Solve
sin^2(2x) 2 sin^2(x) = 0 over [0, 2pi). (Hint: use a double
angle formula, then factorize.)

To find F(x), we integrate the given derivative function. F'(x) = 6 implies that F(x) is the antiderivative of 6 with respect to x, which is 6x + C. To find F"(x), we differentiate F'(x) with respect to x. F"(x) is the derivative of 6x + C, which is simply 6.

To find F(x), we need to integrate the given derivative function F'(x) = 6. Since the derivative of a function gives us the rate of change of the function, integrating F'(x) will give us the original function F(x).

Integrating F'(x) = 6 with respect to x, we obtain:

∫6 dx = 6x + C

Here, C is the constant of integration, which can take any value. So, the antiderivative or the general form of F(x) is 6x + C, where C represents the constant.

To find F"(x), we differentiate F'(x) = 6 with respect to x. Since the derivative of a constant is zero, F"(x) is simply the derivative of 6x, which is 6.

Therefore, the function F(x) is given by F(x) = 6x + C, and its second derivative F"(x) is equal to 6.

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Answer:

㏑ [a² / y^4]

Step-by-step explanation:

2 ㏑a = ㏑ a²

4 ㏑ y = ㏑ y^4

so, 2 ㏑ a - 4 ㏑ y

= ㏑a² - ㏑y^4

= ㏑ [a² / y^4]

Based on the information, the volume of this box is 65776 cm³.

The volume of a box is given by the formula:

V = lwh

We are given that the sum of the width, height, and length must be less than or equal to 258 cm. This can be written as:

l + w + h <= 258

We are given that the sum of l, w, and h must be less than or equal to 258. This means that each of l, w, and h must be less than or equal to 258/3 = 86 cm.

Therefore, the dimensions of the box with maximum volume are 86 cm by 86 cm by 86 cm.

The volume of this box is:

V = 86 cm * 86 cm * 86 cm

= 65776 cm³

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The area of the region enclosed by the curves y = x - 5 and y = 4 is 4.5 square units.

To find the area enclosed by the curves, we need to determine the points where the curves intersect. By setting the equations equal to each other, we find x - 5 = 4, which gives x = 9.

To find the area, we integrate the difference between the curves over the interval [0, 9].

[tex]∫(x - 5 - 4) dx from 0 to 9 = ∫(x - 9) dx from 0 to 9 = [0.5x^2 - 9x] from 0 to 9 = (0.5(9)^2 - 9(9)) - (0.5(0)^2 - 9(0)) = 40.5 - 81 = -40.5 (negative area)[/tex]

Since the area cannot be negative, we take the absolute value, giving us an area of 40.5 square units. Rounding to the nearest thousandth, we get 40.500, which is approximately 40.5 square units.

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Answer:

20 students brought their lunch on Thursday.

Step-by-step explanation:

5% of 400 = 20 students

400 x .05 = 20

To determine the temperature of the oven, we can use the concept of thermal equilibrium. When two objects are in thermal equilibrium, they are at the same temperature.

In this case, the thermometer and the oven reach thermal equilibrium when their temperatures are the same.

Let's denote the initial temperature of the oven as T (in °C). According to the information given, the thermometer initially reads 19°C and then reads 27°C after 26 seconds and 28°C after 52 seconds.

Using the data provided, we can set up the following equations:

Equation 1: T + 26k = 27 (after 26 seconds)

Equation 2: T + 52k = 28 (after 52 seconds)

where k represents the rate of temperature change per second.

To find the value of k, we can subtract Equation 1 from Equation 2:

(T + 52k) - (T + 26k) = 28 - 27

26k = 1

k = [tex]\frac{1}{26}[/tex]

Now that we have the value of k, we can substitute it back into Equation 1 to find the temperature of the oven:

T + 26(\frac{1}{26}) = 27

T + 1 = 27

T = 27 - 1

T = 26°C

Therefore, the temperature of the oven is 26°C.

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The statement "d^2y/dx^2 = (dy/dx)^2" is false.

The correct statement is that "d^2y/dx^2" represents the second derivative of y with respect to x, while "(dy/dx)^2" represents the square of the first derivative of y with respect to x.

The second derivative, d^2y/dx^2, represents the rate of change of the slope of a function or the curvature of the graph. It measures how the slope of the function is changing.

