A firm's demand curve is Q = 2 - 0.01P, where Q is measured in millions. The firm's output, when marginal revenue is equal to zero, is ____million.a. 1b. 2c. 3d. 4

The solution to the equation 30 = w/2 + 10 for w is w = 40

From the question, we have the following parameters that can be used in our computation:

30=w/2 (fraction) +10

Express properly

So, we have the following representation

30 = w/2 + 10

Subtract 10 from both sides

This gives

w/2 = 20

Multiply through by 2

w = 40

Hence, the solution to the equation is w = 40

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The area of the green shaded region would be,

⇒ Area = 27 (7 - π)

Since, The circle is a closed two dimensional figure , in which the set of all points is equidistance from the center.

We have to given that;

A solid figure is shown in image.

Since, The figure is shape of a trapezoid and it contain a circle.

Hence, The area of trapezoid is,

Area = (a + b) h / 2

Where, a and b are base and h is height of trapezoid.

Hence, We get;

The area of trapezoid is,

⇒ Area = (a + b) h / 2

⇒ A = (15 + 20) × 10/2

⇒ A = 35 × 5

⇒ A = 175

And, Area of circle is,

⇒ A = πr²

⇒ A = π × (10/2)²

⇒ A = π × 25

⇒ A = 25π

Hence, The area of the green shaded region would be,

⇒ Area = 175 - 25π

⇒ Area = 27 (7 - π)

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The equation representing the total cost of getting the computer fixed is T = 0.10V + 20H.

Let's define the variables in this situation:

V = Value of the computer (in dollars)

C = Initial charge (in dollars)

S = Service fee (in dollars)

H = Hours of service

According to the given information, there is an initial charge of 10% of the value of the computer, which can be represented as:

C = 0.10V

Additionally, there is a service fee of $20 per hour, so the total service fee can be calculated as:

S = 20H

The value of the computer is given as $525, so we substitute V with 525 in the equation for the initial charge:

C = 0.10 × 525

Now we can summarize the equation to represent the total cost (T) to get the computer fixed:

T = C + S

Substituting the equations for C and S, we have:

T = 0.10V + 20H

Therefore, the equation representing the total cost of getting the computer fixed is:

T = 0.10V + 20H, where V represents the value of the computer in dollars, and H represents the hours of service.

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The value of y is 5/4.

To solve the ratio equation 12:3 = 5:y, we need to find the value of y.

The ratio 12:3 can be simplified by dividing both sides by their greatest common divisor (GCD), which is 3.

Dividing 12 by 3 gives us 4, and dividing 3 by 3 gives us 1. So, the simplified ratio becomes 4:1.

Now we have the equation 4:1 = 5:y. To solve for y, we can set up a proportion:

4/1 = 5/y

Cross-multiplying, we get:

4y = 5 × 1

Simplifying further:

4y = 5

To isolate y, we divide both sides of the equation by 4:

y = 5/4

Therefore, the value of y is 5/4.

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The area of the region R is 24.

To find the area of the region R bounded by the curves f(x) = -x^2 - 4x + 5, g(x) = 2x + 10, and the x-axis over the interval [-5, 1], we need to integrate the difference between the two curves over that interval.

First, let's find the x-values where the two curves intersect:

-f(x) = g(x)

-x^2 - 4x + 5 = 2x + 10

Rearranging the equation:

x^2 + 6x - 15 = 0

Factoring the quadratic equation:

(x + 5)(x - 3) = 0

So the intersection points are x = -5 and x = 3.

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

Area = ∫[a, b] (g(x) - f(x)) dx

Since the region is bounded by the x-axis, we take the absolute value of the difference between g(x) and f(x).

Area = ∫[-5, 1] |(2x + 10) - (-x^2 - 4x + 5)| dx

Simplifying:

Area = ∫[-5, 1] |3x^2 + 6x - 5| dx

Since the expression inside the absolute value is non-negative over the interval [-5, 1], we can simplify the integral further:

Area = ∫[-5, 1] (3x^2 + 6x - 5) dx

Integrating term by term:

Area = [x^3 + 3x^2 - 5x] evaluated from -5 to 1

Evaluating the integral at the limits:

Area = (1^3 + 3(1^2) - 5(1)) - (-5^3 + 3(-5^2) - 5(-5))

Calculating the values:

Area = (1 + 3 - 5) - (-125 + 3(25) + 5(5))

Simplifying:

Area = -1 - (-125 + 75 + 25)

Area = -1 + 25

Area = 24

Therefore, the area of the region R is 24.

