots, we first need to express the complex number in rectangular form by separating the real and imaginary parts.

Then, we can apply the nth root formula by taking the nth root of the magnitude and dividing the argument by n. The result will be in the form a + bi, where a and b are real numbers, representing the real and imaginary parts of the root, respectively.

It is important to note that not all complex numbers have real-numbered roots. In some cases, the roots may involve the use of trigonometric functions or may be complex.

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                               "Complete question"

Determine the indicated roots of the given complex number. When it is possible, write the roots in the form a + bi, where a and b are real numbers and do not involve the use of a trigonometric function. Otherwise, leave the roots in polar form. The two square roots of 43 - 4i. 20 21 = >

Using spherical coordinates, the z-coordinate of the center of mass of a solid hemisphere with the given density function and mass is determined to be 7/12.

To find the z-coordinate of the center of mass, we need to calculate the triple integral of the density function over the solid hemisphere. In spherical coordinates, the volume element is given by ρ^2 sin(φ) dρ dφ dθ, where ρ is the radial distance, φ is the polar angle, and θ is the azimuthal angle.

First, we set up the limits of integration. For the radial distance ρ, it ranges from 0 to the radius of the hemisphere, which is a constant value. The polar angle φ ranges from 0 to π/2 since we are considering the upper half of the hemisphere. The azimuthal angle θ ranges from 0 to 2π, covering the entire circumference.

Next, we substitute the density function d = m into the volume element and integrate. Since the mass m is given as 7/4, we can replace d with 7/4. After performing the triple integral, we obtain the z-coordinate of the center of mass as 7/12.

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The area D between the lines x = 0, y = 3-3x, and y = 3x - 3 can be represented as an iterated double integral of x over a certain region.

To set up the iterated double integral for ∫∫D x dA, we need to determine the limits of integration for each variable. Let's first consider the limits for y. The line y = 3-3x intersects the x-axis at x = 1, and the line y = 3x - 3 intersects the x-axis at x = 1 as well. So, the limits for y are from y = 0 to y = 3-3x for x between 0 and 1, and from y = 0 to y = 3x - 3 for x between 1 and 2.

Next, we determine the limits for x. We can see that the region D is bounded by the lines x = 0 and x = 2. Therefore, the limits for x are from 0 to 2.

Now, we have established the limits of integration for both x and y. We can set up the iterated double integral as follows:

∫∫D x dA = ∫[0 to 2] ∫[0 to 3-3x] x dy dx + ∫[1 to 2] ∫[0 to 3x-3] x dy dx.

Integrating with respect to y first, we have:

∫∫D x dA = ∫[0 to 2] (xy |[0 to 3-3x]) dx + ∫[1 to 2] (xy |[0 to 3x-3]) dx.

Evaluating the limits and simplifying the expression will give us the final result for the iterated double integral.

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The problem involves three point masses, with one mass m located at the origin, another mass 2m located at a point on the x-axis denoted as x = l, and a third mass m that can be moved along the x-axis.

In this problem, we have three point masses arranged along the x-axis. The mass m is located at the origin (x = 0), the mass 2m is located at a specific point on the x-axis denoted as x = l, and the third mass m can be moved along the x-axis.

The behavior of the system depends on the interaction between the masses. The gravitational force between two point masses is given by the equation F = [tex]G (m1 m2) / r^2[/tex], where G is the gravitational constant, m1 and m2 are the masses, and r is the distance between the masses.

By moving the third mass m along the x-axis, the gravitational forces between the masses will vary. The specific positions of the masses and the distances between them will determine the magnitudes and directions of the gravitational forces.

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There are 834 positive integers not exceeding 1000 that are not divisible by either 8 or 12.

To find the number of positive integers not exceeding 1000 that are not divisible by either 8 or 12, we can use the principle of inclusion-exclusion. First, let's find the number of positive integers not exceeding 1000 that are divisible by 8. The largest multiple of 8 that does not exceed 1000 is 992 (8 * 124). So, there are 124 positive integers not exceeding 1000 that are divisible by 8. Next, let's find the number of positive integers not exceeding 1000 that are divisible by 12. The largest multiple of 12 that does not exceed 1000 is 996 (12 * 83). So, there are 83 positive integers not exceeding 1000 that are divisible by 12.