On the other hand, (dy/dx)^2 represents the square of the first derivative, which represents the rate of change or the slope of a function at a particular point.

These two expressions have different meanings and convey different information about the behavior of a function. Therefore, the statement that d^2y/dx^2 = (dy/dx)^2 is false.

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Given integrals:

(a) sin x cos x dx

(b) 1 + cos(t/2) dt

(c) ∫y sin(y) dy

(d) ∫(1+2/(1+x)) dx

(a) sin x cos x dx

Integration by substitution:

Let, u = sin x du/dx = cos x dx = du/cos x

We get, ∫sin x cos x dx

= ∫u du= u2/2 + C

= sin2 x / 2 + C

(b) 1 + cos(t/2) dt

Integrating both parts of the sum separately,

we get:

∫1 dt + ∫cos(t/2) dt

= t + 2 sin(t/2) + C

(c) ∫y sin(y) dy

Integration by parts:

Let, u = y dv

= sin(y) du/dy

= 1v = -cos(y)

We get,

∫y sin(y) dy

= -y cos(y) + ∫cos(y) dy

= -y cos(y) + sin(y) + C(d) ∫(1+2/(1+x)) dx

Integration by substitution:

Let, u = 1 + x du/dx = 1dx= du

We get,

∫(1+2/(1+x)) dx

= ∫du + 2 ∫dx/(1+x)

= u + 2 ln(1 + x) + C

Therefore, the above integrals can be evaluated as follows:

(a) sin x cos x dx = sin2 x / 2 + C

(b) 1 + cos(t/2) dt = t + 2 sin(t/2) + C

(c) ∫y sin(y) dy = -y cos(y) + sin(y) + C

(d) ∫(1+2/(1+x)) dx = u + 2 ln(1 + x) + C = (1+x) + 2 ln(1 + x) + C

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The line integral of the given function, ∫_c y²dx+xdy, along the arc of the parabola x = 4 - y² from (-5, -3) to (0, 2), can be evaluated by parameterizing the curve and then calculating the integral using the parameterization.

To evaluate the line integral, we first need to parameterize the given curve. Since the parabola is defined by x = 4 - y², we can choose y as the parameter. Let's denote y as t, where t varies from -3 to 2. Then, we can express x in terms of t as x = 4 - t².

Next, we differentiate the parameterization to obtain dx/dt = -2t and dy/dt = 1. Now, we substitute these values into the line integral expression: ∫_c y²dx + xdy = ∫_c y²(-2t)dt + (4 - t²)dt.

Now, we integrate with respect to t, using the limits of -3 to 2, since those are the parameter values corresponding to the given endpoints. After integrating, we obtain the value of the line integral.

By evaluating the integral, you will find the numerical result for the line integral along the arc of the parabola x = 4 - y² from (-5, -3) to (0, 2), based on the given function ∫_cy²dx + xdy.

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The given angle, 134.1899 degrees, needs to be converted to degrees, minutes, and seconds format.

To convert the angle from decimal degrees to degrees, minutes, and seconds, we can use the following steps.

First, let's extract the whole number of degrees from the given angle. In this case, the whole number of degrees is 134.

Next, we need to determine the minutes portion. To do this, multiply the decimal portion (0.1899) by 60. The result, 11.394, represents the minutes.

Finally, to find the seconds, multiply the decimal portion of the minutes (0.394) by 60. The outcome, 23.64, represents the seconds.

Combining all the values, we have the converted angle as 134 degrees, 11 minutes, and 23.64 seconds.

In conclusion, the given angle of 134.1899 degrees can be converted to degrees, minutes, and seconds format as 134 degrees, 11 minutes, and 23.64 seconds. This conversion allows for a more precise representation of the angle in a commonly used format for measuring angles.

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The limit of (cos(1+y)) as (x,y) approaches (0,0) does not exist.

The limit of (7x^2 + y^4)/(x^2 + 12) as (x,y) approaches (0,0) does not exist.

The limit of (x^2 + y^2)/(x - y) as (x,y) approaches (0,0) does not exist.

