PROBLEM 2: Evaluate the following in maple, first by direct integration, then decompose into separate fractions and integrate. a) ) - 4 od bla+vieta-1 * ſ. 27+51+2 blæ ?)2x+) os dr ) 5-x 3 2x2 5x drd) x-1 dx 2(x+1)

Answer:

True, making multiplication a game can motivate children to learn.

The derivatives of the given functions are :

(a) (f ◦ c)'(t) = et * (-sin(t) + cos(t))

(b) (f ◦ c)'(t) = (5t + 2) * e^(t(5t + 2) * 3t)

(c) (f ◦ c)'(t) = Simplified expression involving exponentials, logarithms, and derivatives of trigonometric functions.

(d) (f ◦ c)'(t) = exp(2t^2) + 2t * exp(2t^2)

To verify the first special case of the chain rule for the compositions, let's calculate the derivatives for each case:

(a) Given f(x, y) = xy and c(t) = (et, cos(t))

The composition is (f ◦ c)(t) = f(c(t)) = f(et, cos(t)) = (et * cos(t))

Taking the derivative, we have:

(f ◦ c)'(t) = (et * -sin(t) + cos(t) * et)

So, (f ◦ c)'(t) = et * (-sin(t) + cos(t))

(b) Given f(x, y) = exy and c(t) = (5t + 2, 3t)

The composition is (f ◦ c)(t) = f(c(t)) = f(5t + 2, 3t) = e^(t(5t + 2) * 3t)

Taking the derivative, we have:

(f ◦ c)'(t) = (5t + 2) * e^(t(5t + 2) * 3t)

(c) Given f(x, y) = (x^2 + y^2) log(x^2 + y^2) and c(t) = (et, e^-t)

The composition is (f ◦ c)(t) = f(c(t)) = f(et, e^-t) = (et^2 + e^-t^2) * log(et^2 + e^-t^2)

Taking the derivative, we have:

(f ◦ c)'(t) = (2et + (-e^-t)) * (et^2 + e^-t^2) * log(et^2 + e^-t^2) + (et^2 + e^-t^2) * (2et + (-e^-t)) * (1/(et^2 + e^-t^2)) * (2et + (-e^-t))

Simplifying the expression will give the final result.

(d) Given f(x, y) = x * exp(x^2 + y^2) and c(t) = (t, -t)

The composition is (f ◦ c)(t) = f(c(t)) = f(t, -t) = t * exp(t^2 + (-t)^2) = t * exp(2t^2)

Taking the derivative, we have:

(f ◦ c)'(t) = exp(2t^2) + 2t * exp(2t^2)

Please note that for case (c), the expression might be more complex due to the presence of logarithmic functions. It requires further simplification.

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The function f(x) = 8 - 6x equals its average value on the interval [0,6) at the point x = 3.

To find the average value of a function on an interval, we need to calculate the definite integral of the function over that interval and divide it by the length of the interval.

The average value of f(x) on the interval [0,6) is given by:

Average value = (1/(6-0)) * ∫[0,6) f(x) dx

The integral of f(x) = 8 - 6x is obtained by using the power rule for integration:

∫[0,6) (8 - 6x) dx = [8x - 3x^2/2] evaluated from 0 to 6

Evaluating the integral, we have:

[8(6) - 3(6^2)/2] - [8(0) - 3(0^2)/2] = 48 - 54 = -6

Therefore, the average value of f(x) on the interval [0,6) is -6.

To find the point(s) at which f(x) equals its average value, we set f(x) equal to -6:

8 - 6x = -6

Simplifying the equation, we have:

6x = 14

x = 14/6 = 7/3

Therefore, the function f(x) = 8 - 6x equals its average value on the interval [0,6) at the point x = 7/3.

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The current price of the unit trust, after 5 years, is approximately £13,863 to the nearest whole number of pounds.

To calculate the current price of the unit trust, we need to consider the two different periods: the last 3 years with a 7% annual increase and the period before that with a 3% annual decrease.

Calculation for the period with a 7% annual increase:

We'll start with the initial investment of £12,000 and calculate the value after each year.

Year 1: £12,000 + (7% of £12,000) = £12,840

Year 2: £12,840 + (7% of £12,840) = £13,759.80

Year 3: £13,759.80 + (7% of £13,759.80) = £14,747.67

Calculation for the period with a 3% annual decrease:

We'll take the value at the end of the third year (£14,747.67) and calculate the decrease for each year.