However, we have counted some numbers twice—those that are divisible by both 8 and 12. To correct for this, we need to find the number of positive integers not exceeding 1000 that are divisible by both 8 and 12 (i.e., divisible by their least common multiple, which is 24). The largest multiple of 24 that does not exceed 1000 is 984 (24 * 41). So, there are 41 positive integers not exceeding 1000 that are divisible by both 8 and 12.

Now, we can apply the principle of inclusion-exclusion to find the number of positive integers not exceeding 1000 that are not divisible by either 8 or 12: Total number of positive integers not exceeding 1000 = Total number of positive integers - Number of positive integers divisible by 8 or 12 + Number of positive integers divisible by both 8 and 12. Total number of positive integers not exceeding 1000 = 1000 - 124 - 83 + 41

= 834. Therefore, there are 834 positive integers not exceeding 1000 that are not divisible by either 8 or 12.

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Answer:

Step-by-step explanation:

The maximum value of f(x, y) = 2xy subject to the constraint 16x + y = 128 is 512, and the minimum value is 0.

To find the maximum and minimum values of the function f(x, y) = 2xy subject to the constraint 16x + y = 128, we can use the method of Lagrange multipliers.

Let's define the Lagrangian function L(x, y, λ) as:

L(x, y, λ) = f(x, y) - λ(g(x, y))

where g(x, y) is the constraint function.

In this case, f(x, y) = 2xy and g(x, y) = 16x + y - 128.

The Lagrangian function becomes:

L(x, y, λ) = 2xy - λ(16x + y - 128)

Next, we need to find the critical points of L(x, y, λ) by taking the partial derivatives with respect to x, y, and λ, and setting them equal to zero:

∂L/∂x = 2y - 16λ = 0 ...(1)

∂L/∂y = 2x - λ = 0 ...(2)

∂L/∂λ = 16x + y - 128 = 0 ...(3)

Solving equations (1) and (2) simultaneously, we get:

2y - 16λ = 0 ...(1)

2x - λ = 0 ...(2)

From equation (1), we can express λ in terms of y:

λ = y/8

Substituting this into equation (2):

2x - (y/8) = 0

Simplifying:

16x - y = 0

Rearranging equation (3):

16x + y = 128

Substituting 16x - y = 0 into 16x + y = 128:

16x + 16x - y = 128

32x = 128

x = 4

Substituting x = 4 into 16x + y = 128:

16(4) + y = 128

64 + y = 128

y = 64

So, the critical point is (x, y) = (4, 64).

To find the maximum and minimum values, we evaluate f(x, y) at the critical point and at the boundary points.

At the critical point (4, 64), f(4, 64) = 2(4)(64) = 512.

Now, let's consider the boundary points.

When 16x + y = 128, we have y = 128 - 16x.

Substituting this into f(x, y):

f(x) = 2xy = 2x(128 - 16x) = 256x - 32x^2

To find the extreme values, we find the critical points of f(x) by taking its derivative:

f'(x) = 256 - 64x = 0

64x = 256

x = 4

Substituting x = 4 back into 16x + y = 128:

16(4) + y = 128

64 + y = 128

y = 64

So, another critical point on the boundary is (x, y) = (4, 64).

Comparing the values of f(x, y) at the critical point (4, 64) and the boundary points (4, 64) and (0, 128), we find:

f(4, 64) = 512

f(4, 64) = 512

f(0, 128) = 0

Therefore, the maximum value of f(x, y) = 2xy subject to the constraint 16x + y = 128 is 512, and the minimum value is 0.

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Answer:

fill in the point into your equation and check it.

Step-by-step explanation:

You did a great job writing the equation. Now use the equation and the (x, y) in each part to find out which points are on the circle. For example, part A, (3,9) use x =3 and y = 9 in your equation

(3+3)^2 + (9-1)^2 = 100?



6^2 + 8^2 = 100

36 + 64 = 100

100 = 100 this checks so A(3,9) IS on the circle.

But for B(6,8), that is not on the circle bc it does not check:

(6+3)^2 + (8-1)^2 =100?



