show thatcos (z w) = coszcoswsinzsinw, assuming the correspondingidentity forzandwreal.

To obtain a Maclaurin series for the given function f(x) = sin x, we can use the known Maclaurin series for sin x, which is:
sin x = x - (x^3)/3! + (x^5)/5! - (x^7)/7! + ...

Multiplying this series by x^3 gives:
sin x 3 = x^3 - (x^6)/3! + (x^8)/5! - (x^10)/7! + ...
Therefore, the Maclaurin series for f(x) = sin x 3 is:
f(x) = x^3 - (x^6)/3! + (x^8)/5! - (x^10)/7! + ...
To find the associated radius of convergence r, we can use the ratio test. The nth term of the series is given by:
a_n = (-1)^(n-1) * (x^3)^(2n-1) / (2n-1)!
Using the ratio test, we have:
lim |a_(n+1) / a_n| = lim |(-1)^n+1 * (x^3)^(2n+1) / (2n+1)!| / |(-1)^n * (x^3)^(2n-1) / (2n-1)!|
= lim |(-1) * x^6 / ((2n+1)(2n))| = 0
Since the limit is less than 1 for all values of x, the series converges for all x. Therefore, the radius of convergence is infinity, which is consistent with the fact that sin x has an infinite radius of convergence.

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Our guess was correct and any monic quartic polynomial can be the characteristic polynomial of some 4x4 matrix using this companion matrix.

To show that p(lambda) is the characteristic equation of the companion matrix, we need to construct the companion matrix and then calculate its characteristic polynomial. The companion matrix for p(lambda) is given by:

A =
[ 0   0  -a ]
[ 1   0  -b ]
[ 0   1  -c ]

The characteristic polynomial of A is the determinant of the matrix (lambdaI - A), where I is the identity matrix of size 3. This gives:

det(lambdaI - A) =
| lambda   0       a      |
| -1      lambda   b      |
| 0       -1      lambda+c|

Expanding the determinant along the first row, we get:

p(lambda) = lambda^3 + clambda^2 + blambda + a

Thus, p(lambda) is indeed the characteristic polynomial of the companion matrix A.

For a monic quartic polynomial, a guess for the companion matrix is:

A =
[ 0   0   0  -a ]
[ 1   0   0  -b ]
[ 0   1   0  -c ]
[ 0   0   1  -d ]

Calculating the determinant of (lambdaI - A), we get:

p(lambda) = lambda^4 + dlambda^3 + clambda^2 + blambda + a

In general, for an n-degree monic polynomial, the companion matrix will be an (n-1) x (n-1) matrix with the coefficients arranged in a particular way. The determinant of (lambdaI - A) will give the characteristic polynomial of the matrix, which will be the same as the given polynomial.

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An approximation of 3√29 accurate to 0.0001 is 3.1058 (rounded to four decimal places).

We can use the binomial series expansion to approximate the function f(x) = 3√x as follows:

f(x) = x^(1/3) = (1 + (x - 1))^(1/3)

Using the binomial series expansion for (1 + t)^n, where t = x - 1 and n = 1/3, we have:

f(x) = (1 + (x - 1))^(1/3) = 1 + (1/3)(x - 1) - (1/9)(x - 1)^2 + (4/81)(x - 1)^3 - (14/243)(x - 1)^4 + ...

Now, we can substitute x = 29 and truncate the series at the term involving (x - 1)^4, since we want an accuracy of 0.0001. We get:

f(29) ≈ 1 + (1/3)(28) - (1/9)(28)^2 + (4/81)(28)^3 - (14/243)(28)^4

f(29) ≈ 3.105835

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The correct option is d. The day on which Elizabeth first gets charged an overdraft fee is Saturday. Her account balance first becomes negative on this day.

From the given data, we can calculate the balance on each day as shown:

Balance on Monday = $252 - $114.60 + $150.00 = $287.40

Balance on Tuesday = $287.40 - $79.68 = $207.72

Balance on Wednesday = $207.72 - $161.39 = $46.33

Balance on Thursday = $46.33 - $57.40 = -$11.07

Balance on Friday = -$11.07 - $22.85 + $75.00 = $41.08

Balance on Saturday = $41.08 - $140.55 = -$99.47

We see that Elizabeth's balance first becomes negative on Saturday, so she will be charged an overdraft fee on that day.Answer: d. Saturday

The day on which Elizabeth first gets charged an overdraft fee is Saturday. Her account balance first becomes negative on this day.

