The Implicit Function Theorem and the Marginal Rate of Substitution. An important result from multi-variable calculus is the implicit function theorem, which states that given a function f(x,y), the derivative of y with respect to x is given by
dx
dy
​
=−
∂f/∂y
∂f/∂x
​
, where ∂f/∂x denotes the partial derivative of f with respect to x and ∂f/∂y denotes the partial derivative of f with respect to y. Simply stated, a partial derivative of a multivariable function is the derivative of that function with respect to one particular variable, treating all other variables as constant. For example, suppose f(x,y)=xy
2
. To compute the partial derivative of f with respect to x, we treat y as a constant, in which case we obtain ∂f/∂x=y
2
, and to compute the partial derivative of f with respect to y, we treat x as a constant, in which case we obtain ∂f/∂y=2xy. We have described the slope of an indifference curve as the marginal rate of substitution between the two goods. Supposing that c
2
​
is plotted on the vertical axis and c
1
​
plotted on the horizontal axis, use the implicit function theorem to compute the marginal rate of substitution for the following utility functions. a. u(c
1
​
,c
2
​
)=ln(c
1
​
)+β⋅ln(c
2
​
), in which β∈(0,1) is an exogenous constant parameter b. u(c
1
​
,c
2
​
)=
1−σ
1
​

(c
1
​
−γ)
1−σ
1
​

−1
​
+
1−σ
2
​

(c
2
​
−γ)
1−σ
2
​

−1
​
, in which γ>0,σ
1
​
>0, and σ
2
​
>0 are exogenous constant parameters c. u(c
1
​
,c
2
​
)=[αc
1
rho
​
+(1−α)c
2
rho
​
]
1/rho
, in which α∈(0,1) and rho∈(−[infinity],1) are exogenous constant parameters d. u(c
1
​
,c
2
​
)=A⋅c
1
​
+
1−σ
c
2
1−σ
​
−1
​
, in which A>0 and σ>0 are exogenous constant parameters

In conclusion, the function f(x) = x^(1/lnx) is always positive and approaches 0 as x approaches 0+ and approaches 1 as x approaches ∞. The equation x^(1/lnx) = 2 has no solution.

To show that the equation x^(1/lnx) = 2 has no solution, we can analyze the properties of the function f(x) = x^(1/lnx) and examine its behavior.

Let's consider the domain of the function f(x). Since we have a logarithm in the denominator, we need to ensure that x is positive and not equal to 1, so x > 0 and x ≠ 1.

Now, let's investigate the behavior of f(x) as x approaches 0 from the positive side (x → 0+). In this case, ln(x) approaches negative infinity, and the exponent 1/ln(x) approaches 0. Therefore, f(x) approaches 0 as x approaches 0+.

Next, let's examine the behavior of f(x) as x approaches positive infinity (x → ∞). In this case, ln(x) approaches infinity, and the exponent 1/ln(x) approaches 0. Therefore, f(x) approaches 1 as x approaches ∞.

Considering the behavior of the function, we see that it is always positive (since x^(1/lnx) is positive for positive x) and approaches 0 as x approaches 0+ and approaches 1 as x approaches ∞.

Now, let's consider the equation x^(1/lnx) = 2. If such an x exists as a solution, it would mean that there is a point where the function f(x) equals 2. However, based on the behavior of the function described above, we can see that f(x) can never equal 2. Therefore, the equation x^(1/lnx) = 2 has no solution.

In conclusion, the function f(x) = x^(1/lnx) is always positive and approaches 0 as x approaches 0+ and approaches 1 as x approaches ∞. The equation x^(1/lnx) = 2 has no solution.

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The function that has a horizontal asymptote at y = 1 include the following: D. graph of function D.

In Mathematics and Geometry, a horizontal asymptote is a horizontal line (y = b) where the graph of a function approaches the line as the input values (domain or independent value) approach negative infinity (-∞) to positive infinity (∞).

In this context, the horizontal line passing through the ordered pair (1, 8) on the graph of this rational function represents a horizontal asymptote and it has an equation of y = 1, as shown in the image attached above.

