4 gallons of gas used to drive 200 miles

By using the L'Hospital's Rule, we have shown that the limit of x^(3/x) as x approaches infinity is equal to 1.

To find the limit of the function [tex]x^{3/x}[/tex] as x approaches infinity, we can use L'Hopital's Rule, which states that if we have an indeterminate form of a fraction such as 0/0 or infinity/infinity, we can take the derivative of the numerator and the denominator separately until we no longer have an indeterminate form.

First, we can rewrite the function as e^(ln([tex]x^{3/x}[/tex])). Then, we can use the properties of logarithms to simplify it further as e^((3ln(x))/x). Now, we have an indeterminate form of infinity/infinity, and we can apply L'Hopital's Rule.

Taking the derivative of the numerator and denominator, we get:

lim x→∞ [tex]x^{3/x}[/tex] = lim x→∞ e^((3ln(x))/x)

= e^(lim x→∞ (3ln(x))/x)

Using L'Hopital's Rule on the exponent, we get:

= e^(lim x→∞ (3/x²))

Since the denominator is approaching infinity faster than the numerator, the limit of 3/x² as x approaches infinity is zero, and we are left with:

= e^(0)

= 1

Therefore, the limit of [tex]x^{3/x}[/tex] as x approaches infinity is 1.

Alternatively, we can use some algebraic manipulation and the squeeze theorem to find the limit without using L'Hopital's Rule. We can rewrite the function as [tex]x^{3/x}[/tex] = (x^(1/x))³, and notice that as x approaches infinity, 1/x approaches zero, and so x^(1/x) approaches 1 (as the exponential function with base e^(1/x) approaches 1). Therefore, we have:

lim x→∞ [tex]x^{3/x}[/tex]= lim x→∞ (x^(1/x))³

= 1³

= 1

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Complete Question:

Find the limit. Use L'Hospital's Rule if appropriate. If there is a more elementary method, consider using it.

lim x→∞ [tex]x^{3/x}[/tex]

The 95% confidence interval estimate of the population mean μ is 97.53 < μ < 112.47.

We are required to find the 95% confidence interval estimate of the population mean μ, given the sample size n=18, mean x =105, and standard deviation s=15.

1. Determine the t-score for a 95% confidence interval with n-1 degrees of freedom. For a sample size of 18, you have 17 degrees of freedom.

Using a t-table or calculator, you find the t-score to be approximately 2.11.

2. Calculate the margin of error (E) using the formula:

E = t-score × (s / √n)

E = 2.11 × (15 / √18)

E ≈ 7.47

3. Calculate the lower and upper bounds of the confidence interval using the sample mean (x) and margin of error (E):

Lower bound: x - E = 105 - 7.47 = 97.53

Upper bound: x + E = 105 + 7.47 = 112.47

So, the 95% confidence interval estimate is 97.53 < μ < 112.47. Please note that these values are rounded to two decimal places.

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Answer:

630 count each 180 and the add them upp

So, an equivalent expression to [tex]-18 - 64x[/tex] is[tex]-2(9+32x)[/tex].

Expressions are considered equivalent when they have the same value, regardless of their form. There are different methods to determine whether expressions are equivalent depending on the type of expressions involved. Here are some examples:

Numeric expressions:

To determine whether two numeric expressions are equivalent, we can evaluate them and compare their results. For example, the expressions 3 + 4 and 7 have the same value, so they are equivalent.

Algebraic expressions:

To determine whether two algebraic expressions are equivalent, we can simplify both expressions using algebraic rules and compare the results. For example, the expressions 2x + 4 - x and x + 4 have the same value for any value of x, so they are equivalent.

Logical expressions:

To determine whether two logical expressions are equivalent, we can use truth tables to evaluate the expressions and compare the results. For example, the expressions (A ∧ B) ∨ C and (A ∨ C) ∧ (B ∨ C) have the same truth values for any values of A, B, and C, so they are equivalent.

The expression -18 - 64x can be simplified by factoring out a common factor of -2 from both terms. This gives:

[tex]-18 - 64x = -2(9 + 32x)[/tex]

So, an equivalent expression to -18 - 64x is -2(9 + 32x).

