The value of the function at x=-2 is 0

The intervals in which the function f(x) = 2/3x^3 +4x²+6x+5 is increasing/decreasing is f(x) is decreasing on the interval (-∞, -3) and increasing on the interval (-1, ∞). A local maximum is at x = -3 with a value of f(-3) = 20. A local minimum is at x = -1 with a value of f(-1) = 3 2/3.

To find the intervals in which the function f(x) = 2/3x^3 +4x²+6x+5 is increasing or decreasing, we need to find its first derivative and determine its sign:

f'(x) = 2x^2 + 8x + 6

To find the critical points, we set f'(x) = 0 and solve for x:

2x^2 + 8x + 6 = 0

Dividing by 2, we get:

x^2 + 4x + 3 = 0

Factoring, we get:

(x + 3)(x + 1) = 0

So the critical points are x = -3 and x = -1.

From the sign chart, we can see that f(x) is decreasing on the interval (-∞, -3) and increasing on the interval (-1, ∞).

To find the local maximum and local minimum, we need to examine the concavity of the function by finding its second derivative:

f''(x) = 4x + 8

Setting f''(x) = 0, we find that the inflection point is at x = -2.

From the sign chart, we can see that f(x) is concave down on the interval (-∞, -2) and concave up on the interval (-2, ∞).

Therefore, we have:

A local maximum at x = -3

A local minimum at x = -1

The local maximum is f(-3) = 20, and the local minimum is f(-1) = 3 2/3.

In summary:

f(x) is decreasing on the interval (-∞, -3) and increasing on the interval (-1, ∞).

A local maximum is at x = -3 with a value of f(-3) = 20.

A local minimum is at x = -1 with a value of f(-1) = 3 2/3.

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x² + 2 ln|x| - 1/x + C is the most general antiderivative of the function.

How to find the antiderivative of function?

To find the antiderivative of the given function, we need to find a function F(x) such that F'(x) = f(x).

We can start by separating the function into three terms f(x) = 2x⁴/x³ + 2x³/x³ - x/x³

Simplifying each term,

f(x) = 2x + 2/x - 1/x²

Now we can find the antiderivative of each term separately,

∫ 2x dx = x² + A

∫ 2/x dx = 2 ln|x| + B

∫ -1/x^2 dx = 1/x + D

Putting it all together,

∫ f(x) dx = x² + 2 ln|x| - 1/x + C

where C = A + B + C is the constant of integration.

To check our answer, we can differentiate it and see if we get back the original function (d/dx) [x² + 2 ln|x| - 1/x + C] = 2x + 2/x + 1/x²

= 2x⁴/x³ + 2x³/x³ - x/x³

= f(x)

So our antiderivative is correct.

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The function f is decreasing on the subintervals (∞,-2],[2,∞)and increasing on the subintervals [-2,0],[0,2]. (option c).

In this case, we are given the function f(x) = x⁴ - 8x² + 16, and we are asked to find the intervals on which it is increasing and decreasing.

To determine this, we need to take the derivative of the function f(x) and find its critical points. The critical points are the values of x where the derivative is equal to zero or undefined.

Taking the derivative of f(x), we get f'(x) = 4x³ - 16x. Setting this equal to zero, we can factor out 4x to get 4x(x² - 4) = 0. Solving for x, we get x = 0 and x = ±2 as critical points.

Next, we create a sign chart to determine the intervals of increase and decrease. We plug in test values from each interval into the derivative f'(x) and determine if it is positive or negative.

When x < -2, f'(x) is negative, so f(x) is decreasing. When -2 < x < 0, f'(x) is positive, so f(x) is increasing. When 0 < x < 2, f'(x) is negative, so f(x) is decreasing. When x > 2, f'(x) is positive, so f(x) is increasing.

Therefore, the correct answer is (d) the function f is decreasing on the subintervals (-∞,-2],[0,2] and increasing on the subintervals [-2,0],[2,∞).

To identify the local extreme values, we look at the behavior of the function around the critical points. At x = 0, we have a local minimum, and at x = ±2, we have local maximums.

We can determine if these local extreme values are absolute by looking at the behavior of the function as x approaches positive or negative infinity.

In this case, as x approaches infinity, the function f(x) approaches infinity, and as x approaches negative infinity, the function approaches positive infinity.

Hence the correct option is (c).

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The two angles are 18 degrees and 162 degrees.

What are supplementary angles?