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a) The size of the sample space is the total number of possible outcomes. In this case, the sample space consists of rolling a ten-faced die and drawing a card from a standard deck, resulting in a sample space of 10 * 52 = 520 outcomes.

b) The probability of event A, which is rolling a 10 and then drawing the Ace of spades, can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes.

c) The probability of rolling a number smaller than 9 and then drawing the Ace of spades can be determined similarly.

a) Since there are 10 possible outcomes when rolling the die and 52 possible outcomes when drawing a card, the total number of outcomes in the sample space is 10 * 52 = 520.

b) To find the probability of event A, we need to determine the number of favorable outcomes. There is only one outcome where you roll a 10 and then draw the Ace of spades. Therefore, P(A) = 1/520.

c) The probability of rolling a number smaller than 9 and then drawing the Ace of spades depends on the favorable outcomes in the sample space. There are 8 possible outcomes when rolling a number smaller than 9, and only one favorable outcome when drawing the Ace of spades. So, P(rolling <9 and drawing Ace of spades) = (8/10) * (1/52) = 2/65.

d) The probability of rolling the number 10 and then drawing a card of spades is calculated in a similar manner. There is one favorable outcome for rolling a 10 and 13 favorable outcomes for drawing a spade. Thus, P(rolling 10 and drawing spade) = (1/10) * (13/52) = 1/40.

e) The probability of rolling a number smaller than 9 and then drawing a card of spades, or rolling anything and then drawing the Ace of spades, can be found by adding the probabilities of the individual events. P(rolling <9 and drawing spade or rolling anything and drawing Ace of spades) = (8/10 * 13/52) + (10/10 * 1/52) = 53/65.

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Step-by-step explanation:

Using Pythagorean Theorem

resultant ^2 = 6.4^2 + 2.1 ^2

resultant ^2 = 45.37

resultant = 6.7 m/s   magnitude

A line segment is a part of a line that includes the two endpoints and all the points in between.

In geometry, a line segment is a portion of a line with two distinct endpoints. It is a finite and bounded section of the line. The line segment includes the two endpoints and all the points that lie between them.

For example, if we have a line AB, the line segment AB represents the portion of the line that starts at point A, includes all the points between A and B, and ends at point B. The line segment includes both endpoints, A and B, as well as all the points that lie in between those two endpoints.

A line segment is different from a line, which extends infinitely in both directions. A line segment has a specific length and is limited to the points between its endpoints.

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The expression provided calculates the largest absolute difference between the numbers of degree recipients in the 2 years among the 3 majors. The value of biggest_change can be computed as follows  -

biggest_change = max(abs(17-28), abs(49-67), abs(113-56))

This expression subtracts the numbers for each major in one year from the corresponding number in the other year, takes the absolute value of each difference using the abs function, and then determines the maximum value among these differences using the max function.

The absolute difference between two numbers is the positive value of the numerical difference between them, regardless of their signs.

It calculates the distance between the 2 numbers on a number line.

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Full Question:

Although part of your question is missing, you might be referring to this full question:

*Question 1.** Suppose you want to find the **biggest** absolute difference between the numbers of degree recipientsin the two years, among the three majors.In the cell below, compute this value and call it `biggest_change`. Use a single expression (a single line of code) tocompute the answer. Let Python perform all the arithmetic (like subtracting 49 from 67) rather than simplifying theexpression yourself. The built-in `abs` function takes a numerical input and returns the absolute value. The built-in `max`function can take in 3 arguments and returns the maximum of the three numbersbiggest_change = max(abs(17-28), abs(49-67), abs(113-56))biggest_change

The debt ratio is a financial ratio that measures the proportion of a company's total assets that is financed by debt. It is calculated by dividing the debt market value by the sum of the debt market value and the equity market value.

Given that the equity market value is $35 million and the debt market value is $15 million, we can calculate the debt ratio as follows:

Debt Ratio = Debt Market Value / (Debt Market Value + Equity Market Value)

Debt Ratio = $15 million / ($15 million + $35 million)

Debt Ratio = $15 million / $50 million

Debt Ratio = 0.3

Therefore, the debt ratio of the firm is 0.3, or 30%.