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When a₁ = 4, d = 3.5, and n = 14 are given.The value of A(14) is 49.5, the correct answer is option C. The issue appears to be related to number juggling arrangements, where A(n) speaks to the nth term of the arrangement and a₁ speaks to the primary term of the sequence.

Ready to utilize the equation for the nth term of a math grouping:

A(n) = a₁ + (n-1)d

where d is the common contrast between sequential terms.

A(14) = 4 + (14-1)3.5

Streamlining this condition, we get:

A(14) = 4 + 13*3.5

A(14) = 4 + 45.5

A(14) = 49.5

Hence, the esteem of A(14) is 49.5, which is choice C. 

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Answer: Calculate

Step-by-step explanation: How to calculate the margin of error?

1. Get the population standard deviation (σ) and sample size (n).

2. Take the square root of your sample size and divide it into your population standard deviation.

3. Multiply the result by the z-score consistent with your desired confidence interval according to the following table:

The probability of z being less than 0.25 as 0.5987 based on population proportion.

To use the normal approximation, we first need to check if the conditions are met. For this, we need to check if np and n(1-p) are both greater than or equal to 10.

np = 98 x 0.56 = 54.88

n(1-p) = 98 x 0.44 = 43.12

Since both np and n(1-p) are greater than 10, we can use the normal approximation.

Next, we need to find the mean and standard deviation of the sampling distribution of proportion.

Mean = np = 54.88

Standard deviation = sqrt(np(1-p)) = sqrt(98 x 0.56 x 0.44) = 4.43

Now we can standardize the variable X and find the probability:

z = (X - mean) / standard deviation = (56 - 54.88) / 4.43 = 0.25

Using a standard normal table or calculator, we can find the probability of z being less than 0.25 as 0.5987.

Therefore, P(X < 56) = P(Z < 0.25) = 0.5987.

Note that we rounded the mean and standard deviation to two decimal places, but you should keep the full values in your calculations to minimize rounding errors.

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The given statement "A manufacturing manager has developed a table that shows the average production volume which is an example of a numerical measure." is true because it represents central tendency.

The average production level is a numerical measure that represents the central tendency of the production volume for a given period of time. In this case, the manufacturing manager has calculated the average production volume for each day over the past three weeks.

Numerical measures are quantitative values that summarize or describe the data. They are used to provide insights into the characteristics of a dataset, such as the distribution, variability, and central tendency.

Common numerical measures include measures of central tendency, such as the mean, median, and mode, as well as measures of dispersion, such as variance and standard deviation.

The average production level, also known as the mean, is a commonly used measure of central tendency. It is calculated by adding up all the production volumes and dividing by the number of days. The resulting value represents the typical or average production level for the period of time in question.

Therefore, the statement that the average production level is an example of a numerical measure is true.

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Answer: integers 0 <= x <= 60. Determine the mean and variance of X. This problem has been solved!

1 answer

·

Top answer:

Theory : If random variable Y fol

Doesn’t include: < ‎6).

Step-by-step explanation:

Doesn’t include: < ‎6).

Answer:

Step-by-step explanation:

Mode is the one that occurs the most.

Stem is first number repeated for leaf

so your list of numbers is

10,12,12,22,23,24,30

The one that occurs the most is 12.  So your mode is 12.

If there are multiple numbers that are "most" then there is no mode.

1) We are 90% confident that the average boot time of all computers with an SSD is greater than the average of all computers with an HDD.

Option 5) We are 90% confident that the difference between the two sample means falls within the interval. This means that we can say with 90% confidence that the true difference in average boot times between computers with SSDs and HDDs is somewhere between 4.2 and 13 seconds. We cannot make any conclusions about which type of drive has a faster boot time or whether there is a significant difference between them without further analysis or information.
1) We are 90% confident that the average boot time of all computers with an SSD is greater than the average of all computers with an HDD.

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The Maclaurin series are:

f(x) = x^2/(1-x^3) = x^2 + x^4 + x^6 + x^8 + ...

The interval of convergence can be found by testing the endpoint values of the series.

