Determine whether the series is conv 8 4n + 15-n - n = 1

The absolute extreme values on the interval are:

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

Absolute minimum: f(x) = 2 at x = π/2

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

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

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

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

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

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

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

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

At x = 0:

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

At x = π/2:

f(π/2) = 2 + cos^2(π/2) = 2 + 0 = 2

At x = π:

f(π) = 2 + cos^2(π) = 2 + 1 = 3

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

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

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To find the indefinite integral of √(x²√(x) + 6x + 144) dx, we can use the substitution technique. Let's choose the substitution u = x²√(x).

Differentiating both sides with respect to x, we get du/dx = (3/2)x√(x) + 2x²/(2√(x)) = (3/2)x√(x) + x√(x) = (5/2)x√(x).  Rearranging the equation, we have dx = (2/5) du / (x√(x)).  Now, substitute u = x²√(x) and dx = (2/5) du / (x√(x)) into the integral.  ∫ √(x²√(x) + 6x + 144) dx becomes ∫ √(u + 6x + 144) * (2/5) du / (x√(x)).  Simplifying further, we have (2/5) ∫ √(u + 6x + 144) du / (x√(x)).  Now, we can simplify the integrand by factoring out the common term (u + 6x + 144)^(1/2) from the numerator and denominator: (2/5) ∫ du / x√(x) = (2/5) ∫ du / (√(x)x^(1/2)).  Using the power rule of integration, we have (2/5) * 2 (√(x)x^(1/2)) = (4/5) (x^(3/2)).  Therefore, the indefinite integral of √(x²√(x) + 6x + 144) dx is (4/5) (x^(3/2)) + C, where C is the constant of integration.

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1) Null Hypothesis (H0): There is no significant difference in performance (average points) by the fighters in the 3 weight divisions.

Alternative Hypothesis (HA): At least one of the weight divisions has a significantly different performance (average points) than the others.

2) To determine if there is a significant difference in performance by the fighters in the 3 weight divisions, we can use a statistical test such as Analysis of Variance (ANOVA). ANOVA is used to compare the means of three or more groups and determine if there is a significant difference among them.

By performing the ANOVA test with a level of significance (α) of 0.05, we can obtain a p-value. The p-value is a measure that indicates the probability of obtaining the observed data, or data more extreme, assuming the null hypothesis is true. If the p-value is less than the chosen significance level (0.05 in this case), we reject the null hypothesis. Otherwise, if the p-value is greater than or equal to 0.05, we fail to reject the null hypothesis.

To perform the ANOVA test and obtain the p-value, the data points scored by 24 fighters in the 3 weight divisions are required. Unfortunately, the data points are not provided in the given information. Once the data is available, it can be analyzed using Excel or Minitab to obtain the ANOVA results and determine if there is a significant difference in performance among the weight divisions.

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The given matrix is a 2x2 matrix: A = [4 8]. To determine if the matrix is invertible, we need to find the determinant of the matrix.

The determinant of a 2x2 matrix can be calculated using the formula:

det(A) = ad - bc,

where a, b, c, and d are the elements of the matrix.

In this case, a = 4, b = 8, c = 0, and d = 0. Plugging these values into the determinant formula, we have:

det(A) = (4 * 0) - (8 * 0) = 0 - 0 = 0.

The determinant of the matrix is 0.

If the determinant of a matrix is zero, it means that the matrix is not invertible. In other words, the given matrix does not have an inverse.

To summarize, the determinant of the matrix [4 8] is 0, indicating that the matrix is not invertible.

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The series ∑ (-1)^n*n! from n=1 to infinity diverges and the series does not satisfy the conditions for convergence according to the alternating series test.

To determine the convergence of the series ∑ (-1)^n*n! from n=1 to infinity, we can use the alternating series test.

The alternating series test states that if a series satisfies two conditions:

In our case, the terms (-1)^n*n! alternate in sign, as (-1)^n changes sign with each term. However, we need to check the behavior of the absolute values of the terms.

Taking the absolute value of each term, we have |(-1)^n*n!| = n!.

Now, we need to consider the behavior of n! as n increases. We know that n! grows very rapidly as n increases, much faster than any power of n. Therefore, n! does not approach zero as n increases.

Since the absolute values of the terms (n!) do not approach zero, the series does not satisfy the conditions for convergence according to the alternating series test.

Therefore, the series ∑ (-1)^n*n! from n=1 to infinity diverges.

