Do the leading coefficient and degree of the polynomial functions affect the behavior of its graph?.

We are given an economy with two goods and Leontief consumers who have utility functions that depend on the minimum of the two goods. The initial endowments of the consumers are positive, and we are assuming that the aggregate endowment of the economy has more of the first good than the second. If a price system p=(p1,p2) is an equilibrium with market clearing, we need to show that p1=0 and p2>0.

We start by considering the demand function for a Leontief consumer, which is given by xi1(p) = xi2(p) = (p1 + p2)mi/pi, where mi represents the initial endowment of consumer i and pi represents the price of good i.

Since the aggregate endowment ω=(ω1,ω2) of the economy satisfies ω1 > ω2, we know that the total supply of the first good is greater than the total supply of the second good.

Now, if p1 > 0, then for any positive value of p2, the demand for the second good xi2(p) will be positive for all consumers. This implies that the total demand for the second good will be greater than the total supply, leading to a market imbalance.

To achieve market clearing, where total demand equals total supply, we must have p1 = 0. This is because if p1 = 0, the demand for the first good xi1(p) will be zero for all consumers, and the total supply of the first good will match the total supply. Additionally, p2 must be positive to ensure positive demand for the second good.

Therefore, we have shown that in an equilibrium price system with market clearing, p1 = 0 and p2 > 0, given the initial endowment condition ω1 > ω2.

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The cost of one rose bush is $5 and the cost of one pot of ivy is $ 14.

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides. LHS = RHS is a common mathematical formula.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given:

Arjun spent $198 on 7 rose bushes and 13 pots of ivy.

Mei spent $204 on 14 rose bushes and 10 pots of ivy.

let the cost of rose bushes be x and cost of ivy pots be y.

So, 7x+ 12y = 198.....(1)

14x+ 10y = 204........(2)

Now solving above equation

x = 234/49

  ≈ 4.77551

y = 96/7

≈ 13.714286

Hence, cost of one rose bush is $5 and the cost of one pot of ivy is $ 14.

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Question:

Here are five sums. Use the distributive property to write each sum as a product with two factors.

\((a)\ 2a + 7a\)      \((b)\ 5z - 10\)    \((c)\ c - c^2\)      \((d)\ r + r+ r+ r\)    \((e)\ 2x - \frac{1}{2}\)

Answer:

\((a)\ 2a + 7a = 9a\)

\((b)\ 5z - 10 = 5(z - 2)\)

\((c)\ c - c^2 = c(1 - c)\)

\((d)\ r + r + r + r = 4r\)

\((e)\ 2x - \frac{1}{2} = \frac{1}{2}(4x - 1)\)

Step-by-step explanation:

Required

Apply distributive property

\((a)\ 2a + 7a\)      

Apply distributive property [Factor out a]

\(2a + 7a = a(2 + 7)\)

\(2a + 7a = a(9)\)

Open bracket

\(2a + 7a = a*9\)

\(2a + 7a = 9a\)

\((b)\ 5z - 10\)    

Express 10 as 5 * 2

\(5z - 10 = 5z - 5*2\)

Apply distributive property [Factor out 5]

\(5z - 10 = 5(z - 2)\)

\((c)\ c - c^2\)      

Express c as c * c

\(c - c^2 = c - c * c\)

Apply distributive property [Factor out c]

\(c - c^2 = c(1 - c)\)

\((d)\ r + r+ r+ r\)    

Express r as r * 1

\(r + r + r + r = r * 1 + r * 1 + r * 1 + r *1\)

Apply distributive property [Factor out 1]

\(r + r + r + r = r(1+1+1+1)\)

\(r + r + r + r = r * 4\)

\(r + r + r + r = 4r\)

\((e)\ 2x - \frac{1}{2}\)

Express 2x as 4x/2

\(2x - \frac{1}{2} = \frac{4x}{2} - \frac{1}{2}\)

Apply distributive property [Factor out 1/2]

\(2x - \frac{1}{2} = \frac{1}{2}(4x - 1)\)

Answer:

7 stamps

Step-by-step explanation:

4 invitations x three stamps = 12 stamps needed

12 stamps-5 stamps = 7 stamps

Answer: 7

Step-by-step explanation:4x3=12,12-5=7

Answer:

3,975.16 United States Dollar

Step-by-step explanation:

Solution:

If we choose a usual coordinate axis, an object going down would be traveling a negative distance, so a possible answer would be:

C. skiing 161 ft down a mountain.

On the other hand, An expense of money involves a waste of money, which is negative. So another possible answer would be:

D. spending $spending $161 on vacation.

Then, we can conclude that the correct answer would be:

C. skiing 161 ft down a mountain.

D. spending $spending $161 on vacation.

Walking is not safe on the path whenever rabbits have been seen in the area, and berries are ripe along the path. This is formalized by using the →(if-then) and ∧(logical and) operators.