We can use the geometric series formula to find the Maclaurin series of f(x):

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

Differentiating both sides with respect to x gives:

[1/(1-x)]' = 1 + 2x + [tex]3x^{2} +4x^{3}[/tex]+ ...

Multiplying both sides by [tex]x^{2}[/tex] gives:

[tex]x^{2}[/tex]/(1-[tex]x^{3}[/tex]) = [tex]x^{2}[/tex] + [tex]x^{4}[/tex] + [tex]x^{6}[/tex] + [tex]x^{8}[/tex] + ...

Therefore, we have:

f(x) = [tex]x^{2}[/tex]/(1-[tex]x^{3}[/tex]) = [tex]x^{2} +x^{4} +x^{6} +x^{8} +...[/tex]

The interval of convergence can be found by testing the endpoint values of the series. When x = ±1, the series becomes:

1 - 1 + 1 - 1 + ...

This series does not converge, so the interval of convergence is -1 < x < 1.

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The limit of the given sequence is infinity. (e) an = tan(n). This sequence oscillates between -infinity and infinity, so it diverges.

(a) To determine if the sequence converges or diverges, we can use the limit comparison test. We compare the given sequence to a known sequence whose convergence/divergence we know. Let bn = 5n^2/n^3 = 5/n. Then, taking the limit as n approaches infinity of (an/bn), we get:

lim (an/bn) = lim [(3 + (5n^2)/tn^(3/n) + 2)/(5/n)]
= lim [(3n^(3/n) + 5n^2)/5]
= ∞

Since the limit is infinity, the sequence diverges.

(b) bn = 1/n. This is a p-series with p = 1, which diverges. Therefore, the given sequence also diverges.

(c) An = cos(n) + 1. The cosine function oscillates between -1 and 1, so the sequence oscillates between 0 and 2. However, since there is no limit to the oscillation, the sequence diverges.

(d) b) bn = e^(2n)/(n+2). To determine if this sequence converges or diverges, we can use the ratio test. Taking the limit as n approaches infinity of (bn+1/bn), we get:

lim (bn+1/bn) = lim (e^(2(n+1))/(n+3)) * (n+2)/e^(2n)
= lim (e^2/(n+3)) * (n+2)
= 0

Since the limit is less than 1, the sequence converges. To find the limit, we can use L'Hopital's rule to evaluate the limit of (e^(2n)/(n+2)) as n approaches infinity:

lim (e^(2n)/(n+2)) = lim (2e^(2n)/(1))
= ∞

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To show that the amount of glucose A(t) in the bloodstream at any time t is governed by the given differential equation, we need to consider the rates of glucose infusion and removal.

1. Glucose is infused into the bloodstream at a constant rate of C g/min. This means the rate of glucose infusion is simply C.

2. The glucose is converted and removed from the bloodstream at a rate proportional to the amount of glucose present. We can represent this by the equation: removal rate = kA, where k is a constant and A is the amount of glucose at time t.

Now, we can write the differential equation for A(t) by considering the net rate of change of glucose in the bloodstream. The net rate is the difference between the infusion rate and the removal rate:

A'(t) = infusion rate - removal rate

Substitute the values for the infusion rate and removal rate from the steps above:

A'(t) = C - kA

The amount of glucose A(t) in the bloodstream at any time t is governed by the differential equation A'(t) = C - kA, where C is the constant rate of glucose infusion, and k is the constant proportionality factor for glucose removal. This equation represents the net rate of change of glucose in the bloodstream, considering both infusion and removal rates.

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The limit of (2x-1)(3-x)/(x-1)(x+3) as x approaches infinity is 0.

To find the limit of the function (2x-1)(3-x)/(x-1)(x+3) as x approaches infinity, we will divide both the numerator and denominator through the highest power of x. In this case, the highest power of x is x², so we can divide both the numerator & the denominator through x²:

[tex][(2x-1)/(x^2)] * [(3-x)/((x-1)/(x^2)(x+3))][/tex]

Now, as x approaches infinity, every of the fractions within the expression procedures zero except for (2x-1)/(x²). This fraction techniques 0 as x procedures infinity because the denominator grows quicker than the numerator. therefore, the limit of the expression as x strategies infinity is:

0 * 0 = 0

Consequently, When x gets closer to infinity, the limit of (2x-1)(3-x)/(x-1)(x+3) is 0.