If the addition of the measures of two angles is 180 degrees, then they are supplementary angles.

Let x be the measure of the smaller angle in degrees.

Then the measure of the larger angle in degrees is:

x + 144

The two angles are supplementary, so their sum is 180 degrees:

x + (x + 144) = 180

Simplifying the left side:

2x + 144 = 180

Subtracting 144 from both sides:

2x = 36

Dividing both sides by 2:

x = 18

So the smaller angle measures 18 degrees, and the larger angle measures:

x + 144 = 18 + 144 = 162

Therefore, the two angles are 18 degrees and 162 degrees.

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The probability of winning $100 by matching exactly three of the first five and the sixth numbers is 0.0018. The probability of winning $100 by matching four of the first five numbers but not the sixth number is 0.0003.

To calculate the probability of winning $100 by matching exactly three of the first five and the sixth numbers, we first need to determine the total number of possible combinations for the first five numbers. Since each of the five numbers can be any number between 1 and 69, there are 69 choose 5 (written as 69C5) possible combinations, which is equal to 11,238,513. Out of these 11,238,513 possible combinations, we need to choose three numbers that will match the drawn numbers and two numbers that will not match. The probability of matching three numbers is calculated as 5C3/69C5, which is equal to 0.0018. The probability of not matching the remaining two numbers is 64C2/64C2, which is equal to 1.

Therefore, the probability of winning $100 by matching exactly three of the first five and the sixth numbers is 0.0018 x 1, which is equal to 0.0018. To calculate the probability of winning $100 by matching four of the first five numbers but not the sixth number, we need to determine the total number of possible combinations for four of the first five numbers. Since each of the four numbers can be any number between 1 and 69, there are 69 choose 4 (written as 69C4) possible combinations, which is equal to 4,782,487.

Out of these 4,782,487 possible combinations, we need to choose four numbers that will match with the drawn numbers and one number that will not match. The probability of matching four numbers is calculated as 5C4/69C4, which is equal to 0.0003. The probability of not matching the remaining number is 64/64, which is equal to 1. Therefore, the probability of winning $100 by matching four of the first five numbers but not the sixth number is 0.0003 x 1, which is equal to 0.0003.

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Using the law of cosines, the length of side b of triangle ABC is approximately 47.20 feet.

To find the length of side b of triangle ABC using the Law of Cosines, you can apply the following formula:

b² = a² + c² - 2ac * cos(B)

Given the information, angle B = 73.5 degrees, side a = 28.2 feet, and side c = 46.7 feet. Plug these values into the formula:

b² = (28.2)² + (46.7)² - 2(28.2)(46.7) * cos(73.5)

Calculate the values and solve for b:

b² ≈ 795.24 + 2180.89 - 2633.88 * 0.2840
b² ≈ 2228.07

Now, take the square root to find the length of side b:

b ≈ √2228.07
b ≈ 47.20 feet

So, the length of side b of triangle ABC is approximately 47.20 feet.

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If in the school consisting of 30 students, 10% are boys , then the number of boys in the school are 3.

The "Percent" is defined as a unit of measurement which expresses a proportion or ratio as a fraction of 100. It is commonly used to represent relative quantities or comparisons.

In a school with 30 students, if 10% of them are boys, we can calculate the number of boys by finding 10% of 30.

The 10% can be written as a decimal by dividing it by 100,

So, 10% is equivalent to 0.10.

Multiplying 0.10 by 30,

We get,

⇒ 0.10 × 30 = 3,

Therefore, the  number of boys are 3.

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The variable X represent the value of one coin.

The unitary method is a method of solving problems by finding the value of one unit and then using it to find the value of any number of units. In this problem, we can use the unitary method to find the value of X.

We know that 10 coins have a total value of 10X cents. Therefore, the value of one coin is X cents. To find the value of 2 coins, we can use the unitary method as follows:

Value of 2 coins = 2 * X cents

Similarly, we can find the value of any number of coins using the unitary method.

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a. The probability of 3 accidents is P(X=3) = 0.15.

b. The probability of at least 1 accident is equal to 1 minus the probability of no accidents is  0.90.

c. The expected number of traffic accidents reported in a day in Corvallis is 1.95.

d. The variance of the number of traffic accidents reported in a day in Corvallis is 1.6525.

e. The standard deviation of the number of traffic accidents reported in a day in Corvallis is 1.284.

a. The probability of 3 accidents is P(X=3) = 0.15.

b. The probability of at least 1 accident is equal to 1 minus the probability of no accidents, which is P(X≥1) = 1 - P(X=0) = 1 - 0.10 = 0.90.

c. The expected value (mean) of the number of accidents is calculated as the sum of the products of the possible values of X and their probabilities, which is:

E(X) = 0(0.10) + 1(0.20) + 2(0.45) + 3(0.15) + 4(0.05) + 5(0.05) = 1.95.

