what is the second derivative of x^n when n= greater than or equal to 2

A vector x orthogonal to the row space of A, and a vector y orthogonal to column space, and a vector z orthogonal to the null space. The orthogonal vector is :

A = [tex]\left[\begin{array}{ccc}1&2&1\\2&1&0\\1&-2&2\end{array}\right][/tex]

The orthogonal complement of the subspace V contains any vector perpendicular to V. This orthogonal subspace is denoted V⊥. (pronounced "V perp").

By this definition, null space is the orthogonal complement of row space. Every x perpendicular to the line satisfies Ax = 0 and lies in null space.

vice versa. If v is orthogonal to null space, it must be in row space. Otherwise, we can add this v as an extra row of the matrix without changing its null space. The rice space will become larger, breaking the rule of r+(n−r) = n.

The column space extent of A. These two vectors are the basis of col(A) , but they are not normalized.

In this case, the columns of A are already orthonormal, so you don't need to use the Gram-Schmidt procedure. To normalize a vector and then divide it by its norm:

                   [tex]\left[\begin{array}{ccc}1&2&1\\2&4&3\\3&6&4\end{array}\right][/tex]

and the vector after orthogonal process is:

                  [tex]\left[\begin{array}{ccc}1&2&1\\2&1&0\\1&-2&2\end{array}\right][/tex]

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The new estimated value of the trust funds in 2038 would be $2.5 trillion.

Funds are resources of monetary value that can be used to acquire goods, services, or to meet financial obligations. They are usually obtained through the sale of goods and services or by borrowing. Funds can come from a variety of sources, including individuals, businesses, governments, and other organizations. Funds can be used to purchase assets such as stocks, bonds, real estate, or other forms of investments.

This is because a 10% relative increase over the original estimated 7.6% rate of decline means that the actual rate of decline is 8.36% (7.6% x 1.10 = 8.36%). Applying this rate to the original starting value of $4 trillion, the value of the trust funds in 2038 would be $4 trillion x 0.9164 (which is the equivalent of 1 - 0.0836) = $2.5 trillion.

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The new estimated value of the trust funds in 2038 would be $2.4 trillion.

A trust fund is a legal arrangement in which a person or organization (the trustor) transfers assets to another person or organization (the trustee) to be managed and distributed for the benefit of a third party (the beneficiary). The trustor sets out a set of instructions (the trust deed) on how the trust funds should be handled, managed, and distributed. The trustee is legally bound to abide by these instructions, and is responsible for the management and distribution of the trust funds according to the trust deed. Trust funds can be used for a variety of purposes, such as providing for a beneficiary's education, medical care, or retirement.

This is because a 10% increase in the rate of decline would result in a rate of decline of 8.36% (7.6% plus 10% of 7.6%).
Using this new rate of decline, the value of the trust funds in 2038 would be $4 trillion multiplied by (1 - 0.0836)⁵,
which is equal to $2.4 trillion.

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Answer:

3/4 of a bag of dog food will last 3 days.

Answer:

D. y = -2x

Step-by-step explanation:

slope = rise/run = 2/-1 = -2

y-intercept = (0,0)

y = mx + b

y = -2x + 0

y = -2x

Answer:

No.

Step-by-step explanation:

Given two triangles on a coordinate plane, you can check whether they are congruent by using the distance formula to find the lengths of their sides. If three pairs of sides are congruent, then the triangles are congruent by the above theorem.

Over here, they aren't congruent by the SSS.

Preparation of a Corrected set of Financial Statements :

Income Statements :

PARTY WAGON .INC

INCOME STATEMENT

FOR THE YEAR ENDED DECEMBER 31

CURRENT YEAR

Party Revenue Earned

156,000

Total Revenue

156,000

Expenses :

Insurance Expenses

2,160

Office Rent Expense

14,400

Supplies Expense

1,440

Salary Expense

90,000

Repair and Maintainence expense

2,400

Travel Expense

7,200

Miscellaneous Expense

4,320

Interest Expense

5,280

Depreciation - Van

9,600

Depreciation -Music and Equipment

8,400

Total Expenses

145,200

Income Before Income Tax

10,800

Less : Income Tax Expense

2,400

Net Income

8,400

Jack included balance sheet accounts in the income statement, therefore the corrected income statement consisting of revenue and expenses accounts is provided above.

Statement of Retained Earnings :

Retained Earnings (As per adjusted trial balance)  18,000

Add : Net Income  8,400

Less : Dividends  (1,200)

Retained Earnings Dec 31, Current Year  25,200

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Angle PLE is measured at 90°. The options supplied do not contain the solution.

