= Let f(x) = x3, and compute the Riemann sum of f over the interval [7, 8], choosing the representative points to be the midpoints of the subintervals and using the following number of subintervals (n

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Answer 1

To compute the Riemann sum of the function [tex]f(x) = x^3[/tex] over the interval [7, 8], the representative points to be the midpoints of the subintervals. The number of subintervals (n) will determine the accuracy of the approximation.

The Riemann sum is an approximation of the definite integral of a function over an interval using rectangles. To compute the Riemann sum with midpoints, we divide the interval [7, 8] into n subintervals of equal width.

The width of each subinterval is given by Δ[tex]x = (b - a) / n[/tex], where a = 7 and b = 8 are the endpoints of the interval.

The midpoint of each subinterval is given by [tex]x_i = a + (i - 1/2)[/tex]Δx, where i ranges from 1 to n.

Next, we evaluate the function f at each midpoint: [tex]f(x_i) = (x_i)^3[/tex].

Finally, we compute the Riemann sum as the sum of the areas of the rectangles: Riemann sum = Δ[tex]x * (f(x_1) + f(x_2) + ... + f(x_n))[/tex].

The number of subintervals (n) determines the accuracy of the approximation. As n increases, the Riemann sum becomes a better approximation of the definite integral.

In conclusion, to compute the Riemann sum of [tex]f(x) = x^3[/tex] over the interval [7, 8] with midpoints, we divide the interval into n subintervals, compute the representative points as the midpoints of the subintervals, evaluate the function at each midpoint, and sum up the areas of the rectangles. The value of n determines the accuracy of the approximation.

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Related Questions

Question 8 G0/10 pts 3 99 Details 23 Use Simpson's Rule and all the data in the following table to estimate the value of the integral 1 f(a)da. X 5 f(x) 8 3 12 برابر 8 11 14 17 20 23 11 15 6 13 2

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Using Simpson's Rule, the estimated value of the integral ∫f(a)da is 89.

Simpson's Rule is a numerical integration method that approximates the value of an integral by dividing the interval into subintervals and using a quadratic polynomial to interpolate the function within each subinterval. The table provides the values of f(x) at different points. To apply Simpson's Rule, we group the data into pairs of subintervals. Using the formula for Simpson's Rule, we calculate the estimated value of the integral to be 89. This is obtained by multiplying the common interval width (5) by one-third of the sum of the first and last function values (11+15), and adding to it four times one-third of the sum of the function values at the odd indices (6+2+13).

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Find the intervals on which fis increasing and the intervals on which it is decreasing. f(x) = 10-x? Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The function is increasing on the open interval(s) and decreasing on the open interval(s) (Simplify your answers. Type your answers in interval notation. Use a comma to separate answers as needed.) B. The function is increasing on the open interval(s). The function is never decreasing. (Simplify your answer. Type your answer in interval notation. Use a comma to separate answers as needed.) The function is decreasing on the open interval(s). The function is never increasing. (Simplify your answer. Type your answer in interval notation. Use a comma to separate answers as needed.) D. The function is never increasing nor decreasing.

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For the given function f(x) = 10 - x, the function is never increasing. (option c)

To determine the intervals on which the function is increasing or decreasing, we need to examine the slope of the function. The slope of a function represents the rate at which the function is changing. In this case, the slope of f(x) = 10 - x is -1, which means that the function is decreasing at a constant rate of 1 as we move along the x-axis.

Since the slope is negative (-1), the function is always decreasing. This means that the function f(x) = 10 - x is decreasing on the entire domain. Therefore, we can conclude that the function is never increasing.

The correct answer choice for this question is C. The function is never increasing.

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dt Canvas Golden West College MyGWC S * D Question 15 Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. dt © &(a)= (5-5) ° 8(a)= (9-4) © & (9) - (9-9")' (a)=

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The derivative of the given function F(a) = ∫[5 to a] 8(t) dt, using Part 1 of the Fundamental Theorem of Calculus, is F'(a) = (9 - 4a) © (9a).

The derivative of the given function can be found using Part 1 of the Fundamental Theorem of Calculus, which states that if a function is defined as the integral of another function, then its derivative can be found by evaluating the integrand at the upper limit of integration and multiplying by the derivative of the upper limit with respect to the variable. In this case, let's consider the function F(a) = ∫[5 to a] 8(t) dt, where 8(t) = (9 - 4t) © (9t). We want to find F'(a), the derivative of F(a) with respect to a.

By applying Part 1 of the Fundamental Theorem of Calculus, we evaluate the integrand 8(t) at the upper limit of integration, which is a, and then multiply by the derivative of the upper limit with respect to a, which is 1.

Therefore, F'(a) = 8(a) * 1 = (9 - 4a) © (9a).

In summary, the derivative of the given function F(a) = ∫[5 to a] 8(t) dt, using Part 1 of the Fundamental Theorem of Calculus, is F'(a) = (9 - 4a) © (9a).

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.The variables x and y vary inversely. Use the given values to write an equation relating x and y. Then find y when x = 3. x = 1, y = 9

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The given problem states that x and y vary inversely, and by using the given values, an equation is formed (x * y = 9) which can be used to find y when x = 3 (y = 3).

Since x and y vary inversely, we can write the equation as x * y = k, where k is a constant.

Using the given values x = 1 and y = 9, we can substitute them into the equation to find the value of k:

1 * 9 = k

k = 9

Therefore, the equation relating x and y is x * y = 9.

To find y when x = 3, we substitute x = 3 into the equation:

3 * y = 9

y = 9 / 3

y = 3

So, when x = 3, y = 3.

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Question * Let R be the region in the first quadrant bounded above by the parabola y = 4 x² and below by the line y = 1. Then the area of R is: None of these √√3 units squared This option 6 units

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The area of region R is 1/3 units squared. None of the given options match this result, so the correct answer is "None of these."

To find the area of the region R bounded by the parabola y = 4[tex]x^{2}[/tex] and the line y = 1, we need to determine the points of intersection between these two curves.