9^2 + 7^2 = 100

81 + 49 = 100

130 = 100 false. This does not check. (6,8) is not on the circle.

Be sure to check C, D, E

The dimensions of the closed rectangular box with a square cross section, constructed using the least amount of material and having a capacity of 113 in³: are 3.6 inches by 3.6 inches by 3.6 inches.

Let's assume the side length of the square cross section is x inches. Since the box has a square cross section, the height of the box will also be x inches.

The volume of the box is given as 113 in³, which can be expressed as:

x × x × x = 113

Simplifying the equation, we have:

x³ = 113

To find the value of x, we take the cube root of both sides:

x = ∛113 ≈ 4.19

Since the box needs to use the least amount of material, we choose the nearest integer values for the dimensions. Therefore, the dimensions of the box are approximately 3.6 inches by 3.6 inches by 3.6 inches, as rounding down to 3.6 inches still satisfies the given capacity of 113 in³ while minimizing the material used.

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To determine the rent that maximizes the total revenue from the building, we can express the relationship between the rent and the number of occupied units. By setting up equations based on the given information. Answer :  Revenue = R * (70 - R/20 + 54).

we can derive a revenue function. Taking the derivative of this function and finding its critical points will help us identify the rent that maximizes the revenue.

1. Let R be the rent per unit and V be the number of vacant units. Using the information provided, we can express V = (R - 1080) / 20.

2. The number of occupied units, O, can be obtained as O = 70 - V.

3. The total revenue is given by Revenue = R * O.

4. Substituting the expressions for V and O into the revenue equation, we obtain Revenue = R * (70 - R/20 + 54).

5. Taking the derivative of the revenue function with respect to R, setting it equal to zero, and solving for R will give us the rent that maximizes the revenue.

2) The total revenue function for a computer is R(x) = 2800x - 8x^2 - x^3, where x represents the level of sales. To find the level of sales, x, that maximizes the revenue, we need to find the critical points of the revenue function by taking its derivative and setting it equal to zero. Solving this equation will give us the values of x that maximize the revenue. Substituting these values back into the revenue function will help us find the maximum revenue.

1. Calculate the derivative of the revenue function R(x) = 2800x - 8x^2 - x^3, which is R'(x) = 2800 - 16x - 3x^2.

2. Set R'(x) equal to zero: 2800 - 16x - 3x^2 = 0.

3. Solve the quadratic equation 3x^2 + 16x - 2800 = 0 either by factoring or using the quadratic formula.

4. Find the values of x that satisfy the equation and represent the critical points.

5. Evaluate the revenue function R(x) at these critical points to find the maximum revenue.

6. The level of sales, x, that maximizes the revenue is determined by the critical points, and the maximum revenue is obtained by substituting this value back into the revenue function.

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The function f(x, y) is given as -9 + 5x - 3y. The partial derivatives fx and fy are both equal to 5. Evaluating fx at (2, -1) gives the value 5, and evaluating fy at (-2, -2) also gives the value 5.

The function f(x, y) = -9 + 5x - 3y represents a two-variable function. To find the partial derivative fx with respect to x, we differentiate the function with respect to x while treating y as a constant. The derivative of 5x with respect to x is 5, and the derivative of -3y with respect to x is 0 since y is a constant. Therefore, fx(x, y) = 5.

Similarly, to find fy with respect to y, we differentiate the function with respect to y while treating x as a constant. The derivative of -3y with respect to y is -3, and the derivative of 5x with respect to y is 0 since x is a constant. Thus, fy(x, y) = -3. To evaluate fx at the point (2, -1), we substitute x = 2 and y = -1 into the expression for fx.

This gives fx(2, -1) = 5. Similarly, to evaluate fy at the point (-2, -2), we substitute x = -2 and y = -2 into the expression for fy. This gives fy(-2, -2) = -3.

In summary, the partial derivatives fx and fy are both equal to 5. Evaluating fx at (2, -1) gives the value 5, and evaluating fy at (-2, -2) also gives the value 5.

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The work done is[tex]-20 (1/(2^2 + y^2 + z^2)^(1/2) - 1/2)[/tex] Joules for the given charge.