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If Walmart is excluded from the list, the measure of center that would be more affected is the mean. The median, on the other hand, is less affected by extreme values, as it represents the middle value in the dataset.

If Walmart is excluded from the list, the measure of center that would be more affected is the mean. This is because Walmart is a large retailer and has a significant impact on the overall average of the data. Removing Walmart from the list would decrease the total sales and therefore decrease the mean. The median, on the other hand, would be less affected by the exclusion of Walmart as it only looks at the middle value of the data and is less sensitive to extreme values. The measure of center that would be more affected is the mean. Value like Walmart's revenue would significantly change the average. The median is less affected by extreme values, as it represents the middle value in the dataset.

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The absolute maximum value of f(x) over the interval [-3, 2] is approximately 5.24 and it occurs at x = 2, and the absolute minimum value is 1/8 and it occurs at x = 1/2.

To find the absolute maximum and minimum values of the function:

[tex]f(x) = (x^2-x)^{(2/3)}[/tex] over the interval [-3, 2], we need to follow these steps:

Find the critical points of the function by solving f'(x) = 0:

[tex]f'(x) = (2x - 1)\times{2/3}\times (x^2 - x)^{(-1/3)} = 0[/tex]

Solving for 2x - 1 = 0, we get x = 1/2. This is the only critical point in the interval [-3, 2].

Evaluate the function at the endpoints and the critical point:

f(-3) = ∛36 ≈ 3.301

f(2) = ∛12² ≈ 5.24

f(1/2) = 1/8

Determine which value is the absolute maximum and which is the absolute minimum:

The absolute maximum value is f(2) ≈ 5.24, and it occurs at x = 2.

The absolute minimum value is f(1/2) = 1/8, and it occurs at x = 1/2.

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Answer:

No--of the 52 cards, 13 are hearts. Of the 13 cards that are hearts, there is one card that is also an ace--the ace of hearts.

The new coordinates are A'=(2, -10), B'=(12, -10), C'=(12, -2), D'=(2, -2), and Square ABCD and Square A'B'C'D' are similar, with a scale factor of 2.

What is the scale factor?

A scale factor is a number that represents the amount of magnification or reduction applied to an object, image, or geometrical shape.

Part A:

When Square ABCD is reflected across the x-axis, the y-coordinates of each vertex will change sign while the x-coordinates remain the same. Therefore, the new coordinates of the reflected square will be:

A=(1,-5), B=(6,-5), C=(6,-1), D=(1,-1)

When this reflected square is dilated by a scale factor of 2, each vertex is multiplied by a scalar value of 2, centered at the origin. This means that the new coordinates will be:

A'=(2, -10), B'=(12, -10), C'=(12, -2), D'=(2, -2)

Part B:

To determine whether Square ABCD is similar or congruent to Square A'B'C'D', we need to compare their corresponding side lengths and angles.

First, let's compare the side lengths:

AB = BC = CD = DA = 5 units

A'B' = B'C' = C'D' = D'A' = 10 units

We can see that the side lengths of Square A'B'C'D' are exactly twice as long as the corresponding side lengths of Square ABCD. This tells us that the two squares are similar, with a scale factor of 2.

Next, let's compare the angles:

Square ABCD has four right angles (90 degrees) at each vertex.

Square A'B'C'D' also has four right angles (90 degrees) at each vertex.

Since the corresponding angles of the two squares are congruent, this confirms that the two squares are similar.

Therefore, we can conclude that Square ABCD and Square A'B'C'D' are similar, with a scale factor of 2.