Year 4: £14,747.67 - (3% of £14,747.67) = £14,298.72

Year 5: £14,298.72 - (3% of £14,298.72) = £13,862.75

Therefore, the current price of the unit trust, after 5 years, is approximately £13,863 to the nearest whole number of pounds.

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The mass will reach a maximum height of 0.82 m above its equilibrium position before falling back down due to gravity.

We need to use the principles of Hooke's law and conservation of energy.

Hooke's law states that the force exerted by a spring is proportional to its displacement from equilibrium, and this relationship can be expressed mathematically as F = -kx, where F is the force, k is the spring constant, and x is the displacement.

Given that a mass of 3 kg stretches a spring 5/2, we can determine the spring constant using the formula k = (mg)/x, where m is the mass, g is the acceleration due to gravity, and x is the displacement.

Plugging in the values, we get:
k = (3 kg x 9.8 m/s^2)/(5/2 m) = 58.8 N/m

Now we can use the conservation of energy to find the maximum height that the mass will reach.

At the highest point, all of the potential energy is converted to kinetic energy, and vice versa at the lowest point.

Therefore, we can equate the initial potential energy to the final kinetic energy, using the formulas:
PE = mgh
KE = 1/2 mv^2

where PE is potential energy, KE is kinetic energy, m is the mass, h is the height, and v is the velocity.

Plugging in the values, we get:
PE = (3 kg x 9.8 m/s^2 x 1 m) = 29.4 J
KE = (1/2 x 3 kg x 4 m/s^2) = 6 J

Since energy is conserved, we can equate these two values and solve for h:
PE = KE
mgh = 1/2 mv^2
h = v^2/2g
h = (4 m/s)^2 / (2 x 9.8 m/s^2)
h = 0.82 m

Therefore, the mass will reach a maximum height of 0.82 m above its equilibrium position before falling back down due to gravity.

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The function f(x) = ((x+1)/(x-1))^3 is bijective as it is both injective and surjective, meaning it has a one-to-one correspondence between its domain and codomain.

To prove that f(x) = [tex]((x+1)/(x-1))^3[/tex]is bijective, we need to show that it is both injective and surjective.

Injectivity: To prove injectivity, we assume that f(x1) = f(x2) and show that it implies x1 = x2. So, let's assume f(x1) = f(x2) and substitute the function values:

[tex]((x1+1)/(x1-1))^3 = ((x2+1)/(x2-1))^3[/tex]

Taking the cube root of both sides, we get:

(x1+1)/(x1-1) = (x2+1)/(x2-1)

Cross-multiplying and simplifying, we have:

x1 + 1 = x2 + 1

This implies x1 = x2, which shows that the function is injective.

Surjectivity: To prove surjectivity, we need to show that for every y in the codomain, there exists an x in the domain such that f(x) = y. In this case, the codomain is R - {1}.

Let y be an arbitrary element in R - {1}. We can solve the equation f(x) = y for x:

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

Taking the cube root of both sides, we get:

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

Cross-multiplying and simplifying, we have:

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

Expanding and rearranging terms, we get:

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

Factoring out x, we have:

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

Dividing both sides by (1 - y^(1/3)), we get:

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

This shows that for any y in R - {1}, we can find an x in the domain such that f(x) = y, proving surjectivity.

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The area of the four-leaf rose with the equation r = 2cos(20) is approximately 2.758 square units.

The four-leaf rose is a polar curve represented by the equation r = 2cos(20). To find its area, we can integrate the equation over the desired region. The limits of integration for the angle θ would typically be from 0 to 2π, covering a full revolution. However, since the curve has four petals, we need to evaluate the area for only one-fourth of the curve.

By integrating the equation r = 2cos(20) from 0 to π/10, we can calculate the area of one petal. Using the formula for polar area, A = (1/2)∫[r(θ)]^2dθ, where r(θ) is the polar equation, we can compute the area.

Performing the integration and evaluating the result, we find that the area of one petal is approximately 0.344 square units. Since the four-leaf rose has four identical petals, the total area enclosed by the curve is four times this value, giving us an approximate total area of 2.758 square units.

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The distance between the complex numbers z1 = 2 + 2i and z2 = 1 + 5i is approximately 4.242 units. The complex midpoint between z1 and z2 is located at 1.5 + 3.5i.