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The integral that represents the length of the curve from (-3, 282) to (3, 366) is ∫[a to b] √(1 + (dy/dx)^2) dx.

To find the length of a curve defined by the equation y = f(x) between two points (a, f(a)) and (b, f(b)), we can set up an integral. The integral representing the length of the curve is given by ∫[a to b] √(1 + (dy/dx)^2) dx, where dy/dx represents the derivative of y with respect to x.

In this case, the equation of the curve is y = 4x^4 - 14xy. To find the length of the curve between (-3, 282) and (3, 366), we need to evaluate the integral ∫[-3 to 3] √(1 + (dy/dx)^2) dx.

The expression inside the square root, 1 + (dy/dx)^2, represents an infinitesimal length element along the curve. By summing up these infinitesimal lengths over the interval [a, b], the integral calculates the total length of the curve between the given points.

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The mass of silver that can be plated onto an object is 0.319 g.

The amount of silver plated onto the object can be calculated using Faraday's law of electrolysis, which states that the mass of a substance produced at an electrode during electrolysis is directly proportional to the quantity of electricity passed through the cell.

The formula for calculating the mass of silver plated is:

mass of silver plated = (current x time x atomic mass of silver) / (Faraday's constant x 1000)

current = 8.70 A, time = 33.5 minutes = 2010 seconds

Atomic mass of silver (Ag) = 107.87 g/mol

Faraday's constant = 96,485 C/mol

Substituting the values in the above formula, we get:

mass of silver plated = (8.70 A x 2010 s x 107.87 g/mol) / (96,485 C/mol x 1000)

= 0.319 g

Therefore, the mass of silver plated onto the object in 33.5 minutes at 8.70 A of current is 0.319 g.

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The equation of line is y = -2x - 1 and the slope is m = -2

Given data ,

Let the equation of the line be represented as A

Now , the equation of the line perpendicular to A is B

where y = ( 1/2 )x - 6

So , the slope of the line is given by

m₁ x m₂ = -1

m₂ = -2

Now , the line passes through the point P ( -5 , 9 )

On simplifying , we get

The equation of line is y - y₁ = m ( x - x₁ )

y - 9 = ( -2 ) ( x - ( -5 ) )

y - 9 = ( -2 ) ( x + 5 )

y - 9 = -2x - 10

Adding 9 on both sides , we get

y = -2x - 1

Hence , the equation of line is y = -2x - 1

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Answer:

Step-by-step explanation:

A safety stock level of approximately 23 units would be needed to achieve the required 95% service level.

The safety stock level can be calculated as:

Safety stock = z * σ * sqrt(L)

where z is the z-score corresponding to the desired service level, σ is the standard deviation of daily demand, and L is the lead time (in days).

In this case, z = 1.64, σ = 7, L = 14 (2 weeks x 7 days/week), so:

Safety stock = 1.64 * 7 * sqrt(14) ≈ 22.8

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To determine the price of widgets that a company should sell to maximize profit, you need to find the value of x at which the given equation will produce the highest y value.

Here's how to solve this:

Step 1: Rewrite the equation in standard form y = -44x² + 1375x - 6548 becomes

y = -44(x² - 31.25x) - 6548

Step 2: Complete the square by adding and subtracting the square of half of the coefficient of x:

y = -44(x² - 31.25x + (31.25/2)² - (31.25/2)²) - 6548

y = -44((x - 15.625)² - 244.141) - 6548

y = -44(x - 15.625)² + 10723.564

Step 3: The maximum value of y occurs when

(x - 15.625)² = 244.141/44.

Therefore,

x - 15.625 = ±sqrt(244.141/44)

x = 15.625 ± 2.765

x = 18.39 or 12.86

Since the company cannot sell at a negative price, x must be $12.86 or $18.39.

The company should sell widgets at $12.86 or $18.39 to maximize profit to the nearest cent.

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The inverse function is [tex]\( f^{-1}(x) = x^2 + \frac{1}{4} \)[/tex], where [tex]\( x \geq 0 \)[/tex].

The inverse of the given function can be determined by changing [tex]\( f(x) \)[/tex] to [tex]\( y \)[/tex], switching [tex]\( x \) and \( y \)[/tex], and solving for [tex]y[/tex]. The resulting function can be written as:

[tex]\[ f^{-1}(x) = x^2 + \frac{1}{4} \][/tex]

where [tex]\( x \geq 0 \)[/tex].