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The maximum population density occurs at (10, ∞).

To find the maximum population density, we need to find the critical point of the given function. Taking partial derivatives with respect to x and y, we get:

∂P/∂x = -50x + 500

∂P/∂y = -25

Setting both partial derivatives equal to zero, we get:

-50x + 500 = 0

-25 = 0

Solving for x and y, we get:

x = 10

y = any value

Substituting x = 10 into the original equation, we get:

P(10,y) = -25(10)² - 25y + 500(10) + 600y + 180

P(10,y) = -2500 - 25y + 5000 + 600y + 180

P(10,y) = 575y - 2320

To find the maximum value of P(10,y), we need to take the second partial derivative with respect to y:

∂²P/∂y² = 575 > 0

Since the second partial derivative is positive, we know that P(10,y) has a minimum value at y = -∞ and a maximum value at y = ∞. Therefore, the maximum population density occurs at (10, ∞).

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Answer:

The answer is 20%.

Step-by-step explanation:

Answer:

20%

Step-by-step explanation:

To write the decimal as a percent, we multiply it by 100

0.20 = 0.20 × 100 = 20%

Hence, 0.20 is the same as 20%.

A recurrence relation for the sequence dn which gives the amount of drug in the blood after the nth dose is given by option A. dn+1 = (dn/2) + 90.

The limit of the sequence is given by option A. 180 mg

To find the recurrence relation for the sequence (dn),

Analyze the problem.

Each dose adds 90 mg of the medication to the blood,

and every 24 hours, half of the drug in the blood is eliminated.

Let us assume d0 is the initial amount of drug in the blood,

and di represents the amount of drug in the blood after the ith dose.

d0 = 60 mg.

After the first dose, the amount of drug in the blood will be,

d1 = d0 + 90

After the second dose, the amount of drug in the blood will be,

d2 = (d1/2) + 90

After the third dose, the amount of drug in the blood will be,

d3 = (d2/2) + 90

Observe that for each subsequent dose, the amount of drug in the blood is half of the previous amount plus 90 mg.

The recurrence relation for the sequence (dn) is,

dn+1 = (dn/2) + 90

The correct answer is:

A. dn+1 = (dn/2) + 90

To determine the limit of the sequence (dn),

Analyze what happens as n approaches infinity.

In the long run, the amount of drug in the blood should stabilize, meaning that the limit of the sequence exists.

Let us find the limit of the sequence directly. Start by assuming the limit is L,

L = (L/2) + 90

To solve this equation for L, multiply both sides by 2,

2L = L + 180

Subtracting L from both sides,

L = 180

The limit of the sequence (dn) is 180 mg.

A. The limit of the sequence is 180 mg

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Answer:

explantion

Step-by-step explanation:

exaplantion:

just a little bit but you can either

Other wise that is it

bonus ( in a way )

graphing.

Other wise that is it

                   The answer is this little thing on top↑↑↑↑

the approximate area of the surface obtained by rotating the given curve about the x-axis is 804 square units.

What is Area?

In geometry, the area can be defined as the space occupied by a flat shape or the surface of an object. Generally, the area is the size of the surface

To find the area of the surface obtained by rotating the curve x = t², y = 2t (where 0 ≤ t ≤ 9) about the x-axis, we can use the formula for the surface area of revolution.

The formula for the surface area of revolution is given by:

A = 2π∫[a,b] y(t) √(1 + (dy/dt)²) dt

In this case, we have:

y(t) = 2t

dy/dt = 2

Substituting these values into the formula, we have:

A = 2π∫[0,9] 2t √(1 + 4) dt

A = 2π∫[0,9] 2t √(5) dt

A = 4π√5 ∫[0,9] t dt

A = 4π√5 [t²/2] [0,9]

A = 4π√5 [(9²/2) - (0²/2)]

A = 4π√5 [81/2]

A = 162π√5

Rounding this value to the nearest whole number, we get:

A ≈ 804

Therefore, the approximate area of the surface obtained by rotating the given curve about the x-axis is 804 square units.

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the approximate area of the surface obtained by rotating the given curve about the x-axis is 804 square units.

What is Area?

In geometry, the area can be defined as the space occupied by a flat shape or the surface of an object. Generally, the area is the size of the surface

To find the area of the surface obtained by rotating the curve x = t², y = 2t (where 0 ≤ t ≤ 9) about the x-axis, we can use the formula for the surface area of revolution.