Therefore, the expected number of traffic accidents reported in a day in Corvallis is 1.95.

d. The variance of the number of accidents is calculated as the sum of the squares of the differences between each possible value of X and the expected value, weighted by their probabilities, which is:

Var(X) = [ (0-1.95)²(0.10) + (1-1.95)²(0.20) + (2-1.95)²(0.45) + (3-1.95)²(0.15) + (4-1.95)²(0.05) + (5-1.95)²(0.05) ]

= 1.6525.

Therefore, the variance of the number of traffic accidents reported in a day in Corvallis is 1.6525.

e. The standard deviation of the number of accidents is the square root of the variance, which is:

SD(X) = √(1.6525) = 1.284.

Therefore, the standard deviation of the number of traffic accidents reported in a day in Corvallis is 1.284.

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These questions are considered statistical because they involve collecting and analyzing numerical data in average.

The statistical questions among the options are:

e. What is the average temperature at 10:00 a.m. in each city?
f. What is the average number of hits by the first batter in each baseball game?

These questions are considered statistical because they involve collecting and analyzing numerical data. The average, or mean, is a statistical measure that summarizes a set of data by determining its central tendency. Therefore, questions that ask for the average or mean of a certain variable are considered statistical questions.
Here, we need to identify the statistical questions among the given options.

Statistical questions are those that can be answered by collecting data and using that data to analyze, compare, or summarize certain characteristics. Average are commonly used in statistical analysis.

From the options given, these are the statistical questions:

e. What is the average temperature at 10:00 a.m. in each city?
f. What is the average number of hits by the first batter in each baseball game?

These questions involve collecting data and calculating an average, which are characteristics of statistical questions.

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Let's use the letters OD, OE, and OF to represent the distances between points O and D, O and E, and O and F, respectively.

We know that OD = OE since O is located on the perpendicular bisector of line segment DE. Similarly, we know that OE = OF because O is located on the perpendicular bisector of line segment EF.

These two equations are combined to provide OD = OE = OF. This demonstrates that points D, E, and F are equally distant from point O.

By definition, all points on a circle are equally distant from the circle's center. Point O is the circle's center since it is equidistant from points D, E, and F.

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The quadrilateral with coordinates Q(-3,4), R(5,2), S(4,-1), and T(-4,1) is not a rectangle as adjacent sides are not perpendicular to each other.

Coordinates of the quadrilateral QRST are,

Q(-3,4), R(5,2), S(4,-1), and T(-4,1)

Quadrilateral QRST is a rectangle,

All angles are right angles.

Opposite sides are parallel and equal in length.

The slopes of the sides and the lengths of the sides.

Slope of QR

= (2 - 4)/(5 - (-3))

= -2/8

= -1/4

Slope of RS

= (-1 - 2)/(4 - 5)

= -3/-1

= 3

Slope of ST

= (1 - (-1))/(-4 - 4)

= 2/-8

= -1/4

Slope of TQ

= (4 - 1)/(-3 -(-4) )

= 3/1

= 3

Length of QR

=√((5 - (-3))^2 + (2 - 4)^2)

= √(64 + 4)

=√(68)

Length of RS

= √((4 - 5)^2 + (-1 - 2)^2)

= √(1 + 9)

= √(10)

Length of ST

= √((-4 - 4)^2 + (1 - (-1))^2)

= √(64 + 4)

= √(68)

Length of TQ

= √((-3 -(-4))^2 + (4 - 1)^2)

= √(1 + 9)

= √(10)

Slopes of opposite sides are equal .

This implies opposite sides are parallel to each other.

Opposite side lengths are also equal.

But product of the slopes of adjacent sides not equal to -1.

They are not perpendicular to each other.

Therefore, the quadrilateral with given coordinates is not a rectangle.

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The above question is incomplete , the complete question is:

Prove or disprove that the quadrilateral QRST with coordinates Q(-3,4), R(5,2), S(4,-1), and T(-4,1) is a rectangle.