To show that the limit of (cos(1+y)) as (x,y) approaches (0,0) does not exist, we can consider approaching along different paths. For example, if we approach along the path y = 0, the limit becomes cos(1+0) = cos(1), which is a specific value. However, if we approach along the path y = -1, the limit becomes cos(1+(-1)) = cos(0) = 1, which is a different value. Since the limit depends on the path taken, the limit does not exist.

To find the limit of (7x^2 + y^4)/(x^2 + 12) as (x,y) approaches (0,0), we can try approaching along different paths. For example, approaching along the x-axis (y = 0), the limit becomes (7x^2 + 0)/(x^2 + 12) = 7x^2/(x^2 + 12). Taking the limit as x approaches 0, we get 0/12 = 0. However, if we approach along the path y = x, the limit becomes (7x^2 + x^4)/(x^2 + 12). Taking the limit as x approaches 0, we get 0/12 = 0. Since the limit depends on the path taken and gives a consistent value of 0, we conclude that the limit exists and is equal to 0.

To find the limit of (x^2 + y^2)/(x - y) as (x,y) approaches (0,0), we can again approach along different paths. For example, approaching along the x-axis (y = 0), the limit becomes (x^2 + 0)/(x - 0) = x^2/x = x. Taking the limit as x approaches 0, we get 0. However, if we approach along the path y = x, the limit becomes (x^2 + x^2)/(x - x) = 2x^2/0, which is undefined. Since the limit depends on the path taken and gives inconsistent results, we conclude that the limit does not exist.

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The set of all solutions to the homogeneous system ax = 0, where 'a' is a scalar, is the null space or kernel of the matrix 'a'. To find the inverse of 'a', we need to check if 'a' is invertible. If 'a' is non-zero, then its inverse 'a^-1' exists and is equal to 1/a. However, if 'a' is zero, it does not have an inverse.

To describe the set of all solutions to the homogeneous system ax = 0, we consider the equation in the form of a matrix-vector multiplication: A*x = 0, where A is a matrix consisting of 'a' as its scalar entry and x is the vector. The homogeneous system ax = 0 represents a linear equation in which the right-hand side is the zero vector.

The solution to this system, x, is the null space or kernel of the matrix 'a'. The null space is the set of all vectors x such that Ax = 0. If 'a' is a non-zero scalar, the null space consists only of the zero vector since any non-zero vector multiplied by 'a' would not equal zero. However, if 'a' is zero, then any vector can be a solution since the equation would always yield zero.

To find the inverse of 'a', we need to check if 'a' is invertible. If 'a' is a non-zero scalar, then it has an inverse 'a^-1' which is equal to 1/a. Multiplying 'a' by its inverse would yield the identity matrix. However, if 'a' is zero, it does not have an inverse. The concept of an inverse is defined for non-zero values only.

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The required value of x is 18.4.

Given the right-angled triangle with hypotenuse is x and one side is equal to 13 and angle is 45°.

To find the one side of the triangle by using the trigonometric functions  tan a and then use Pythagoras theorem to find the value of x.

Pythagoras theorem states that [tex]hypotenuse^2 = base^2 + perpendicular^2[/tex].

In triangle, tan a = perpendicular / base.

That implies, tan 45° = 13/x

On evaluating the value tan 45° = 1 gives,

1 = 13/ x

on cross multiplication gives,

x = 13.

By using Pythagoras theorem, find the base of the triangle,

[tex]hypotenuse^2 = base^2 + perpendicular^2[/tex].

[tex]x^{2} = 13^2 +13^2[/tex]

[tex]x^{2}[/tex] = 2 ×[tex]13^{2}[/tex]

take square root on both sides gives,[tex]\sqrt{2}[/tex]

x = 13 [tex]\sqrt{2}[/tex]

x = 13 × 1.141

x  = 18.38

Rounding off to tenths gives,

x = 18.4.

Hence, the required value of x is 18.4.

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The cosine of the angle C between the position vectors of A and B satisfies cos C = cos 6 cos 0' + sin 0 sin ' cos(0 - 0).

Let A be the point on the unit sphere with colatitude 0 and longitude ; let B be the point on the unit sphere with colatitude ' and longitude ¢'.

Write down the position vectors of A and B with respect to the origin, and by considering A·B, show that the cosine of the angle C between the position vectors of A and B satisfies cos C = cos 6 cos 0' + sin 0 sin ' cos(0 - 0).