To find the distance between two complex numbers, we can use the formula:

distance = |z2 - z1|, where z1 and z2 are the given complex numbers.

For z1 = 2 + 2i and z2 = 1 + 5i:

z2 - z1 = (1 + 5i) - (2 + 2i)

       = -1 + 3i

The magnitude or absolute value of -1 + 3i can be calculated as:

|z2 - z1| = sqrt((-1)^2 + (3)^2)

         = sqrt(1 + 9)

         = sqrt(10)

         ≈ 3.162

Therefore, the distance between z1 and z2 is approximately 3.162 units.

To find the complex midpoint, we can use the formula:

midpoint = (z1 + z2) / 2

For z1 = 2 + 2i and z2 = 1 + 5i:

midpoint = ((2 + 2i) + (1 + 5i)) / 2

        = (3 + 7i) / 2

        = 1.5 + 3.5i

Hence, the complex midpoint between z1 and z2 is located at 1.5 + 3.5i.

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The chain rule of differentiation must be applied to calculate dI/dt, the derivative of the current with respect to time, in order to ascertain the rate at which the current I is changing when R = 3 and V = 12.

The following change rates are provided:

(Voltage dropping rate) dV/dt = -0.1 volts/sec

The resistance is growing at a rate of 2 ohms/sec.

V = IR is what we get from Ohm's Law. With regard to time t, we can differentiate this equation as follows:

d(IR)/dt = dV/dt

When we use the chain rule, we obtain:

R(dI/dt) + I(dR/dt) = dV/dt

Since R = 3 and V = 12 are the quantities we are most interested in, we insert these values into the equation and solve for dI/dt:

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The integral represents the volume of the solid obtained by rotating the region bounded by the curves y = sin(x), y = 0, x = 0, and x = π/2 about the y-axis.

To find the volume of the solid, we can use the method of cylindrical shells. Since we are rotating the region bounded by the curves y = sin(x), y = 0, x = 0, and x = π/2 about the y-axis, each cross section of the solid will be a cylindrical shell with thickness dy and radius x.

The volume of a single cylindrical shell is given by the formula V = 2πx * h * dy, where x represents the radius and h represents the height of the shell.

The height of each shell can be represented as h = f(x) - g(x), where f(x) is the upper curve (y = sin(x)) and g(x) is the lower curve (y = 0). In this case, h = sin(x) - 0 = sin(x).

Substituting x = x(y) into the formula for the volume of a cylindrical shell, we have V = 2πx(y) * sin(x) * dy.

To determine the limits of integration for y, we need to find the range of y-values that correspond to the region bounded by y = sin(x), y = 0, x = 0, and x = π/2. In this case, the limits of integration are y = 0 to y = 1.

Now, we can set up the integral for the volume:

V = ∫[0,1] 2πx(y) * sin(x) * dy

By evaluating this integral, we can find the volume of the solid obtained by rotating the given region about the y-axis.

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a. The vectors ü, 5ū, and -5ū are related in terms of magnitude and direction. The vectors 5ū and -5ū have the same magnitude as ü but differ in direction.

Specifically, the vector 5ū is in the same direction as ü, while -5ū is in the opposite direction. Both 5ū and -5ū are scalar multiples of the vector ü, with the scalar being 5 and -5 respectively.

In vector algebra, multiplying a vector by a scalar result in a new vector with the same direction as the original vector but with a different magnitude. When we multiply the vector ü by 5, we obtain a new vector 5ū with a magnitude five times greater than ü.

The direction of 5ū remains the same as that of ü. On the other hand, multiplying ü by -5 gives us a new vector -5ū, which has the same magnitude as ü but points in the opposite direction.

b. No, it is not possible for the sum of 3 parallel vectors to be equal to the zero vector, except when all three vectors have zero magnitude.

Parallel vectors have the same or opposite direction but can have different magnitudes. When adding vectors, the resultant vector is determined by the vector's magnitude and direction.

In the case of parallel vectors, their magnitudes add up, resulting in a vector with a magnitude equal to the sum of the magnitudes of the individual vectors.

Since the zero vector has zero magnitude, the sum of three non-zero parallel vectors will always have a non-zero magnitude. However, if all three parallel vectors have zero magnitude, their sum will also be the zero vector since adding zero vectors does not change their magnitude or direction.

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The faster cyclist's speed is 46 km/hr and the slower cyclist's speed is 36 km/hr.

