In the triangle below, what is the measure (in degrees) of angle X?
66°
55°
X

Matthew can make a total of 7 bowls with the 3.5 pounds of clay he has.

To find the number of bowls Matthew can make, we need to divide the total amount of clay he has by the amount of clay needed to make one bowl. Matthew has 3.5 pounds of clay, and he needs 1/2 of a pound to make one bowl. To divide these two values, we can write the division equation as:

3.5 pounds ÷ 1/2 pound per bowl

To simplify this division, we can multiply the numerator and denominator by the reciprocal of 1/2, which is 2/1. This gives us:

3.5 pounds ÷ 1/2 pound per bowl × 2/1

Multiplying across, we get:

3.5 pounds × 2 ÷ 1 ÷ 1/2 pound per bowl

Simplifying further, we have:

7 pounds ÷ 1/2 pound per bowl

Now, to divide by a fraction, we multiply by its reciprocal. So we can rewrite the division equation as:

7 pounds × 2/1 bowl per 1/2 pound

Multiplying across, we get:

7 pounds × 2 ÷ 1 ÷ 1/2 pound

Simplifying gives us:

14 bowls ÷ 1/2 pound

Dividing by 1/2 is the same as multiplying by its reciprocal, which is 2/1. So we have:

14 bowls × 2/1

Multiplying across, we find:

28 bowls

Therefore, Matthew can make a total of 28 bowls with the 3.5 pounds of clay he has.

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The line integral is: ∫_c F · dr = ∬_D (curl F) · dA = -70/3.

To apply Green's theorem, we need to find the curl of the vector field:

curl F = (∂Q/∂x - ∂P/∂y) = (-16x^2 - 6, 0, 5)

where F = (P, Q) = (3cos(x) - 6y^2, sin(5y) + 16x^3).

Now, we can apply Green's theorem to evaluate the line integral over the boundary of the square:

∫_c F · dr = ∬_D (curl F) · dA

where D is the region enclosed by the square [0, 1] × [0, 1].

Since the curl of F has only an x and z component, we can simplify the double integral by integrating with respect to y first:

∬_D (curl F) · dA = ∫_0^1 ∫_0^1 (-16x^2 - 6) dy dx

= ∫_0^1 (-16x^2 - 6) dx

= (-16/3) - 6

= -70/3

Therefore, the line integral is:

∫_c F · dr = ∬_D (curl F) · dA = -70/3.

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The line integral of F over the unit circle C is zero:

∮C F · dr = ∬D curl(F) · dA = 0

Hence, the answer is zero.

To use Green's theorem to evaluate the line integral of the given function around the unit circle, we need to first find its equivalent double integral over the region enclosed by the circle.

Green's theorem relates the line integral of a vector field over a closed curve to the double integral of the curl of the same vector field over the region enclosed by the curve.

Let's consider the vector field [tex]F = (0, 0, xy^2).[/tex]

Its curl is given by:

curl(F) = (∂Q/∂x - ∂P/∂y) i + (∂P/∂x + ∂Q/∂y) j + (∂R/∂x - ∂Q/∂y) k

= (0 - 0) i + (0 + 0) j + (0 - 2xy) k

= -2xy k

Here, P = 0, Q = 0, and[tex]R = xy^2[/tex] are the components of the vector field F.

Now, we can apply Green's theorem to evaluate the line integral of F over the unit circle C:

∮C F · dr = ∬D curl(F) · dA

where D is the region enclosed by the unit circle C and dA is the area element in the xy-plane.

Since the unit circle is given by[tex]x^2 + y^2 = 1,[/tex]  we can use polar coordinates to evaluate the double integral:

∬D curl(F) · dA = ∬D (-[tex]2r^3[/tex] sin θ cos θ) r dr dθ

= -2 ∫[0,2π] ∫[0,1] [tex]r^4[/tex]sin θ cos θ dr dθ

= 0 (since the integrand is odd in sin θ).

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Answer:

Step-by-step explanation:

a.