The position vector of A with respect to the origin is given by the unit vector [x, y, z] which is such that

x = cos 0 sin y = sin 0 sin z = cos 0.

Position vector of A = [cos 0 sin, sin 0 sin , cos 0].

The position vector of B with respect to the origin is given by the unit vector [x, y, z] which is such that:

x = cos ¢' sin 'y = sin ¢' sin 'z = cos '.

Position vector of B = [cos ' sin ¢', sin ' sin ¢', cos '].

Now, A·B = |A| |B| cos C cos C = A·B/|A| |B|= [cos 0 sin ¢' + sin 0 sin 'cos(0 - ¢')] / 1 = cos 6 cos 0' + sin 0 sin 'cos(0 - ¢').

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The volume under the elliptic paraboloid over the rectangle R=[−1,1]×[−1,1] is 32/5 cubic units.

To calculate the volume under the elliptic paraboloid over the given rectangle, we need to set up a double integral. The volume can be calculated as the double integral of the function z=3x^2+5y^2 over the rectangle R=[−1,1]×[−1,1].

∫∫R (3x^2 + 5y^2) dA

Using the properties of double integrals, we can rewrite the integral as:

∫∫R 3x^2 + ∫∫R 5y^2 dA

The integration over each variable separately gives:

(3/3)x^3 + (5/3)y^3

Evaluating the above expression over the rectangle R=[−1,1]×[−1,1], we get:

[(3/3)(1^3 - (-1)^3)] + [(5/3)(1^3 - (-1)^3)]

Simplifying further:

(2/3) + (10/3)

Which equals 32/5 cubic units. Therefore, the volume under the elliptic paraboloid over the given rectangle is 32/5 cubic units.

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The function is not a solution of the differential equation xy' - 7y - xe*, x > 0. y = x

To determine if y = x(15+ e^x) is a solution of the differential equation xy' - 7y - xe^x = 0, we need to substitute y and y' into the left-hand side of the equation and see if it simplifies to 0.

First, we find y':

y' = (15 + e^x) + xe^x

Next, we substitute y and y' into the equation and simplify:

x(15 + e^x) + x(15 + e^x) - 7x(15 + e^x) - x^2 e^x

= x(30 + 2e^x - 105 - 7e^x - xe^x)

= x(-75 - 6e^x - xe^x)

Since this expression is not equal to 0 for all x > 0, y = x(15 + e^x) is not a solution of the differential equation xy' - 7y - xe^x = 0.

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[tex]\textit{arc's length}\\\\ s = r\theta ~~ \begin{cases} r=radius\\ \theta =\stackrel{radians}{angle}\\[-0.5em] \hrulefill\\ r=9\\ \theta =\frac{2\pi }{3} \end{cases}\implies s=(9)\cfrac{2\pi }{3}\implies s=(9)\cfrac{2(3.14) }{3}\implies s=18.84 \\\\[-0.35em] ~\dotfill[/tex]

[tex]\textit{area of a sector of a circle}\\\\ A=\cfrac{\theta r^2}{2} ~~ \begin{cases} r=radius\\ \theta =\stackrel{radians}{angle}\\[-0.5em] \hrulefill\\ r=9\\ \theta =\frac{2\pi }{3} \end{cases}\implies A=\cfrac{2\pi }{3}\cdot \cfrac{9^2}{2} \\\\\\ A=\cfrac{2(3.14) }{3}\cdot \cfrac{9^2}{2}\implies A=84.78[/tex]

To evaluate the limit using L'Hôpital's rule, we need to take the derivative of the numerator and denominator separately and then evaluate the limit again.

Given the expression: lim (6x / e^2 + 6x) as x approaches 0

Taking the derivative of the numerator and denominator separately:

The derivative of 6x with respect to x is simply 6.

The derivative of e^2 + 6x with respect to x is 6.

Now we have the new expression:

lim (6 / 6) as x approaches 0

Simplifying, we get:

lim 1 as x approaches 0

Therefore, the limit of the expression is equal to 1.