In this equation, [tex]\( f^{-1}(x) \)[/tex] represents the inverse function, [tex]\( x \)[/tex] is the input value, and the term [tex]\( x^2 + \frac{1}{4} \)[/tex] represents the corresponding output value of the inverse function. Additionally, the condition [tex]\( x \geq 0 \)[/tex] indicates that the inverse function is defined only for non-negative values of [tex]x[/tex].

In conclusion, the inverse function of the given function is [tex]\( f^{-1}(x) = x^2 + \frac{1}{4} \)[/tex], indicating a relationship where the input values squared are added to a constant term.

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Let g be the function defined by x

g(x) = ∫ ( -1/2 + cos (t^3 + 2t)) dt for 0 < x < π/3. At what value of x does g attain a relative maximum is option D, 1.489.

To arrive at this answer, we need to find the derivative of the function g(x) and set it equal to zero to determine the critical points. Then, we need to test the values of g(x) at the critical points and the endpoints of the interval to determine where the function attains a relative maximum.

Taking the derivative of g(x) with respect to x, we get:

g'(x) = -1/2 + cos((x^3)+(2x)) * (3x^2)

Setting g'(x) equal to zero, we get:

-1/2 + cos((x^3)+(2x)) * (3x^2) = 0

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

We can see from this equation that cos((x^3)+(2x)) must be positive for the equation to hold. This means that (x^3)+(2x) must be in the range [0, π/2] or [2π, 5π/2] (since cos is positive in these ranges).

Using a graphing calculator or software, we can find that there are two solutions in the interval [0, π/3]: approximately 0.471 and 1.489.

To determine which of these values corresponds to a relative maximum, we can test the values of g(x) at these points and the endpoints of the interval.

g(0) ≈ 0.322
g(0.471) ≈ 0.783
g(π/3) ≈ 0.111
g(1.489) ≈ 0.782

We can see that g(0.471) and g(1.489) are both greater than g(0) and g(π/3), and that g(1.489) is slightly greater than g(0.471). Therefore, the function attains a relative maximum at x = 1.489.

In conclusion, the main answer to the question is option D, 1.489. We arrived at this answer by finding the derivative of the function g(x), setting it equal to zero, and testing the values of g(x) at the critical points and endpoints of the interval.

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32% of students in the sample do not use Fac/ebook.

The percentage of the students not using Faceb/ook is solved using the formula

= (number of students not using fa/cebook) / (total number of students) * 100

The number of students who do not use Fa/cebook is the sum of the values in the "Does not use Fac/ebook" column, which is 4 + 67 = 71.

The total number of students in the sample is the sum of the values in the "TOTAL" row, which is 219.

hence we have that

(71 / 219) × 100 ≈ 32.42

Rounding to the nearest percent  approximately 32% of students in the sample do not use Fa/cebook.

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We can't make a triangle with these angles, the correct option is A.

Remember that the sum of the interior angles of any triangle must always be equal to 180 degrees.

In this case, we know that the measures of the angles must be 60, 60, and 40. Adding these angles we will get:

60 + 60 + 40 = 160

That is different than 180, then these angles can't be the interior angles of a triangle, then the correct option is A, none.

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f is well-defined but not onto or one-to-one, g is well-defined and one-to-one but not onto, and his is well-defined but not onto or one-to-one.

I. A function is well-defined if each element in the domain is assigned a unique element in the range.

Looking at the definitions of the functions given, we can see that each element in the domain is assigned a unique element in the range for all three functions.

Therefore, all three functions f, g, and h are well-defined.

II. A function is onto if every element in the range is mapped to by at least one element in the domain.

To determine if a function is onto, we need to check if every element in the range {a, b, c, d} is assigned to at least one element in the domain {1, 2, 3, 4}.

For f, we can see that a and b are both assigned to elements in the domain, but c and d are not.

Therefore, f is not onto.

For g, we can see that every element in the range is assigned to an element in the domain.

Therefore, g is onto.

For h, we can see that a and d are both assigned to elements in the domain, but b and c are not.

Therefore, h is not onto.

III. A function is one-to-one if each element in the domain is assigned to a unique element in the range.

To determine if a function is one-to-one, we need to check if no two elements in the domain are assigned to the same element in the range.

For f, we can see that both 1 and 3 are assigned to a. Therefore, f is not one-to-one.

For g, we can see that no two elements in the domain are assigned to the same element in the range. Therefore, g is one-to-one.

For h, we can see that both 3 and 4 are assigned to a. Therefore, h is not one-to-one.