The formula for the surface area of revolution is given by:

A = 2π∫[a,b] y(t) √(1 + (dy/dt)²) dt

In this case, we have:

y(t) = 2t

dy/dt = 2

Substituting these values into the formula, we have:

A = 2π∫[0,9] 2t √(1 + 4) dt

A = 2π∫[0,9] 2t √(5) dt

A = 4π√5 ∫[0,9] t dt

A = 4π√5 [t²/2] [0,9]

A = 4π√5 [(9²/2) - (0²/2)]

A = 4π√5 [81/2]

A = 162π√5

Rounding this value to the nearest whole number, we get:

A ≈ 804

Therefore, the approximate area of the surface obtained by rotating the given curve about the x-axis is 804 square units.

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a. To find the dot product of vectors v and u, we multiply their corresponding components and sum the results:

v · u = (6i + 3j - 2k) · (7i + 24j)

= 6(7) + 3(24) + (-2)(0)

= 42 + 72 + 0

= 114

Therefore, the dot product of v and u is 114.

b. To find the length (magnitude) of vector v, we use the formula:

|v| = √(v · v)

Substituting the components of v into the formula, we have:

|v| = √((6i + 3j - 2k) · (6i + 3j - 2k))

= √(6^2 + 3^2 + (-2)^2)

= √(36 + 9 + 4)

= √49

= 7

Therefore, the length of vector v is 7.

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To evaluate the integral ∫(7 - 7x)/(1 - √x) dx, we can start by rationalizing the denominator and simplifying the expression.

First, we multiply both the numerator and denominator by the conjugate of the denominator, which is (1 + √x): ∫[(7 - 7x)/(1 - √x)] dx = ∫[(7 - 7x)(1 + √x)/(1 - √x)(1 + √x)] dx

Expanding the numerator:∫[(7 - 7x - 7√x + 7x√x)/(1 - x)] dx Simplifying the expression:

∫[(7 - 7√x)/(1 - x)] dx

Now, we can split the integral into two separate integrals: ∫(7/(1 - x)) dx - ∫(7√x/(1 - x)) dx The first integral can be evaluated using the power rule for integration: ∫(7/(1 - x)) dx = -7ln|1 - x| + C1

For the second integral, we can use a substitution u = 1 - x, du = -dx: ∫(7√x/(1 - x)) dx = -7∫√x du Integrating √x:

-7∫√x du = -7(2/3)(1 - x)^(3/2) + C2

Combining the results: ∫(7 - 7x)/(1 - √x) dx = -7ln|1 - x| - 14/3(1 - x)^(3/2) + C Therefore, the evaluated integral is -7ln|1 - x| - 14/3(1 - x)^(3/2) + C.

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The revenue function is given by R(p) = (7000 - 0.2p) * (-5).

The demand for wireless headphones is given by the equation x = 7000 - 0.1p, where x represents the quantity demanded and p represents the price in dollars.

To find the revenue function R(p), we multiply the price p by the quantity demanded x:

R(p) = p * x

Substituting the given demand equation into the revenue function, we have:

R(p) = p * (7000 - 0.1p)

Simplifying further:

R(p) = 7000p - 0.1p²

Now, we can find the associated revenue function R'(p) by differentiating R(p) with respect to p:

R'(p) = 7000 - 0.2p

To find the rate at which revenue is changing with respect to time, we need to consider the rate at which the price is changing. Given that the price is decreasing at a rate of $5 per week, we have dp/dt = -5.

Finally, we can find the rate of change of revenue with respect to time (dR/dt) by multiplying R'(p) by dp/dt:

dR/dt = R'(p) * dp/dt

= (7000 - 0.2p) * (-5)

This equation represents the rate of change of revenue with respect to time, considering the given price decrease rate.

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For a factorable trinomial x² + bx + c, the absolute value of b can be less than, equal to, or greater than the absolute value of c, depending on the specific values of b and c.

The quadratic trinomial formula in one variable has the general form ax2 + bx + c, where a, b, and c are constant terms and none of them are zero.

For any factorable trinomial of the form x² + bx + c, the absolute value of b can sometimes be less than, equal to, or greater than the absolute value of c. The relationship between the absolute values of b and c depends on the specific values of b and c.