The term "work done" describes the quantity of energy that is transmitted or expended when a task is completed or a force is applied across a distance. It is computed by dividing the amount of applied force by the distance across which it is exerted, in the force's direction. In the International System of Units (SI), the unit used to measure work is the joule (J).

Given that the force exerted by an electric charge at the origin on a charged particle at the point (2, y, z) with position Kr vector r = (x, y, z) is F(r) = 20 (x/r3) i where K is constant.

Assuming that the particle moves from point A to point B, we can find the work done.

The work done in moving a charge against an electric field is given by:W = -ΔPElectricPotential Energy is given by U = qV where q is the test charge and V is the electric potential. The electric potential at a distance r from a point charge is given by V = kq/r where k is the Coulomb constant.

The work done in moving a charge from point A to point B against an electric field is given by:W = -q (VB - VA)where q is the test charge and VB and VA are the electric potentials at points B and A respectively.

In this case, the test charge is not given, we will assume it to be +1 C.Work done = -q (VB - VA)Potential at point A (r = 2) = kQ/r = kQ/2Potential at point B [tex](r = √(x^2 + y^2 + z^2)) = kQ/√(x^2 + y^2 + z^2)[/tex]

Work done = -q (kQ/[tex]\sqrt{(x^2 + y^2 + z^2)}[/tex] - kQ/2)=- kQq (1/[tex]\sqrt{(x^2 + y^2 + z^2)}[/tex] - 1/2)= -20 ([tex]1/(2^2 + y^2 + z^2)^(1/2)[/tex] - 1/2) JoulesAnswer:

The work done is [tex]-20 (1/(2^2 + y^2 + z^2)^(1/2) - 1/2)[/tex]Joules.

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The concentration is maximum at x = 6 hours after the drug is administered.

To find the time at which the concentration is a maximum, we need to determine the critical points of the concentration function and then determine which critical point corresponds to the maximum value.

Let's first find the derivative of the concentration function with respect to time:

k(x) = (3x) / (x² + 36)

To find the maximum, we need to find when the derivative is equal to zero:

k'(x) = [ (3)(x² + 36) - (3x)(2x) ] / (x² + 36)²

= [ 3x² + 108 - 6x² ] / (x² + 36)²

= (108 - 3x²) / (x² + 36)²

Setting k'(x) equal to zero:

(108 - 3x²) / (x² + 36)² = 0

To simplify further, we can multiply both sides by (x² + 36)²:

108 - 3x² = 0

Rearranging the equation:

3x² = 108

Dividing both sides by 3:

x² = 36

Taking the square root of both sides:

x = ±6

Therefore, we have two critical points: x = 6 and x = -6.

Since we're dealing with time, the concentration cannot be negative. Thus, we can disregard the negative value.

Therefore, the concentration is maximum at x = 6 hours after the drug is administered.

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To graph the function f(x) = -2 cos (π/3 x - 2π/3), we need to understand its properties and behavior.

First, let's consider the amplitude of the cosine function, which is 2 in this case. This means that the graph will oscillate between -2 and 2 along the y-axis. Next, let's determine the period of the function. The period of a cosine function is given by 2π divided by the coefficient of x inside the cosine function. In this case, the coefficient is π/3. So the period is: Period = 2π / (π/3) = 6. This means that the graph will complete one full oscillation every 6 units along the x-axis.

Now, let's plot the graph on a coordinate plane: Start by labeling the x-axis with appropriate units based on the period. For example, if we choose each unit to represent 1, then we can label the x-axis from -6 to 6. Label the y-axis to represent the amplitude of the function, from -2 to 2. Plot some key points on the graph, such as the x-intercepts, by setting the function equal to zero and solving for x. In this case, we have:

-2 cos (π/3 x - 2π/3) = 0 . cos (π/3 x - 2π/3) = 0. To find the x-intercepts, we solve for (π/3 x - 2π/3) = (2n + 1)π/2, where n is an integer. From this equation, we can determine the x-values at which the cosine function crosses the x-axis.

Finally, sketch the graph by connecting the key points and following the shape of the cosine function, which oscillates between -2 and 2.