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a. The first derivative of f(x) is f'(x) = 28x, and the second derivative is f''(x) = 28.

b. The first derivative of g(t) = f'(t^2 + 2) is 56t(t^2 + 2)

c. The first derivative of v(s) is v'(s) = 2s / (s^2 - 4), and the second derivative is v''(s) = (-2s^2 - 8) / (s^2 - 4)^2.

d.  The first derivative of h(x) is h'(x) = -3, and the second derivative is h''(x) = 0.

(a) To compute the first and second derivatives of the function f(x) = 14x^2 - 12, we'll differentiate each term separately.

First derivative:

f'(x) = d/dx (14x^2 - 12)

= 2(14x)

= 28x

Second derivative:

f''(x) = d^2/dx^2 (14x^2 - 12)

= d/dx (28x)

= 28

Therefore, the first derivative of f(x) is f'(x) = 28x, and the second derivative is f''(x) = 28.

(b) To find the first derivative of g(t) = f'(t^2 + 2), we need to apply the chain rule. The chain rule states that if h(x) = f(g(x)), then h'(x) = f'(g(x)) * g'(x).

Let's start by finding the derivative of f(x) = 14x^2 - 12, which we computed earlier as f'(x) = 28x.

Now, we can apply the chain rule:

g'(t) = d/dt (t^2 + 2)

= 2t

Therefore, the first derivative of g(t) = f'(t^2 + 2) is:

g'(t) = f'(t^2 + 2) * 2t

= 28(t^2 + 2) * 2t

= 56t(t^2 + 2)

(c) To compute the first and second derivatives of v(s) = ln(s^2 - 4), we'll apply the chain rule and the derivative of the natural logarithm.

First derivative:

v'(s) = d/ds ln(s^2 - 4)

= 1 / (s^2 - 4) * d/ds (s^2 - 4)

= 1 / (s^2 - 4) * (2s)

= 2s / (s^2 - 4)

Second derivative:

v''(s) = d/ds (2s / (s^2 - 4))

= (2(s^2 - 4) - 2s(2s)) / (s^2 - 4)^2

= (2s^2 - 8 - 4s^2) / (s^2 - 4)^2

= (-2s^2 - 8) / (s^2 - 4)^2

Therefore, the first derivative of v(s) is v'(s) = 2s / (s^2 - 4), and the second derivative is v''(s) = (-2s^2 - 8) / (s^2 - 4)^2.

(d) To compute the first and second derivatives of h(x) = 523 - 3x + 14, note that the derivative of a constant is zero.

First derivative:

h'(x) = d/dx (523 - 3x + 14)

= -3

Second derivative:

h''(x) = d/dx (-3)

= 0

Therefore, the first derivative of h(x) is h'(x) = -3, and the second derivative is h''(x) = 0.

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If the two odometer readings were 48,592 and 48,892, and the amount of gas was 8.5 gallons then his miles per gallon will be 35.

To calculate Eric's miles per gallon (MPG), we need to determine the number of miles he traveled on 8.5 gallons of gas.

Given that the odometer readings were 48,592 and 48,892, we can find the total number of miles traveled by subtracting the initial reading from the final reading:

Total miles traveled = Final odometer reading - Initial odometer reading

                   = 48,892 - 48,592

                   = 300 miles

To calculate MPG, we divide the total miles traveled by the amount of gas used:

MPG = Total miles traveled / Amount of gas used

   = 300 miles / 8.5 gallons

Performing the division gives us:

MPG = 35.2941176...

Rounding the MPG to the nearest whole number, we get:

MPG ≈ 35

Therefore, Eric's miles per gallon is approximately 35.

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According to the information we can infer that the probability of drawing a triangle is 0.2.

To identify the probability of each figure we must perform the following procedure:

triangle

The probability of drawing a triangle would be 0.2.

Circle

The probability of drawing a circle would be 0.14.

Square

The probability of drawing a square would be 0.25.

Based on the information, we can infer that the probability of drawing a triangle would be 0.2.

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The following is the response to the initial value problem:

y(t) = e^(-t) * (7 * cos(t) + sin(t)) - 6 * cos(t)

To solve the given initial value problem for a damped mass-spring system with a sinusoidal force, we'll start by finding the complementary solution of the homogeneous equation y" + 2y' + 2y = 0. Then we'll use the method of undetermined coefficients to find the particular solution for the forced term r(t).