Given information and corresponding atomic propositions:

We need to formalize the given statements in terms of atomic propositions r, b, and w, which are defined as follows:

r: Rabbits have been seen in the area.

b: Berries are ripe along the path.

w: Walking on the path is safe.

Now, let us formalize each of the given statements in terms of these atomic propositions:

a) Berries are ripe along the path, but rabbits have not been seen in the area.

b: Rabbits have not been seen in the area, and walking on the path is safe, but berries are ripe along the path.

c: If berries are ripe along the path, then walking is safe if and only if rabbits have not been seen in the area.

d: It is not safe to walk along the path, but rabbits have not been seen in the area, and the berries along the path are ripe.

e) For walking on the path to be safe, it is necessary but not sufficient that berries not be ripe along the path and for rabbits not to have been seen in the area.

Walking is not safe on the path whenever rabbits have been seen in the area, and berries are ripe along the path.

The formalizations in terms of atomic propositions are:

a) b ∧ ¬r.b) ¬r ∧ w ∧

b.c) (b → w) ∧ (¬r → w).

d) ¬w ∧ ¬r ∧

b.e) (¬r ∧ ¬b) → w.b ∧

Berries are ripe along the path, but rabbits have not been seen in the area.

This is formalized by using the ∧(logical and) operator.

(¬r ∧ ¬b) → w: It means For walking on the path to be safe, it is necessary but not sufficient that berries not be ripe along the path and for rabbits not to have been seen in the area.

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Answer:

The diameter of the circle is 7/2.

Step-by-step explanation:

⭐What is the radius of a circle?

⭐What is the diameter of a circle?

Thus, the diameter of said circle is twice the radius of said circle.

diameter = 2(radius)

diameter = 2(7/4)                   . . . . substitute known values

diameter = 14/4                      . . . . multiply the numerator & denominator

diameter = 7/2                       . . . . simplify

please mark brainliest! :)

Answer:

20 , Hope it Helps

Step-by-step explanation:

have a good day

A theorem which is used to find the rational solutions of a polynomial equation is known as the rational root theorem.

A polynomial expression is given and after solving them we get the two values known as the roots of the function.

These are also known as the zeros of polynomial as after putting these values in the equation we get the result as 0 .

We should use the rational root theorem only when the value of coefficients are small.

The theorem states that each rational solution x = p ⁄ q, when on simplified satisfies the below mentioned points,

p is an integer factor of the constant term a0, and

q is an integer factor of the leading coefficient an.

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Hmmm... uhh true? I think sorry I don't know either:(

Answer:

True

Step-by-step explanation:

a. To find a vector normal to the plane with the equation 9x - 4y - 112 = 2, we can take the coefficients of x, y, and z as the components of the normal vector. Therefore, a vector normal to the plane is \(\(\mathbf{N} = \begin{bmatrix} 9 \\ -4 \\ 0 \end{bmatrix}\)\).

b. To find a vector normal to the plane with the equation x - z = 0, we can take the coefficients of x, y, and z as the components of the normal vector. Therefore, a vector normal to the plane is \(\(\mathbf{N} = \begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}\)\).

In both cases, the vector normal to the plane represents the direction perpendicular to the plane.

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Given:

The given endpoints are ( -2,10) and (-10, -12).

Required:

We need to find the midpoint of te given endpoints.

Explanation:

Consider the formua to find the midpoint.

Consider the points ( -2,10) and (-10, -12).

Final answer:

Answer:

<A = 76°

<B = 63°

<C = 41°

Plug in the value of x

<C = 34 + 7

<C = 41°

Step-by-step explanation:

\( 63 + (2x + 8) + (x + 7) = 180 \) (sum of ∆)

Solve for x

\( 63 + 2x + 8 + x + 7 = 180 \)

Add like terms

\( 78 + 3x = 180 \)

Subtract 78 from each side

\( 78 + 3x - 78 = 180 - 78 \)

\( 3x = 102 \)

Divide both sides by 3

x = 34

<A = 2x + 8

Plug in the value of x

<A = 2(34) + 8 = 76°

<B = 63°

<C = x + 7

Plug in the value of x

<C = 34 + 7

<C = 41°

The volume of the rectangular piece with sides of lengths of 7.08 × 10–3 m, 2.18 × 10–2 m, and 4.51 × 10–3m is 6. 96 × 10^ -7 m

Volume is defined as the amount of space or capacity occupied by a three-dimensional shape.

It is represented in cubic units.