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The existence of two relative maximum points on the smooth curve signifies changes in the curve's slope and indicates a non-monotonic behavior in the corresponding interval.

Consider a smooth curve with no undefined points. If it has two relative maximum points, it implies that there are two distinct points on the curve where the slope changes from positive to negative.

A relative maximum point occurs when the curve reaches a local maximum value in a specific interval. At these points, the slope of the curve changes from positive to negative, indicating that the curve is increasing before the point and decreasing after the point.

The presence of two relative maximum points suggests that the curve undergoes an increase in slope, reaches a maximum value, then decreases in slope, reaches a lower value, and then increases in slope again, reaching a second maximum value.

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Answer:

2.25 in²

Step-by-step explanation:

area of square = L X W

= 1.5 X 1.5

= 2.25 (in²)

To determine which function has a larger maximum, we need to compare the vertex points of both functions.

For Function 1, the equation is f(x) = -3x^2 + 4x + 2. The coefficient of the x^2 term is negative, indicating that the parabola opens downward. The vertex of the parabola can be found using the formula x = -b / (2a), where a is the coefficient of x^2 (-3) and b is the coefficient of x (4).

Using the formula, we find that the x-coordinate of the vertex is x = -4 / (2 * -3) = 2/3. Substituting this value into the equation, we can find the corresponding y-coordinate: f(2/3) = -3(2/3)^2 + 4(2/3) + 2 = 4/3.

For Function 2, the given information states that the graph is a parabola that opens downward and passes through the points (-1/2, 0), (1/2, 4), and (2, 0). The vertex of this parabola can be found as the midpoint between the x-coordinates of the two known points with equal y-values. In this case, it would be the midpoint between (-1/2, 0) and (2, 0), which is (3/4, 0). Therefore, the maximum value is y = 0.

Comparing the y-values of the vertices, we see that f(2/3) = 4/3 and f(3/4) = 0. Since 4/3 is greater than 0, we can conclude that Function 1 has a larger maximum.

Therefore, the correct answer is:

a) Function 1 has a larger maximum.

The Laplace transform of the solution y(t) to the given initial value problem y"+y=g(t), y(0) = -4, y'(0) = 0, where g(t) = t, t<6; 5, t>6 is represented by the function Y(s) (in terms of s).

To solve the initial value problem y"+y=g(t) with the given initial conditions, we can take the Laplace transform of both sides of the equation. This transforms the differential equation into an algebraic equation in the Laplace domain.

Applying the initial conditions to the transformed equation, we can find the Laplace transform, Y(s), of the solution y(t). The exact expression for Y(s) can be obtained by using the table of Laplace transforms and the properties of Laplace transforms.

By substituting the Laplace transform of the input function, g(t), into the transformed equation, we can solve for Y(s) in terms of the Laplace variable s.

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True. The statistics of each vehicle's random process will be represented by filter coefficients and a noise variance.

In this project, the statistics of each vehicle's random process will be represented by filter coefficients and a noise variance. This means that the filter coefficients will be used to model the vehicle's behavior over time, while the noise variance will represent the variability of the data. These values will be used to generate predictions about the vehicle's future behavior.

The project in question is likely related to modeling and predicting the behavior of a system that involves multiple vehicles. To do this, it is necessary to first collect data about each vehicle's behavior over time. This data can then be used to develop a statistical model that can be used to predict future behavior. One common approach to modeling vehicle behavior is to use a random process model. This involves modeling the vehicle's behavior as a stochastic process, which means that it is subject to random fluctuations over time. The goal is to estimate the statistical properties of this process, such as its mean and variance, in order to make predictions about future behavior.

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The missing variable is given as follows:

B) 105º.

The sum of the interior angle measures of a polygon with n sides is given by the equation presented as follows:

S(n) = 180 x (n - 2).

The polygon in this problem has 4 sides, hence the sum of the interior angle measures is given as follows:

S(4) = 180 x (4 - 2)

S(4) = 360º.

The interior angle measures are given as follows:

Then the value of x is obtained as follows:

85 + 110 + 90 + x = 360

285 + x = 360

x = 75º.

Then, applying the exterior angle theorem, the value of b is given as follows:

b = 180 - 75

b = 105º.

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The statement is False.

While the Renaissance was indeed a period of great cultural and intellectual advancements, it would be inaccurate to categorize it solely as a flowering of mathematics and science.