Let the speed of the faster cyclist be x km/hr. Then the speed of the slower cyclist is x-10 km/hr.
As they are travelling towards each other, their relative speed will be the sum of their speeds. So,
Relative speed = x + (x-10) = 2x - 10 km/hr
Time taken to meet = 5 hours
Distance travelled = relative speed x time taken
210 = (2x-10) x 5
Solving for x, we get x = 46 km/hr (approx.)
Therefore, the faster cyclist's speed is 46 km/hr and the slower cyclist's speed is 36 km/hr.

To solve this problem, we need to use the formula Distance = Speed x Time. Since the two cyclists are travelling towards each other, we need to find their relative speed by adding their speeds. Then we can use the distance and time given to calculate their speeds individually using the formula Speed = Distance / Time.

The faster cyclist is travelling at a speed of 46 km/hr, while the slower cyclist is travelling at a speed of 36 km/hr.

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Based on the calculations like multiplication, subtraction, we conclude that, Team Olympus could have tied either 28 times or 19 times.

What is subtraction?

Subtraction is one of the basic arithmetic operations in mathematics. It is a process of finding the difference or the result of taking away one quantity from another.

To determine how many times Team Olympus could have tied, we need to consider the total number of points they obtained and the points awarded for wins and ties.

In each game, Team Olympus can either win, lose, or tie. If they win a game, they receive 3 points, and if they tie a game, they receive 1 point.

Since Team Olympus played 19 games, the maximum number of points they could have earned if they won every game would be 19 * 3 = 57 points. However, they obtained a total of 28 points, which is less than the maximum possible.

To calculate the number of wins, we can subtract the number of points obtained from wins (3 points each) from the total points (28 points). The remaining points would be the number of points obtained from ties.

Number of points from ties = Total points - Number of wins * Points per win

Number of points from ties = 28 - Number of wins * 3

To find the possible number of ties, we need to determine the values of Number of wins that result in a non-negative number of points from ties.

Let's calculate the possible values:

Number of wins = 0:

Number of points from ties = 28 - 0 * 3 = 28 points

28 points can be obtained from 28 ties.

Number of wins = 1:

Number of points from ties = 28 - 1 * 3 = 25 points

25 points cannot be obtained from ties since it is not divisible by 1.

Number of wins = 2:

Number of points from ties = 28 - 2 * 3 = 22 points

22 points cannot be obtained from ties since it is not divisible by 1.

Number of wins = 3:

Number of points from ties = 28 - 3 * 3 = 19 points

19 points can be obtained from 19 ties.

Based on the calculations, Team Olympus could have tied either 28 times or 19 times.

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The derivative dy/dx using implicit differentiation dy/dx = (10*9e^(9x) - m*u^(m-1) * du/dx) / (3y^2).
.

To find dy/dx by implicit differentiation, we need to differentiate both sides of the equation with respect to x.
Starting with the given equation:

1u^m + 1y^3 = 10e^(9x)

We first take the derivative of each term separately using the chain rule:

d/dx (1u^m) = m*u^(m-1) * du/dx
d/dx (1y^3) = 3y^2 * dy/dx
d/dx (10e^(9x)) = 10*9e^(9x)

Now, putting it all together using the chain rule and solving for dy/dx:

m*u^(m-1) * du/dx + 3y^2 * dy/dx = 10*9e^(9x)
dy/dx = (10*9e^(9x) - m*u^(m-1) * du/dx) / (3y^2)

And there you have it, the derivative dy/dx using implicit differentiation.

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To approximate the integral of x² [tex]e^{(-x^2)}[/tex] dx using a series, expand  [tex]e^{(-x^2)}[/tex] as a power series and integrate each term. The number of terms needed depends on the desired accuracy.

To approximate the integral of x²  [tex]e^{(-x^2)}[/tex] dx using a series, we can express the function  [tex]e^{(-x^2)}[/tex] as a power series expansion and then integrate it term by term.

The power series expansion of  [tex]e^{(-x^2)}[/tex] is given by:

 [tex]e^{(-x^2)}[/tex] = 1 - x² + (x² * x²)/2! - (x² * x² * x²)/3! + ...