First, let's set the equations equal to each other and solve for x:

4[tex]x^{2}[/tex]=1

Divide both sides by 4:

[tex]x^{2}[/tex] = 1/4

Taking the square root of both sides, we get:

x = ±1/2

Since we're only interested in the region in the first quadrant, we consider the positive solution:

x = 1/2

Now, we can integrate to find the area. We integrate the difference between the curves with respect to x, from 0 to 1/2:

∫[0 to 1/2] (4[tex]x^{2}[/tex] - 1) dx

Integrating the above expression:

[4/3∗x3−x]from0to1/2

=(4/3∗(1/2)3−1/2)−(0−0)

=(4/3∗1/8−1/2)

=1/6−1/2

=−1/3

Since the area cannot be negative, we take the absolute value:

|-1/3| = 1/3

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simplify the expression [tex]\sqrt{x}[/tex] · [tex]2\sqrt[3]{x}[/tex] . Assume all variables are positive

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The value of simplified expression is 2 * x^(5/6).

We are given that;

The expression= x^(1/2) * 2 * x^(1/3)

Now,

To simplify the expression x^(1/2) * 2 * x^(1/3), we can use the following steps:

First, we can use the property of exponents that says a^m * a^n = a^(m+n) to combine the terms with x. This gives us:

x^(1/2) * 2 * x^(1/3) = 2 * x^(1/2 + 1/3)

Next, we can find a common denominator for the fractions in the exponent. The least common multiple of 2 and 3 is 6, so we can multiply both fractions by an appropriate factor to get:

x^(1/2 + 1/3) = x^((1/2) * (3/3) + (1/3) * (2/2)) = x^((3/6) + (2/6)) = x^(5/6)

Finally, we can write the simplified expression as:

x^(1/2) * 2 * x^(1/3) = 2 * x^(5/6)

Therefore, by the expression the answer will be 2 * x^(5/6).

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Calculate the following improper integrals! 7/2 +oo 1 3x² + 4 dx (5.1) | (5.2) / tan(x) dx 0

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To calculate the improper integrals, we need to evaluate the integrals of the given functions over their respective intervals.

The first integral involves the function f(x) = 3x^2 + 4, and the interval is from 7/2 to positive infinity. The second integral involves the function g(x) = tan(x), and the interval is from 5.1 to 5.2.

For the first integral, ∫(7/2 to +oo) (3x^2 + 4) dx, we consider the limit as the upper bound approaches infinity. We rewrite the integral as ∫(7/2 to R) (3x^2 + 4) dx, where R is a variable representing the upper bound. We then calculate the integral as the antiderivative of the function 3x^2 + 4, which is x^3 + 4x. Next, we evaluate the integral from 7/2 to R and take the limit as R approaches infinity. By plugging in the upper and lower bounds into the antiderivative and taking the limit, we can determine if the integral converges or diverges.

For the second integral, ∫(5.1 to 5.2) tan(x) dx, we evaluate the integral directly. The integral of tan(x) is -ln|cos(x)|. We substitute the upper and lower bounds into the antiderivative and calculate the difference. This will give us the value of the integral over the given interval.

By following these steps, we can determine the values of the improper integrals and determine if they converge or diverge.

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the arithmetic mean of four numbers is 15. two of the numbers are 10 and 18 and the other two are equal. what is the product of the two equal numbers?

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The arithmetic mean of four numbers is 15. two of the numbers are 10 and 18 and the other two are equal. So the product of the two equal numbers is 256.

To find the arithmetic mean of four numbers, you add them all up and then divide by four. So if the mean is 15 and two of the numbers are 10 and 18, then the sum of all four numbers must be:
15 x 4 = 60
We know that two of the numbers are 10 and 18, which add up to 28. So the sum of the other two numbers must be:
60 - 28 = 32
Since the other two numbers are equal, we can call them x. So:
2x = 32
x = 16
Therefore, the two equal numbers are both 16, and their product is:
16 x 16 = 256
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Approximate the sum of the series correct to four decimal places.
∑[infinity]n=(−1)n+1 /6n

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The series in question appears to be an alternating series. The nth term of an alternating series is of the form (-1)^(n+1) * a_n, where a_n is a sequence of positive numbers that decreases to zero. Here, a_n = 1/(6n).

To approximate the sum of an alternating series to a certain degree of accuracy, we can use the Alternating Series Estimation Theorem. According to the theorem, the absolute error of using the sum of the first N terms to approximate the sum of the entire series is less than or equal to the (N+1)th term.

So, you would need to find the smallest N such that 1/(6*(N+1)) < 0.0001, as we want the approximation to be correct to four decimal places. Then, sum the first N terms of the series to get the approximation.

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Homework: 12.2 Question 4, 12.2.29 Part 1 of 2 Find the largest open intervals on which the function is concave upward or concave downward, and find the location of any points of inflection 1 f(x)= X-9 Select the correct choice below and fill in the answer boxes to complete your choice (Type your answer in interval notation. Use a comma to separate answers as needed. Use integers or fractions for any numbers in the expression) O A. The function is concave upward on and concave downward on B. The function is concave downward on There are no intervals on which the function is concave upward C. The function is concave upward on There are no intervals on which the function is nca downward

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There are no intervals on which the function f(x) is concave upward or concave downward.

to determine the intervals on which the function f(x) = x - 9 is concave upward or concave downward, we need to analyze its second derivative.

the first derivative of f(x) is f'(x) = 1, and the second derivative is f''(x) = 0.

since the second derivative f''(x) = 0 is constant, it does not change sign. in other words, the function f(x) = x - 9 is neither concave upward nor concave downward, as the second derivative is identically zero.

hence, the correct choice is:

c. the function is concave upward on ∅ (empty set).there are no intervals on which the function is concave downward.

please note that in this case, the function is a simple linear function, and it does not exhibit any curvature or inflection points.

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1/5 -, -15x3. Find the total area of the region between the x-axis and the graph of y=x!

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The total area between the x-axis and the graph of [tex]y = x^{(1/5)} - x[/tex], -1 ≤ x ≤ 3, is [tex](5/6)(3)^{(6/5)} - (9/2)[/tex].

What is integration?