The equation to graph the left side of the letter "V" is y = -3x - 6.

b.  The points for the right side are then (-4, -6) and (0, 6).

c. The equation to graph the right side of the letter "V" is y = 3x + 6.

a.

The slope-intercept form of a linear equation is  y = mx + b.

The  points (-4, 6) and (0, -6):

m = (change in y) / (change in x)

= (-6 - 6) / (0 - (-4))

= -12 / 4

= -3

the y-intercept (b):

6 = -3(-4) + b

6 = 12 + b

b = 6 - 12

b = -6

b.

We will use the points (-4, 6) and (0, -6) and reverse the sign of the y-values. The points for the right side will be  (-4, -6) and (0, 6).

c.

We find slope (m) using the points (-4, -6) and (0, 6):

m = (change in y) / (change in x)

= (6 - (-6)) / (0 - (-4))

= 12 / 4

= 3

The y-intercept (b):

-6 = 3(-4) + b

-6 = -12 + b

b = -6 + 12

b = 6

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The predicted number of times the pointer will land on section B in 1,000 trials can be determined by calculating the relative frequency of B based on the recorded results of 50 spins.

To find the relative frequency, we divide the number of times the spinner landed on B by the total number of spins. In this case, let's assume that the spinner landed on section B, say, 10 times out of the 50 recorded spins.

To predict the number of times the pointer will land on B in 1,000 trials, we can use the ratio of the number of spins for B in 50 trials to the total number of spins in 1,000 trials.

Thus, the predicted number of times the pointer will land on section B in 1,000 trials would be:

Predicted number of times on B = (Number of times on B in 50 trials / Total number of spins in 50 trials) * Total number of spins in 1,000 trials

Let's assume the spinner landed on B 10 times in the 50 recorded spins. The calculation would be:

Predicted number of times on B = (10 / 50) * 1,000 = 200

Therefore, the predicted number of times the pointer will land on section B in 1,000 trials is 200.

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The derivative of the function g(s) is:

g'(s) = 2 ∫(t - t^9)^6 (1 - 9t^8)^(-1) dt

To apply Part 1 of the Fundamental Theorem of Calculus, we need to first express the function as an integral with a variable upper limit of integration.

We can do this by letting u = t - t^9, so du/dt = 1 - 9t^8. Solving for dt, we get dt = du / (1 - 9t^8).

Substituting this into the integral, we have:

g(s) = 2s ∫(t - t^9)^6 dt

= 2s ∫u^6 (1 - 9t^8)^(-1) du

Now we can differentiate g(s) with respect to s using the chain rule and Part 1 of the Fundamental Theorem of Calculus:

g'(s) = d/ds [2s ∫u^6 (1 - 9t^8)^(-1) du]

= 2 ∫u^6 (1 - 9t^8)^(-1) du

Note that since the integral is with respect to u, we can treat (1 - 9t^8)^(-1) as a constant with respect to u, so we can pull it out of the integral.

Taking the derivative of the integral with respect to s just leaves us with the constant factor of 2.

Therefore, the derivative of the function g(s) is:

g'(s) = 2 ∫(t - t^9)^6 (1 - 9t^8)^(-1) dt

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The value of k'(-2) = 41

Using the product rule, k′(−2)=f(−2)g′(−2)h(−2)+f(−2)g(−2)h′(−2)+f′(−2)g(−2)h(−2). Substituting the given values, we get k′(−2)=(-5)(8)(3)+(-5)(-7)(-10)+(9)(-7)(3)= -120+350-189= 41.

The product rule states that the derivative of the product of two or more functions is the sum of the product of the first function and the derivative of the second function with the product of the second function and the derivative of the first function.

Using this rule, we can find the derivative of k(x) with respect to x. We are given the values of f(−2), f′(−2), g(−2), g′(−2), h(−2), and h′(−2). Substituting these values in the product rule, we can calculate k′(−2). Therefore, the derivative of the function k(x) at x=-2 is equal to 41.

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The work required to compress the 1 kg propane gas adiabatically from 100 kPa to 800 kPa is -325.3 kJ.