The Renaissance, which spanned roughly from the 14th to the 17th century, witnessed significant developments in various fields including art, literature, philosophy, politics, and religion, in addition to advancements in mathematics and science.

The Renaissance was characterized by a revival of interest in classical knowledge and a shift towards humanism, emphasizing the value of human potential, creativity, and individuality. This led to remarkable achievements in art, with the emergence of renowned artists such as Leonardo da Vinci and Michelangelo, who revolutionized painting and sculpture during this era.

Literature also flourished, with influential works produced by authors like William Shakespeare and Miguel de Cervantes.

While mathematics and science did experience notable advancements during the Renaissance, including contributions from figures like Nicolaus Copernicus, Galileo Galilei, and Johannes Kepler, these developments were not the sole focus of the period.

Instead, the Renaissance represented a broader cultural and intellectual awakening characterized by a renewed interest in the humanities, the exploration of new ideas, and a questioning of traditional beliefs.

The Renaissance was a multidimensional movement that encompassed advancements in various disciplines and fields of knowledge, not limited to mathematics and science alone. It was a period marked by innovation and creativity across a wide range of intellectual pursuits, making it inaccurate to reduce it to merely a flowering of mathematics and science.

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Answer:

Step-by-step explanation: 22

tan30° = [tex]\frac{12}{x}[/tex] ⇒ [tex]\frac{\sqrt{3} }{3}[/tex] = [tex]\frac{12}{x}[/tex] ⇒ x = 12√3

c = √(12√3)² + 12² = 24

P= a+b+c = 12√3 + 12 + 24 = 36+12√3

The work required to stretch the spring from 7 cm to 15 cm past equilibrium is 0.268 J (joules).

What is work?

Work is a physical quantity that measures the amount of energy transferred to or from an object due to the application of a force over a displacement. It is the product of the magnitude of the force applied to an object and the distance over which the force is exerted.

The work done to stretch or compress a spring can be calculated using the formula: W = (1/2) * k * Δx².

Given that the spring constant (k) is 140 kg/s² and the displacement (Δx) is 15 cm - 7 cm = 8 cm = 0.08 m, we can substitute these values into the formula:

W = (1/2) * 140 kg/s² * (0.08 m²)

W = 0.5 * 140 kg/s² * 0.0064 m²

W = 0.448 J

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Answer:

(x – 4)^2 + (y +10)^2 = 64

Step-by-step explanation:

Recall that the formula for a circle is:  (x – h)^2 + (y – k)^2 = r^2

1. Find the Radius (r): Luckily, we can count the distance since the coordinates have the same y-value. 12 - 4 = 8. So, r^2 = 8^2 = 64

2.  Find h and k. These are the x and y coordinates of the center of the circle. So, h = 4, k = -10

3. Substitute the values in the equation:

(x – 4)^2 + (y – (-10))^2 = 8^2

(x – 4)^2 + (y +10)^2 = 64

Answer:

  (x -4)² +(y +10)² = 64

Step-by-step explanation:

Given a circle through point (12, -10) with center (4, -10), you want its equation.

The equation of a circle with center (h, k) and radius r is ...

  (x -h)² +(y -k)² = r²

You are given the center (h, k) = (4, -10). You only need to know the radius to finish the equation. That will be the value of r that makes the equation true at the given point:

  (x -4)² + (y +10)² = r²

  (12 -4) + (-10 +10)² = r² . . . . . . with (x, y) = (12, -10), the point on the circle

  8² + 0 = r²

The equation of the circle is (x -4)² +(y +10)² = 64.

__

Additional comment

The equation of a circle is essentially a statement of the distance formula. It is telling you that the circle consists of all points that are distance r from the center.

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Answer: 1,250,000

Step-by-step explanation: To find the 10th term of the given geometric sequence, we need to first determine the common ratio (r) between consecutive terms. We can do this by dividing any term by the previous term. Let's use the second and first terms for this:

r = 20/4 = 5

Now that we know the common ratio, we can use the formula for the nth term of a geometric sequence to find the 10th term:

an = a1 * r^(n-1)

where:

an = the nth term

a1 = the first term

r = the common ratio

n = the term we want to find

Substituting the values we know, we get:

a10 = 4 * 5^(10-1)

Simplifying, we get:

a10 = 4 * 5^9

a10 = 1,250,000

Therefore, the 10th term of the given geometric sequence is 1,250,000.