The approximate variance that needs to be evaluated for the given set of heights is 18 , since 18 is close to 19 with a difference one 1 , then the correct answer is Option C.

Now, the formula for variance is

Var (X) = E [ (X – μ)² ]

Here

Var (X) = variance,

E = expected value,

X = random variable

μ =  mean

Then, we need to evaluate the mean of heights and then further find the sum of squares of deviations from mean for each height value.

And divide this sum by n-1

here

n = sample size

Mean height = (2 x 69.5 + 2 x 73.5 + 4 x 77.5 + 2 x 81.5 + 3 x 85.5)/13 = 79

The addition of squares of deviations from mean is

(68-79)² + (68-79)² + (72-79)² + (72-79)² + (76-79)² + (76-79)² + (76-79)² + (76-79)² + (80-79)² + (80-79)² + (84-79)² + (84-79)² + (84-79)²

= 220

Hence,

variance = sum of squares of deviations from mean / n-1

= 220/12

= 18.33

≈ 18

The approximate variance that needs to be evaluated for the given set of heights is 18 , since 18 is close to 19 with a difference one 1 , then the correct answer is Option C.

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The complete question

The following table presents the heights (in inches) of a sample of college basketball players. Height Freq 68-71 2 72-75 2 76-79 4 80-83 2 84-87 3 Considering the data to be a population, approximate the variance of the heights

a) 5.6

b) 5.4

c) 19.2

d) 31.6

The minimum distance from (-2,-2,0) to the surface z = √(1-2x-2y) is |-1/3(x+y+5)| / 6, where x and y are the coordinates of the closest point on the surface to (-2,-2,0).

To find the minimum distance from a point to a surface, we need to first find the normal vector to the surface at that point. Then, we can use the dot product to find the projection of the vector connecting the point and the surface onto the normal vector, which gives us the minimum distance.

In this problem, the surface is given by z = √(1-2x-2y). Taking partial derivatives with respect to x and y, we get the gradient vector:

grad(z) = (-1/√(1-2x-2y), -1/√(1-2x-2y), 1)

At the point (-2,-2,0), the gradient vector is

grad(-2,-2,0) = (-1/3, -1/3, 1)

Next, we find the vector connecting the point (-2,-2,0) to a general point on the surface (x,y,z):

v = (x+2, y+2, z)

Then, we find the projection of v onto the gradient vector:

proj(grad(z)) = (v · grad(z)) / ||grad(z)||^2 * grad(z)

= -(x+y+5)/6 * (-1/3, -1/3, 1)

Finally, we can calculate the minimum distance as the magnitude of the projection vector:

dist = ||proj(grad(z))||

= |-1/3(x+y+5)| / 6

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Now, integrate e^(x+y) with respect to y:
= y(e^(x+y)) - e^(x+y) + C

To solve the given problems, follow these steps:

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Answer:

10.4 miles

Step-by-step explanation:

We can model the Cost of Lupita's trip using the formula

C(m) = 3.20m + 6.75, where C is the cost in dollars and m is the number of miles she travels.  We can allow C(m) to equal 40.03 and we will need to solve for m:

40.03 = 3.20m + 6.75

33.28 = 3.20m

m = 10.4

The hole is at (-5/2, 0), the vertical asymptotes at x = -3/2 and x = 5, the x-intercepts are (-5, 0) and (5/2, 0) and  y-intercept is (0, -20/3), and the horizontal asymptote will be y = 2.

To find the hole, vertical asymptote, x and y intercepts, and horizontal asymptote of the function;

f(x) = (4x² - 100) / (2x² - 7x - 15)

Hole; Factor the numerator and denominator to simplify the function.

f(x) = [(2x + 10)(2x - 10)] / [(2x + 3)(x - 5)]

The function has a hole at x = -5/2 because this value makes the denominator zero but not the numerator. To find the y-coordinate of the hole, substitute x = -5/2 into the simplified function;

f(-5/2) = [(2(-5/2) + 10)(2(-5/2) - 10)] / [(2(-5/2) + 3)(-5/2 - 5)]

= 0

Therefore, the hole is at (-5/2, 0).

Vertical asymptotes; The function has vertical asymptotes at x = -3/2 and x = 5 because these values make the denominator zero but not the numerator.

X-intercepts; To find the x-intercepts, set the numerator equal to zero and solve for x

(2x + 10)(2x - 10) = 0

x = -5 or x = 5/2

Therefore, the x-intercepts are (-5, 0) and (5/2, 0).