In summary, f is well-defined but not onto or one-to-one, g is well-defined and one-to-one but not onto, and h is well-defined but not onto or one-to-one.

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The probability that the secret code is composed of numbers with a GCD of 4 is approximately 0.68%.

a) The probability that the secret code is composed of numbers with a greatest common divisor (GCD) of 4 can be determined by finding the total number of favorable outcomes and dividing it by the total number of possible outcomes.

To have a GCD of 4, the numbers must be divisible by 4. Out of the 23 available cards, there are 5 numbers (4, 8, 12, 16, and 20) that are divisible by 4.

Since each student picks one card, the first student has a 5/23 chance of selecting a card divisible by 4. Once the first card is selected, there are 4/22 cards remaining for the second student, 3/21 for the third student, and 2/20 for the fourth student.

To calculate the overall probability, we multiply the probabilities of each student's selection:

P(GCD 4) = (5/23) * (4/22) * (3/21) * (2/20) ≈ 0.0068 or 0.68%

Therefore, the probability that the secret code is composed of numbers with a GCD of 4 is approximately 0.68%.

b) If the top four students picked the numbers, we need to determine the probability of getting at least 3 prime numbers.

There are 9 prime numbers between 1 and 23 (2, 3, 5, 7, 11, 13, 17, 19, 23). We will calculate the probability of picking 3 prime numbers and 4 prime numbers separately, and then add them together.

P(3 prime numbers) = (9/23) * (8/22) * (7/21) * (14/20)

P(4 prime numbers) = (9/23) * (8/22) * (7/21) * (6/20)

To find the probability of getting at least 3 prime numbers, we add these two probabilities:

P(at least 3 prime numbers) = P(3 prime numbers) + P(4 prime numbers)

The result will give us the probability of obtaining at least 3 prime numbers when the top four students pick the numbers.

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1. The output size of the first layer is 28 x 28, and the number of parameters in the first layer is 156.

2. By using parameter sharing, we can reduce the number of parameters to 78 in the proposed scheme.

The output size of a convolutional layer can be calculated using the formula:

output_size = (input_size - filter_size + 2 × padding) / stride + 1

In this case, the input size is 32 x 32, the filter size is 5 x 5, padding is 0, and stride is 1. Substituting the values into the formula, we get:

output_size = (32 - 5 + 2 × 0) / 1 + 1 = 28 x 28.

The number of parameters in a convolutional layer is determined by the number of filters and the size of each filter, including the biases. Each filter has a size of 5 x 5, so the total number of parameters in the first layer is (6 filters x 5 x 5) + (6 biases) = 156 parameters.

To reduce the number of parameters, we can use parameter sharing. By sharing weights between adjacent filters, we can reduce the number of independent sets of weights. In this case, we can share weights between filters 1 and 2, and between filters 3 and 4. This reduces the number of independent sets of weights from 6 to 3.

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Therefore, the long-run fraction of time the hemoglobin molecule is in the free state is 3/17, the fraction of time it is attached to an oxygen molecule is 6/17, and the fraction of time it is attached to a carbon monoxide molecule is 8/17.

We can formulate the Markov chain model as follows:

State 0: the hemoglobin molecule is free

State +: the hemoglobin molecule is attached to an oxygen molecule

State -: the hemoglobin molecule is attached to a carbon monoxide molecule

Let Pij be the transition probability from state i to state j. Then:

P00 = 1 - (1/3) - (2/4) = 1/6 (the hemoglobin molecule remains free)

P0+ = 1/3 (an oxygen molecule attaches)

P0- = 2/4 = 1/2 (a carbon monoxide molecule attaches)

P++ = 1/3 (the attached oxygen molecule remains)

P+- = 2/4 = 1/2 (the attached oxygen molecule is replaced by a carbon monoxide molecule)

P+0 = 1 - (1/3) - (2/4) = 1/6 (the oxygen molecule detaches)

P-+ = 1/3 (the attached carbon monoxide molecule is replaced by an oxygen molecule)

P-- = 1/4 (the attached carbon monoxide molecule remains)

P-0 = 2/4 = 1/2 (the carbon monoxide molecule detaches)

The transition matrix is:

   [1/6   1/3   1/2]

P = [1/6   1/3   1/2]

   [1/3   1/3   1/4]

To find the long-run fraction of time the hemoglobin molecule is in each of its three states, we need to solve the equation:

πP = π

where π = [π0, π+, π-] is the vector of long-run state probabilities. This gives us the system of equations:

π0/6 + π+/6 + π+/3 = π0

π0/3 + π+/3 + π-/3 = π+

π0/2 + π+/2 + π-/4 = π-

Solving this system, we get:

π0 = 3/17

π+ = 6/17

π- = 8/17

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Answer: 6:01

If it ended at 6:01, then it ended at 6:01. If it started at 6:01, then it would've ended at 9:20

Step-by-step explanation:

Hello! I'm happy to help you with your question. Here's the notation for each sequence:

(a) 2 + 4 + 6 + 8 + ... + 2n can be rewritten as:
∑(2i) where i goes from 1 to n.