Let's consider a few cases:

1. If both b and c are positive or both negative: In this case, the absolute value of b can be less than, equal to, or greater than the absolute value of c. For example:

  - In the trinomial x² + 2x + 3, the absolute value of b (|2|) is less than the absolute value of c (|3|).

  - In the trinomial x² + 4x + 3, the absolute value of b (|4|) is greater than the absolute value of c (|3|).

  - In the trinomial x² + 3x + 3, the absolute value of b (|3|) is equal to the absolute value of c (|3|).

2. If b and c have opposite signs: In this case, the absolute value of b can also be less than, equal to, or greater than the absolute value of c. For example:

  - In the trinomial x² - 4x + 3, the absolute value of b (|4|) is greater than the absolute value of c (|3|).

  - In the trinomial x² - 2x + 3, the absolute value of b (|2|) is less than the absolute value of c (|3|).

  - In the trinomial x² - 3x + 3, the absolute value of b (|3|) is equal to the absolute value of c (|3|).

Therefore, for a factorable trinomial x² + bx + c, the absolute value of b can be less than, equal to, or greater than the absolute value of c, depending on the specific values of b and c.

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It seems like you have a question related to a cone and its parametric equations. Based on the given information, the parametric equations for the cone are:

x = u * v * cos(v)
y = u * v * sin(v)
z = u

where u ranges from 0 to 1, and v ranges from 0 to 2π.

These equations describe the coordinates (x, y, z) of points on the surface of the cone as functions of the parameters u and v. The parameter u determines the height along the cone, while v represents the angle around the central axis of the cone.

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Answer:

He will make 55,000 dollars a year

Step-by-step explanation:

[tex]300[/tex] × [tex]50 = 15000[/tex]

[tex]15000 + 40000 = 55000[/tex]

Among the given choices, the basis (fundamental system) of solutions for the ODE is:

(a) [tex]e^{4x}[/tex]

(c) [tex]e^{2x}[/tex]

(f) [tex]xe^{2x}[/tex]

(g) [tex]e^{4x}+x[/tex]

The given differential equation is a second-order homogeneous linear ODE with constant coefficients. The characteristic equation associated with this ODE is obtained by substituting [tex]y = e^{4x}[/tex]into the ODE:

[tex](D^2 - 4D + 4)y = 0,[/tex]

where D denotes the derivative operator.

The characteristic equation is [tex](D - 2)^2 = 0[/tex], which has a repeated root of 2. This means that the basis (fundamental system) of solutions will consist of functions of the form [tex]e^{2x}[/tex] and [tex]xe^{2x}[/tex].

Among the given choices, the basis (fundamental system) of solutions for the ODE is:

(a) [tex]e^{4x}[/tex]

(c) [tex]e^{2x}[/tex]

(f) [tex]xe^{2x}[/tex]

(g) [tex]e^{4x}+x[/tex]

These functions satisfy the differential equation and are linearly independent, thus forming a basis of solutions for the given ODE.

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The correct answer is: The volume of the solid obtained by rotating the region enclosed by the equations y = e = 0 and x = 5 about the y-axis is 125πe.

The region which is enclosed by the equations y = e = 0 and x = 5 needs to be rotated about the y-axis. Thus, to find the volume of the solid obtained in the process of rotation of this region about the y-axis, one can use the method of cylindrical shells. The formula for the method of cylindrical shells is given as:

∫(from a to b)2πrh dr,

where "r" is the distance of the cylindrical shell from the axis of rotation, "h" is the height of the cylindrical shell, and "a" and "b" are the lower and upper limits of the region respectively.

Using the given conditions, we have a = 0 and b = 5The height "h" of the cylindrical shell is given by the equation

h = e - 0 = e = 2.71828 (approx.)

Now, the distance "r" of the cylindrical shell from the axis of rotation (y-axis) can be calculated using the equation

r = x

The lower limit of the integral is "a" = 0 and the upper limit of the integral is "b" = 5.

Substituting all the values in the formula of the method of cylindrical shells, we get:

V = ∫(from 0 to 5)2πrh dr= ∫(from 0 to 5)2π(re) dr= 2πe ∫(from 0 to 5)r dr= 2πe [(5²)/2 - (0²)/2]= 125πe

Thus, the volume of the solid obtained by rotating the region enclosed by the equations y = e = 0 and x = 5 about the y-axis is 125πe, where "e" is the value of Euler's number, which is approximately equal to 2.71828.

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Both series converge, the sum of the given series is the sum of their individual sums is 22/3.