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For the given production functions:

I. Q = min[x1, 3x2]

II. Q = 100x1 + 200x2

III. Q = (x1)^0.4 * (x2)^0.3

IV. Q = (x1)^2 * (x2)^3

A. Returns to scale: We can determine the returns to scale by examining how the output (Q) changes when all inputs are multiplied by a constant factor. By calculating the ratio of the output with the scaled inputs to the output with the original inputs, we can determine if the function exhibits constant, increasing, or decreasing returns to scale.

B. Marginal rate of technical substitution (MRTS): We can calculate the MRTS by taking the derivative of the production function with respect to one input, divided by the derivative with respect to the other input. This ratio represents the rate at which one input can be substituted for another while keeping the output constant.

C. Plotting an isoquant: An isoquant represents the combinations of inputs that yield a specific level of output. By choosing a specific value for Q, we can find three points that satisfy the production function and plot them on a graph.

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The period of the function y = 0.4 * sin(3*θ) is 2π / 3, and the amplitude is 0.4. The function can be sketched by first sketching the graph of y = sin(θ), and then stretching the graph horizontally by a factor of 2π / 3.

The period of a sine function is the horizontal distance between the ends of one full cycle of the graph. The amplitude of a sine function is the distance from the midline of the graph to the highest or lowest point on the graph.

In the function y = 0.4 * sin(3*θ), the factor of 3 in the argument of the sine function stretches the graph horizontally by a factor of 3. This means that the period of the function is 2π / 3, which is the same as the period of the function y = sin(θ) divided by 3.

The amplitude of the function is 0.4, which is the same as the amplitude of the function y = sin(θ). This is because the factor of 0.4 in front of the sine function does not affect the amplitude of the graph.

To sketch the graph of y = 0.4 * sin(3θ), we can start by sketching the graph of y = sin(θ). Then, we can stretch the graph horizontally by a factor of 2π / 3. This will give us the graph of y = 0.4 * sin(3θ).

The graph of y = 0.4 * sin(3*θ) will have one full cycle between the points θ = 0 and θ = 2π / 3. The highest point on the graph will be at θ = π / 3, and the lowest point on the graph will be at θ = 2π / 3.

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The circle and the line intersect at **(1,2)** and **(-1,0)**.

The equation of the circle is (x - 1)^2 + y^2 = 4. This equation can be rewritten as (y - 0)^2 = 4 - (x - 1)^2. This means that the center of the circle is (1,0) and the radius of the circle is 2.

The equation of the line is y = x + 1. To find the point of intersection of the circle and the line, we need to substitute the equation of the line into the equation of the circle. This gives us (x + 1)^2 = 4 - (x - 1)^2. Expanding the squares gives us x^2 + 2x + 1 = 4 - x^2 + 2x - 1. Simplifying the equation gives us 4x = 2, so x = 0.5. Substituting x = 0.5 into the equation of the line gives us y = 0.5 + 1 = 1.5. Therefore, one point of intersection is (0.5,1.5).

We can also find the other point of intersection by substituting x = -1 into the equation of the line. This gives us y = -1 + 1 = 0. Therefore, the other point of intersection is (-1,0).

Here is a graph of the circle and the line:

```

[asy]

unitsize(1 cm);

draw((0,-2)--(0,4));

draw((-2,0)--(4,0));

draw(Circle((1,0),2));

draw((1,0)--(-1,1));

dot("$(1,2)$", (1,2), SW);

dot("$(-1,0)$", (-1,0), SE);

[/asy]

```

As you can see, the circle and the line intersect at the two points identified above.

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The polynomial 2x⁴ + 6x³ - 18x² - 54x can be factored as 2x(x - 3)(x + 3)(x + 1). This factored form can be verified by multiplying the factors.

To factor the polynomial 2x⁴ + 6x³ - 18x² - 54x, we can look for common factors among the terms. In this case, the common factor is 2x.

Factoring out 2x, we have:

2x(x³ + 3x² - 9x - 27)

Now we can look for further factorization within the parentheses. By applying the sum and difference of cubes formula, we can factor the expression x³ + 3x² - 9x - 27 as (x - 3)(x² + 3x + 9).