(b) 1 + 5 + 9 + 13 + ... + 425 can be rewritten as:
∑(4j-3) where j goes from 1 to 106. (Note: 425 is the 106th term in this sequence)

(c) 1 + 12 + 13 + 14 + ... + 150 can be rewritten as:
1 + ∑(k) ,where k goes from 12 to 150

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(a) The mean of x is 3.450

To determine the mean of x, where x has a continuous uniform distribution over the interval [1.7, 5.2], you need to follow these steps:

Step 1: Identify the lower limit (a) and upper limit (b) of the interval. In this case, a = 1.7 and b = 5.2.

Step 2: Calculate the mean (μ) using the formula: μ = (a + b) / 2.

Step 3: Plug in the values of a and b into the formula: μ = (1.7 + 5.2) / 2.

Step 4: Calculate the mean: μ = 6.9 / 2 = 3.45.

Therefore, the mean of x is 3.450 when rounded to 3 decimal places.

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The correct expression is,

⇒ $1.8 + 0.25y = $2.25

Where, y is number of quarters.

We have to given that;

You have 18 dimes and Quarters that are worth $2.25.

Since, We know that;

1 dimes = 0.10 dollar

1 quarters = 0.25 dollar

Hence, We get;

18 dimes = 18 x 0.10

              = 1.8 dollars

So, We can formulate the correct expression which represent the situation is,

⇒ $1.8 + 0.25y = $2.25

Where, y is number of quarters.

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The two parts of a rational number are the numerator and the denominator.

The numerator is the top number in a fraction, and it represents the number of parts being considered. The denominator is the bottom number in a fraction, and it represents the total number of equal parts into which the whole is divided. Together, the numerator and denominator define the value of the rational number, which is expressed as a ratio of two integers. For example, in the fraction 3/4, the numerator is 3 and the denominator is 4, so the rational number represents three out of four equal parts.

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The line integral is zero: ∫C 4sin(y)dx + 4xcos(y)dy = 0.

To apply Green's Theorem, we need to find the curl of the vector field F = (4sin(y), 4xcos(y)). We have:

∂F2/∂x = 4cos(y)

∂F1/∂y = 4cos(y)

So the curl of F is:

curl(F) = ∂F2/∂x - ∂F1/∂y = 0

Since the curl of F is zero, we can apply Green's Theorem to find the line integral along the ellipse C:

∫C F · dr = ∬R curl(F) dA = 0

where R is the region enclosed by C, and dA is an infinitesimal area element.

Therefore, the line integral is zero:

∫C 4sin(y)dx + 4xcos(y)dy = 0

So the answer is 0.

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The probability that Chauncey Billups misses his first free throw and makes the second is 0.1056. This probability is obtained by multiplying the probability of missing a free throw (0.12) with the probability of making a free throw (0.88). Answer is c) 0.1056.

To calculate the probability, we first determine that the probability of missing a free throw is 1 - 0.88 = 0.12, as Billups makes free throws 88% of the time.The probability that Chauncey Billups misses his first free throw and makes the second can be calculated by multiplying the probabilities of each event.

Given that he makes free throw shots 88% of the time, the probability of missing a free throw is 1 - 0.88 = 0.12.

To find the probability of missing the first shot and making the second, we multiply the probabilities: 0.12 * 0.88 = 0.1056.

Therefore, the correct answer is c) 0.1056.

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Answer: Let's evaluate the expression "(9 + 3) · 4" step by step:

Parentheses/Brackets: Calculate the expression inside the parentheses.

(9 + 3) = 12

Multiplication: Multiply the result from step 1 by 4.

12 · 4 = 48

Therefore, the correct step-by-step explanation is:

The expression "(9 + 3) · 4" simplifies to 48.

A circular swimming pool with a diameter of 64 feet would have a radius of 32 feet. This means that the distance from the center of the pool to any point on the edge (or circumference) would be 32 feet.