To find the first four terms of the series, we substitute n = 0, 1, 2, and 3 into the expression.

The first four terms are:

n = 0: (2 / [tex]2^0[/tex]) + (2 / [tex]5^0[/tex]) = 2 + 2 = 4

n = 1: (2 / [tex]2^1[/tex]) + (2 / [tex]5^1[/tex]) = 1 + 0.4 = 1.4

n = 2: (2 / [tex]2^2[/tex]) + (2 / [tex]5^2[/tex]) = 0.5 + 0.08 = 0.58

n = 3: (2 / [tex]2^3[/tex]) + (2 / [tex]5^3[/tex]) = 0.25 + 0.032 = 0.282

To determine if the series converges or diverges, we can split it into two separate geometric series: ∑(2 / [tex]2^n[/tex]) and ∑(2 / [tex]5^n[/tex]).

The first series converges with a sum of 4, and the second series also converges with a sum of 10/3.

Since both series converge, the sum of the given series is the sum of their individual sums: 4 + 10/3 = 22/3.

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The question is -

Write out the first four terms of the series to show how the series starts. Then find the sum of the series or show that it diverges.

∑ n=0 to ∞ ((2 / 2^n) + (2 / 5^n))

Answer:

Step-by-step explanation:

The answer is B. 15m

The formula for Volume is V=lwh (l stands for length, w stands for width, and h stands for height). However, in this problem yo need to find the length. - this can be found by multiplying width times height and then dividing that result with 3600.

  -         3600/20*12 = l

             3600/240 = l

              15 = l

Hope it helps!

Step-by-step explanation:

5(3x+2) - 2(4x-1)

To expand and simplify the expression 5(3x+2) - 2(4x-1), you can apply the distributive property of multiplication over addition/subtraction. Let's break it down step by step:

First, distribute the 5 to both terms inside the parentheses:

5 * 3x + 5 * 2 - 2(4x-1)

This simplifies to:

15x + 10 - 2(4x-1)

Next, distribute the -2 to both terms inside its parentheses:

15x + 10 - (2 * 4x) - (2 * -1)

This simplifies to:

15x + 10 - 8x + 2

Combining like terms:

(15x - 8x) + (10 + 2)

This simplifies to:

7x + 12

Therefore, the expanded and simplified form of 5(3x+2) - 2(4x-1) is 7x + 12.

To identify the appropriate convergence test for each series, we need to examine the behavior of the terms in the series as n approaches infinity. For the series (−1)n +7 a), we can use the alternating series test,

It states that if a series has alternating positive and negative terms and the absolute value of the terms decrease to zero, then the series converges. For the series 2n e³n n=1 00 n' + 2 ο Σ, we can use the ratio test, which compares the ratio of successive terms in the series to a limit. If this limit is less than one, the series converges.  For series 3n, we can use the divergence test, which states that if the limit of the terms in a series is not zero, then the series diverges. Performing these tests, we find that (−1)n +7 a) converges, 2n e³n n=1 00 n' + 2 ο Σ converges, and 3n diverges. In summary, we need to choose the appropriate convergence test for each series based on the behavior of the terms, and performing these tests helps us determine whether a series converges or diverges.

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The simplified difference quotient for the function

f(x) = -4x² - x - 1 is -8x - 4h - 1.

To compute the difference quotient for the function f(x) = -4x² - x - 1, we need to find the value of f(x + h) and subtract f(x), all divided by h. Let's proceed with the calculations step by step.

First, we substitute x + h into the function f(x) and simplify:

f(x + h) = -4(x + h)² - (x + h) - 1

        = -4(x² + 2xh + h²) - x - h - 1

        = -4x² - 8xh - 4h² - x - h - 1

Next, we subtract f(x) from f(x + h):

f(x + h) - f(x) = (-4x² - 8xh - 4h² - x - h - 1) - (-4x² - x - 1)

                = -4x² - 8xh - 4h² - x - h - 1 + 4x² + x + 1

                = -8xh - 4h² - h

Finally, we divide the above expression by h to get the difference quotient:

(f(x + h) - f(x)) / h = (-8xh - 4h² - h) / h

                      = -8x - 4h - 1

The simplified difference quotient for the function f(x) = -4x² - x - 1 is -8x - 4h - 1. This expression represents the average rate of change of the function f(x) over the interval [x, x + h]. As h approaches zero, the difference quotient approaches the derivative of the function.