To approximate the integral, we can integrate each term of the series individually. The integral of x²  [tex]e^{(-x^2)}[/tex] dx is therefore:

∫(x²  [tex]e^{(-x^2)}[/tex]dx) = ∫(x² * (1 - x² + (x² * x²)/2! - (x² * x² * x²)/3! + ...)) dx

Integrating each term, we get:

∫(x² * (1 - x² + (x² * x²)/2! - (x² * x² * x²)/3! + ...)) dx = ∫(x² - x⁴ + (x⁶)/2! - (x⁸)/3! + ...) dx

We can now integrate each term term by term. The integral of x² dx is (x³)/3, the integral of -x⁴ dx is -(x⁵)/5, the integral of (x⁶)/2! dx is (x⁷)/7, and so on.

Continuing this process, we can evaluate the integral term by term until we reach the desired level of precision. The number of terms needed will depend on the desired accuracy of the approximation.

By using this series approximation method, we can estimate the value of the integral of x²  [tex]e^{(-x^2)}[/tex] dx to three decimal places.

The complete question is:

"Use a series to approximate the integral of x²[tex]e^{(-x^2)}[/tex] dx to three decimal places."

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Using a Riemann sum with right-hand endpoints and 8 rectangles, we can approximate the vehicle's displacement, distance traveled, and average velocity over the two-hour period.

(a) To approximate the vehicle's displacement over the two hours, we can use a Riemann sum. The displacement is given by the change in position, which can be estimated by summing the areas of the rectangles formed by the function values at the right-hand endpoints. Each rectangle has a width of Δt = (2-0)/8 = 0.25 hours. The height of each rectangle is given by the function y = t³ - 1 evaluated at the right-hand endpoint. By calculating the sum of the areas of these rectangles, we can approximate the displacement over the two-hour period.

(b) To approximate the distance traveled by the vehicle over the two hours, we need to consider the absolute values of the function values. Distance is a scalar quantity and does not take into account the direction. By using the absolute values of the function values, we ensure that negative displacements are accounted for. Therefore, the process is similar to part (a), but with the absolute values of the function values.

(c) The average velocity of the vehicle over the two-hour period can be approximated by dividing the total displacement (part a) by the time interval (2 hours). This provides an estimate of the average velocity over the given time period.

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Option B: 33(x+y) is not a linear equation in one variable.

The linear equation in one variable is an equation that can be written in the form ax + b = 0, where x represents the variable and a and b are constants.

Among the given options, option B: 33(x+y) is not a linear equation in one variable.

In option B, the equation contains two variables, x and y, which means it is a linear equation in two variables. To be a linear equation in one variable, there should be only one variable present in the equation.

On the other hand, options A, C, and D can all be written in the form ax + b = 0, where x is the variable, and a and b are constants. Therefore, options A, C, and D are linear equations in one variable.

Hence, option B: 33(x+y) is not a linear equation in one variable.

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The vector−2028is a linear combination of the vectors 132 and −6−9−6if and only if the matrix equation = has a solution .

To determine if the vector −2028is a linear combination of the vectors 132 and −6−9−6, we can construct a matrix using these vectors as columns:

1  -6

3  -9

2  -6

Let's denote this matrix as A. We can write the matrix equation as A=, where is the coefficient vector we are looking for, and ⃗ is the given vector −2028.

For this matrix equation to have a solution, the matrix A must be invertible, meaning it has a unique solution. If A is invertible, we can solve the equation by multiplying both sides by the inverse of A: A⁻¹A = A⁻¹, which simplifies to = A⁻¹.

If the matrix A is not invertible, it means that the columns of A are linearly dependent, and the equation A=does not have a unique solution. In this case, the vector −2028cannot be expressed as a linear combination of the given vectors 132 and−6−9−6.

Therefore, the vector −2028 is a linear combination of the vectors 132 and −6−9−6 if and only if the matrix equation= has a solution .

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Answer:

If this limit exists, then m'(x) is differentiable at x = 0. Otherwise, it is not differentiable at x = 0.

Step-by-step explanation:

To determine if m'(x) is differentiable at x = 0, we need to consider the continuity and differentiability conditions for the derivative.

Given that m' is continuous at x = 0, we know that the limit of m'(x) as x approaches 0 exists, and m'(0) is well-defined.

To determine if m'(x) is differentiable at x = 0, we need to check if the derivative of m'(x) exists at x = 0. The derivative of m'(x) is denoted as m''(x).

Given that m''(0) = 0, it suggests that the second derivative of m(x) has a critical point at x = 0. However, this information alone is not sufficient to conclude whether m'(x) is differentiable at x = 0.