In this case, we have a cylinder/piston containing 1 kg of propane gas, so we can use the mass of propane to calculate the number of moles of gas. The molar mass of propane is approximately 44 g/mol, so the number of moles of propane is:

n = m/M = 1000 g / 44 g/mol = 22.73 mol

We can also use the given initial pressure and temperature to find the initial volume of the gas.

Therefore, we can rearrange the ideal gas law to solve for the initial volume:

V = nRT/P = (22.73 mol)(8.31 J/(mol*K))(300 K)/(100 kPa) = 6.83 m³

Now, let's consider the work done on the gas during the compression process.

We can use the first law of thermodynamics to relate the change in internal energy to the initial and final states of the gas:

ΔU = Q - W

where ΔU is the change in internal energy, Q is the heat transferred to the gas, and W is the work done on the gas.

Since the process is adiabatic, Q = 0. Therefore, we can simplify the equation to:

ΔU = -W

The change in internal energy can be related to the pressure and volume of the gas using the adiabatic equation:

[tex]PV^{\gamma}[/tex] = constant

where γ is the ratio of specific heats, which is approximately 1.3 for propane. Since the process is reversible, we can use the adiabatic equation to find the final temperature of the gas:

[tex]T_f = T_i (P_f/P_i)^{(\gamma -1)/\gamma}[/tex] = (300 K)(800 kPa/100 kPa)[tex]^{(1.3-1)/1.3}[/tex] = 680.8 K

Now we can use the adiabatic equation and the initial and final temperatures to find the work done on the gas:

W = [tex](\gamma/(\gamma -1))P_i(V_f - V_i)[/tex]= (1.3/(1.3-1))(100 kPa)(V - 6.83 m³)

We can solve for V by rearranging the adiabatic equation:

[tex]V_f = V_i(P_i/P_f)^{1/\gamma}[/tex] = 6.83 m³ (100 kPa/800 kPa)[tex]^{1/1.3}[/tex] = 1.84 m³

Substituting into the expression for work, we get:

W = (1.3/(1.3-1))(100 kPa)(1.84 m³ - 6.83 m³) = -325.3 kJ

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The shear diagram for the beam with m0 = 200 lb-ft and l = 20 ft can be represented as a piecewise linear function with two segments: a downward linear segment from x = 0 to x = 20, and a constant segment at -200 lb from x = 20 onwards.

The shear diagram provides a visual representation of how the shear force varies along the length of the beam. In this case, we are given that the beam has a fixed moment at the left end (m0 = 200 lb-ft) and a length of 20 ft (l = 20 ft).

Starting from the left end of the beam (x = 0), we observe a downward linear segment in the shear diagram. This segment represents a gradual decrease in shear force from the fixed moment until it reaches the right end of the beam at x = 20 ft.

At x = 20 ft, we encounter a change in behavior. The shear force remains constant at -200 lb, indicating that the beam experiences a continuous downward shear force of 200 lb from this point onwards.

By plotting the shear diagram, engineers and analysts can gain insights into the distribution of shear forces along the beam, which is crucial for understanding the structural behavior and designing appropriate supports and reinforcements.

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Given that 14% of students speak Spanish, 7% speak French, and 4% speak both languages, the probability can be determined as 0.286 (option B).

Let's denote the event "speaks Spanish" as S and the event "speaks French" as F. We want to find the probability of F given S, denoted as P(F|S).

Using conditional probability, we have the formula:

P(F|S) = P(F ∩ S) / P(S)

Given that 14% speak Spanish (P(S) = 0.14), 7% speak French (P(F) = 0.07), and 4% speak both languages (P(F ∩ S) = 0.04), we can substitute these values into the formula:

P(F|S) = P(F ∩ S) / P(S) = 0.04 / 0.14 = 0.286

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The probability that z lies between -2.41 and 0 is approximately 0.9911.

To find the probability that a standard normal variable, z, falls within a specific range, we can use the standard normal distribution table or a statistical calculator.

In this case, we want to find the probability that z lies between -2.41 and 0. By referencing the standard normal distribution table or using a calculator, we can determine the area under the curve corresponding to this range. The resulting value represents the probability of z falling within that range.