The acute angle is defined as being less than 90 degrees. Right angles are 90 degrees in length. Angles that are obtuse have a greater angle than 90 degrees. Discover the various sorts of angles and examples of each.

As complementary angles, ZPLA and ZELA, we can infer that:

ZPLA + ZELA = 90°

Using the following expressions in place of ZPLA and ZELA, we obtain:

5x-2 + x+8 = 90

When we simplify the equation, we obtain:

6x + 6 = 90

6 is subtracted from both sides to yield:

6x = 84

Dividing by 6, we get:

x = 14

With the knowledge of x, we can determine the dimensions of ZPLA and ZELA:

ZPLA = 5x-2 = 5(14)-2 = 68°

ZELA = x+8 = 14+8 = 22°

Therefore, we can find the measure of angle PLE by subtracting the measures of ZPLA and ZELA from 180°:

PLE = 180 - ZPLA - ZELA

PLE = 180 - 68 - 22

PLE = 90

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Answer:

p=O-12

Step-by-step explanation:

original price:O

12 less than O is "O-12"

therefore p=O-12

Answer:

(2, -4)

Step-by-step explanation:

[tex]{ \sf{f(x) = {x}^{3} - {3x}^{2} }} \\ { \sf{ {f}^{l}(x) = {3x}^{2} - 6x }} \\ \\ { \sf{0 = x(3x - 6)}} \\ { \sf{0 = 3x - 6}} \: \: \: \: \: \\ { \sf{3x = 6}} \\ { \sf{x = 2}}[/tex]

For x = 2

[tex]{ \sf{f(2) = {2}^{3} - 3 {(2)}^{2} }} \\ { \sf{f(2) = 8 - 12}} \\ { \sf{f(2) = - 4}}[/tex]

Point is (2, -4)

Answer:

I'm pretty confident that the answer is E

The answer will of given mathematical logic will be T F T T F respectivelly.

A negation, conjunction, and disjunction are the fundamental mathematical logics. The symbols for negation, conjunction, and disjunction in mathematical logic are "," "," and "v," respectively.

Logical proofs frequently employ mathematical logic. Proofs are legitimate arguments that establish the veracity of mathematical assertions. A series of statements make up an argument. The conclusion is the last assertion, and the premises are all the statements that came before it (or hypothesis).

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The complete truth table is shown in the below diagram.

≈ q → ≈ p: True  False  True  True

The contrapositive of a conditional statement is a new conditional statement that is formed by negating both the hypothesis (the "if" part) and the conclusion (the "then" part) of the original statement, and switching their positions. The truth table for the contrapositive of a conditional statement has the same number of rows as the truth table for the original statement.

For example, if the original statement is "If it is raining, then the ground is wet", then the contrapositive would be "If the ground is not wet, then it is not raining."

According to the given table the contrapositive of a conditional statement q and p is defines as;

True

False

True

True

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Answer:

Equation: [tex]P=10n[/tex]

[tex]P=40[/tex] when n = 4

Step-by-step explanation:

Given

Direct Variation: P ∝ n

When n = 15, P = 150

Required

Determine the equation

Her earnings if she sold 4 necklaces

P ∝ n

Convert variation to equation

[tex]P=kn[/tex]

When P = 150, n = 15 (Substitute these values in the given equation)

[tex]150=k\times15[/tex]

Divide both sides by 15

[tex]\dfrac{150}{15} =\dfrac{k\times15}{15}[/tex]

[tex]10=k[/tex]

[tex]k=10[/tex]

Substitute k = 10 in [tex]P=kn[/tex]

[tex]P=10\times n[/tex]

[tex]P=10n[/tex] ------- Equation that relates P and n

When she sold 4 necklaces, n = 4

Substitute n = 4 in the formula above

[tex]P=10\times4[/tex]

[tex]P=40[/tex]

The value of Zeff for a valence electron in carbon using slater's rule is equals to 3.60.

Effective nuclear charge (Zeff) for a valence electron in carbon using Slater's rules,

Consider the shielding effect of the inner electrons.

Slater's rules state that the effective nuclear charge felt by an electron in an atom = the actual nuclear charge minus the shielding constant for that electron.

The shielding constant for an electron in a particular shell is determined by summing the contributions of all the electrons in inner shells.

Slater assigned the following shielding constants for each shell,

Electrons in the same shell contribute 0.35

Electrons in the next inner shell contribute 0.85

Electrons in all inner shells beyond that contribute 1.00

Carbon has an atomic number of 6, so it has six electrons.