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the vector projection of v onto w is -1i + 3j.To get the vector projection of v onto w, you'll need to use the following formula:

proj_w(v) = (v • w / ||w||^2) * w
Where v = 2i + 4j, w = -2i + 6j, "•" represents the dot product, and ||w|| is the magnitude of w.
Step 1: Find the dot product (v • w)
v • w = (2 * -2) + (4 * 6) = -4 + 24 = 20
Step 2: Find the magnitude of w (||w||)
||w|| = √((-2)^2 + (6)^2) = √(4 + 36) = √40
Step 3: Square the magnitude of w (||w||^2)
||w||^2 = (40)
Step 4: Calculate the scalar value (v • w / ||w||^2)
Scalar value = (20) / (40) = 0.5
Step 5: Multiply the scalar value by w to get the vector projection of v onto w
proj_w(v) = 0.5 * (-2i + 6j) = -1i + 3j

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The limit of the given sequence is 7.

From the first three terms, we can see that the sequence is alternating between adding 5 and dividing by 6.

As we continue down the sequence, we can see that the terms approach 7.

To prove this, we can use the formula for the sum of an infinite geometric series, which is:
S = a / (1 - r)
Where S is the sum of the series, a is the first term, and r is the common ratio. In this case, a = 1 and r = 5/6. Plugging in these values, we get:
S = 1 / (1 - 5/6)
S = 1 / (1/6)
S = 6

Hence, we need to add the last term, which is 6, to get the actual sum of the sequence. Therefore, the limit of the sequence is 7.

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The 18th term of the arithmetic sequence -13, -9, -5, -1, 3,... is 55. Thus, the right answer is option A. which is A(18) = 55

Arithmetic Progression is a sequence of numbers in which the difference between two numbers in the series is a fixed definite value.

The specific number in the arithmetic progression is calculated by

[tex]a_n=a_1+(n-1)d[/tex]

where [tex]a_n[/tex] is the term in arithmetic progression at the nth term

[tex]a_1[/tex] is the initial term in the arithmetic progression

d is the difference between two consecutive terms

In the given question, [tex]a_1[/tex] = -13

d = -9 - (-13) = 4

Therefore, to calculate the 18th term,

[tex]a_{13}=a_1+(18-1)d[/tex]

= -13 + (17) 4

= -13 + 68

= 55

Hence, the 18th term of the above AP is 55

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The algebraic expression with fractional exponents that is equivalent to the given radical expression is [tex]x^{\frac{3}{5} }[/tex].

Variables, constants, and mathematical operations (such as addition, subtraction, multiplication, division, and exponentiation) can all be found in an algebraic equation.

In algebra and other areas of mathematics, algebraic expressions are used to depict connections between quantities and to resolve equations and issues. A radical expression is any mathematical formula that uses the radical (also known as the square root symbol) sign.

To convert the radical expression [tex]\sqrt[5]{x^{3} }[/tex] into an algebraic expression with fractional exponents, we use the following rule:

[tex]a^{1/n}[/tex] = (n-th root of a)

In this case, we have:

[tex]\sqrt[5]{x^{3} }[/tex] = [tex](x^{3})^{\frac{1}{5} }[/tex]

= [tex]x^{\frac{3}{5}}[/tex]

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Expressing the regression equation in terms of the x variable instead of the y variable will cause the y-intercept and slope to change.

When the regression equation is expressed in terms of the y variable, it takes the form of: [tex]u=ax+b[/tex]                                where a is the y-intercept (the value of y when x = 0) and b is the slope (the rate at which y changes with respect to x).

If we express the same regression equation in terms of the x variable, it becomes:

[tex]x=\frac{y-a}{b}[/tex]

In this form, the y-intercept becomes [tex](0,\frac{a}{b} )[/tex], which is a point on the x-axis, and the slope becomes [tex]\frac{1}{b}[/tex] , which is the reciprocal of the slope in the original equation.

Therefore, expressing the regression equation in terms of the x variable instead of the y variable will cause the y-intercept and slope to change

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The distance between the Pluto and the sun is about 5,905,800,000 kilometers.

Distance between Mercury  and the sun is = 57,900,000 kilometers.

The distance of Pluto from the sun

=  Pluto is about 1.02 x 10^2 times farther away from the sun than Mercury.

⇒Distance of Pluto from the sun

= Distance of Mercury from the sun x 1.02 x 10^2

⇒Distance of Pluto from the sun = 57,900,000 km x 1.02 x 10^2

⇒Distance of Pluto from the sun = 57,900,000 km x 102

⇒ Distance of Pluto from the sun = 5,905,800,000 kilometers

Therefore, the distance of Pluto is about 5,905,800,000 kilometers away from the sun.