The area of a circle can be calculated using the formula A = πr²,

where A represents the area and r represents the radius. In this case, the radius is 32 feet, so the area of the pool would be:

A = π × (32 feet)²

A = π × 1024 square feet

A ≈ 3.14 × 1024 square feet

A ≈ 3,210.24 square feet

So, the approximate area of the circular community swimming pool would be around 3,210.24 square feet.

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The new circular community swimming pool has a diameter of 64 feet. A. What is the area of the community pool?

If z is a standard normal variable, the probability that z lies between -0.55 and 0.55 is D. 0.4176.

The probability that z lies between -0.55 and 0.55 can be found by using the standard normal distribution table or a calculator with a built-in normal distribution function.

Using a standard normal distribution table, we can find the area under the curve between -0.55 and 0.55, which is equivalent to the probability we are trying to find.

The table gives us the area to the left of a z-score, so we need to subtract the area to the left of -0.55 from the area to the left of 0.55.

Looking at the table, we can find that the area to the left of -0.55 is 0.2912, and the area to the left of 0.55 is 0.7088.

Therefore, the area between -0.55 and 0.55 is:
0.7088 - 0.2912 = 0.4176

So the answer is D. 0.4176.

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The x-coordinates of all local minima using the second derivative test is [tex](27/11)^(^1^/^2^).[/tex]

First, we need to find the critical points by setting the first derivative equal to zero:

g'(x) = [tex]11x^10 - 27x^8[/tex] = 0

Factor out x^8 to get:

[tex]x^8(11x^2 - 27)[/tex] = 0

So the critical points are at x = 0 and x =  ±[tex](27/11)^(^1^/^2^).[/tex]

Next, we need to use the second derivative test to determine which critical points correspond to local minima. The second derivative of g(x) is:

g''(x) =[tex]110x^9 - 216x^7[/tex]

Plugging in x = 0 gives g''(0) = 0, so we cannot use the second derivative test at that critical point.

For x = [tex](27/11)^(^1^/^2^)[/tex], we have g''(x) = [tex]110x^9 - 216x^7 > 0[/tex], so g(x) has a local minimum at x =[tex](27/11)^(^1^/^2^).[/tex]

For x = -[tex](27/11)^(^1^/^2^)[/tex], we have g''(x) = [tex]-110x^9 - 216x^7 < 0[/tex], so g(x) has a local maximum at x = -[tex](27/11)^(^1^/^2^)[/tex]

Therefore, the x-coordinates of the local minima of g(x) are [tex](27/11)^(^1^/^2^).[/tex]

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Answer:

The value of the line integral s F .dr is -1/4 + 2/3j.

To evaluate the line integral s F .dr, where C is given by the vector function r(t) = ⟨x(t), y(t)⟩, we need to find the limits of integration and express F in terms of r(t).

First, let's find the limits of integration. We are not given any specific values of t, so we need to find the range of t that corresponds to the curve C. Since C is not explicitly defined, we can use the parameterization r(t) = ⟨t, t^2⟩ as a possible representation of C. We can see that as t varies, r(t) traces out a parabola in the xy-plane. Therefore, we can take the limits of integration to be the range of t that corresponds to this parabolic segment. One way to find this range is to solve the quadratic equation y = x^2 for x in terms of y, which gives x = ±√y. Since we are only interested in the part of the parabola that lies in the first quadrant, we take x = √y. Thus, the limits of integration are t = 0 to t = 1.

Next, let's express F in terms of r(t). We have F(x, y) = ⟨-xy, -x⟩ = -xy⟨1, 0⟩ - x⟨0, 1⟩ = -xyi - xj. To express F in terms of r(t), we substitute x = t and y = t^2, which gives F(r(t)) = -t^3i - tj.

Now we can evaluate the line integral using the formula

s F .dr = ∫a^b F(r(t)) . r'(t) dt,

where r'(t) = ⟨dx/dt, dy/dt⟩ is the derivative of r(t). In our case, r'(t) = ⟨1, 2t⟩.

Thus, we have

s F .dr = ∫0^1 F(r(t)) . r'(t) dt
= ∫0^1 (-t^3i - tj) . ⟨1, 2t⟩ dt
= ∫0^1 (-t^3 + 2t^2j) dt
= [-1/4t^4 + 2/3t^3j]0^1
= (-1/4 + 2/3j) - (0 + 0j)
= -1/4 + 2/3j.

Therefore, the value of the line integral s F .dr is -1/4 + 2/3j.

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