The first two electrons are in the inner shell, leaving four valence electrons in the outer shell.

Zeff for a valence electron in carbon,

Consider the contributions of the inner electrons.

Two electrons in the 1s orbital contribute

2× 0.35 = 0.70 to the shielding constant.

The two electrons in the 2s orbital contribute

2 × 0.85 = 1.70 to the shielding constant.

There are no electrons in the 2p orbitals,

This implies,

They do not contribute to the shielding constant.

Thus, the effective nuclear charge felt by a valence electron in carbon is,

Zeff

= 6 - 0.70 - 1.70

= 3.60

Rounding to two decimal places, we get,

Zeff = 3.60

Therefore, the Zeff for a valence electron in carbon is equals to 3.60.

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Answer: y = 3x-2 (for first question)

Step-by-step explanation:

Equation of the line: y = mx+c. For A, it passes through (0, -2) and has a gradient of 3. So, you substitute the values inside the equation.

x = 0

y = -2

m = 3

-2 = 0+c

c= -2

ans: y = 3x-2

Use the same method to complete the rest of the questions. GL

Note: I am just a student, if I get this wrong I hope someone corrects me, thanks

The difference between the two formulas is that nPr takes order into account, while nCr does not. To determine which formula to use, consider whether order matters in the problem. If order matters, use nPr. If order does not matter, use nCr.

a) An example of an application problem that uses nPr could be the following: In how many ways can a committee of 3 people be chosen from a group of 7 if the order of selection matters?

b) An example of an application problem that uses nCr could be the following: In how many ways can a committee of 3 people be chosen from a group of 7 if the order of selection does not matter?

c) The formula for nPr is n!/(n-r)!, where n is the total number of items and r is the number of items being selected in a specific order. The formula for nCr is n!/r!(n-r)!, where n is the total number of items and r is the number of items being selected without regard to order.

a) The tree diagram for the problem is:

     H          T

    / \        / \

   H   T      H   T

  / \ / \    / \ / \

 H  T H  T  H  T H  T

b) The probability that all the coins will turn up heads is 1/8 or 0.125, since there is only one outcome out of the eight possible outcomes that satisfies this condition.

c) The probability that there will be at least one tails is 7/8 or 0.875, since there are seven outcomes out of the eight possible outcomes that satisfy this condition (all except HHH).

Therefore, All the possible outcomes are:  [tex]HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.[/tex]

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Hence, at the end οf year 2, the machine's bοοk value is $70,000.

Example οf Calculating Residual Value fοr a Business-Owned Vehicle. Hence, if the cοrpοratiοn sells the car after 6 years, 60% οf the οriginal cοst οf the car is depreciated οver 6 years, and the residual value is 40%. Remaining Value Befοre Tax is anοther name fοr this value.

The cοst οf an asset is evenly dispersed acrοss the asset's useful life with the straight-line technique οf depreciatiοn. Cοmpute the annual depreciatiοn cοsts and deduct it frοm the machine's οriginal cοst tο arrive at the machine's bοοk value at the end οf year twο.

Yοu can cοmpute the yearly depreciatiοn expense as fοllοws:

Cοst οf Asset - Residual Value / Useful Life = Depreciatiοn Cοsts

Inputting the values prοvided yields:

Depreciatiοn Expense = ($86,000 - $6,000) / 10 = $8,000 per year

The machine's οriginal cοst must be subtracted frοm the bοοk value at the end οf year twο in οrder tο determine the machine's value:

Bοοk Value at End οf Year 2 = Cοst οf Asset - (Depreciatiοn Expense x Number οf Years)

Bοοk Value at End οf Year 2 = $86,000 - ($8,000 x 2) = $70,000

Thus, the machine's bοοk value at the end οf year twο is $70,000.

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Answer:

g(5)=69

Step-by-step explanation:

g(5)=3(5)^2-2(5)+4

g(5)=75-10+4

g(5)=69

The required area of the irregular quadrilateral as given in the attached diagram is equal to 37.5 square units.

In the attached diagram,

Area of each square grid is equal to 1 square units.

Area of the irregular quadrilateral

= Area of the triangle A + Area of the triangle B + Area of the triangle C + Area of the rectangle D  ___(1)

Area of triangle A = ( 1/2 ) × Base × height

                              = ( 1/2 ) × 4 × 8

                              = 16 square units

Area of triangle B = ( 1/2 ) × Base × height

                              = ( 1/2 ) × 5 × 5

                              = 12.5 square units

Area of triangle C = ( 1/2 ) × Base × height

                              = ( 1/2 ) × 3 × 4

                              = 6 square units

Area of rectangle D = length × width

                                 = 3 × 1

                                 = 3 square units

Substitute all the values in (1 ) we have,

Area of the irregular quadrilateral

= 16 + 12.5 + 6 + 3

= 37.5 square units.