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In experimental studies, the researcher manipulates the exposure, that is he or she allocates subjects to the intervention or exposure group.

In experimental studies, the researcher manipulates the exposure or intervention by allocating subjects to the intervention or exposure group. This allows for the comparison of outcomes between the intervention/exposure group and the control group, which did not receive the intervention/exposure.

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The solution to the difference equation Yt+1 + 3y = 16; Yo = 100,t 20.

Hence, the correct option is A.

To solve the difference equation Yt+1 + 3Yt = 16, with initial condition Y0 = 100 and t ≥ 0, we can first find the homogeneous solution, which is

Yh(t) = [tex]A(-1/3)^t[/tex]

Where A is a constant determined by the initial condition. Put in the initial condition Y0 = 100, we get

Yh(0) = A = 100

Therefore, the homogeneous solution is

Yh(t) = [tex]100(-1/3)^t[/tex]

Next, we find the particular solution by assuming a constant value for Yt+1 and Yt, which gives us

Yp(t) = 4

This is because we have

Yt+1 + 3Yt = 16

Yp(t+1) + 3Yp(t) = 16

4 + 3Yp(t) = 16

Yp(t) = 4

So the particular solution is

Yp(t) = 4

Finally, the general solution is the sum of the homogeneous and particular solution

Y(t) = Yh(t) + Yp(t) = [tex]100(-1/3)^t[/tex] + 4

Using the initial condition Y20 = 100, we can solve for the constant A

Y20 = [tex]100(-1/3)^20 + 4 = A(-1/3)^20 + 4[/tex]

100 = [tex]A(-1/3)^20 + 4[/tex]

A = 96

Therefore, the solution to the difference equation Yt+1 + 3Yt = 16 with initial condition Y0 = 100 and t ≥ 0 is

Y(t) = [tex]96(-1/3)^t + 4[/tex]

Yt = [tex]96(-3)^t + 4[/tex]

Hence, the correct option is A.

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For an automobile service center with average of 12 cars per hour, the probability of the service center being totally empty in a given hour is equals to 0.000335.

The Poisson Probability Distribution is use to determine the probability for the number of events that occur in a period when the average number of events is known. Formula is written as following :[tex]P( \lambda,x) = \frac{e^{ -\lambda } \lambda^{x}}{x!}[/tex], where

Now, we have an automobile service center take care of 12 cars per hour.

Rate on average of car survice hour ,[tex] \lambda[/tex] = 8

We have to determine the value of probability of the service center when it being totally empty in a hour, P(8, 0). So,

[tex]P( 8, 0) = \frac{e^{ -8 } 8^{0}}{0!}[/tex]

= [tex]e^{ -8 } [/tex]

= (2.7182)⁻⁸

= 0.000335

Hence, required probability value is 0.000335.

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The second derivative of the function is zero.

The given function is -

y = 4 cos(x) sin(x)

We can write the first derivative as -

y' = dy/dx = 4 {cos(x) cos(x) - sin(x) sin(x)}

y' = dy/dx = = 4{cos²(x) - sin²(x)}

We can write the second derivative as -

y'' = d²y/dx² = 4{-2cos(x)sin(x) + 2sin(x)cos(x)}

y" = d²y/dx² = 4 x 0

y" = 0

So, the second derivative of the function is zero.

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The particular antiderivative of X(t) that satisfies the given condition X(0) = 1 is:

[tex]X(t) = 6e^t - 6t + 1[/tex].

The given derivative is X(t) = 6e^t.

To find the particular antiderivative of X'(t), we need to integrate X'(t) with respect to t:

[tex]\int6e^t)dt = 6e^t + C[/tex]

where C is the constant of integration.

Now, we use the given condition X(0) = 1 to find the value of C:

X(0) = 6(1) - 6 + C = 1

Simplifying, we get:

C = 1 + 6 - 6 = 1

Therefore, the particular antiderivative of X(t) that satisfies the given condition X(0) = 1 is:

[tex]X(t) = 6e^t - 6t + 1[/tex].

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