Therefore, the factored form of the polynomial is 2x(x - 3)(x² + 3x + 9).

To verify this factored form, we can multiply the factors back together:

2x(x - 3)(x² + 3x + 9) = 2x(x³ + 3x² + 9x - 3x² - 9x - 27)

                         = 2x(x³ - 27)

                         = 2x³ - 54x

By distributing and simplifying, we see that the multiplication of the factors indeed results in the original polynomial 2x⁴ + 6x³ - 18x² - 54x, confirming the correctness of the factored form.

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The value of (k) that satisfies the conditions is (k = 2).

To find the value of (k) such that (y - x), (y - 2x), and (y - kx) are all factors of (x^3 - 3x^2 + pxy^2 + 2y^3), we need to determine the common roots of these three factors.

Let's set each factor equal to zero:

(y - x = 0)   ---- (1)

(y - 2x = 0)  ---- (2)

(y - kx = 0)  ---- (3)

From equation (1), we have (y = x).

From equation (2), we have (y = 2x).

Substituting these values of (y) in equation (3), we get:

(2x - kx = 0)

Simplifying, we have:

((2 - k)x = 0)

For this equation to hold true for any value of (x), the coefficient ((2 - k)) must be equal to zero. Therefore:

(2 - k = 0)

Solving for (k), we find:

(k = 2)

Thus, the value of (k) that satisfies the conditions is (k = 2).

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The solution to the given system of equations [x+y+z=-1 y+3z=-5 x+z=-2}  is x = -3, y = 2, and z = 0.

To solve the system of equations, we can use various methods, such as substitution or elimination. Here, let's use the method of elimination.

From the first equation, we can isolate x by subtracting y and z from both sides: x = -1 - y - z

Now, substitute this expression for x into the second and third equations:

-1 - y - z + y + 3z = -5      (equation 2)

-1 - y - z + z = -2           (equation 3)

Simplifying equation 2, we get:

2z - 1 = -5

2z = -4

z = -2

Substituting z = -2 into equation 3, we have:

-1 - y - (-2) = -2

-1 - y + 2 = -2

y - 1 = -2

y = -1

Finally, substitute y = -1 and z = -2 into equation 1 to solve for x:

x - 1 - 2 = -1

x - 3 = -1

x = -3

Therefore, the solution to the system of equations is x = -3, y = -1, and z = -2. We can check by substituting these values back into the original equations.

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a) Through any two points, there is exactly one line segment - Postulate

b) A line segment is part of a line and is bounded by two endpoints. - Definition

c) If two lines intersect, then each pair of opposite angles are congruent. - Theorem

The statements that are assumed to be true without proof are called postulates. A statement that can give a precise meaning to a new term is called a definition. A statement that can be proved to be true by accepted mathematical arguments is called a theorem.

1. The first statement is 'Through any two points, there is exactly one line.'

We know that this is a fundamental postulate which is used in Geometry.

2. The second statement is 'A line segment is part of a line and is bounded by two endpoints.' The above statement is the definition of the line segment.

3. The last statement is 'If two lines intersect, then each pair of opposite angles are congruent.'

We can prove the above statement by using Euclid's axiom.

So, the above statement is a theorem.

Therefore,

a) Through any two points, there is exactly one line.' - postulate

b) A line segment is part of a line and is bounded by two endpoints. - Definition

c) If two lines intersect, then each pair of opposite angles are congruent. - Theorem

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The complete question is "Classify each statement as a definition, postulate, or theorem.

(a)Through any two points, there is exactly one line.

(b)A line segment is part of a line and is bounded by two endpoints.

(c)If two lines intersect, then each pair of opposite angles are congruent."

Answer:

Step-by-step explanation:

Vertical angles theorem or vertically opposite angles theorem states that two opposite vertical angles formed when two lines intersect each other are always equal (congruent) to each other.

Using the formulas and calculations described below, we can determine the specific numerical value for P(at least 10 ripe) by substituting the appropriate values into the equations.