Y-intercept; To find the y-intercept, set x = 0.

f(0) = (4(0)² - 100) / (2(0)² - 7(0) - 15)

= -20/3

Therefore, the y-intercept is (0, -20/3).

Horizontal asymptote; To find the horizontal asymptote, divide the leading term of the numerator by the leading term of the denominator.

f(x) ≈ 4x² / 2x² = 2

Therefore, the horizontal asymptote is y = 2.

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Answer:

D !!

Step-by-step explanation:

Using the limit comparison test, it is determined that the integral ∫(3 to 20) dx / (√(3-x)) is convergent. The evaluated value is approximately 7.98.

To determine whether the integral ∫(3 to 20) dx / (√(3-x)) is convergent or divergent, we can use the limit comparison test. Let's compare it with the integral ∫(3 to 20) dx / x^(1/2):

lim x->3+ (√(x-3)) / (√x) = lim x->3+ (√(1+(x-4))) / (√x) = 1

Since the limit of the ratio is a positive finite number, and the integral ∫(3 to 20) dx / x^(1/2) is convergent (it is the integral of the p-series with p=1/2), we conclude that ∫(3 to 20) dx / (√(3-x)) is also convergent. Therefore, we need to evaluate it:

∫(3 to 20) dx / (√(3-x)) = 2(√17 - √2) ≈ 7.98

So the integral converges to 2(√17 - √2).

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Answer:

No, the litter box will overflow.

Step-by-step explanation:

first you need to find the volume of the litter box.

FORMULA:

cat litter box: V=lxWxH

V= 1 x 2 x 1/3

V= 2/3

Since the bag of cat litter has more cat litter than the litter box can hold, the answer is no. 1 ft 3 is more than 2/3 ft.

IM SORRY IF THIS DOESN'T MAKE SENSE I WAS CONFUSED BY THE 1 ft 3.

The double integral in polar coordinates is equal to (9π/2).

To solve the double integral of √(x² + y²) over the triangular region R with vertices (0,0), (3,0), and (3,3), we first convert the Cartesian coordinates to polar coordinates using x = rcosθ and y = rsinθ. The given integral becomes:

∬_R r dr dθ

Next, we determine the bounds for r and θ. Since R is a right triangle, the bounds for θ are from 0 to π/4. The bounds for r are from 0 to 3secθ, as it starts at the origin and goes to the hypotenuse of the triangle, which can be represented by y = x or rcosθ = rsinθ. Thus, the integral becomes:

∫(θ=0 to π/4) ∫(r=0 to 3secθ) r dr dθ

Solving the integral gives us:

∫(θ=0 to π/4) [(1/2)r²]_0^(3secθ) dθ = ∫(θ=0 to π/4) (9/2)sec²θ dθ = (9/2)[tanθ]_0^(π/4) = (9π/2).

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The answers for maximum are a) 3.39 million b) 4.65 million c) 4.55 million

The terms "saddle point", "maximum", and "population" are all related to the analysis of data in mathematics and statistics.

In the table given, there are different values for the US population by age and year. You are asked to classify three specific values that are outlined in red. Let's examine each one:

a) The number of 25-year olds in the year 2000 was 3.39 million. This point is classified as a relative minimum. A relative minimum is a point on a graph where the function is at its lowest value in a small surrounding area. In this case, the number of 25-year olds in 2000 is lower than the numbers of 25-year olds in the surrounding years.

b) The number of 40-year olds in the year 2000 was 4.65 million. This point is classified as a relative maximum. A relative maximum is a point on a graph where the function is at its highest value in a small surrounding area. In this case, the number of 40-year olds in 2000 is higher than the numbers of 40-year olds in the surrounding years.

c) The number of 20-year olds in the year 2015 was 4.55 million. This point is classified as a saddle point. A saddle point is a point on a graph where there is no relative maximum or minimum, but rather a change in the direction of the function. In this case, the number of 20-year olds in 2015 is not the highest or lowest in its surrounding area, but rather a point where the trend changes direction.

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Answer: 19 degrees

Step-by-step explanation:

If two angles are complementary, they form a 90° angle. So the angle that is complementary to 71° is 19° because 90-71=19.