[tex]\displaystyle \int x^2\cos(4x)dx\hspace{5em}\textit{let's use integration by parts} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ u=x^2\implies \cfrac{du}{dx}=2x\hspace{7em}v=\displaystyle \int \cos(4x)dx\implies v=\cfrac{\sin(4x)}{4} \\\\[-0.35em] ~\dotfill\\\\ \displaystyle \int x^2\cos(4x)dx\implies \cfrac{x^2\sin(4x)}{4}-\cfrac{1}{2}\int x\sin(4x)dx\leftarrow \stackrel{ \textit{now let's again for this, use} }{\textit{integration by parts}} \\\\[-0.35em] ~\dotfill[/tex]

[tex]~~ \hspace{5em}\displaystyle \int x\sin(4x)dx \\\\[-0.35em] ~\dotfill\\\\ u_1=x\implies \cfrac{du}{dx}=1\hspace{7em}\displaystyle v_1=\int \sin(4x)dx\implies v_1=\cfrac{-cos(4x)}{4} \\\\[-0.35em] ~\dotfill\\\\ \displaystyle \int x\sin(4x)dx\implies \cfrac{-x\cos(4x)}{4}+\cfrac{1}{4}\int \cos(4x)dx \\\\\\ \displaystyle \int x\sin(4x)dx\implies \cfrac{-x\cos(4x)}{4}+\cfrac{1}{4}\left( \cfrac{\sin(4x)}{4} \right)[/tex]

[tex]\displaystyle \int x\sin(4x)dx\implies \cfrac{-x\cos(4x)}{4}+\cfrac{\sin(4x)}{16} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{now let's put together both the outer and nested integration by parts}}{\displaystyle \int x^2\cos(4x)dx\implies \cfrac{x^2\sin(4x)}{4}-\cfrac{1}{2}\left[ ~~ \cfrac{-x\cos(4x)}{4}+\cfrac{\sin(4x)}{16} ~~ \right]} \\\\\\ \displaystyle \int x^2\cos(4x)dx\implies \cfrac{x^2\sin(4x)}{4}+\cfrac{x\cos(4x)}{8}-\cfrac{\sin(4x)}{32}+C[/tex]

Answer: The answer above is true.

Step-by-step explanation:

Both can help you to visualize  time series data.

Have a great day! I hope that this helps! :)

In a village, the number of cases of a particular infectious disease has been decreasing exponentially at a rate of 90% per year since the vaccine was introduced. Starting with 100 cases, the predicted number of cases after 2 years is 1.

The exponential decrease in cases can be calculated using the formula A = P * [tex](1 - r)^t[/tex], where A is the final number of cases, P is the initial number of cases, r is the rate of decrease as a decimal (90% = 0.9), and t is the number of years. Plugging in the values, we have A = 100 * (1 - 0.9)^2. Simplifying, we get A = 100 * [tex](0.1)^2[/tex] = 100 * 0.01 = 1. Therefore, after 2 years, the predicted number of cases in the village would be 1.

This significant decrease in cases highlights the effectiveness of the vaccine in controlling the spread of the infectious disease. The 90% annual reduction rate indicates a rapid decline in the number of infected individuals. Over the course of two years, the original 100 cases have dwindled down to just one case, showcasing the successful impact of the vaccination efforts. It is important to continue monitoring the situation and maintain vaccination campaigns to ensure sustained protection and prevent any potential resurgence of the disease in the future.

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Answer:

Step-by-step explanation o:

50% of the prospective customers surveyed would be willing to pay a maximum of $7.01 to $8.

To determine the percentage of prospective customers willing to pay a maximum of $7.01 to $8,

Number of respondents willing to pay $7.01 to $8: 10

Total number of respondents: 20

To find the percentage, we can use the formula:

Percentage

= (Number of respondents willing to pay $7.01 to $8 / Total number of respondents) x 100

= (10 / 20) x 100

= 50%

Therefore, 50% of the prospective customers surveyed would be willing to pay a maximum of $7.01 to $8.

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Long answer: According to the International Telecommunication Union (ITU), there were an estimated 5 billion unique mobile phone users in the world as of January 2018. However, it is important to note that this number does not necessarily equate to the number of individual subscribers, as some people may have more than one mobile phone or SIM card. Additionally, the number of cell phone subscribers can vary by country and region. For example, according to Statista, the number of mobile phone users in the United States was approximately 266 million in 2018. Overall, while there is not a definitive global estimate for the number of cell phone subscribers in 2018, it is clear that mobile technology continues to have a significant impact on the way people communicate and access information worldwide.

The slope of the line forming the smallest right triangle, when a line is drawn through the point (1, 2), is 2.