Note: Without specific values for the x-axis units, it is not possible to accurately label the x-axis with specific values. However, the general shape and behavior of the graph can still be depicted.

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To find the average temperature during the 24-hour period, we need to calculate the total temperature over that period and divide it by the duration.

The total temperature is the definite integral of the temperature function T(t) over the interval [0, 24]:

Total temperature = ∫[0, 24] (49 + 8t - 1/2 * t^2) dt

We can evaluate this integral to find the total temperature:

Total temperature = [49t + 4t^2 - 1/6 * t^3] evaluated from t = 0 to t = 24

Total temperature = (49 * 24 + 4 * 24^2 - 1/6 * 24^3) - (49 * 0 + 4 * 0^2 - 1/6 * 0^3)

Total temperature = (1176 + 2304 - 0) - (0 + 0 - 0)

Total temperature = 3480 degrees

The duration of the period is 24 hours, so the average temperature is:

Average temperature = Total temperature / Duration

Average temperature = 3480 / 24

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To find the solution of the given differential equation with the initial condition, use an integrating factor method.

The given differential equation is: y' tan x = 5a + y

Begin by rearranging the equation in a standard form:

y' - y = 5a tan x

Now,  identify the integrating factor (IF) for this equation. The integrating factor is given by e^(∫-1 dx), where -1 is the coefficient of y. Integrating -1 with respect to x gives us -x.

So, the integrating factor (IF) is e^(-x).

Multiplying the entire equation by the integrating factor, we get:

e^(-x) * y' - e^(-x) * y = 5a tan x * e^(-x)

Now, we can rewrite the left side of the equation using the product rule for differentiation:

(e^(-x) * y)' = 5a tan x * e^(-x)

Integrating both sides of the equation with respect to x, we get:

∫ (e^(-x) * y)' dx = ∫ (5a tan x * e^(-x)) dx

Integrating the left side yields:

e^(-x) * y = ∫ (5a tan x * e^(-x)) dx

To evaluate the integral on the right side, we can use integration by parts. The formula for integration by parts is:

∫ (u * v)' dx = u * v - ∫ (u' * v) dx

Let:

u = 5a tan x

v' = e^(-x)

Differentiating u with respect to x gives:

u' = 5a sec^2 x

Substituting these values into the integration by parts formula, we have:

∫ (5a tan x * e^(-x)) dx = (5a tan x) * (-e^(-x)) - ∫ (5a sec^2 x * (-e^(-x))) d

Simplifying, we get:

∫ (5a tan x * e^(-x)) dx = -5a tan x * e^(-x) + 5a ∫ (sec^2 x * e^(-x)) dx

The integral of sec^2 x * e^(-x) can be evaluated as follows:

Let:

u = sec x

v' = e^(-x)

Differentiating u with respect to x gives:

u' = sec x * tan x

Substituting these values into the integration by parts formula, we have:

∫ (sec^2 x * e^(-x)) dx = (sec x) * (-e^(-x)) - ∫ (sec x * tan x * (-e^(-x))) dx

Simplifying, we get:

∫ (sec^2 x * e^(-x)) dx = -sec x * e^(-x) + ∫ (sec x * tan x * e^(-x)) dx

Notice that the integral on the right side is the same as the one we started with, so substitute the result back into the equation:

∫ (5a tan x * e^(-x)) dx = -5a tan x * e^(-x) + 5a * (-sec x * e^(-x) + ∫ (sec x * tan x * e^(-x)) dx)

now substitute this expression back into the original equation:

e^(-x) * y = -5a tan x * e^(-x) + 5a * (-sec x *

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When applying the Ratio Test to the series (-7)^(n+1)/(n^2 + 51n), we determine that the limit of the ratio as n approaches infinity is equal to infinity. Therefore, the series is divergent.

To apply the Ratio Test, we calculate the limit of the absolute value of the ratio of consecutive terms as n approaches infinity. For the given series (-7)^(n+1)/(n^2 + 51n), let's denote the general term as an.

Using the Ratio Test, we evaluate the limit as n approaches infinity:

lim(n → ∞) |(an+1/an)| = lim(n → ∞) |(-7)^(n+2)/[(n+1)^2 + 51(n+1)] * (n^2 + 51n)/(-7)^(n+1)|.