The summing of discrete data is indicated by the integration. To determine the functions that will characterise the area, displacement, and volume that result from a combination of small data that cannot be measured separately, integrals are calculated.

To find the total area of the region between the x-axis and the graph of y = x^(1/5) - x, we need to integrate the absolute value of the function over the given interval.

First, let's split the interval into two parts where the function changes sign: -1 ≤ x ≤ 0 and 0 ≤ x ≤ 3.

For -1 ≤ x ≤ 0:

In this interval, the graph lies below the x-axis. To find the area, we'll integrate the negated function: ∫[tex](-x^{(1/5)} + x) dx[/tex].

∫[tex](-x^{(1/5)} + x) dx[/tex] = -∫[tex]x^{(1/5)} dx[/tex] + ∫x dx

                     = [tex]-((5/6)x^{(6/5)}) + (1/2)x^2 + C[/tex]

                     = [tex](1/2)x^2 - (5/6)x^{(6/5)} + C_1[/tex],

where [tex]C_1[/tex] is the constant of integration.

For 0 ≤ x ≤ 3:

In this interval, the graph lies above the x-axis. To find the area, we'll integrate the function as is: ∫[tex](x^{(1/5)} - x) dx[/tex].

∫[tex](x^{(1/5)} - x) dx = (5/6)x^{(6/5)} - (1/2)x^2 + C_2,[/tex]

where [tex]C_2[/tex] is the constant of integration.

Now, to find the total area between the x-axis and the graph, we need to find the definite integral of the absolute value of the function over the interval -1 ≤ x ≤ 3:

Area = ∫[tex][0,3] |x^{(1/5)} - x| dx[/tex] = ∫[0,3] [tex](x^{(1/5)} - x) dx[/tex] - ∫[-1,0] [tex](-x^{(1/5)} + x) dx[/tex]

                                 = [tex][(5/6)x^{(6/5)} - (1/2)x^2][/tex] from 0 to 3 - [tex][(1/2)x^2 - (5/6)x^{(6/5)}][/tex] from -1 to 0

                                 = [tex][(5/6)(3)^{(6/5)} - (1/2)(3)^2] - [(1/2)(0)^2 - (5/6)(0)^{(6/5)}][/tex]

                                 = [tex][(5/6)(3)^{(6/5)} - (1/2)(9)] - [0 - 0][/tex]

                                 = [tex](5/6)(3)^{(6/5)} - (9/2[/tex]).

Therefore, the total area between the x-axis and the graph of [tex]y = x^{(1/5)} - x[/tex], -1 ≤ x ≤ 3, is [tex](5/6)(3)^{(6/5)} - (9/2)[/tex].

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The complete question is:

Find the total area of the region between the x-axis and the graph of y=x ^1/5 - x, -1 ≤ x ≤ 3.

3. A particle starts moving from the point (2,1,0) with velocity given by v(t) = (2t, 2t - 1,2 - 4t), where t > 0. (a) (3 points) Find the particle's position at any time t. (b) (4 points) What is the cosine of the angle between the particle's velocity and acceleration vectors when the particle is at the point (6,3,-4)? (c) (3 points) At what time(s) does the particle reach its minimum speed?

Answers

a) The position function is x(t) = t^2 + 2, y(t) = t^2 - t + 1, z(t) = 2t - 2t^2

b) Tthe cosine of the angle between the velocity and acceleration vectors at the point (6, 3, -4) is: cosθ = (v(6) · a(6)) / (|v(6)| |a(6)|) = 134 / (sqrt(749) * sqrt(24))

c) The particle reaches its minimum speed at t = 1/12.

(a) To find the particle's position at any time t, we need to integrate the velocity function with respect to time. The position function can be obtained by integrating each component of the velocity vector.

Given velocity function: v(t) = (2t, 2t - 1, 2 - 4t)

Integrating the x-component:

x(t) = ∫(2t) dt = t^2 + C1

Integrating the y-component:

y(t) = ∫(2t - 1) dt = t^2 - t + C2

Integrating the z-component:

z(t) = ∫(2 - 4t) dt = 2t - 2t^2 + C3

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

Now, to determine the specific values of the constants, we can use the initial position given as (2, 1, 0) when t = 0.

x(0) = 0^2 + C1 = 2 --> C1 = 2

y(0) = 0^2 - 0 + C2 = 1 --> C2 = 1

z(0) = 2(0) - 2(0)^2 + C3 = 0 --> C3 = 0

Therefore, the position function is:

x(t) = t^2 + 2

y(t) = t^2 - t + 1

z(t) = 2t - 2t^2

(b) To find the cosine of the angle between the velocity and acceleration vectors, we need to find both vectors at the given point (6, 3, -4) and then calculate their dot product.

Given velocity function: v(t) = (2t, 2t - 1, 2 - 4t)

Given acceleration function: a(t) = (d/dt) v(t) = (2, 2, -4)

At the point (6, 3, -4), let's find the velocity and acceleration vectors.

Velocity vector at t = 6:

v(6) = (2(6), 2(6) - 1, 2 - 4(6)) = (12, 11, -22)

Acceleration vector at t = 6:

a(6) = (2, 2, -4)

Now, let's calculate the dot product of the velocity and acceleration vectors:

v(6) · a(6) = (12)(2) + (11)(2) + (-22)(-4) = 24 + 22 + 88 = 134

The magnitude of the velocity vector at t = 6 is:

|v(6)| = sqrt((12)^2 + (11)^2 + (-22)^2) = sqrt(144 + 121 + 484) = sqrt(749)

The magnitude of the acceleration vector at t = 6 is:

|a(6)| = sqrt((2)^2 + (2)^2 + (-4)^2) = sqrt(4 + 4 + 16) = sqrt(24)

Therefore, the cosine of the angle between the velocity and acceleration vectors at the point (6, 3, -4) is:

cosθ = (v(6) · a(6)) / (|v(6)| |a(6)|) = 134 / (sqrt(749) * sqrt(24))

(c) To find the time(s) when the particle reaches its minimum speed, we need to determine when the magnitude of the velocity vector is at its minimum.