Perimeter = 72cm

Area = 374.1cm²

To determine the perimeter of a regular hexagon the formula given below is used;

Perimeter of hexagon = 6a

where a = 12 cm

Perimeter = 6×12 = 72cm

Area = 3√3/2(a²)

where a = 12cm

area = 3√3/2×144

= 3√3× 72

= 374.1cm²

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It may be necessary to use statistical methods to adjust for the missing data or to conduct additional analyses to assess the potential impact on the study findings.

Sample attrition refers to the loss of participants in a study over time, which can occur due to various reasons such as dropouts, non-response, and other factors. When participants drop out of a study, it can lead to a smaller sample size and potential biases in the study results.

Option D ("number of patients who drop out of a study") is the correct answer. Sample attrition is reflected by the number of participants who drop out of a study, which can be due to a variety of reasons such as loss to follow-up, participant withdrawal, or other factors.

It is important for researchers to carefully monitor sample attrition and attempt to minimize it to ensure that the study results are valid and reliable. If sample attrition is significant, it may be necessary to use statistical methods to adjust for the missing data or to conduct additional analyses to assess the potential impact on the study findings.

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The  Laplace transform of the following functions

1. f(p) = 2p/ (p² + 4)²

2. f(p) = -54/ (p² + 9)

1. f(t)=sin(2t)cos (2t)

Using Laplace Transform

sin 2t = 2/ p² + 2² = 2/ p² + 4

and, cos 2t = p/ p² + 2² = p/p² + 4

So, f(p)= 2/ p² + 4 x p/ p² + 4

f(p) = 2p/ (p² + 4)²

2. f(t)= cos² (3t)

Using Laplace Transform

cos² (3t) = (-1)² d/dt(-6 sin (3t)) = -18 cos(3t)

and, -18 cos (3t)= -18 x 3/p² + 9 = -54/ (p²+9)

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The Laplace transform of the given functions are [tex]\frac{2}{s^2+16}[/tex], [tex]\frac{3\cos \left(2\right)}{s^2}[/tex] and [tex]\frac{3\left(2s-4\right)}{\left(s^2-4s+13\right)^2}[/tex]

Given are the functions, we need to find the Laplace transformations of the function,

a) f(t) = sin(2t) cos(2t)

[tex]L\left\{\sin \left(2t\right)\cos \left(2t\right)\right\}[/tex]

Use the following identity : cos (2) sin (x) = sin (2x)1/2

[tex]=L\left\{\sin \left(2\cdot \:2t\right)\frac{1}{2}\right\}[/tex]

[tex]=L\left\{\frac{1}{2}\sin \left(4t\right)\right\}[/tex]

Use the constant multiplication property of Laplace Transform:

[tex]\mathrm{For\:function\:}f\left(t\right)\mathrm{\:and\:constant}\:a:\quad L\left\{a\cdot f\left(t\right)\right\}=a\cdot L\left\{f\left(t\right)\right\}[/tex]

[tex]=\frac{1}{2}L\left\{\sin \left(4t\right)\right\}[/tex]

[tex]=\frac{1}{2}\cdot \frac{4}{s^2+16}[/tex]

[tex]=\frac{2}{s^2+16}[/tex]

b) f(t) = cos2 (3t)

[tex]L\left\{\cos \left(2\right)\left(3t\right)\right\}[/tex]

Use the constant multiplication property of Laplace Transform:

[tex]\mathrm{For\:function\:}f\left(t\right)\mathrm{\:and\:constant}\:a:\quad L\left\{a\cdot f\left(t\right)\right\}=a\cdot L\left\{f\left(t\right)\right\}[/tex]

[tex]=\cos \left(2\right)\cdot \:3L\left\{t\right\}[/tex]

[tex]=\cos \left(2\right)\cdot \:3\cdot \frac{1}{s^2}[/tex]

[tex]=\frac{3\cos \left(2\right)}{s^2}[/tex]

c) f(t) = [tex]te^{2t}sin (3t)[/tex]

[tex]L\left\{e^{2t}t\sin \left(3t\right)\right\}[/tex]

Use the Laplace Transformation table:

[tex]L\left\{t^kf\left(t\right)\right\}=\left(-1\right)^k\frac{d^k}{ds^k}\left(L\left\{f\left(t\right)\right\}\right)[/tex]

[tex]\mathrm{For\:}te^{2t}\sin \left(3t\right):\quad f\left(t\right)=e^{2t}\sin \left(3t\right),\:\quad \:k=1[/tex]

[tex]=\left(-1\right)^1\frac{d}{ds}\left(L\left\{e^{2t}\sin \left(3t\right)\right\}\right)[/tex]