The slope of the line forming the smallest right triangle with the positive x and y axes, when a line is drawn through the point (1, 2), can be determined as follows.

First, let's consider the two axes as the legs of the right triangle, and the line drawn through (1, 2) as the hypotenuse. The slope of the hypotenuse can be calculated by finding the difference in y-coordinates divided by the difference in x-coordinates between the two endpoints.

Since the x-coordinate of the point where the line intersects the x-axis is 0 (positive x-axis), and the y-coordinate of the point where the line intersects the y-axis is 0 (positive y-axis), the difference in y-coordinates is 0 - 2 = -2, and the difference in x-coordinates is 0 - 1 = -1.

Therefore, the slope of the line forming the smallest right triangle is -2/-1 = 2.

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The values of p for which the series ∑(n=1)^(∞) ((-1)^n / (n^p)) converges conditionally are p > 0.

To determine the values of p for which the series ∑(n=1)^(∞) ((-1)^n / (n^p)) converges conditionally, we can apply the alternating series test.

According to the alternating series test, a series of the form ∑((-1)^n * b_n) converges conditionally if:

1. The terms b_n are positive and decreasing (|b_n+1| ≤ |b_n|), and

2. The limit of b_n as n approaches infinity is 0 (lim(n→∞) b_n = 0).

In this case, our terms are b_n = 1 / (n^p). Let's check these conditions:

1. The terms are positive and decreasing:

  To satisfy this condition, we need to show that |(1 / ((n+1)^p))| ≤ |(1 / (n^p))| for all n.

  Taking the ratio of consecutive terms:

  |(1 / ((n+1)^p)) / (1 / (n^p))| = (n^p) / ((n+1)^p) = (n / (n+1))^p.

Since (n / (n+1)) is less than 1 for all n, raising it to the power p will still be less than 1 for p > 0. Therefore, the terms are positive and decreasing.

2. The limit of the terms as n approaches infinity is 0:

  lim(n→∞) (1 / (n^p)) = 0 for p > 0.

Based on the conditions of the alternating series test, the series converges conditionally for p > 0.

Therefore, the values of p for which the series ∑(n=1)^(∞) ((-1)^n / (n^p)) converges conditionally are p > 0.

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the local minimum value of f is obtained at x = 3.

To find the intervals on which the function f(x) = 5x^4 - 20x^3 + 2 is increasing or decreasing, we need to analyze the behavior of the derivative of the function.

(a) Finding the interval(s) on which f is increasing:

To determine where the function is increasing, we need to find the intervals where the derivative is positive (greater than zero).

First, let's find the derivative of f(x):

f'(x) = 20x^3 - 60x^2

Now, we set f'(x) > 0 and solve for x:

20x^3 - 60x^2 > 0

20x^2(x - 3) > 0

We have two critical points: x = 0 and x = 3.

We create a sign chart to analyze the intervals:

Intervals: (-∞, 0), (0, 3), (3, +∞)

Test point: x = 1

20(1)^2(1 - 3) > 0

-40 < 0 (negative)

From the sign chart, we see that f(x) is increasing on the interval (0, 3).

(b) Finding the interval(s) on which f is decreasing:

To determine where the function is decreasing, we need to find the intervals where the derivative is negative (less than zero).

Using the sign chart from part (a), we see that f(x) is decreasing on the intervals (-∞, 0) and (3, +∞).

(c) Finding the local minimum and maximum values of f:

To find the local minimum and maximum values, we need to examine the critical points and the behavior of the function at those points.

The critical points are x = 0 and x = 3.

To determine if these critical points correspond to local minimum or maximum values, we can analyze the second derivative of f(x).

Taking the second derivative:

f''(x) = 60x^2 - 120x

For x = 0:

f''(0) = 0

For x = 3:

f''(3) = 60(3)^2 - 120(3) = 180 > 0

Since f''(3) > 0, we can conclude that x = 3 corresponds to a local minimum.

As for a local maximum, since f''(0) = 0, we cannot determine if x = 0 corresponds to a local maximum or another type of point (such as an inflection point).

In summary:

(a) The interval on which f is increasing: (0, 3)

(b) The intervals on which f is decreasing: (-∞, 0) and (3, +∞)

(c) The local minimum value of f: x = 3

(d) The local maximum value: DNE (cannot be determined without further information)

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To prove these limits using the ε – δ definition, we need to show that for any given ε > 0, there exists a δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε, where L is the desired limit.