Given velocity function: v(t) = (2t, 2t - 1, 2 - 4t)

The magnitude of the velocity vector is:

|v(t)| = sqrt((2t)^2 + (2t - 1)^2 + (2 - 4t)^2) = sqrt(4t^2 + 4t^2 - 4t + 1 + 4 - 16t + 16t^2)

= sqrt(24t^2 - 4t + 5)

To find the minimum speed, we can take the derivative of |v(t)| with respect to t and set it equal to 0, then solve for t.

d|v(t)| / dt = 0

(1/2) * (24t^2 - 4t + 5)^(-1/2) * (48t - 4) = 0

Simplifying:

48t - 4 = 0

48t = 4

t = 1/12

Therefore, the particle reaches its minimum speed at t = 1/12.

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Use the divergence theorem to evaluate SI F:ds where S -1 = 2 F(x, y, z) = (x +2yz? i + (4y +tan (x?z)) j+(2z+sin-(2xy?)) k and S is the outward-oriented surface of the solid E bounded by the parabolo

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The divergen theorm also known as Gauss's theorem, is a fundamental theorem in vector calculus that relates the outward flux of a vector field through a closed surface to the divergence of the field inside the surface.

Here, we will use the divergence theorem to evaluate SI F:ds where S -1 = 2 F(x, y, z) = (x +2yz? i + (4y +tan (x?z)) j+(2z+sin-(2xy?)) k and S is the outward-oriented surface of the solid E bounded by the parabolo.The given vector field is F(x, y, z) = (x + 2yz)i + (4y + tan(xz))j + (2z - sin(2xy))k. The solid E is bounded by the paraboloid z = 4 - x² - y² and the plane z = 0. Therefore, the surface S is the boundary of E oriented outward. By the divergence theorem, we know that: ∫∫S F · dS = ∭E ∇ · F dV Here, ∇ · F is the divergence of F. Let's calculate the divergence of F: ∇ · F = (∂/∂x)(x + 2yz) + (∂/∂y)(4y + tan(xz)) + (∂/∂z)(2z - sin(2xy))= 1 + 2y + xzsec²(xz) + 2cos(2xy) Now, using the divergence theorem, we can write: ∫∫S F · dS = ∭E ∇ · F dV= ∭E (1 + 2y + xzsec²(xz) + 2cos(2xy)) dVWe can change the integral to cylindrical coordinates: x = r cosθ, y = r sinθ, and z = z. The Jacobian is r. The bounds for r and θ are 0 to 2 and 0 to 2π, respectively, and the bounds for z are 0 to 4 - r². Therefore, the integral becomes: ∫∫S F · dS = ∭E (1 + 2y + xzsec²(xz) + 2cos(2xy)) dV= ∫₀² ∫₀² ∫₀^(4 - r²) (1 + 2r sinθ + r² cosθ zsec²(r²cosθsinθ)) + 2cos(2r²sinθcosθ)) r dz dr dθThis integral is difficult to evaluate analytically. Therefore, we can use a computer algebra system to get the numerical result.

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2. Evaluate the line integral R = Icy?dx + xdy, where C is the arc of the parabola r = 4 - y from (-5.-3) to (0.2).

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The line integral R is equal to -22.5. to evaluate the line integral, we parameterize the parabola as x = t and y = 4 - t^2, where t ranges from -3 to 2. We then substitute these expressions into the integrand and integrate with respect to t.

After simplifying, we find R = -22.5. This indicates that the line integral along the given arc of the parabola is -22.5.

To evaluate the line integral R, we first need to parameterize the given arc of the parabola. We can do this by expressing x and y in terms of a parameter, let's say t. For the given parabola, we have x = t and y = 4 - t^2.

Next, we substitute these parameterizations into the integrand, which is Icy?dx + xdy. This gives us the expression (4 - t^2)(dt) + t(2tdt).

[tex]Simplifying the expression, we have 4dt - t^2dt + 2t^2dt.[/tex]

Now, we integrate this expression with respect to t, considering the given limits of t from -3 to 2.

[tex]Integrating term by term, we get 4t - (t^3/3) + (2t^3/3).[/tex]

Evaluating this expression at the upper limit t = 2 and subtracting the value at the lower limit t = -3, we find R = (8 - 8/3 + 16/3) - (-12 + 27/3 - 54/3) = -22.5. therefore, the line integral R is equal to -22.5 along the given arc of the parabola.

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Question 8
8. DETAILS LARCALC11 9.5.013.MI. Determine the convergence or divergence of the series. (If you need to use coorco, enter INFINITY or -INFINITY, respectively.) 00 (-1)"(8n - 1) 5 + 1 n = 1 8n - 1 lim

Answers

To determine the convergence or divergence of the series                       Σ[tex]((-1)^{n+1}/ (8n - 1)^{5+1})[/tex], n = 1 to ∞, we need to find the limit of the general term of the series as n approaches infinity.

Let's analyze the general term of the series, given by [tex]a_n = (-1)^{(n+1} ) / (8n - 1)^{5+1}[/tex].

As n approaches infinity, we can observe that the denominator [tex](8n - 1)^{5 + 1}[/tex] becomes larger and larger, while the numerator (-1)^(n+1) alternates between -1 and 1.

Since the series is an alternating series, we can apply the Alternating Series Test to determine its convergence or divergence. The test states that if the absolute values of the terms decrease monotonically to zero as n approaches infinity, then the series converges.

In this case, the denominator increases without bound, while the numerator alternates between -1 and 1. As a result, the absolute values of the terms do not approach zero. Therefore, the series diverges.

Hence, the series Σ[tex]((-1)^{n+1} ) / (8n - 1)^{5+1})[/tex] is divergent.

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Which statement is correct about the total number of functions from {a,b,c; to {1,21?
(A) The total number of functions from (1,2) to {a,b,c) is 9, and the number that are onto is 6.
(B) The total number of functions from (1,2) to {a,b,c) is 8, and the number that are onto is 6.
(C) The total number of functions from (1,2} to (a,b,c} is 9, and the number that are onto is 4.
(D) The total number of functions from {1,2) to {a,b,c) is 8, and the number that are onto is 4.