Approximately 0.9911 is the probability that z lies between -2.41 and 0 when rounded to four decimal places. This means that there is a high likelihood (approximately 99.11%) that a randomly chosen value of z from a standard normal distribution falls within this range.

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So, -7π/9, -19π/9, and -31π/9 are three negative angles coterminal with 5π/9.

To find angles coterminal with 5π/9, we need to add or subtract a multiple of 2π until we reach another angle with the same terminal side.

To find a positive coterminal angle, we can add 2π (one full revolution) repeatedly until we get an angle between 0 and 2π:

5π/9 + 2π = 19π/9

19π/9 - 2π = 11π/9

11π/9 - 2π = 3π/9 = π/3

So, 19π/9, 11π/9, and π/3 are three positive angles coterminal with 5π/9.

To find a negative coterminal angle, we can subtract 2π (one full revolution) repeatedly until we get an angle between -2π and 0:

5π/9 - 2π = -7π/9

-7π/9 - 2π = -19π/9

-19π/9 - 2π = -31π/9

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The interval of hours for which Juan is closer to the stadium than Rajani is t < 6, which means within the first 6 hours after noon.

To graph the functions representing Juan's and Rajani's distances from the stadium, we can use the equations:

J(t) = 260 - 30t (Juan's distance from the stadium)

R(t) = 380 - 50t (Rajani's distance from the stadium)

The functions represent the distance remaining (in miles) as a function of time (in hours) afternoon.

To determine the interval of hours for which Juan is closer to the stadium than Rajani, we need to find the values of t where J(t) < R(t).

Let's solve the inequality:

260 - 30t < 380 - 50t

-30t + 50t < 380 - 260

20t < 120

t < 6

Thus, the inequality shows that for t < 6, Juan is closer to the stadium than Rajani.

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it's 30

The correct answer is "b. t-test for a single sample". This hypothesis-testing procedure is used to determine whether a sample mean is significantly different from a known population mean when the population variance is unknown.

The correct answer is "b. t-test for a single sample". This hypothesis-testing procedure is used to determine whether a sample mean is significantly different from a known population mean when the population variance is unknown. The t-test for a single sample is a statistical test that compares the sample mean to a hypothetical population mean, using the t-distribution. It helps researchers determine whether the sample mean is a reliable estimate of the population mean, or whether the difference between the two means is due to chance.

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The lengths in the square are VU = 15, SU = 15√2, TV = 15√2 and SW = (15√2)/2

From the question, we have the following parameters that can be used in our computation:

The square (see attachment)

The side length of the square is

Length = 15

So, we have

VU = 15

For the diagonal, we have

TV = VU * √2

So, we have

TV = 15 * √2

Evaluate

TV = 15√2

This also means that

SU = 15√2

This is because

SU = TV

Lastly, we have

SW = SU/2

So, we have

SW = (15√2)/2

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Using linear approximation, we estimate that f(2.85) is approximately equal to 1.1.

Using linear approximation, we can estimate the value of a function near a known point by using the tangent line at that point.

The equation of the tangent line at x = 3 is given by:

y - f(3) = f'(3)(x - 3)

Plugging in f(3) = 2 and f'(3) = 6, we get:

y - 2 = 6(x - 3)

Simplifying, we get:

y = 6x - 16

To estimate f(2.85), we plug in x = 2.85 into the equation for the tangent line:

f(2.85) ≈ 6(2.85) - 16

f(2.85) ≈ 1.1

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Linear approximation is a method used to estimate a function value based on its linear equation. In this case, we can use the linear equation of the tangent line at x=3 to approximate f(2.85). Using the point-slope formula, we have:
y - 2 = 6(x - 3)

Simplifying this equation, we get:

y = 6x - 16

Now, substituting x=2.85 in this equation, we get:

f(2.85) ≈ 6(2.85) - 16 = -2.9

Therefore, the estimated value of f(2.85) using linear approximation is -2.9. It is important to note that this method gives an approximation and may not be completely accurate, but it is useful in situations where an estimate is needed quickly and easily.
Hi! To use linear approximation to estimate f(2.85), we'll apply the formula: L(x) = f(a) + f'(a)(x-a), where L(x) is the linear approximation, f(a) is the function value at a, f'(a) is the derivative at a, and x is the input value.