(a) To compute lim(x² + 2x)/(x - 3), we can factor the numerator as x(x + 2) and simplify the expression. The limit can be evaluated by substituting the value x = 3 into the simplified expression, which results in an indeterminate form. To prove the limit using the ε – δ definition, we need to show that for any given ε > 0, there exists a δ > 0 such that if 0 < |x - 3| < δ, then |(x² + 2x)/(x - 3) - L| < ε.

(b) To compute lim(x² + 2x), we can simplify the expression by factoring out x and evaluating the limit as x approaches infinity. To prove the limit using the ε – δ definition, we need to show that for any given ε > 0, there exists a δ > 0 such that if x > δ, then |x² + 2x - L| < ε.

(c) To compute lim(x - 3)/9, we can simplify the expression and evaluate the limit directly. To prove the limit using the ε – δ definition, we need to show that for any given ε > 0, there exists a δ > 0 such that if 0 < |x - 3| < δ, then |(x - 3)/9 - L| < ε.

In each case, the specific values of L and the corresponding ε and δ values will depend on the given function and limit. By following the ε – δ definition and finding appropriate δ values for a given ε, we can rigorously prove the limits using the limit definition.

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The missing step in solving the inequality is Subtract 8x from both sides of the inequality is

The given inequality is 8x<2x+3

eight times of x less than two times of x plus three

x is the variable in the inequality which we have to solve

Subtract 2x from both sides

8x-2x<3

6x<3

Divide both sides by 6

x<1/2

Where as they solved inequality by Subtract 3 from both sides of the inequality:

8x-3<2x

Now to isolate the x term we have Subtract 8x from both sides of the inequality:

Hence, Subtract 8x from both sides of the inequality is the missing step in solving the inequality

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The point-slope form of a line, we can write the equation of the tangent line with slope -1/2 passing through the point (2,1):

y - 1 = (-1/2)(x - 2)

To find the equation of the tangent line to the curve x^2 + 2xy + 4y^2 = 12 at the point (2,1) on the ellipse, we can use implicit differentiation. By taking the derivative of the equation with respect to x and solving for dy/dx, we can obtain the slope of the tangent line. Then, using the point-slope form of a line, we can determine the equation of the tangent line.

To find the derivative dy/dx, we differentiate both sides of the equation x^2 + 2xy + 4y^2 = 12 with respect to x. Using the chain rule and product rule, we get:

2x + 2y + 2x(dy/dx) + 8y(dy/dx) = 0

Next, we can solve this equation for dy/dx by isolating the terms involving dy/dx:

2x(dy/dx) + 8y(dy/dx) = -2x - 2y

Factoring out dy/dx, we have:

(2x + 8y)(dy/dx) = -2x - 2y

Dividing both sides by (2x + 8y), we get:

dy/dx = (-2x - 2y) / (2x + 8y)

Now, we can substitute the coordinates of the given point (2,1) into the derivative to find the slope of the tangent line at that point:

m = dy/dx = (-2(2) - 2(1)) / (2(2) + 8(1)) = -6/12 = -1/2

Using the point-slope form of a line, we can write the equation of the tangent line with slope -1/2 passing through the point (2,1):

y - 1 = (-1/2)(x - 2)

Simplifying this equation gives the final equation of the tangent line to the ellipse at the point (2,1).

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intervals is the (A) (−∞, 0] and [0, ∞) only.

To determine the intervals on which the function y = g(x/6x) is increasing, we need to analyze the derivative of the function.

Let's find the derivative of g(x/6x) with respect to x:

g'(x) = d/dx [g(x/6x)]

Using the chain rule, we have:

g'(x) = g'(x/6x) * d/dx (x/6x)

Since the function g(x) is differentiable and increasing for all real numbers, g'(x) > 0 for all x.

Now let's analyze the interval (0, ∞):

For x > 0, the expression x/6x simplifies to 1/6, which is a constant.

So, g'(x) = g'(1/6) * d/dx (1/6)

Since g'(x) > 0, the derivative g'(1/6) is also positive.

Thus, for x > 0, g'(x) > 0, which means the function y = g(x/6x) is increasing on the interval (0, ∞).

Similarly, we can analyze the interval (-∞, 0):

For x < 0, the expression x/6x simplifies to -1/6, which is a constant.

So, g'(x) = g'(-1/6) * d/dx (-1/6)

Again, since g'(x) > 0, the derivative g'(-1/6) is positive.

Thus, for x < 0, g'(x) > 0, which means the function y = g(x/6x) is increasing on the interval (-∞, 0).

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