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To determine the values of x, y, and z that satisfy the given equations, we can use row operations on the augmented matrix representing the system of equations.

We start by writing the system of equations as an augmented matrix:

| 1 1 1 | 5 |

| 2 1 0 | -25 |

| 0 1 -4 | -4 |

We can perform row operations to simplify the augmented matrix and solve for the values of x, y, and z. Applying row operations, we can subtract twice the first row from the second row and subtract the second row from the third row:

| 1 1 1 | 5 |

| 0 -1 -2 | -55 |

| 0 0 -2 | -29 |

Now, we can divide the second row by -1 and the third row by -2 to simplify the matrix further:

| 1 1 1 | 5 |

| 0 1 2 | 55 |

| 0 0 1 | 29/2 |

From the simplified matrix, we can see that x = 5, y = 55, and z = 29/2. Therefore, the values of x, y, and z that satisfy the given equations are x = 5, y = 55, and z = 29/2.

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The derivative of the function F(x) = ∫[a to x] 6tcos(cos(t²)) dt is given by F'(x) = 6cos(cos(x²)) + 12x²*sin(cos(x²))*sin(x²).

To find the derivative of the function F(x) = ∫[a to x] 6t*cos(cos(t²)) dt using the Fundamental Theorem of Calculus, we can apply Part I of the theorem.

According to Part I of the Fundamental Theorem of Calculus, if we have a function F(x) defined as the integral of another function f(t) with respect to t, then the derivative of F(x) with respect to x is equal to f(x).

In this case, the function F(x) is defined as the integral of 6t*cos(cos(t²)) with respect to t. Let's differentiate F(x) to find its derivative F'(x):

F'(x) = d/dx ∫[a to x] 6t*cos(cos(t²)) dt.

Since the upper limit of the integral is x, we can apply the chain rule of differentiation. The chain rule states that if we have an integral with a variable limit, we need to differentiate the integrand and then multiply by the derivative of the upper limit.

First, let's find the derivative of the integrand, 6t*cos(cos(t²)), with respect to t. We can apply the product rule here:

d/dt [6tcos(cos(t²))]

= 6cos(cos(t²)) + 6t*(-sin(cos(t²)))(-sin(t²))2t

= 6cos(cos(t²)) + 12t²sin(cos(t²))*sin(t²).

Now, we multiply this derivative by the derivative of the upper limit, which is dx/dx = 1:

F'(x) = d/dx ∫[a to x] 6tcos(cos(t²)) dt

= 6cos(cos(x²)) + 12x²*sin(cos(x²))*sin(x²).

It's worth noting that in this solution, the lower limit 'a' was not specified. Since the lower limit is not involved in the differentiation process, it does not affect the derivative of the function F(x).

In conclusion, we have found the derivative F'(x) of the given function F(x) using Part I of the Fundamental Theorem of Calculus.

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Answer:

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The perimeter is the sum of all 5 sides.

Set up equation and solve for h:

The given system of differential equations can be written using matrices as follows:

X' = AX + B,

where X = [x, y, z] is the vector of variables, X' represents the derivative of X with respect to some independent variable, A is the coefficient matrix, and B is the constant matrix.

In this case, the coefficient matrix A is [[1, 2, 0], [0, 0, 2], [4, -4, 0]], and the constant matrix B is [-2, 1, 52].

To solve the system, we can find the eigenvalues and eigenvectors of the coefficient matrix A.

These eigenvalues and eigenvectors help in diagonalizing the coefficient matrix, allowing us to solve the system using the diagonalized form.

Once we have the diagonalized form, we can solve each equation individually to obtain the solutions for x, y, and z. Finally, we combine these solutions using linear combinations to form the general solution for the system.

However, without specific eigenvalues, eigenvectors, or initial conditions, it is not possible to provide the numerical solution.

If you have the eigenvalues, eigenvectors, or initial conditions, please provide them, and I can assist you in solving the system using the given matrices.

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To find the volume of the solid bounded by the given surfaces, we'll set up the integral using cylindrical coordinates. The closest option from the given choices is (C) 67.

The cylinder x^2 + y^2 = 4 can be expressed in cylindrical coordinates as r^2 = 4, where r is the radial distance from the z-axis.

We need to determine the limits for r, θ, and z to define the region of integration.

Limits for r:

Since the cylinder is bounded by r^2 = 4, the limits for r are 0 to 2.