1. Complementary Solution:

The characteristic equation for the homogeneous equation is obtained by substituting y = e^(rt) into the equation:

r^2 + 2r + 2 = 0

Using the quadratic formula, we find the roots:

r = (-2 ± √(-4)) / 2

r = -1 ± i

The characteristic roots are complex conjugates, which yield the following complementary solution:

y_c(t) = e^(-t) * (c1 * cos(t) + c2 * sin(t))

2. Particular Solution:

To find the particular solution, we need to consider the sinusoidal force r(t). In this case, r(t) can be represented as r(t) = A * cos(t), where A is a constant.

We assume the particular solution has the form:

y_p(t) = B * cos(t) + C * sin(t)

Substituting this into the original equation, we find:

-2B * sin(t) + 2C * cos(t) + 2(B * cos(t) + C * sin(t)) = A * cos(t)

Equating coefficients of like terms, we have:

-2B + 2C + 2B = 0  => C = 0

2C - 2B = A     => B = -A/2

Therefore, the particular solution is:

y_p(t) = -A/2 * cos(t)

3. Complete Solution:

The complete solution is the sum of the complementary and particular solutions:

y(t) = y_c(t) + y_p(t)

    = e^(-t) * (c1 * cos(t) + c2 * sin(t)) - A/2 * cos(t)

4. Applying Initial Conditions:

Given y(0) = 1 and y'(0) = -5, we can substitute these values into the solution to determine the values of c1, c2, and A.

At t = 0:

y(0) = e^0 * (c1 * cos(0) + c2 * sin(0)) - A/2 * cos(0)

    = c1 - A/2 = 1     => c1 = 1 + A/2

Differentiating y(t):

y'(t) = -e^(-t) * (c1 * cos(t) + c2 * sin(t)) + e^(-t) * (-c2 * cos(t) + c1 * sin(t)) + A/2 * sin(t)

At t = 0:

y'(0) = -c1 + A/2 = -5    => c1 = A/2 - 5

Setting the two expressions for c1 equal to each other:

1 + A/2 = A/2 - 5

A = 12

Therefore, c1 = 1 + A/2 = 1 + 12/2 = 7 and c2 = A/2 - 5 = 12/2 - 5 = 1.

The final solution for the given initial value problem is:

y(t) = e^(-t) * (7 * cos(t) + sin(t)) - 6 * cos(t)

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To verify the identity sin x - 2 + sin x / (sin x - sin x - 1) = (sin x + 1) / (sin x - 1), we'll follow the steps: Multiply the numerator and denominator by sin x: (sin x - 2 + sin x) * sin x / [(sin x - sin x - 1) * sin x]

Simplifying the numerator: (2 sin x - 2) * sin x

Simplifying the denominator: (-1) * sin x^2

The expression becomes: (2 sin^2 x - 2 sin x) / (-sin x^2)

Factor the expression in the numerator: 2 sin x (sin x - 1) / (-sin x^2)

Simplify further by canceling out common factors: -2 (sin x - 1) / sin x

Distribute the negative sign: -2sin x / sin x + 2 / sin x

The expression becomes: -2 + 2 / sin x

Simplify the expression: -2 + 2 / sin x = -2 + 2csc x

The final result is: -2 + 2csc x, which is not equivalent to (sin x + 1) / (sin x - 1).Therefore, the given identity is not verified by the simplification.

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Answer:

a. The initial condition is that there is no salt in the vat at time t = 0, so S(0) = 0.

b. the amount of salt in the vat at time t is S(t) = 3 - 3e^(-t/2) pounds.

a. The rate of change of the amount of salt in the vat can be expressed as the difference between the amount of salt entering and leaving the vat per unit time. The amount of salt entering the vat per unit time is the concentration of salt in the water entering the vat multiplied by the rate of water entering the vat, which is (1/2) * 3 = 3/2 pounds per minute. The amount of salt leaving the vat per unit time is the concentration of salt in the vat multiplied by the rate of water leaving the vat, which is (S(t)/6) * 3 = (1/2)S(t) pounds per minute. Thus, we have the differential equation:
dS/dt = (3/2) - (1/2)S(t)
The initial condition is that there is no salt in the vat at time t = 0, so S(0) = 0.