Here, we have a = 3, f(a) = f(3) = 2, f'(a) = f'(3) = 6, and x = 2.85.

Step 1: L(x) = f(a) + f'(a)(x-a)
Step 2: L(2.85) = 2 + 6(2.85-3)
Step 3: L(2.85) = 2 + 6(-0.15)
Step 4: L(2.85) = 2 - 0.9

The linear approximation to estimate f(2.85) is L(2.85) = 1.1.

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The random variable described is discrete, as the number of pets a family can have can only take on whole number values.

It cannot take on non-integer values such as 2.5 pets or 3.7 pets. The possible values for this random variable are 0, 1, 2, 3, and so on, up to some maximum number of pets that a family might have.

Since the number of pets can only take on a countable number of possible values, this is a discrete random variable.

In contrast, a continuous random variable can take on any value within a range, such as the height or weight of a person, which can vary continuously.

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Rounding up to the nearest whole number, you would need to survey approximately 753 randomly selected air passengers to be 90% confident that the sample percentage is within 3.5 percentage points of the true population percentage.

To determine the sample size needed for estimating a population percentage with a specified margin of error and confidence level, we can use the formula for sample size calculation for proportions. The formula is:

n = (Z^2 * p * (1-p)) / E^2

Where:

n is the required sample size,

Z is the Z-score corresponding to the desired confidence level (for a 90% confidence level, Z ≈ 1.645),

p is the estimated population proportion (since we don't have an estimate, we can use 0.5 for maximum sample size),

E is the desired margin of error (in decimal form).

In this case, the desired margin of error is 3.5 percentage points, which is 0.035 in decimal form.

Plugging in the values, we have:

n = (1.645^2 * 0.5 * (1-0.5)) / 0.035^2

Calculating this expression gives us:

n ≈ 752.93

Rounding up to the nearest whole number, you would need to survey approximately 753 randomly selected air passengers to be 90% confident that the sample percentage is within 3.5 percentage points of the true population percentage.

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To find the surface area of a triangular prism, you need to find the area of the triangular bases and add them to the areas of the rectangular sides.

Surface area of the triangular prism can be found out using the following steps:

Find the area of the triangle which is A, by the following formula.

A = 1/2 × b × hA

= 1/2 × 4 × 5A

= 10m²

Find the perimeter of the base (P) which can be calculated by adding the three sides of the triangle.

P = a + b + cP = 3 + 4 + 5P = 12m

Now find the area of each rectangle which can be calculated by multiplying the adjacent sides.A = LW = 5 × 3 = 15m²

Since there are two rectangles, multiply the area by 2.2 × 15 = 30m²Add the areas of the triangle and rectangles to get the surface area of the triangular prism:

Surface area = A + 2 × LW = 10 + 30 = 40m²

Therefore, the surface area of the given triangular prism is 40m².

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To find the Maclaurin series for f(x) = xe3x, we can start by taking the derivative of the function:

f'(x) = (3x + 1)e3x

Taking the derivative again, we get:

f''(x) = (9x + 6)e3x

And one more time:

f'''(x) = (27x + 18)e3x

We can see a pattern emerging here, where the nth derivative of f(x) is of the form:

f^(n)(x) = (3^n x + p_n)e3x

where p_n is a constant that depends on n. Using this pattern, we can write out the Maclaurin series for f(x):

f(x) = f(0) + f'(0)x + f''(0)x^2/2! + f'''(0)x^3/3! + ... + f^(n)(0)x^n/n! + ...

Plugging in the values we found for the derivatives at x=0, we get:

f(x) = 0 + (3x + 1)x + (9x + 6)x^2/2! + (27x + 18)x^3/3! + ... + (3^n x + p_n)x^n/n! + ...