Limits for θ:

Since we want to consider the entire cylinder, the limits for θ are 0 to 2π.

Limits for z:

The planes z = 0 and y + z = 3 intersect at z = 1. Therefore, the limits for z are 0 to 1.

Now, let's set up the integral to find the volume:

V = ∫∫∫ dV

Using cylindrical coordinates, the volume element dV is given by: dV = r dz dr dθ

Therefore, the volume integral becomes:

V = ∫∫∫ r dz dr dθ

Integrating with respect to z first:

V = ∫[0 to 2π] ∫[0 to 2] ∫[0 to 1] r dz dr dθ

Integrating with respect to z: ∫[0 to 1] r dz = r * [z] evaluated from 0 to 1 = r

Now, the volume integral becomes:

V = ∫[0 to 2π] ∫[0 to 2] r dr dθ

Integrating with respect to r: ∫[0 to 2] r dr = 0.5 * r^2 evaluated from 0 to 2 = 0.5 * 2^2 - 0.5 * 0^2 = 2

Finally, the volume integral becomes:

V = ∫[0 to 2π] 2 dθ

Integrating with respect to θ: ∫[0 to 2π] 2 dθ = 2 * [θ] evaluated from 0 to 2π = 2 * 2π - 2 * 0 = 4π

Therefore, the volume of the solid bounded by the given surfaces is 4π.

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The equation of the line perpendicular to [tex]y = 4x + 5[/tex] and passing through the point (-12, 4) is [tex](1/4)x + 4y = 13.[/tex]

To find the equation of a line that is perpendicular to the line y = 4x + 5 and passes through the point (-12, 4), we can use the fact that perpendicular lines have slopes that are negative reciprocals of each other.

The given line has a slope of 4. The negative reciprocal of 4 is -1/4. Therefore, the slope of the perpendicular line is -1/4.

Using the point-slope form of a linear equation, we can write the equation of the line as:

[tex]y - y₁ = m(x - x₁)[/tex]

where (x₁, y₁) is the point (-12, 4) and m is the slope (-1/4).

Substituting the values into the equation:

[tex]y - 4 = (-1/4)(x - (-12))y - 4 = (-1/4)(x + 12)[/tex]

Multiplying both sides by -4 to eliminate the fraction:

[tex]-4(y - 4) = -4(-1/4)(x + 12)-4y + 16 = (1/4)(x + 12)[/tex]

Simplifying the equation:

[tex]-4y + 16 = (1/4)x + 3[/tex]

Rearranging the terms to get the equation in the standard form:

[tex](1/4)x + 4y = 13[/tex]

Therefore, the equation of the line perpendicular to [tex]y = 4x + 5[/tex]and passing through the point (-12, 4) is [tex](1/4)x + 4y = 13.[/tex]

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(a) The speed of the boat is √208 km/h, which simplifies to p√13 km/h, where p is a constant.

(b) The position vector of point X, denoted as (x, y), is (12, 8) km.

(a) To find the speed of the boat, we need to calculate the distance traveled divided by the time taken. Given that the boat travels in a straight line at a constant speed, we can use the distance formula:

Distance = ||position final - position initial||

Using the given information, the initial position of the boat is (-11, -2) km, and the final position after 90 minutes (1.5 hours) is (1, 6) km. Let's calculate the distance:

Distance = ||(1, 6) - (-11, -2)||

= ||(1 + 11, 6 + 2)||

= ||(12, 8)||

= √(12^2 + 8^2)

= √(144 + 64)

= √208

Now, we divide the distance by the time taken:

Speed = Distance / Time

= √208 / 1.5

= (√(208) / √(1.5^2)) * (1.5 / 1.5)

= (√208 / √(1.5^2)) * (1.5 / 1.5)

= (√208 / 1.5) * (1.5 / 1.5)

= (√208 * 1.5) / 1.5

= √208

(b) Given that point X is due northeast of O, we can infer that the displacement in the x-direction is equal to the displacement in the y-direction. Let's denote the position vector of X as (x, y).

From the given information, we know that the boat starts at (-11, -2) km and ends at (1, 6) km. Therefore, the displacement in the x-direction is:

x = 1 - (-11) = 12 km.

Since X is due northeast, the displacement in the y-direction is the same as the displacement in the x-direction:

y = 6 - (-2) = 8 km.

Hence, the position vector of X is (12, 8) km.

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