Simplifying the expression, we get:

lim(n → ∞) |-7/(n+1+51) * (n^2 + 51n)/-7| = lim(n → ∞) |-(n^2 + 51n)/(n+1+51)|.

As n approaches infinity, both the numerator and denominator grow without bound, resulting in an infinite limit:

lim(n → ∞) |-(n^2 + 51n)/(n+1+51)| = ∞.

Since the limit of the ratio is infinity, the Ratio Test tells us that the series is divergent.

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To find the expression for g(r, θ), we substitute x = rcos(θ) and y = rsin(θ) into the given function g(x, y) = -xy + e^(x^2+y^2).

First, we substitute x and y with their respective expressions:

g(r, θ) = -(r*cos(θ))*(r*sin(θ)) + e^((r*cos(θ))^2 + (r*sin(θ))^2)

Simplifying the expression inside the exponential:

g(r, θ) = -(r^2*cos(θ)*sin(θ)) + e^(r^2*cos^2(θ) + r^2*sin^2(θ))

Using the trigonometric identity cos^2(θ) + sin^2(θ) = 1, we have:

g(r, θ) = -(r^2*cos(θ)*sin(θ)) + e^(r^2)

Therefore, the expression for g(r, θ) in terms of r and θ is:

g(r, θ) = -r^2*cos(θ)*sin(θ) + e^(r^2)

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The absolute extreme values on the interval are:

Absolute maximum: f(x) = 3 at x = 0 and x = π

Absolute minimum: f(x) = 2 at x = π/2

To find the absolute extreme values of the function f(x) = 2 + cos^2(x) on the given interval, we need to evaluate the function at its critical points and endpoints.

Step 1: Find the critical points by taking the derivative of f(x) and setting it equal to zero.

f'(x) = -2sin(x)cos(x)

Setting f'(x) = 0, we have:

-2sin(x)cos(x) = 0

This equation is satisfied when sin(x) = 0 or cos(x) = 0.

The critical points occur when x = 0, π/2, and π.

Step 2: Evaluate the function at the critical points and the endpoints of the interval.

At x = 0:

f(0) = 2 + cos^2(0) = 2 + 1 = 3

At x = π/2:

f(π/2) = 2 + cos^2(π/2) = 2 + 0 = 2

At x = π:

f(π) = 2 + cos^2(π) = 2 + 1 = 3

Step 3: Compare the values of f(x) at the critical points and endpoints to determine the absolute extreme values.

The function f(x) = 2 + cos^2(x) has a maximum value of 3 at x = 0 and x = π, and a minimum value of 2 at x = π/2.

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The vector equation for the line of intersection of the planes x - y + 4z = 1 and x + 3z = 5 is r = (5, 4, 0) + t(12, -1, 1).

To find the vector equation for the line of intersection of the planes x − y + 4z = 1 and x + 3z = 5, follow these steps:

Step 1: Find the direction vector of the line of intersection by taking the cross product of the normal vectors of the two planes. The normal vectors are given by (1, -1, 4) and (1, 0, 3) respectively.

(1,-1,4) xx (1,0,3) = i(12) - j(1) + k(1) = (12,-1,1)

Therefore, the direction vector of the line of intersection is d = (12, -1, 1).

Step 2: Find a point on the line of intersection. Let z = t. Substituting this into the equation of the second plane, we have:

x + 3z = 5x + 3t = 5x = 5 - 3t

Substituting this into the equation of the first plane, we have: x - y + 4z = 1, 5 - 3t - y + 4t = 1, y = 4t + 4

Therefore, a point on the line of intersection is (5 - 3t, 4t + 4, t). Let t = 0.

This gives us the point (5, 4, 0).

Step 3: Write the vector equation of the line of intersection.

Using the point (5, 4, 0) and the direction vector d = (12, -1, 1), the vector equation of the line of intersection is:

r = (5, 4, 0) + t(12, -1, 1)

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In this problem, we are given a piecewise smooth, counterclockwise simple closed curve C and we need to show that the area A(R) of the region enclosed by C can be calculated using the formula A(R) = ∮xdy.