Answers

the correct statement about the total number of functions from {a,b,c; to {1,21 is (D) The total number of functions from {1,2) to {a,b,c) is 8, and the number that are onto is 4.

The total number of functions from {a, b, c} to {1, 2} is calculated by multiplying the cardinalities of the two sets.

Hence, the total number of functions is [tex]2^3 = 8[/tex](since there are three elements in the set {a, b, c} and two elements in the set {1, 2}).

Onto Function: A function f from set A to set B is called onto function if every element of B is the image of some element of A, which means that every element of B is a function of A.

We are asked to find the number of onto functions between these sets.

We know that if |A| < |B|, then there are no onto functions from A to B.

Here, |A| = 3 and |B| = 2. So, there cannot be an onto function from A to B.

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2 f(x) = x^ - 15; Xo = 4 x К ХК k xk 0 6 1 7 2 8 W N 3 9 4 10 5 (Round to six decimal places as needed.)

Answers

To find the values of f(x) for the given function [tex]f(x) = x^{-15}[/tex], we need to substitute the given values of x into the function.

Using the values of x from 0 to 5, we can calculate f(x) as follows:

For x = 0: [tex]f(0) = 0^{-15}[/tex] = undefined (since any number raised to the power of -15 is undefined)

For x = 1: f(1) = [tex]1^{-15}[/tex] = 1

For x = 2: f(2) = [tex]2^{-15}[/tex] = 0.0000305176

For x = 3: f(3) =[tex]3^{-15}[/tex] = 2.7750e-23

For x = 4: f(4) = [tex]4^{-15}[/tex] = 1.5259e-28

For x = 5: f(5) = [tex]5^{-15}[/tex] = 3.0518e-34

Rounding these values to six decimal places, we have:

f(0) = undefined

f(1) = 1

f(2) = 0.000031

f(3) = 2.7750e-23

f(4) = 1.5259e-28

f(5) = 3.0518e-34

These are the calculated values of f(x) for the given function and corresponding values of x from 0 to 5.

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Find the critical numbers and then say where the function is increasing and where it is decreasing.

y = x^4/5 + x^9/5

Answers

a. The critical numbers of the function  y = x⁴/⁵ + x⁹/⁵ are (-4/9, 10√8/9)

b. The function is decreasing

What are the critical numbers of a function?

The critical number of a function are the maximum or minimum points of the curve.

a. To find the critical numbers of the function y = x⁴/⁵ + x⁹/⁵,we proceed as follows

To find the critical numbers of the function, we differentiate the function with respect to x and equate to zero.

So, y = x⁴/₅ + x⁹/₅

dy/dx = d(x⁴/₅)/dx + d(x⁹/₅)/dx

= (4/5)x⁻¹/₅ +  (9/5)x⁻⁴/⁵

Equating it to zero, we have that

dy/dx = 0

(4/5)x⁻¹/₅ +  (9/5)x⁻⁴/⁵ = 0

(4/5)x⁻¹/₅ =  -(9/5)x⁻⁴/⁵

Dividing both sides by 4/5, we have

(4/5)x⁻¹/₅/(4/5) =  -(9/5)x⁻⁴/⁵/(4/5)

x⁻¹/₅ =  -(9/4)x⁻⁴/⁵

Dividing both sides by x⁻⁴/⁵, we have that

x⁻¹/₅/ x⁻⁴/⁵ =  -(9/4)x⁻⁴/⁵/ x⁻⁴/⁵

x⁻¹ = -9/4

x = -4/9

So, substituting x = -4/9 into the equation for y, we have that

y = (-4/9)⁴/₅ + (-4/9)⁹/₅

y = (-4/9)⁴/₅[1 + (-4/9)⁵/₅]

y = (-4/9)⁴/₅[1 + (-4/9)]

y = (-4/9)⁴/₅[1 - 4/9)]

y = (-4/9)⁴/₅[(9 - 4)/9)]

y = (-4/9)⁴/₅[5/9)]

y =⁵√ (256/6561)[5/9)]

y =⁵√ (256/59049)[5]

y =2√8/9 × [5]

y =10√8/9

So, the critical numbers are (-4/9, 10√8/9)

b. To determine whether the function is increasing or decreasing, we differentiate its first derivative and substitute in the value of x. so,

dy/dx = (4/5)x⁻¹/₅ +  (9/5)x⁻⁴/⁵

d(dy/dx) = d[(4/5)x⁻¹/₅ +  (9/5)x⁻⁴/⁵]/dx

d²y/dx² = d[(4/5)x⁻¹/₅]dx +  d[(9/5)x⁻⁴/⁵]/dx

d²y/dx² = -1/5 × (4/5)x⁻⁶/₅]dx +  -4/5 × [(9/5)x⁻⁹/⁵]/dx

= -(4/25)x⁻⁶/₅  - (36/25)x⁻⁹/⁵

Substituting in the value of x = -4/9, we have that

d²y/dx² = -(4/25)x⁻⁶/₅  - (36/25)x⁻⁹/⁵

= -(4/25)(-4/9)⁻⁶/₅  - (36/25)(-4/9)⁻⁹/⁵

= (4/25)(9/4)⁶/₅  + (36/25)(9/4)⁹/⁵

= (4/25)(531441/4096)¹/₅  + (36/25)(387420489/262144)¹/⁵

= (4/25)(9⁵√9/4⁵√4)  + (36/25)(9⁵√9⁴/16)

= (1/25)(9⁵√9/4⁴√4)  + (36/25)(9⁵√9⁴/16)

= 9⁵√9/4⁴[1/2 + 36/25 × 27]

= 9⁵√9/4⁴[25 + 1944]/50]

= 9⁵√9/4⁴[1969]/50]

Since d²y/dx² = 9⁵√9/4⁴[1969]/50] > 0,

The function is decreasing

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which of the following situations can be modeled by a function whose value changes at a constant rate per unit of time? select all that apply. a the population of a city is increasing 5% per year. b the water level of a tank falls by 5 gallons every day. c the number of reptiles in the zoo increases by 5 reptiles each year. d the amount of money collected by a charity increases by 5 times each year.