To determine differentiability at x = 0, we need to analyze the behavior of m'(x) in the vicinity of x = 0. Specifically, we need to examine the limit of the difference quotient of m'(x) as x approaches 0:

lim┬(h→0)⁡〖(m'(0+h) - m'(0))/h〗

If this limit exists, then m'(x) is differentiable at x = 0. Otherwise, it is not differentiable at x = 0.

The given information does not provide any specific details about the behavior of m'(x) in the vicinity of x = 0 or any additional conditions that would allow us to determine the differentiability of m'(x) at x = 0.

Therefore, without further information, we cannot determine whether m'(x) is differentiable at x = 0 based solely on the given conditions of m''(0) = 0 and the continuity of m' at x = 0.

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To find the volume of the solid obtained by rotating the region between the curves y = 2cos(θ) - 2 and y = -9 around the x-axis from x = -1.925 to x = 1.975, we can use the disk method.Evaluating this integral will give you the volume of the solid.

The volume V can be calculated using the formula:

V = [tex]∫[a to b] π[R(x)^2 - r(x)^2] dx[/tex],

where R(x) is the outer radius and r(x) is the inner radius.

In this case, the outer radius R(x) is given by the top equation: R(x) = 2cos(θ) - 2,

and the inner radius r(x) is given by the bottom equation: r(x) = -9.

Since the given equations are in terms of θ, we need to express them in terms of x. Let's do the conversion:

For the top equation: y = 2cos(θ) - 2,

we can rewrite it as x = 2cos(θ) - 2, and solving for cos(θ) gives cos(θ) = (x + 2) / 2.

Substituting this into the equation, we get [tex]R(x) = 2[(x + 2) / 2] - 2 = x[/tex].

Now we can calculate the volume:

[tex]V = ∫[-1.925 to 1.975] π[(x)^2 - (-9)^2] dx.[/tex]

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The Mean Value Theorem(MVT) does not apply to the function f(x) = √x - 2 on the interval [1, 9] because this function is not continuous on [1, 9].

The Mean Value Theorem states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point c in (a, b) where the derivative of the function is equal to the average rate of change of the function over the interval [a, b].

In this case, the function f(x) = √x - 2 is not continuous on the interval [1, 9]. The square root function √x is not defined for negative values of x, and since the interval [1, 9] includes the point x = 0, the function is not defined at that point. Therefore, the function is not continuous on the interval [1, 9], and as a result, the Mean Value Theorem does not apply.

For the Mean Value Theorem(MVT) to be applicable, it is necessary for the function to satisfy the conditions of continuity and differentiability on the given interval. Since f(x) = √x - 2 is not continuous at x = 0, it fails to meet the conditions required by the Mean Value Theorem. Consequently, we cannot apply the theorem to make any conclusions about the existence of a point where the derivative of the function equals the average rate of change on the interval [1, 9].

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8) The derivative of

[tex]F(x) = e^(cosh(x^2)) is f'(x) = 2x * sinh(x^2) * e^(cosh(x^2)).[/tex]

9) The derivative of

[tex]F(x) = tanh^(-1)(3x^2) is f'(x) = 6x / (1 + 9x^4).[/tex]

In both cases, we can find the derivative by applying the chain rule and the derivative of the inner function.

In the first case, to find the derivative of [tex]F(x) = e^(cosh(x^2))F(x) = e^(cosh(x^2))[/tex], we use the chain rule. Let's denote the inner function as u = cosh(x^2). The derivative of u with respect to x is du/dx = sinh(x^2) * 2x by applying the chain rule. Then, we can find the derivative of F(x) by multiplying the derivative of the outer function, which is e^u[tex]e^u[/tex], by the derivative of the inner function. Therefore, f'(x) = 2x * sinh(x^2) * e^(cosh(x^2)).[tex]f'(x) = 2x * sinh(x^2) * e^(cosh(x^2)).[/tex]

In the second case, to find the derivative of

[tex]F(x) = tanh^(-1)(3x^2),[/tex] we again use the chain rule.

Let's denote the inner function as u = 3x². The derivative of u with respect to x is du/dx = 6x. Then, we can find the derivative of F(x) by multiplying the derivative of the outer function, which is tanh^(-1)(u), by the derivative of the inner function. The derivative of tanh^(-1)(u) can be written as 1 / (1 + u²). Therefore, [tex]f'(x) = 6x / (1 + 9x^4).[/tex]

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The approximation for f'(9.4) is approximately 8.2. To find the differential dy when x = 1 and dx = 0.3, we can use the formula for the differential: dy = f'(x) * dx.