Simplifying this expression, we get:

f(x) = x(1 + 3x + 9x^2/2! + 27x^3/3! + ... + 3^n x^n/n! + ...)

This is the Maclaurin series for f(x) = xe3x. To find the radius of convergence, we can use the ratio test:

lim |a_n+1/a_n| = lim |3x(n+1)/(n+1)! / 3x/n!|
= lim |3/(n+1)| |x| -> 0 as n -> infinity

So the radius of convergence is infinity, which means that the series converges for all values of x.

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Answer:[tex]\sqrt{3}[/tex]/2

Step-by-step explanation:

Substitute the value of the variable into the expression and simplify.

Answer:

yes

Step-by-step explanation:

The ball will return to the ground after approximately 4.5 seconds.

To find the time it takes for the ball to return to the ground, we need to determine when the height of the ball is zero. In other words, we need to solve the equation f(t) = 54t - 12t² = 0.

Let's set the equation equal to zero and solve for t:

54t - 12t² = 0

Factoring out common terms:

t(54 - 12t) = 0

Now, we have two possible solutions for t:

t = 0

This solution represents the initial time when the ball was thrown.

54 - 12t = 0

Solving this equation for t:

54 - 12t = 0

12t = 54

t = 54 / 12

t = 4.5

So, the ball will return to the ground after approximately 4.5 seconds.

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The values of x, y and z that correspond to the critical point of the function f(x,y) 4x2 + 7x + 6y + 2y are  (-7/8, -3/2).

To find the values of x, y, and z that correspond to the critical point of the function f(x, y) = 4x^2 + 7x + 6y + 2y^2, we need to find the partial derivatives with respect to x and y, and then solve for when these partial derivatives are equal to 0.

Step 1: Find the partial derivatives
∂f/∂x = 8x + 7
∂f/∂y = 6 + 4y

Step 2: Set the partial derivatives equal to 0 and solve for x and y
8x + 7 = 0 => x = -7/8
6 + 4y = 0 => y = -3/2

Now, we need to find the value of z using the given equation c = za. Since we do not have any information about c, we cannot determine the value of z. However, we now know the critical point coordinates for the function are (-7/8, -3/2).

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Therefore, the general power series solution of the differential equation y'' + 3y' = 0, expanded at t0 = 0, is: y(t) = c_0 + c_1 t - (3/2) c_1 t^2 + (9/8) c_1 t^3 - (15/48) c_1 t^4 + ... + (-1)^n (3/(n+2)) c_(n+1) t^(n+2) + ... where c_0 and c_1 are arbitrary constants.

To find the power series solution of the given differential equation, we assume that the solution can be expressed as a power series:

y(t) = ∑(n=0 to ∞) c_n t^n

where c_n is the nth coefficient to be determined.

Taking first and second derivatives of y(t) with respect to t, we get:

y'(t) = ∑(n=1 to ∞) n c_n t^(n-1)

y''(t) = ∑(n=2 to ∞) n(n-1) c_n t^(n-2)

Substituting these expressions into the differential equation, we get:

∑(n=2 to ∞) n(n-1) c_n t^(n-2) + 3∑(n=1 to ∞) n c_n t^(n-1) = 0

Shifting the index of the first summation to start from n=0, we get:

∑(n=0 to ∞) (n+2)(n+1) c_(n+2) t^n + 3∑(n=0 to ∞) (n+1) c_(n+1) t^n = 0

We can simplify this expression by setting the coefficients of each power of t to zero:

(n+2)(n+1) c_(n+2) + 3(n+1) c_(n+1) = 0, for n ≥ 0

Simplifying this expression further, we get:

c_(n+2) = -(3/((n+2)(n+1))) c_(n+1), for n ≥ 0

This gives us a recursive formula for the coefficients c_n in terms of c_0 and c_1:

c_(n+2) = -(3/(n+2)) c_(n+1), for n ≥ 0

c_0 and c_1 are arbitrary constants.

To find the power series solution expanded at t0 = 0, we need to set c_0 = y(0) and c_1 = y'(0) and solve for the remaining coefficients using the recursive formula.