Therefore, the area of irregular quadrilateral is equal to 37.5 square units.

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The above question is incomplete, the complete question is:

Each square in the grid below has area 1. Find the area of the irregular quadrilateral below.

Diagram is attached.

Answer:

Integral

Step-by-step explanation:

explain integral calculus

The probability that a randomly selected statistics student is a business major or likes grilled cheese is 0.78.

Probability is defined as the ratio of the number of favorable outcomes to the total number of outcomes in other words the probability is the number that shows the happening of the event.

Given that the statistics class has 50 students and among those students, 35 are business majors, and 7 like grilled cheese. Of the business majors, 3 like grilled cheese.

The probability will be calculated as:-

P(AUB) = P(A) + P(B) - P(AB)

[tex]P(AUB)=\dfrac{35}{50}+\dfrac{7}{50}-\dfrac{3}{50}[/tex]

[tex]P(AUB)=\dfrac{42-3}{50}[/tex]

[tex]P(AUB)=\dfrac{39}{50}[/tex]

P(AUB) = 0.78

Therefore the probability that a randomly selected statistics student is a business major or likes grilled cheese is 0.78.

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the probability of indentifying at least 11 shirts correctly by random guessing is 0.0002

The ratio of favaourable outcomes to all possible outcomes of an event is known as the probability. The total number of positive results for an experiment with 'n' outcomes can be represented by the symbol x. The probability of an occurrence can be calculated using the following formula.

Probability(Event) = Positive Results/Total Results = x/n

Total no.of event=12

the probability that Joy will identify at least 11 shirts correctly by random guessing=

P(X≥11)=P(X=11)+P(X=12)

=¹²C₁₁(½)¹(½)¹¹+¹²C₁₂×(½)¹²(½)⁰

=12×½×(1/2)¹¹+(½)¹²

=0.00024414062≈0.0002

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Answer:

(a) Find (f + g)(x)

To find (f + g)(x), we add the two functions f(x) and g(x):

(f + g)(x) = f(x) + g(x) = (4x + 9) + (9x - 5) = 13x + 4

The domain of (f + g)(x) is all real numbers, since there are no restrictions on x that would make (f + g)(x) undefined.

(b) Find (f - g)(x)

To find (f - g)(x), we subtract the function g(x) from f(x):

(f - g)(x) = f(x) - g(x) = (4x + 9) - (9x - 5) = -5x + 14

The domain of (f - g)(x) is all real numbers, since there are no restrictions on x that would make (f - g)(x) undefined.

(c) Find (f * g)(x)

To find (f * g)(x), we multiply the two functions f(x) and g(x):

(f * g)(x) = f(x) * g(x) = (4x + 9)(9x - 5) = 36x^2 + 11x - 45

The domain of (f * g)(x) is all real numbers, since there are no restrictions on x that would make (f * g)(x) undefined.

(d) Find (f / g)(x)

To find (f / g)(x), we divide the function f(x) by g(x):

(f / g)(x) = f(x) / g(x) = (4x + 9) / (9x - 5)

The domain of (f / g)(x) is all real numbers except x = 5/9, since this value would make the denominator of (f / g)(x) equal to zero, resulting in division by zero, which is undefined.

(e) Find f(g(x))

To find f(g(x)), we substitute g(x) into the expression for f(x):

f(g(x)) = 4g(x) + 9

Substituting the expression for g(x), we get:

f(g(x)) = 4(9x - 5) + 9 = 36x - 11

The domain of f(g(x)) is all real numbers, since there are no restrictions on x that would make f(g(x)) undefined.

(f) Find g(f(x))

To find g(f(x)), we substitute f(x) into the expression for g(x):

g(f(x)) = 9f(x) - 5

Substituting the expression for f(x), we get:

g(f(x)) = 9(4x + 9) - 5 = 36x + 76

The domain of g(f(x)) is all real numbers, since there are no restrictions on x that would make g(f(x)) undefined.

(g) Find f(f(x))

To find f(f(x)), we substitute f(x) into the expression for f(x):

f(f(x)) = 4f(x) + 9

Substituting the expression for f(x), we get:

f(f(x)) = 4(4x + 9) + 9 = 16x + 45

The domain of f(f(x)) is all real numbers, since there are no restrictions on x that would make f(f(x)) undefined.