Formula:

Volume, V = length × breadth × height

Given the values;

7. 08 × 10 ^-3m , 2.18 × 10^ -2m and 4. 51 × 10^-3m

Multiply all the lengths

Volume = 7. 08 × 10 ^-3 × 2.18 × 10^ -2 × 4. 51 × 10^-3

= 69.6 × 10^-8

= 6. 96 × 10^ -7 m

Therefore, the volume of the rectangular piece is 6. 96 × 10^ -7 m

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Answer:

A

Step-by-step explanation:

trust

Answer:

The 1st term is -67

Step-by-step explanation:

I. When Sn = \(\frac{n}{2}\)(\(a_{1} + a_{n}\))

   If  Sn = 124, n = 8 and \(a_{n}\) = 98

                124 = 8/2(a + 98)

                124 = 4(a + 98)

                124 = 4a + 392

                124 - 392 = 4a

                -268/4 = a

                -67 = \(a_{1}\)

Hope that help :D

False. If f and g are polynomials of degree n, then f * g is a polynomial of degree at most 2n, not necessarily at most n.

When two polynomials f and g are multiplied, the degree of the resulting polynomial is equal to the sum of the degrees of f and g. In other words, if f has a degree of n and g has a degree of m, then f * g will have a degree of n + m.

Considering that f and g are both polynomials of degree n, we can rewrite them as f(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0 and g(x) = b_nx^n + b_(n-1)x^(n-1) + ... + b_1x + b_0, where a_i and b_i are coefficients.

Now, when we multiply f and g, we get f(x) * g(x) = (a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0) * (b_nx^n + b_(n-1)x^(n-1) + ... + b_1x + b_0).

Expanding this expression results in a polynomial of degree 2n, as the highest degree term will be (a_n * b_n) * x^(2n).

Therefore, the statement "f * g is also a polynomial of degree at most n" is false. The degree of the product of two polynomials is the sum of their individual degrees, not necessarily limited to the maximum degree among them.

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Answer: A. Mean = 9.56, Median = 11, Mode = 7

Step-by-step explanation:

The mean, median and the mode for the given data is 9.56, 9, and 7 respectively.

Given data below:

In order to find the mean, it can be calculated by dividing a sum of all the data points with the number of data points in the data set.

The formula is:

\(\bar{M}=\dfrac{\sum M}{N}\)

\(\bar{M}=\dfrac{4+6+7+7+9+11+13+14+15}{9}\)

\(\bar{M}=\dfrac{86}{9}\)

\(\bar{M}=9.56\)

Next, we will find the median.

In order to find the median, we got to place the numbers in value order and find the middle.

So,

\(\text{Median}=9\)

Lastly, we will find the mode.

To find the mode, order the numbers lowest to highest and see which number appears the most often.

So,

\(\text{Mode}=7\)

Thus, the mean, median and the mode for the given data is 9.56, 9, and 7 respectively.

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Answer:

y = x + 7

Step-by-step explanation:

Pick 2 points on the line to calculate the slope, m:  (3, 10) and (5, 12)

m = Δy/Δx = (12 - 10) / (5 - 3) = 2/2 = 1

Use the point-slope equation and the point (3, 10):   (y - y1) = m(x - x1)

(y - 10) = 1(x - 3)

y - 10 = x - 3

y = x - 3 + 10

y = x + 7

Therefore, the correct answer is that these two events are dependent.

The given events are dependent events. To check whether the two events are dependent or independent, check the following case.

If A and B are independent events, then the conditional probability of A given B is:

P(A|B) = P(A)If A and B are dependent events, then the conditional probability of A given B is:

P(A|B) = P(A and B) / P(B)Now let's solve the problem stated in the question:

A card is drawn off a 52-card deck. Let A be the event "The card is a heart" and let B be the event "The card is a queen".

Now, let us calculate the P(A and B):

We know that,

P(B) = 4/52 = 1/13. (because there are 4 queens in a deck of cards)

For P(A and B), we need to find the probability of drawing a Queen of Hearts. That is,

P(A and B) = 1/52. (Because there is only one queen of hearts in the deck)

Now let us calculate the P(A|B):P(A|B) = P(A and B) / P(B) = (1/52) / (1/13) = 1/4 ≠ P(A)Therefore, P(A|B) ≠ P(A).

This means that events A and B are dependent events.

Therefore, the correct answer is that these two events are dependent.

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Fig. 2.6 alone does not provide sufficient information to calculate the percent elongation of materials. Percent elongation is a measure of the increase in length of a material after it has been subjected to a tensile load, and is calculated by dividing the increase in length by the original length and multiplying by 100.

Fig. 2.6 shows stress-strain curves for different materials, which provide information about the relationship between stress and strain for those materials. While the curve shape can provide some insight into the behavior of the material under load, it does not provide information about the original length or the change in length of the material, which are necessary to calculate percent elongation.

In order to calculate percent elongation, additional information about the material's original length and the change in length after a tensile load is required. This information can be obtained through laboratory testing or from material specifications provided by the manufacturer.