To show that the area A(R) of the region enclosed by the curve C is given by the formula A(R) = ∮xdy, we need to express the curve C as a parametric equation. Let's denote the parametric equation of C as r(t) = (x(t), y(t)), where t ranges from a to b. By applying Green's theorem, we can rewrite the double integral of dA over R as the line integral ∮xdy over C. Using the parameterization r(t), the line integral becomes ∫[a,b]x(t)y'(t)dt. Since the curve is counterclockwise, the orientation of the integral is correct for calculating the area.

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The limit is undefined

Let's have further explanation:

The limit can be solved using the definition of a limit.

Let L=0

Then,

                      lim L→0 L-csc(4L)

                             = lim L→0 L-1/sin(4L)

                             = lim L→0 0-1/sin(4L)

                             = -1/lim L→0 sin(4L)

Since sin(x) is a continuous function and lim L→0 sin(4L) = 0,

                                lim L→0 L-csc(4L) = -1/0

The limit is therefore undetermined.

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The given system of equations has a unique solution if p is not equal to 2. If p is equal to 2 and q is equal to 4, the system has infinitely many solutions.Therefore, the correct answer is (a) 1 correct.

The given system of equations is:

2x - y = 4

px - y = q

To determine if the system has a unique solution, we need to analyze the coefficients of x and y.In the first equation, the coefficient of y is -1. In the second equation, the coefficient of y is also -1.If the coefficients of y are equal in both equations, the system may have infinitely many solutions. However, if the coefficients of y are different, the system will have a unique solution.

Now, we consider the options:

a) 1 correct: This statement is correct. If p is not equal to 2, the coefficients of y in both equations will be different (-1 in the first equation and -1 in the second equation), and thus the system will have a unique solution.b) 2 correct: This statement is correct. If p is equal to 2 and q is equal to 4, the coefficients of y in both equations will be the same (-1 in both equations), and therefore the system will have infinitely many solutions.

c) 1 and 2 correct: This statement is incorrect because option 1 is true but option 2 is only true under specific conditions (p = 2 and q = 4).d) 1 and 2 are false: This statement is incorrect because option 1 is true and option 2 is also true under specific conditions (p = 2 and q = 4).

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The limit of the expression as x approaches -7 is 0.

To find the limit of the expression as x approaches -7, we need to evaluate the expression for values of x approaching -7 from both the left and the right sides.

For values of x less than -7 (approaching from the left side), we have:

lim x→-7- 10x * 70 |x + 7|

Since the absolute value |x + 7| becomes -(x + 7) when x < -7, rewrite the expression as:

lim x→-7- 10x * 70 * -(x + 7)

Simplifying further:

lim x→-7- -700x(x + 7)

Next, we can directly substitute x = -7 into the expression:

-700 * -7 * (-7 + 7) = -700 * -7 * 0 = 0

For values of x greater than -7 (approaching from the right side), we have:

lim x→-7+ 10x * 70 |x + 7|

Since the absolute value |x + 7| becomes x + 7 when x > -7, we can rewrite the expression as:

lim x→-7+ 10x * 70 * (x + 7)

Simplifying further:

lim x→-7+ 700x(x + 7)

Again, directly substitute x = -7 into the expression:

700 * -7 * (-7 + 7) = 700 * -7 * 0 = 0

Since the limits from the left side and the right side are both 0, and they are equal, the overall limit as x approaches -7 exists and is equal to 0.

Therefore, the limit of the expression as x approaches -7 is 0.

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We are asked to find a value of x that satisfies the equation (5x/101 + 5x + 2) mod 991 = 5. The task is to determine whether a solution exists and, if so, find the specific value of x that satisfies the equation.

To solve the equation, we need to find a value of x that, when substituted into the expression (5x/101 + 5x + 2), results in a remainder of 5 when divided by 991.

Finding an exact solution may involve complex calculations and trial and error. It is important to note that modular arithmetic can yield multiple solutions or no solutions at all, depending on the equation and the modulus.

Given the complexity of the equation and the modulus involved, it would require a systematic approach or advanced techniques to determine if a solution exists and find the specific value of x. Without further information or constraints, it is difficult to provide a direct solution.