First, we need to find the derivative of the function y = 5x^2 + 6x + 2. Taking the derivative, we have: y' = 10x + 6. Now we can substitute the values x = 1 and dx = 0.3 into the formula for the differential: dy = (10x + 6) * dx = (10 * 1 + 6) * 0.3 = 4.8. Therefore, the differential dy when x = 1 and dx = 0.3 is dy = 4.8.

Similarly, to find the differential dy when x = 1 and dx = 0.6, we can substitute these values into the formula: dy = (10x + 6) * dx= (10 * 1 + 6) * 0.6= 9.6. Thus, the differential dy when x = 1 and dx = 0.6 is dy = 9.6. To approximate f'(9.4), we can use the given information that f(9.4) = 0.6 and f(9.9) = 4.7. We can use the average rate of change to approximate the derivative: f'(9.4) ≈ (f(9.9) - f(9.4)) / (9.9 - 9.4)= (4.7 - 0.6) / 0.5= 8.2. Therefore, the approximation for f'(9.4) is approximately 8.2.

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Both [tex]y_1 = r^2[/tex] and [tex]y_2 = z_o[/tex]  satisfy the homogeneous form of the differential equation.

A) To verify that [tex]y_1 = r^2[/tex] and [tex]y_2 = z_o[/tex] satisfy the homogeneous form of the differential equation (a'y" - 6xy' + 10y = 0), we need to substitute these functions into the equation and check if the equation holds.

Given differential equation: zy" - 6xy' + 10y = 3.24 + 62%

Homogeneous form: a'y" - 6xy' + 10y = 0

Substituting  [tex]y_1 = r^2[/tex] and [tex]y_2 = z_o[/tex]  into the homogeneous form:

For  [tex]y_1 = r^2[/tex] :

a'([tex]r^2[/tex])'' - 6x([tex]r^2[/tex])' + 10([tex]r^2[/tex]) = 0

a'(2r) - 6x(2r) + 10([tex]r^2[/tex]) = 0

2a'r - 12xr + 10[tex]r^2[/tex] = 0

For y2 = zo:

a'([tex]z_o[/tex])'' - 6x([tex]z_o[/tex])' + 10([tex]z_o[/tex]) = 0

a'(0) - 6x(0) + 10[tex]z_o[/tex] = 0

10[tex]z_o[/tex] = 0

Since 10[tex]z_o[/tex] = 0, it satisfies the homogeneous form.

Therefore, both [tex]y_1 = r^2[/tex] and [tex]y_2 = z_o[/tex]  satisfy the homogeneous form of the differential equation.

B) To solve the given non-homogeneous differential equation using variation of parameters, we assume the particular solution as

[tex]y = u_1(x)y_1 + u_2(x)y_2[/tex], where [tex]y_1[/tex] and [tex]y_2[/tex] are the solutions to the homogeneous equation and [tex]u_1(x)[/tex] and [tex]u_2(x)[/tex] are functions to be determined.

The particular solution is given by:

[tex]y_{p(x)} = u_1(x)y_1 + u_2(x)y_2[/tex]

Taking derivatives:

[tex]y_{p'(x)} = u_1'(x)y_1 + u_2'(x)y_2 + u_1(x)y_1' + u_2(x)y_2'[/tex]

[tex]y_{p''(x)} = u_1''(x)y_1 + u_2''(x)y_2 + 2u_1'(x)y_1' + 2u_2'(x)y_2' + u_1(x)y_1'' + u_2(x)y_2''[/tex]

Substituting these derivatives into the original non-homogeneous equation:

[tex]z(y_1u_1'' + y_2u_2'') + 2z(y_1'u_1' + y_2'u_2') + z(y_1u_1 + y_2u_2) - 6x(y_1'u_1 + y_2'u_2) + 10(y_1u_1 + y_2u_2) = 3.24 + 62\%[/tex]

Matching coefficients of like terms:

[tex]zu_1'' + 2zu_1' + zu_1 = 0[/tex]

[tex]zu_2'' + 2zu_2' + zu_2 = 3.24 + 62\%[/tex]

Now, we can solve these two differential equations for u1(x) and u2(x) using variation of parameters. This involves finding the Wronskian and then solving a system of linear equations.