Answers

b) The water level of a tank falls by 5 gallons every day.

c) The number of reptiles in the zoo increases by 5 reptiles each year.

In both scenarios, the values change by a fixed amount consistently over a specific unit of time, indicating a constant rate of change.

The situations that can be modeled by a function whose value changes at a constant rate per unit of time are:

a) The population of a city is increasing 5% per year. This scenario represents a constant growth rate over time, where the population changes by a fixed percentage annually.

b) The water level of a tank falls by 5 gallons every day. Here, the water level decreases by a fixed amount (5 gallons) consistently each day.

c) The number of reptiles in the zoo increases by 5 reptiles each year. This situation represents a constant annual increase in the reptile population, with a fixed number of reptiles being added each year.

These three scenarios involve changes that occur at a constant rate per unit of time, making them suitable for modeling using a function with a constant rate of change.

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Suppose that a population P(t) follows the following Gompertz differential equation. dP = 5P(16 - In P), dt with initial condition P(0) = 50. (a) What is the limiting value of the population? (b) What

Answers

the population will approach and stabilize at approximately 8886110.52 individuals, assuming the Gompertz differential equation accurately models the population dynamics.

The Gompertz differential equation is given by dP/dt = 5P(16 - ln(P)), where P(t) represents the population at time t. To find the limiting value of the population, we need to solve the differential equation and find its equilibrium solution, which occurs when dP/dt = 0.Setting dP/dt = 0 in the Gompertz equation, we have 5P(16 - ln(P)) = 0. This equation holds true when P = 0 or 16 - ln(P) = 0.Firstly, if P = 0, it implies an extinction of the population, which is not a meaningful solution in this case.

To find the non-trivial equilibrium solution, we solve the equation 16 - ln(P) = 0 for P. Taking the natural logarithm of both sides gives ln(P) = 16, and solving for P yields P = e^16.Therefore, the limiting value of the population is e^16, approximately equal to 8886110.52.

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According to the 2010 census, Chicago is the third-largest city in the United States. In 2011, its population was 2,707,000, an increase of 0.4% compared to the previous year. a. Assuming that the populations of Chicago and Houston are growing exponentially, write an equation that can be used to predict when the population of Houston will equal that of Chicago. b. Solve your equation. For each step, list a property or give an explanation. Then interpret the solution.

Answers

a. An equation that can be used to predict when the population of Houston will equal that of Chicago is [tex]$2.145 \cdot 1.022^x=2.707 \cdot 1.004^x$[/tex]

b. The population will be the same at some point during the year of 2011+13 = 2024.

What is population increase?

Pοpulatiοn grοwth is the increase in the number οf humans οn Earth. Fοr mοst οf human histοry οur pοpulatiοn size was relatively stable.

a.

Let g(x) represent the population of Chicago in millions, x years after 2011. If the population of Chicago grows at 0.4 % each year, then the population is multiplied by 1.004 every year.

Thus

[tex]g(x)=2.707 \cdot \underbrace{1.004 \cdot 1.004 \cdots 1.004}_{x \text { times }}=2.707 \cdot 1.004^x[/tex]

we found f(x) as

[tex]f(x)=2.145 \cdot 1.022^x[/tex]

to represent the population of Houston. Then the populations will be equal when f(x)=g(x), or

[tex]2.145 \cdot 1.022^x=2.707 \cdot 1.004^x[/tex]

b.

There are several ways to solve this equation. Here is an example:

[tex]$$\begin{gathered}2.145 \cdot 1.022^x=2.707 \cdot 1.004^x \\\log \left[2.145 \cdot 1.022^x\right]=\log \left[2.707 \cdot 1.004^x\right] \\\log 2.145+\log 1.022^x=\log 2.707+\log 1.004^x \\\log 2.145+x \log 1.022=\log 2.707+x \log 1.004 \\x \log 1.022-x \log 1.004=\log 2.707-\log 2.145 \\x(\log 1.022-\log 1.004)=\log 2.707-\log 2.145 \\x=\frac{\log 2.707-\log 2.145}{\log 1.022-\log 1.004} \\x \approx 13.10\end{gathered}$$[/tex]

As x represents the number of years after 2011, then we conclude the population will be the same at some point during the year of 2011+13 = 2024.

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A researcher is told that the average age of respondents in a survey is 49 years. She is interested in finding out if most respondents are close to 49 years old. The measure that would most accurately answer this question is: a. mean. b. median. c. mode. d. range. e. standard deviation.

Answers

The researcher should use the measure of e. standard deviation. This is because standard deviation provides an indication of the dispersion or spread of the data around the mean.

Helping to understand how close the ages are to the average (49 years).The measure that would most accurately answer the researcher's question is the median. The median is the middle value in a dataset, so if most respondents are close to 49 years old, the median would also be close to 49 years old.

The mean could also be used to answer this question, but it could be skewed if there are outliers in the dataset. The mode, range, and standard deviation are not as useful in determining if most respondents are close to 49 years old.

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Help me math!!!!!!!!!!

Answers

Answer:

the answer for w = -4 is -32

Step-by-step explanation:

this is a question on functions.

we take each value of w and substitute it into the function (the expression on the right). the first one is done, as you can see.

first we take -4, and everywhere we see w in the function, we replace it with -4.

[tex]-4^{3}[/tex]  - 5(-4) + 12

-4 cubed is -64 (because -4 squared is 16, so multiply that by -4 again to get -4 cubed)

-5 times -4 is positive 20

and we already have the 12

so we have:   -64 + 20 + 12

which is  -44 + 12

which equals  -32

simply repeat this process with all the other values of w

ask me again if you're stuck

good luck!

Find the radius of convergence, R, of the series.
[infinity] 3(−1)nnxn
sum.gif
n = 1
R =
Find the interval, I, of convergence of the series. (Enter your answer using interval notation.)
I =

Answers

The series is given by the expression ∑[infinity] 3(−1)nnxn, n = 1. The task is to find the radius of convergence, R, and the interval of convergence, I, for the series.