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Step-by-step explanation:

The diagonal is bisected by the other diagonal

Soooo:

5x = 6x -7

x = 7

a) An upper bound for error |f (0.5) − P2(0.5)| using the error formula is 0.0208

b) On the interval [0, 1], we have |R2(x)| <= (e/6) √10 x³

c) The maximum value of |f(x) - P2(x)| on the interval [0, 1] occurs at x = π/2, and is approximately 0.1586.

a. As per the given polynomial, to approximate f(0.5) using P2(x), we simply plug in x = 0.5 into P2(x):

P2(0.5) = 1 + 0.5 - (1/2)(0.5)^2 = 1.375

To find an upper bound for the error |f(0.5) - P2(0.5)|, we can use the error formula:

|f(0.5) - P2(0.5)| <= M|x-0|³ / 3!

where M is an upper bound for the third derivative of f(x) on the interval [0, 0.5].

Taking the third derivative of f(x), we get:

f'''(x) = ex (-3cos x + sin x)

To find an upper bound for f'''(x) on [0, 0.5], we can take its absolute value and plug in x = 0.5:

|f'''(0.5)| = e⁰°⁵(3/4) < 4

Therefore, we have:

|f(0.5) - P2(0.5)| <= (4/6)(0.5)³ = 0.0208

b. For n = 2, we have:

R2(x) = (1/3!)[f'''(c)]x³

To find an upper bound for |R2(x)| on the interval [0, 1], we need to find an upper bound for |f'''(c)|.

Taking the absolute value of the third derivative of f(x), we get:

|f'''(x)| = eˣ |3cos x - sin x|

Since the maximum value of |3cos x - sin x| is √10, which occurs at x = π/4, we have:

|f'''(x)| <= eˣ √10

Therefore, on the interval [0, 1], we have:

|R2(x)| <= (e/6) √10 x³

c. To approximate the maximum value of |f(x) - P2(x)| on the interval [0, 1], we need to find the maximum value of the function R2(x) on this interval.

To do this, we can take the derivative of R2(x) and set it equal to zero:

R2'(x) = 2eˣ (cos x - 2sin x) x² = 0

Solving for x, we get x = 0, π/6, or π/2.

We can now evaluate R2(x) at these critical points and at the endpoints of the interval:

R2(0) = 0

R2(π/6) = (e/6) √10 (π/6)³ ≈ 0.0107

R2(π/2) = (e/48) √10 π³ ≈ 0.1586

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The singular points of the given differential equation: (t² - t - 6)x"' + (t+2)x' – (t-3)x= 0 is  t = -2,3 . So the correct answer is option A. The singular point(s) is/are t = -2,3.  Singular points refer to the values of the independent variable where the solution of the differential equation becomes singular.

To find the singular points of the given differential equation, we need to first write it in standard form:
(t²- t - 6)x"' + (t + 2)x' – (t - 3)x= 0
Dividing both sides by t² - t - 6, we get:
x"' + (t + 2) / (t²- t - 6)x' – (t - 3) / (t²- t - 6)x = 0

Now we can see that the coefficients of x" and x' are both functions of t, and so the equation is not in the standard form for identifying singular points. However, we can use the fact that singular points are locations where the coefficients of x" and x' become infinite or undefined.

The denominator of the coefficient of x' is t²- t - 6, which has roots at t = -2 and t=3. These are potential singular points. To check if they are indeed singular points, we need to check the behavior of the coefficients near these points.

Near t=-2, we have:
(t + 2) / (t²- t - 6) = (t + 2) / [(t + 2)(t - 3)] = 1 / (t - 3)
This expression becomes infinite as t approaches -2 from the left, so -2 is a singular point.

Near t=3, we have:
(t + 2) / (t²- t - 6) = (t + 2) / [(t - 3)(t + 2)] = 1 / (t - 3)
This expression becomes infinite as t approaches 3 from the right, so 3 is also a singular point.

Therefore, the singular points of the given differential equation are t=-2 and t=3. The correct answer is A. The singular point(s) is/are t = -2,3.

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