b. This is a first-order linear differential equation, which can be solved using an integrating factor. The integrating factor is e^(t/2), so multiplying both sides of the equation by e^(t/2) yields:
e^(t/2) * dS/dt - (1/2)e^(t/2) * S(t) = (3/2)e^(t/2)
This can be written as:
d/dt [e^(t/2) * S(t)] = (3/2)e^(t/2)
Integrating both sides with respect to t gives:
e^(t/2) * S(t) = 3(e^(t/2) - 1) + C
where C is the constant of integration. Using the initial condition S(0) = 0, we can solve for C to get:
C = 0
Substituting this back into the previous equation gives:
e^(t/2) * S(t) = 3(e^(t/2) - 1)
Dividing both sides by e^(t/2) gives:
S(t) = 3 - 3e^(-t/2)
Therefore, the amount of salt in the vat at time t is S(t) = 3 - 3e^(-t/2) pounds.

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the expression for S(t) is:

S(t) = 3 - 2e^[(t/2) + ln (3/2)] if 3/2 - S/2 > 0

S(t) = 3 + 2e^[(t/2) + ln (3/2)] if 3/2 - S/2 < 0

a. To set up the differential equation for the amount of salt in the vat, we can consider the rate of change of salt in the vat over time. The change in salt in the vat can be expressed as the difference between the salt added and the salt drained.

Let's denote S(t) as the amount of salt in the vat at time t.

The rate of salt added to the vat is given by the concentration of salt in the incoming water (1/2 pound per gallon) multiplied by the rate of water added (3 gallons per minute). Therefore, the rate of salt added is (1/2) * 3 = 3/2 pounds per minute.

The rate of salt drained from the vat is given by the concentration of salt in the vat, S(t), multiplied by the rate of water drained (3 gallons per minute). Therefore, the rate of salt drained is S(t) * (3/6) = S(t)/2 pounds per minute.

Combining these, the differential equation for the amount of salt in the vat is:

dS/dt = (3/2) - (S(t)/2)

The initial condition is given as S(0) = 0, since the vat starts with pure water.

b. To solve the differential equation, we can separate variables and integrate:

Separating variables:

dS / (3/2 - S/2) = dt

Integrating both sides:

∫ dS / (3/2 - S/2) = ∫ dt

Applying the integral and simplifying:

2 ln |3/2 - S/2| = t + C

where C is the constant of integration.

To find C, we can use the initial condition S(0) = 0:

2 ln |3/2 - 0/2| = 0 + C

2 ln (3/2) = C

Substituting C back into the equation:

2 ln |3/2 - S/2| = t + 2 ln (3/2)

Now we can solve for S(t):

ln |3/2 - S/2| = (t/2) + ln (3/2)

Taking the exponential of both sides:

|3/2 - S/2| = e^[(t/2) + ln (3/2)]

Considering the absolute value, we have two cases:

Case 1: 3/2 - S/2 > 0

3/2 - S/2 = e^[(t/2) + ln (3/2)]

3 - S = 2e^[(t/2) + ln (3/2)]

S = 3 - 2e^[(t/2) + ln (3/2)]

Case 2: 3/2 - S/2 < 0

S/2 - 3/2 = e^[(t/2) + ln (3/2)]

S = 3 + 2e^[(t/2) + ln (3/2)]

Therefore, the expression for S(t) is:

S(t) = 3 - 2e^[(t/2) + ln (3/2)] if 3/2 - S/2 > 0

S(t) = 3 + 2e^[(t/2) + ln (3/2)] if 3/2 - S/2 < 0

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The given limit can be expressed as the definite integral: ∫[2 to 8] 5(x^3 - 3x) dx. To express the limit as a definite integral, we can rewrite it in the form: lim [n→∞] Σ[1 to n] f(x_i) Δx where f(x) is the function inside the limit, x_i represents the points in the interval, and Δx is the width of each subinterval.

In this case, the limit is:

lim [n→∞] Σ[1 to n] 5(x^3 - 3x) dx

We can rewrite the sum as a Riemann sum:

lim [n→∞] Σ[1 to n] 5(x_i^3 - 3x_i) Δx

To express this limit as a definite integral, we take the limit as n approaches infinity and replace the sum with the integral:

lim [n→∞] Σ[1 to n] 5(x_i^3 - 3x_i) Δx = ∫[2 to 8] 5(x^3 - 3x) dx

Therefore, the given limit can be expressed as the definite integral:

∫[2 to 8] 5(x^3 - 3x) dx.