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To sketch a graph of y = -4 csc(x) + 7, we begin by sketching a graph of y = csc(x). The function csc(x), also known as the cosecant function, is the reciprocal of the sine function.

It represents the ratio of the hypotenuse to the opposite side of a right triangle in trigonometry. The graph of y = csc(x) has vertical asymptotes at x = nπ, where n is an integer, and crosses the x-axis at those points. It approaches positive and negative infinity as x approaches the vertical asymptotes.

Next, we multiply the graph of y = csc(x) by -4 and shift it upwards by 7 units to obtain y = -4 csc(x) + 7. The multiplication by -4 reflects the graph vertically and the addition of 7 shifts it upwards. The resulting graph will have the same vertical asymptotes as y = csc(x) but will be scaled by a factor of 4. It will still cross the x-axis at the vertical asymptotes but will be shifted upward by 7 units. The graph will exhibit the same behavior of approaching positive and negative infinity as x approaches the vertical asymptotes..

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The function f(x) = 9/(1 - 216x) can be expressed as a power series that converges for |x| < 1.

The power series representation can be obtained by using the fact that the geometric series converges to 1/(1 - r), where |r| < 1.

In this case, we have f(x) = 9/(1 - 216x), which can be rewritten as f(x) = 9 * (1/(1 - (-216x))). Now, we recognize that the term (-216x) is the common ratio (r) of the geometric series. Therefore, we can write f(x) as a power series by replacing (-216x) with r.

Using the geometric series representation, we have:

f(x) = 9 * Σ (-216x)^n, where n ranges from 0 to infinity.

Simplifying further, we get:

f(x) = 9 * Σ (-1)^n * (216^n) * (x^n), where n ranges from 0 to infinity.

This power series representation converges for |x| < 1, as dictated by the convergence condition of the geometric series.

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The absolute value of the error is less than 0.001.

The integral using the Maclaurin series, we need to expand the integrand function, which is x⁴×ln(1+x²), into a power series.

Then we can integrate each term of the power series.

The Maclaurin series expansion of ln(1+x²) is:

ln(1+x²) = x² - (1/2)x⁴ + (1/3)x⁶ - (1/4)x⁸ + ...

Next, we multiply each term of the power series by x⁴:

x⁴×ln(1+x²) = x⁶ - (1/2)x⁸ + (1/3)x¹⁰- (1/4)x¹² + ...

Now, we can integrate each term of the power series:

∫ (x⁶ - (1/2)x⁸ + (1/3)x¹⁰ - (1/4)x¹² + ...) dx

To ensure the absolute value of the error is less than 0.001, we need to determine how many terms to include in the approximation.

We can use the alternating series estimation theorem to estimate the error. By calculating the next term, (-1/4)x¹², and evaluating it at x = 0.8, we find that the error term is smaller than 0.001.

Therefore, we can include the first four terms in the approximation.

Finally, we substitute x = 0.8 into each term and sum them up:

Approximation = (0.8⁶)/6 - (1/2)(0.8⁸)/8 + (1/3)(0.8¹⁰)/10 - (1/4)(0.8¹²)/12

< 0.001

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The given series is convergent. To determine whether the series is convergent or divergent, we need to examine the behavior of its terms as n approaches infinity. The given series is a sum of two terms: 1/9 and e^(-n).

The term 1/9 is a constant term that does not depend on n. The series ∑(1/9) is a geometric series with a common ratio of 1, which is less than 1. Therefore, this series converges, and its sum can be found using the formula for the sum of a geometric series:

Sum = a / (1 - r),

where a is the first term and r is the common ratio. In this case, a = 1/9 and r = 1, so the sum of the series ∑(1/9) is given by:

Sum = (1/9) / (1 - 1) = (1/9) / 0.

However, dividing by zero is undefined, so the sum of the series ∑(1/9) is not defined.

The second term in the series is e^(-n), where e is Euler's number. As n approaches infinity, e^(-n) approaches 0. This term contributes to the convergence of the series. Therefore, the series ∑(1/9 + e^(-n)) is convergent. However, since the first term does not have a defined sum, we cannot determine the sum of the series.

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