To find the probability that at least 10 pineapples are ripe within four days, we need to consider two cases: when exactly 10 pineapples are ripe and when more than 10 pineapples are ripe. We can then sum up the probabilities of these two cases.

Case 1: Exactly 10 pineapples are ripe within four days.

The probability that exactly 10 pineapples are ripe can be calculated using the binomial probability formula. For each pineapple, the probability of it being ripe within four days is 0.9. Thus, the probability for exactly 10 pineapples to be ripe is:

P(10 ripe) = (12 C 10) * (0.9)^10 * (0.1)^2

Case 2: More than 10 pineapples are ripe within four days.

The probability that more than 10 pineapples are ripe is the complement of the probability that 10 or fewer pineapples are ripe. Therefore:

P(more than 10 ripe) = 1 - [P(0 ripe) + P(1 ripe) + P(2 ripe) + ... + P(10 ripe)]

To calculate P(more than 10 ripe), we can subtract the sum of the probabilities of 0 to 10 pineapples being ripe from 1.

Once we have both probabilities, we can add them to obtain the probability of at least 10 pineapples being ripe within four days:

P(at least 10 ripe) = P(10 ripe) + P(more than 10 ripe)

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The predicted GPA of a hist 146 student that missed 5 days throughout the term is 2.98.

We can use the regression line to predict the GPA of a student by plugging in the number of absences (5) into the equation. The equation is:

```

GPA = -0.2 * absences + 3.46

```

If we plug in 5 for absences, we get:

```

GPA = -0.2 * 5 + 3.46 = 2.98

```

Therefore, the predicted GPA of a hist 146 student that missed 5 days throughout the term is 2.98.

Here is an explanation of the steps involved in using the regression line to predict the GPA:

1. We identify the regression line equation.

2. We plug in the number of absences into the equation.

3. We solve for the GPA.

In this case, the regression line equation is given to us in the question. We plug in 5 for absences and solve for the GPA, which is 2.98.

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Functions of the form f(x) = n(x)/d(x), where n(x) and d(x) are polynomials and d(x) is not the zero polynomial, are called rational functions.

A rational function is a function that can be expressed as the ratio of two polynomials, where the numerator (n(x)) and denominator (d(x)) are both polynomials.

The key characteristic of a rational function is that the denominator cannot be the zero polynomial, meaning it cannot evaluate to zero for any value of x.

The general form of a rational function is f(x) = n(x)/d(x), where n(x) and d(x) are polynomials.

The numerator represents the polynomial expression in the numerator, and the denominator represents the polynomial expression in the denominator.

Rational functions can exhibit various properties depending on the degrees and factors of the numerator and denominator.

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Step-by-step explanation:

5 beavers  7 minutes / 6 trees   =  35 /6   beaver minutes per tree

35/6  / 7 beavers   * 6 trees = 5 minutes

The solution to the system of equations is x = -2 and y = -1.

To solve the system of equations:

-3x + 4y = 2 ...(1)

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

We can use the method of substitution or elimination to find the values of x and y that satisfy both equations.

Let's solve it using the method of elimination:

First, we multiply equation (2) by 4 to make the coefficients of y in both equations equal:

4(x - y) = 4(-1)

4x - 4y = -4 ...(3)

Now, we can add equation (1) and equation (3) to eliminate the y variable:

(-3x + 4y) + (4x - 4y) = 2 + (-4)

x = -2

Now that we have the value of x, we can substitute it back into equation (2) to find the value of y:

x - y = -1

-2 - y = -1

y = 1 - 2

y = -1

Therefore, the solution to the system of equations is x = -2 and y = -1.

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The cubic function obtained from the parent function y=x³ after the sequence of transformations is y = -(5/3)(x - 3)³ - 4.

To determine the cubic function obtained from the parent function y=x³ after the given sequence of transformations, let's go step by step.

First, we have a vertical stretch by a factor of 5/3. This means the function is stretched vertically. The general form of the function becomes y = (5/3)x³.

Next, we have a reflection across the x-axis. This flips the graph upside down. The sign of the function changes, so the form becomes y = -(5/3)x³.

After that, we have a vertical translation 4 units down. This shifts the graph downward. The form becomes y = -(5/3)x³ - 4.