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The correct change of variables from x to u for the given integral is [tex]u = x² + 3x[/tex].

To determine the appropriate change of variables, we look for a transformation that simplifies the integrand and makes it easier to evaluate. In this case, we want to eliminate the quadratic term (x²) and have a linear term instead.

By letting [tex]u = x² + 3x,[/tex] we have a quadratic expression that simplifies to a linear expression in terms of u.

To confirm that this substitution is correct, we can differentiate u with respect to x:

[tex]du/dx = (d/dx)(x² + 3x) = 2x + 3.[/tex]

Notice that du/dx is a linear expression in terms of x, which matches the integrand 6x² + 3 after multiplying by the differential dx.

Therefore, the correct change of variables is [tex]u = x² + 3x.[/tex]

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The smallest number that, when divided by 21, 45, and 56, leaves a remainder of 7 is 2527.

The remaining 7 must be added after determining the least common multiple (LCM) of the numbers 21, 45, and 56.

Find the LCM of 21, 45, and 56 first:

21 = 3 * 7

45 = 3^2 * 5

56 = 2^3 * 7

The LCM is the product of the highest powers of all the prime factors involved:

[tex]LCM = 2^3 * 3^2 * 5 * 7 = 8 * 9 * 5 * 7 = 2520[/tex]

Now, let's add the remainder of 7 to the LCM:

Smallest number = LCM + Remainder = 2520 + 7 = 2527

Therefore, the smallest number that, when divided by 21, 45, and 56, leaves a remainder of 7 is 2527.

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Given the function f(x) = x² - 6x - 95, we are to calculate the coordinates of the y-intercept and the x-intercepts of the graph of the function in question 10.

We are also to find the interval in which the function is increasing or decreasing.10.1.

Calculation of the y-intercept We recall that the y-intercept is the point at which the graph of the function intersects the y-axis.

At the y-intercept, x = 0.

Therefore, substituting x = 0 in the equation of the function,

we have y = f(0) = (0)² - 6(0) - 95 = -95

Therefore, the coordinates of the y-intercept are (0, -95).10.2.

Calculation of the x-intercepts

We recall that the x-intercepts are the points at which the graph of the function intersects the x-axis.

At the x-intercept, y = 0.

Therefore, substituting y = 0 in the equation of the function,

we have:0 = x² - 6x - 95Applying the quadratic formula,

we have:x = (-b ± √(b² - 4ac)) / 2aWhere a = 1, b = -6, and c = -95.

Substituting the values of a, b, and c, we have:

x = (6 ± √(6² - 4(1)(-95))) / 2(1)x

= (6 ± √(36 + 380)) / 2x = (6 ± √416) / 2x

= (6 ± 8√26) / 2x

= 3 ± 4√26

Therefore, the coordinates of the x-intercepts are (3 + 4√26, 0) and (3 - 4√26, 0).

The interval of Increase or Decrease of the function to find the interval of increase or decrease, we have to first find the critical points.

Critical points are points at which the derivative of the function is zero or undefined.

Therefore, we have to differentiate the function f(x) = x² - 6x - 95.

Applying the power rule of differentiation,

we have f'(x) = 2x - 6Setting f'(x) = 0, we have:

2x - 6 = 0x = 3At x = 3, the function attains a minimum.

Therefore, we have the following intervals:

The function is decreasing on the interval (-∞, 3) and is increasing on the interval (3, ∞).

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The completed statement with regards to the areas of the triangle and rectangle can be presented as follows;

The length of the triangle is 9 units. The width of the rectangle is 2 units. The area of the rectangle is 18 square units.

A triangle is a three sided polygon.

The area of the triangle can be found by forming a rectangle with the original triangle and the copy of the triangle rotated 180°, to combining with the original triangle to form a rectangle that is a composite figure consisting of two triangles

The length of the rectangle is 9 units

The width of the rectangle is 2 units

The area of the rectangle is; A = 9 × 2 = 18 square units

The rectangle is formed by two triangles, therefore, the area of the triangle is half of the area of the rectangle, which is; Area of triangle = 18/2 = 9 square units

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