Note: Without the specific forms of y1 and y2, it is not possible to provide the exact solution in this format. The solution will involve integrating and manipulating the equations involving u1(x) and u2(x) to find the particular solution.

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The values of all sub-parts have been obtained.

(a). The value of bₙ = ((-1)ⁿ 2n) / (n + 2).

(b). The value of limit is Lim bₙ = 2.

What is series for convergence or divergence?

The term "convergent series" refers to a series whose partial sums tend to a limit. A divergent series is one whose partial sums, in contrast, do not approach a limit. The Divergent series often reach, reach, or don't reach a particular number.

As given series is,

-(2/3) + (4/4) - (6/5) + (8/6) - (10/7) + ...

Assume b₁ = (-2/3), b₂ = (4/4), b₃ = (-6/5), b₄ = (8/6), and b₅ = (-10/7).

Since mod-bi < mod-b(i + 1) for all i implies that mode of the series.

(a). Evaluate the value of bₙ:

From given series,

-(2/3) + (4/4) - (6/5) + (8/6) - (10/7) + ...

Then, b₁ = (-2/3), b₂ = (4/4), b₃ = (-6/5), b₄ = (8/6), and b₅ = (-10/7).

So, bₙ = alpha ∑ (n = 1) {(-1)ⁿ 2n) / (n + 2)}

Thus, bₙ = {(-1)ⁿ 2n) / (n + 2)}.

(b). Evaluate the value of Limit:

lim (n = alpha) mod- bₙ = lim (n = alpha) {(2n) / (n + 2)}

                                      = lim (n = alpha) {(2n) / n(1 + 2/n)}

                                      = 2

Since, lim (n = alpha) bₙ = 2.

Hence, the values of all sub-parts have been obtained.

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The third derivative of f(x) = 2x(x - 1) is 12.the third derivative of the given function is 0, indicating that the rate of change of the slope of the original function is constant at all points

To find the third derivative, we need to differentiate the function successively three times. Let's start by finding the first derivative:f'(x) = 2(x - 1) + 2x(1) = 2x - 2 + 2x = 4x - 2Next, we differentiate the first derivative to find the second derivative:f''(x) = 4

Since the second derivative is a constant, differentiating it again will yield a zero value: f'''(x) = 0

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The final answer to the given function is f′′(x)=3x² +14x.

What is the polynomial equation?

A polynomial equation is an equation in which the variable is raised to a power, and the coefficients are constants. A polynomial equation can have one or more terms, and the degree of the polynomial is determined by the highest power of the variable in the equation.

To find f′′(x) given f′(x) = (t³ +7t² +4), we need to differentiate f(x) twice with respect to x.

Let's start by finding the first derivative, f′(x), using the Fundamental Theorem of Calculus:

[tex]f'(x) = (t^3 +7t^2 +4)]^x_0[/tex]

The derivative of the integral is the integrand evaluated at the upper limit minus the integrand evaluated at the lower limit. Evaluating the integrand at

f′(x) = (x³ +7x² +4) - (03+7(02)+4)

f′(x) = (x³ +7x² +4)

Now, let's differentiate f′(x) to find the second derivative, f′′(x)

f′′(x)= dx/d (x³ +7x² +4)

f'′(x)=3x² +14x

Therefore,

f′′(x)=3x² +14x.

hence, the final answer to the given function is f′′(x)=3x² +14x.

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First, let's find the derivative of y with respect to x. We can use the chain rule for this:

dy/dx = d(tan^(-1)(6(2x+1)))/d(6(2x+1)) * d(6(2x+1))/dx

The derivative of tan^(-1)(u) with respect to u is 1/(1+u^2). Therefore, the derivative of tan^(-1)(6(2x+1)) with respect to (6(2x+1)) is 1/(1+(6(2x+1))^2).

The derivative of 6(2x+1) with respect to x is simply 12.

Now, let's substitute these values into the chain rule:

dy/dx = 1/(1+(6(2x+1))^2) * 12

Simplifying this expression:

dy/dx = 12/(1+(6(2x+1))^2)

Next, we evaluate dy/dx at x = 0:

dy/dx |x=0 = 12/(1+(6(2(0)+1))^2)

        = 12/(1+(6(1))^2)

        = 12/(1+36^2)

        = 12/(1+36)

        = 12/37

Therefore, the value of dy/dx at x = 0 is 12/37.

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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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