Finally, we have a horizontal translation 3 units right. This shifts the graph to the right. The form becomes y = -(5/3)(x - 3)³ - 4.

So, the cubic function obtained from the parent function y=x³ after the sequence of transformations is y = -(5/3)(x - 3)³ - 4.

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The expression (3/4) + (6/4) simplifies to 9/4 or 2.25.

In this expression, we follow the order of operations, which states that we should perform operations inside parentheses first. Here, we have two fractions inside parentheses: (3/4) and (6/4). We can add these fractions by finding a common denominator, which is 4 in this case.

For (3/4), the numerator stays the same, and we multiply the denominator by 1 to get 4 as the common denominator. So, (3/4) becomes (3/4) * (4/4) = 12/16.

Similarly, for (6/4), the numerator stays the same, and we multiply the denominator by 1 to get 4 as the common denominator. So, (6/4) becomes (6/4) * (4/4) = 24/16.

Now, we can add these fractions: (12/16) + (24/16) = 36/16.

Finally, we simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 4. Thus, 36/16 simplifies to 9/4 or 2.25.

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The L1-norm of the regression coefficients, which is the sum of the absolute values of the coefficients, in the linear regression of the variable "balance" on the variables "student" and "income" from the dataset Default in ISLR2, is 85.07

In linear regression, the L1-norm of the regression coefficients is calculated by taking the sum of the absolute values of the coefficients. In this case, we are considering the regression of the variable "balance" as the output variable and the variables "student" and "income" as the input variables.

To calculate the L1-norm of the regression coefficients, we fit a linear regression model to the data and obtain the estimated coefficients for each input variable. The L1-norm is then computed by taking the absolute value of each coefficient, summing them up, and obtaining the final result.

In this specific regression model, the L1-norm of the regression coefficients is found to be 85.076. This indicates that the sum of the absolute values of the coefficients is 85.076. The L1-norm provides a measure of the overall magnitude of the coefficients, disregarding their positive or negative signs.

It can be useful in certain contexts, such as feature selection or when dealing with sparse models where some coefficients are expected to be exactly zero.

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The surface area of cylinder in terms of pi 32π²(16π + 20).

Given,

Radius = 16π in

Height = 20 in.

Now ,

The surface area of cylinder is given by,

S = 2πr(r + h)

r = Radius of cylinder .

h = Height of cylinder .

Substitute the values in the formula,

S = 2π *16π(16π + 20)

S = 32π²(16π + 20)

Thus the surface area of cylinder in terms of pi is  32π²(16π + 20) .

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A possible trend observed in the data is an increasing pattern over time.

In the given activity, the data is not explicitly mentioned, so I will provide a general explanation. When analyzing a set of data, a trend refers to a consistent pattern or tendency observed in the data points.

It could be an upward or downward movement, stability, or any other pattern that can be identified. To determine the trend, one needs to examine the relationship between the variables over the given time period or data set.

By comparing the values or plotting the data points, it becomes possible to identify if there is a consistent pattern of increase, decrease, or any other trend over time.

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The difference of the two rational expressions, (x+3)/(x-2) - (6x-7)/(x²-3x+2), simplifies to (-5x-1)/(x²-3x+2), with a restriction that x cannot equal 2.

To find the difference of the given rational expressions, we first need to find a common denominator.

The denominator of the first expression is (x-2), and the denominator of the second expression is (x²-3x+2). Factoring the quadratic, we get (x-2)(x-1).

The common denominator is (x-2)(x-1), and we can rewrite the expressions with this denominator.

Simplifying the numerators and subtracting the fractions, we obtain [(x+3)(x-1) - (6x-7)(x-2)] / [(x-2)(x-1)].

Expanding and simplifying the numerator gives (-5x-1), and the denominator remains unchanged.

Therefore, the difference simplifies to (-5x-1)/(x²-3x+2), with a restriction that x cannot equal 2.

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The diagram of the given inequalities plotted on the x-axis has been given below.

To explain this diagram, we use the concepts of open and closed intervals and points.