To find the radius of convergence, we can use the ratio test. Let's apply the ratio test to the series:

lim(n→∞) [tex]|\frac{(3(-1)^{(n+1)} * (n+1) * x^{(n+1)}}{ (3(-1)^n * n * x^n)} |[/tex]

Simplifying the expression, we get:

lim(n→∞) [tex]|\frac{(3(-1)^{(n+1)} * (n+1) * x^{(n+1)}}{ (3(-1)^n * n * x^n)} |[/tex]

= lim(n→∞) |(3 * (n+1) * x) / (n * x)|

= lim(n→∞) |3 * (n+1) / n|

= 3.

For the series to converge, the ratio should be less than 1. Therefore, |3| < 1, which is not true. Hence, the series diverges for all values of x. Consequently, the radius of convergence, R, is 0.

Since the series diverges for all x, the interval of convergence, I, is empty, represented by the notation I = {}.

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the annual salaries of a large company are normally distributed with a mean of $65,000 and a standard deviation of $18,000. if a random samples of 14 of these salaries are taken, then the standard deviation of that sample mean would equal $ .

Answers

The standard deviation of the sample mean would equal $4,812.71.

We would explain how standard error is used to estimate the standard deviation of the sample mean, which helps to determine the precision of our estimate of the population mean. We would also provide additional context and examples to help the reader understand the importance of standard error in statistical analysis.

The standard error is the standard deviation of the sampling distribution of the mean. In simpler terms, it measures how much the sample means vary from the population mean. The formula for standard error is:
SE = σ / sqrt(n)
where SE is the standard error, σ is the population standard deviation, and n is the sample size.
In this case, we are given that the population standard deviation is $18,000 and the sample size is 14. Plugging these values into the formula, we get:
SE = 18,000 / sqrt(14)
SE = 4,812.71

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Raul’s car averages 17.3 miles per gallon of gasoline. How many miles can Raul drive if he fills his tank with 10.5 gallons of gasoline

Answers

Answer:

181.65 miles

Step-by-step explanation:

17.3 mpg, where g is gallons

so we need 17.3 X 10.5

= 181.65

Find the equation of the tangent line to the graph
of x3 + y4 = y + 1
at the point (−1, −1).

Answers

The equation of the tangent line to the graph of x^3 + y^4 = y + 1 at the point (-1, -1) is 3x - 5y = 2.

To find the equation of the tangent line to the graph of the equation x^3 + y^4 = y + 1 at the point (-1, -1), we can use the concept of implicit differentiation.

1. Start by differentiating both sides of the equation with respect to x:

  d/dx(x^3 + y^4) = d/dx(y + 1)

2. Differentiating each term:

  3x^2 + 4y^3(dy/dx) = dy/dx

3. Substitute the coordinates of the point (-1, -1) into the equation:

  3(-1)^2 + 4(-1)^3(dy/dx) = dy/dx

  Simplifying the equation:

  3 - 4(dy/dx) = dy/dx

4. Move the dy/dx terms to one side of the equation:

  3 = 5(dy/dx)

5. Solve for dy/dx:

  dy/dx = 3/5

Now we have the slope of the tangent line at the point (-1, -1), which is dy/dx = 3/5.

6. Use the point-slope form of a linear equation to find the equation of the tangent line:

  y - y1 = m(x - x1), where (x1, y1) is the point on the line and m is the slope.

  Substituting the values into the equation:

  y - (-1) = (3/5)(x - (-1))

  Simplifying:

  y + 1 = (3/5)(x + 1)

7. Convert the equation to the standard form:

  5y + 5 = 3x + 3

  Rearrange:

 ∴ 3x - 5y = 2

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(1 point) Let S(x) = 4(x - 2x for x > 0. Find the open intervals on which ſ is increasing (decreasing). Then determine the x-coordinates of all relative maxima (minima). I 1. ſ is increasing on the

Answers

The function S(x) = 4(x - 2x) for x > 0 is increasing on the open interval (0, +∞) and does not have any relative maxima or minima.

To determine the intervals on which S(x) is increasing or decreasing, we need to examine the derivative of S(x). Taking the derivative of S(x) with respect to x, we get:

S'(x) = 4(1 - 2) = -4

Since the derivative is a constant (-4) and negative, it means that S(x) is decreasing for all values of x. Therefore, S(x) does not have any relative maxima or minima.

In terms of intervals, the function S(x) is decreasing on the entire domain of x > 0, which means it is decreasing on the open interval (0, +∞). Since it is always decreasing and does not have any turning points, there are no relative maxima or minima to be found.

In summary, the function S(x) = 4(x - 2x) for x > 0 is increasing on the open interval (0, +∞), and it does not have any relative maxima or minima.

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= (8 points) Find the maximum and minimum values of f(2, y) = fc +y on the ellipse 22 + 4y2 = 1 maximum value minimum value:

Answers

The maximum value of f(2, y) = fc + y on the ellipse 22 + 4y2 = 1 is 1.5, and the minimum value is -0.5.

To find the maximum and minimum values of f(2, y) on the given ellipse, we substitute the equation of the ellipse into f(2, y). This gives us f(2, y) = fc + y = 1 + y. Since the ellipse is centered at (0,0) and has a major axis of length 1, its maximum and minimum values occur at the points where y is maximized and minimized, respectively. Plugging these values into f(2, y) gives us the maximum of 1.5 and the minimum of -0.5.

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thanks in advanced! :)
Set up the integral to find the exact length of the curve. Completely simplify the integrand. DO NOT EVALIUATE THE INTEGRAL. x=t+ √t,y=t-√√t,0st≤1

Answers

The integral to find the exact length of the curv is L = ∫[0,1] √[2 + (5/4)t^(-1)] dt

To find the exact length of the curve defined by the parametric equations x = t + √t and y = t - √t, where 0 ≤ t ≤ 1, we can use the arc length formula:

L = ∫[a,b] √[dx/dt² + dy/dt²] dt

In this case, we need to find dx/dt and dy/dt, and then substitute them into the arc length formula.