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Answer:

Step-by-step explanation:

6x^2 - x + 3 - 5x^2 - 10x + 1

x^2 - 11x + 4

The original polynomial, \(x^2 + 7x + 12\), it can be seen that the two are the same, thus confirming that the polynomial was correctly factored.

To check the result after factoring a polynomial, one can use the FOIL method to ensure that the factored polynomial is equivalent to the original polynomial. The FOIL method stands for First, Outer, Inner, Last, and is used to multiply two binomials. To check the result, one would use the FOIL method to multiply the factored polynomial, and compare the result to the original polynomial. For example, let’s consider the polynomial\(x^2+7x+12\). After factoring, this can be written as (x+4)(x+3). To check the result, one would multiply the two binomials together, (x+4)\((x+3) = x^2 + 3x + 4x + 12\). Comparing this to the original polynomial, \(x^2 + 7x + 12\), it can be seen that the two are the same, thus confirming that the polynomial was correctly factored.

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The value of the double integral \(\int\int_B \frac{x}{y} e^ydA\) where, B is the region 0 ≤ x ≤ 1, x² ≤ y ≤ x by using iteration, is zero.

The given double integral is:

\(\int\int_B \frac{x}{y} e^ydA\)

where B is the region defined by 0 ≤ x ≤ 1 and x² ≤ y ≤ x.

To evaluate this double integral by iteration, we first need to set up the order of integration.

Since the limits of y depend on the value of x, we will integrate with respect to y first and then with respect to x.

Let's start by integrating with respect to y.

The limits of integration for y are x² ≤ y ≤ x.

Therefore, the inner integral becomes:

∫_(x²)^(x) (x/y)\(e^y\) dy

The value of the given double integral is 0.

To solve this integral, we can simplify the integrand:

(x/y)\(e^y\) = x\(e^y\)/y

Using the property of exponentials, we can rewrite this as:

x\(e^y\)/y = x(\(e^y\)/y)

Integrating this expression with respect to y, we get:

x∫_(x²)^(x) (\(e^y\)/y) dy

Next, we integrate this expression with respect to x.

The limits of integration for x are 0 ≤ x ≤ 1.

Therefore, the outer integral becomes:

∫_(0)^(1) [x∫_(x²)^(x) (\(e^y\)/y) dy] dx

Now we can evaluate this double integral by performing the integration in two steps: first with respect to y and then with respect to x.

To evaluate the given double integral, we will perform the integration in two steps: integrating with respect to y and then integrating with respect to x.

First, let's integrate with respect to y:

∫_(x²)^(x) (\(e^y\)/y) dy

To evaluate this integral, we can use the natural logarithm function.

The antiderivative of \(e^y\)/y is ln|y|.

Therefore, the inner integral becomes:

[x ln|y|]_(x²)^(x)

Now, we substitute the limits of integration:

[x ln|x| - x ln|x²|]

Simplifying this expression, we have:

[x ln|x| - 2x ln|x|]

Next, we integrate this expression with respect to x.

The limits of integration for x are 0 ≤ x ≤ 1.

Therefore, the outer integral becomes:

∫_(0)^(1) [x ln|x| - 2x ln|x|] dx

Integrating term by term, we get:

[1/2 x² ln|x| - 2/3 x³ ln|x|]_(0)^(1)

Substituting the limits of integration, we have:

(1/2 - 2/3) ln|1| - (0 - 0)

Simplifying further, we obtain:

(-1/6) ln(1)

Since ln(1) = 0, the final result is:

0

Therefore, the value of the given double integral is 0.

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The complete question is:

Evaluate the double integrals by iteration.

\(\int\int_B \frac{x}{y} e^ydA\)

where, B is the region 0 ≤ x ≤ 1, x² ≤ y ≤ x

Answer:

The answer is 3x + 8.

Step-by-step explanation:

1) Collect like terms.

\(3x + (42 - 34)\)

2) Simplify.

\(3x + 8\)

Hence, the answer is 3x + 8.

The  Trigonometric Identity sin² x + cos²x = 1 is: A. Pythagorean Identity.

The Pythagorean Identity which  tend to asserts that for every angle x, the sum of the squares of the sine and cosine of x is equal to one  is known as or called a  trigonometric identity.

The  Pythagorean identity can be expressed as:

sin² x + cos² x = 1

This identity is crucial to understanding trigonometry and tend to have several uses in numerous branches of science and engineering.

Therefore the correct option is A.

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Answer:

A) 3 2/12

Step-by-step explanation:

6 11/12 - 3 9/12 = 3 2/12

Answer:

Jack earns $37

Mark earns $21

Step-by-step explanation:

m + j = 58

11m + 10j = 601

j = 58 - m

11m + 10(58 - m) = 601

11m + 580 - 10m = 601

1m = 601 - 580

m = 21

j = 58 - m

j = 58 - 21

j = 37