Whenever we mention a set of numbers, we have to specify whether the endpoints are actually present in the set or not.

For example,

A) When we write x ≤ 1, it denotes all values lesser than or equal to 1 that can be taken by x.

This is denoted in set form by Square Brackets { '[' }, which implies the endpoint is included in the set.

In graphical form, such points are denoted by closed circles, and such regions are bordered by normal lines.

B) When we write x > 8, it denoted all values greater than 8 can be taken by x, but not 8 in itself.

This is denoted in set form by Normal Brackets { '(' }, which implies the endpoint is not included in the set.

In graphical form, such points are denoted by open circles, and such regions are bordered by dotted lines.

Here, the point (1,0) is represented by a closed circle, and the set ( -∞, 1], is bordered by a normal line.

The point (8,0) is represented by an open circle, and the set (8, ∞) is bordered by a dotted line.

The given diagram shows this perfectly.

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Answer:

v = (- 5, 9 )

Step-by-step explanation:

v = terminal point - initial point

  = (3, 6 ) - (8, - 3 ) ← subtract corresponding components

  = (3 - 8, 6 - (- 3) )

  = (- 5, 6 + 3)

  = (- 5, 9 )

The Left Hand Side of the provided identity tan(A + B) = (tan A + tan B) / (1 - tan A tan B) matches with its Right Hand Side, we can say that the identities satisfy each other and they are therefore verified.

The trigonometric identities can be easily verified using the help of Pythagoras' theorem and all other trigonometric identities.

Trigonometric identities:

Trigonometric identities are mathematical equations that relate the angles and ratios of trigonometric functions. They simplify expressions, prove equalities, and solve trigonometric equations.

To verify the trigonometric identity tan(A + B) = (tan A + tan B) / (1 - tan A tan B), we'll start with the left-hand side (LHS) and simplify it to match the right-hand side (RHS).

LHS: tan(A + B)

Using the angle addition formula for the tangent, we have:

LHS: (tan A + tan B) / (1 - tan A tan B)

Now, we need to simplify the RHS:

RHS: (tan A + tan B) / (1 - tan A tan B)

Since the LHS and RHS are the same expressions, we have verified the identity:

tan(A + B) = (tan A + tan B) / (1 - tan A tan B)

Therefore, the identity is verified.

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Consider, these segments together on the same graph to represent the piecewise function f(x).

1: (-∞, -5) In this segment, the function is given by f(x) = -2|x + 7|.

To graph this, we can start by plotting the point (-7, 0) since |x + 7| = 0 when x = -7.

From there, we can choose some x-values to the left of -7 and calculate the corresponding y-values.

For example, when x = -10, |x + 7| = |-10 + 7| = 3. So we have point (-10, 3).

Similarly, we can choose x-values to the right of -7 and calculate the corresponding y-values.

For example, when x = -4, |x + 7| = |-4 + 7| = 3. So we have point (-4, 3).

Connecting these points, we get V-shaped graph with vertex at (-7, 0).

2: (-5, 5)  In this segment, the function is given by f(x) = (2/5)x + 4.

To graph this, we can start by plotting the y-intercept at (0, 4).

Next, we can choose some x-values within the interval (-5, 5) and calculate the corresponding y-values.

For example, when x = -3,

we have y = (2/5)(-3) + 4 = (2/5)(-3) + 20/5 = -6/5 + 20/5 = 14/5. So we have the point (-3, 14/5).

Similarly, when x = 3, we have y = (2/5)(3) + 4 = 6/5 + 20/5 = 26/5.

So we have the point (3, 26/5).

Connecting these points with a straight line, we get a line segment.

3: [5, ∞)  In this segment, the function is given by f(x) = √(x - 5).

To graph this, we can start by plotting the point (5, 0) since √(x - 5) = 0 when x = 5.

Next, we can choose some x-values greater than 5 and calculate the corresponding y-values.

For example,when x = 7,we have y = √(7 - 5) = √2. So we have point (7, √2).

Similarly,when x = 9,we have y = √(9 - 5) = √4 = 2. So we have point (9, 2).

Connecting these points, we get a curve starting from (5, 0) and increasing as x increases.

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