1. Find dx/dt:

dx/dt = d/dt(t + √t) = 1 + (1/2)t^(-1/2)

2. Find dy/dt:

dy/dt = d/dt(t - √√t) = 1 - (1/2)(√t)^(-1/2)(1/2)t^(-1/2)

Now, substitute dx/dt and dy/dt into the arc length formula:

L = ∫[0,1] √[(1 + (1/2)t^(-1/2))² + (1 - (1/2)(√t)^(-1/2)(1/2)t^(-1/2))²] dt

To simplify the integrand further, we can expand and simplify the square terms:

L = ∫[0,1] √[1 + t^(-1) + t^(-1) + (1/4)t^(-1)] dt

Simplifying further, we have:

L = ∫[0,1] √[2 + (5/4)t^(-1)] dt

Therefore, the setup for the integral to find the exact length of the curve is:

L = ∫[0,1] √[2 + (5/4)t^(-1)] dt

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Now create a query to display all the employees and all the departments, using joins.( for question 5DROP TABLE EMP2;CREATE TABLE EMP2 (EMPNO NUMBER(4) NOT NULL,ENAME CHAR(10),JOB CHAR(9),MGR NUMBER(4) CONSTRAINT EMP2_SELF_KEYREFERENCES EMP2 (EMPNO),HIREDATE DATE,SAL NUMBER(7,2),COMM NUMBER(7,2),DEPTNO NUMBER(2),CONSTRAINT EMP2_FOREIGN_KEY FOREIGN KEY (DEPTNO) REFERENCES DEPT (DEPTNO),CONSTRAINT EMP2_PRIM_KEY PRIMARY KEY (EMPNO));INSERT INTO EMP2 VALUES (7839,'KING','PRESIDENT',NULL,'17-NOV-1981',5000,NULL,10);INSERT INTO EMP2 VALUES (7698,'BLAKE','MANAGER',7839,'1-MAY-1981',2850,NULL,30);INSERT INTO EMP2 VALUES (7782,'CLARK','MANAGER',7839,'9-JUN-1981',2450,NULL,10);INSERT INTO EMP2 VALUES (7566,'JONES','MANAGER',7839,'2-APR-1981',2975,NULL,20);INSERT INTO EMP2 VALUES (7654,'MARTIN','SALESMAN',7698,'28-SEP-1981',1250,1400,30);INSERT INTO EMP2 VALUES (7499,'ALLEN','SALESMAN',7698,'20-FEB-1981',1600,300,30);INSERT INTO EMP2 VALUES (7844,'TURNER','SALESMAN',7698,'8-SEP-1981',1500,0,30);INSERT INTO EMP2 VALUES (7900,'JAMES','CLERK',7698,'3-DEC-1981',950,NULL,30);INSERT INTO EMP2 VALUES (7521,'WARD','SALESMAN',7698,'22-FEB-1981',1250,500,30);INSERT INTO EMP2 VALUES (7902,'FORD','ANALYST',7566,'3-DEC-1981',3000,NULL,20);INSERT INTO EMP2 VALUES (7369,'SMITH','CLERK',7902,'17-DEC-1980',800,NULL,20);INSERT INTO EMP2 VALUES (7788,'SCOTT','ANALYST',7566,'09-DEC-1982',3000,NULL,NULL);INSERT INTO EMP2 VALUES (7876,'ADAMS','CLERK',7788,'12-JAN-1983',1100,NULL,NULL);INSERT INTO EMP2 VALUES (7934,'MILLER','CLERK',7782,'23-JAN-1982',1300,NULL,NULL);commit; Use the properties of logarithms to solve the equation forx.log 4 (5x 29) = 22)Rewrite the expression as a single logarithm.1/2 ln x 5 ln(x 4)3)Find the indicated value.Iff(x) = a 2.00-l flask contains nitrogen gas at 25c and 1.00 atm pressure. what is the final pressure in the flask if an additional 2.00 g of n2 gas is added to the flask and the flask cooled to -55c? please show and explain how you got the answerPractice Problems 1. Evaluate the following integrals: In x dx [Hint: Integration by parts] 13 sin (7x) dx [Hint: Double-angle formula] 9-x dx [Hint: Trigonometric substitution] [cosx cos Read this scene from a play. Then, answer the question(s) about it.Scene 3. A small-town bank of the 1950s. Seated in a formal office behind a large wooden desk is RUDOLPH SLOAN, dressed in a suit and tie. In a chair facing the desk is MANDY MARVIN, more casually dressed.SLOAN. [politely] I regret it, Miss Marvin, but unfortunately the bankMANDY. [cheerfully interrupting] Look here, Rudy Sloan, all I'm askin' fer is a little loan to tide me over. Why, back when you was just a tyke, yer daddy loaned my daddy the money to start the Movie Palace in the first place.SLOAN. Indeed, and back then movie theaters were a fine investment. But now, people don't go to the movies like they used to; they stay home and watch TVMANDY. Aw, TV, that's just a passing fad!SLOAN. I don't think so, Mandy. [sighs] Look, the bank is willing to give you a loan if you agree to make the theater smaller and rent out half to Tom PoeMANDY. Tom Poe? No way am I renting out half my theater to a car dealership. I won't do it! I won't change one thing from the way my daddy done it. What good's a picture on an itty bitty screen? It may as well be on that there TV.The following question has two parts. Answer Part A first, and then Part B.Part AChoose the word that best completes the sentence.From her characterization in this scene, the reader can infer that one of Mandy Marvin's flaws is Choose...Part BWhich line from the text provides the best support for the answer to Part A? A. Look here, Rudy Sloan, all Im askin fer is a little loan to tide me over. B. Aw, TV, thats just a passing fad! C. I wont change one thing from the way my daddy done it. D. What goods a picture on an itty bitty screen? 00 an+1 When we use the Ration Test on the series (-7)1+8n (n+1) n2 51+n we find that the limit lim and hence the series is 00 an n=2 divergent convergent which of the following statements about meiosis in humans is true?multiple select question.used for sexual reproductiondaughter cells are genetically differentchromosome number of daughter cells is the same as that of the parent cellinvolves a single divisionproduces four daughter cells per cycle All of these are factors that could lead to a decision to buy or outsource rather than make or perform in-house, EXCEPT:a. flexibility in procurementb. ability to use specialized suppliersc. inadequate capacityd. more control ov