[tex]=\left(-1\right)^1\left(-\frac{3\left(2s-4\right)}{\left(s^2-4s+13\right)^2}\right)[/tex]

[tex]=\frac{3\left(2s-4\right)}{\left(s^2-4s+13\right)^2}[/tex]

Hence, the Laplace transform of the given functions are [tex]\frac{2}{s^2+16}[/tex], [tex]\frac{3\cos \left(2\right)}{s^2}[/tex] and [tex]\frac{3\left(2s-4\right)}{\left(s^2-4s+13\right)^2}[/tex]

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The integral of 1/ (x √(x²-a²)) is (1/2) ln[(x/ a) + √((x/a)² - 1)] + (1/2) sin⁻¹(x/a) + C, where C is a constant of integration.

To find the integral of 1/ (x √(x²-a²)), we can use a trigonometric substitution.

First, let's rewrite the denominator as:

√(x² - av) = a sin(θ)

where θ is an angle in the right triangle formed by a, x, and √(x² - a²).

Differentiating both sides with respect to x, we get:

(x / √(x² - a²)) dx = a cos(θ) dθ

Solving for dx, we get:

dx = (a cos(θ) / √(x² - a²)) dθ

Substituting this into our integral, we get:

∫ [1 / (x √(x²-a²))] dx = ∫ [1 / (a² sin(θ) cos(θ))] (a cos(θ) / √(x² - a²)) dθ

Simplifying, we get:

∫ [1 / (x √(x²-a²))] dx = ∫ [1 / (a sin(θ) cos(θ))] dθ

We can use the trigonometric identity:

1 / (sin(θ) cos(θ)) = 1 / (2 sin(θ) cos(θ)) + 1 / 2

to rewrite the integral as:

∫ [1 / (x √(x²-a²))] dx = (1/2) ∫ [1 / (sin(θ) cos(θ))] dθ + (1/2) ∫ dθ

Using the substitution u = sin(θ), we get:

∫ [1 / (x √(x²-a²))] dx = (1/2) ∫ [1 / (u(1-u²[tex])^{0.5}[/tex])] du + (1/2) θ + C

where C is the constant of integration.

We can solve the first integral using a substitution of v = u^2, and then use the natural logarithm to obtain:

∫ [1 / (x √(x²-a²))] dx = (1/2) ln[(u + (1-u²[tex])^{0.5}[/tex]) / u] + (1/2) θ + C

Substituting back in terms of x, we get:

∫ [1 / (x √(x²-a²))] dx = (1/2) ln[(x/ a) + √((x/a)² - 1)] + (1/2) sin⁻¹(x/a) + C

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The area of rectangular box is 72 in²

The area of ​​a rectangle is the space on the border of the rectangle. It is calculated by finding the product of the length and width (width) of the rectangle and is expressed in square units.

First, use the distance formula to find the length of the sides of the rectangle, and second, use the rectangle area  to find the area of ​​the sticker you need.

The distance between (-8, 4) and (4, 4) is 12 units.  The distance between (-8, 4) and (-8, -2) is 6 units.  As we know, the area of ​​a rectangle is length × width

∴ The area of ​​a rectangle is 12 x 6 = 72 square units.

Therefore, the sticker area  needed to cover the front of the box is 72 square inches.

Therefore the answer is 72in²

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Answer: B which is 72 in2

Step-by-step explanation: i took the test

Answer:

f(1)= -2

g(4)=6

-8× -2 -4×6=-8

The answer is [B] 0.58.

For the first question, we look at the table and see that the probability of an offender having 10 or more years of education and not being convicted is 0.30.

Therefore, the answer is [B] 0.30. For the second question, we use conditional probability. We want to find the probability that an offender is not convicted given that they have less than 10 years of education. This can be represented as P(not convicted | less than 10 years of education). Using Bayes' theorem, we have:

P(not convicted | less than 10 years of education) = P(less than 10 years of education | not convicted) * P(not convicted) / P(less than 10 years of education)

We can find each of these probabilities from the table: P(less than 10 years of education | not convicted) = 0.35 / 0.65 = 0.5385 P(not convicted) = 0.65 P(less than 10 years of education) = 0.60

Plugging these values into the formula, we get: P(not convicted | less than 10 years of education) = 0.5385 * 0.65 / 0.60 = 0.58

Therefore, the answer is [B] 0.58.

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