(h) Find g(g(x))

To find g(g(x)), we substitute g(x) into the expression for g(x):

g(g(x)) = 9

Step-by-step explanation:

Answer:

Step-by-step explanation:

(a) Find f(g(x)).

To find f(g(x)), we first need to find g(x) and then substitute it into f(x).

g(x) = 9x - 5

f(g(x)) = f(9x - 5) = 4(9x - 5) + 9 = 36x - 11

Therefore, f(g(x)) = 36x - 11.

(b) Find g(f(x)).

To find g(f(x)), we first need to find f(x) and then substitute it into g(x).

f(x) = 4x + 9

g(f(x)) = g(4x + 9) = 9(4x + 9) - 5 = 36x + 76

Therefore, g(f(x)) = 36x + 76.

(c) Find f(f(x)).

To find f(f(x)), we need to substitute f(x) into f(x).

f(f(x)) = 4(4x + 9) + 9 = 16x + 45

Therefore, f(f(x)) = 16x + 45.

(d) Find g(g(x)).

To find g(g(x)), we need to substitute g(x) into g(x).

g(g(x)) = 9(9x - 5) - 5 = 81x - 50

Therefore, g(g(x)) = 81x - 50.

Domain of f(x) and g(x): Since both f(x) and g(x) are linear functions, their domains are all real numbers.

(e) Find the inverse of f(x).

To find the inverse of f(x), we need to switch the roles of x and f(x) and solve for f(x).

y = 4x + 9

x = 4y + 9

x - 9 = 4y

y = (x - 9) / 4

Therefore, the inverse of f(x) is f^(-1)(x) = (x - 9) / 4.

(f) Find the inverse of g(x).

To find the inverse of g(x), we need to switch the roles of x and g(x) and solve for g(x).

y = 9x - 5

x = 9y - 5

x + 5 = 9y

y = (x + 5) / 9

Therefore, the inverse of g(x) is g^(-1)(x) = (x + 5) / 9.

(g) Find the domain of f^(-1)(x).

The domain of f^(-1)(x) is the range of f(x). Since f(x) is a linear function, its range is all real numbers. Therefore, the domain of f^(-1)(x) is also all real numbers.

(h) Find the domain of g^(-1)(x).

The domain of g^(-1)(x) is the range of g(x). Since g(x) is a linear function, its range is all real numbers. Therefore, the domain of g^(-1)(x) is also all real numbers.

The point on the paraboIa y = x² that is cIosest to the point (9, 1/2) is approximateIy (2.2913, 5.2491). The distance between the two points is approximateIy 6.0291.

CaIcuIus optimization probIems incIude the quest for the point on a curve that is cIosest to a given point. Finding the shortest path between two Iocations on a surface or figuring out how much materiaI is required to make a certain form are just two exampIes of its numerous practicaI uses.

Finding the greatest or Ieast vaIue of a function under specific constraints is a common task in optimization issues. These kinds of issues can be found in a variety of discipIines, such as engineering, economics, physics, and bioIogy. We can increase judgements and process effectiveness by finding soIutions to optimization difficuIties.

For the given paraboIa to fund the cIosest point to (9, 1/2) we use the distance formuIa:

d = √((x2 - x1)² + (y2 - y1)²)

Let, (x, x²), be a point on the paraboIa thus:

d = √((x - 9)²  + (x² - 1/2)²)

The minimum distance is found using the derivative:

d' = (x - 9) + 2(x² - 1/2)(2x) = 0

4x³ - 4x - 17 = 0

x ≈ 2.2913

Substituting the vaIue of x we have:

d = √((2.219 - 9)²  + ((2.219)² - 1/2)²)

d ≈ 6.0291

Hence, the point on the paraboIa y = x²  that is cIosest to the point (9, 1/2) is approximateIy (2.2913, 5.2491). The distance between the two points is approximateIy 6.0291.

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The probability that a customer will actually make a valid warranty claim is 0.075.

A probability is a numerical representation of the likelihood or chance that a specific occurrence will take place. Both proportions ranging from 0 to 1 and percentages ranging from 0% to 100% can be used to describe probabilities.

The probability that a customer will make a valid warranty claim is the probability that the customer both uses the bike in the first year (which has a probability of 0.75) and discovers a defect (which has a probability of 0.10).

Using the multiplication rule of probability, the probability that both of these events occur is:

0.75 x 0.10 = 0.075

Therefore, the probability that a customer will actually make a valid warranty claim is 0.075.

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The constant k that makes f(x) continuous everywhere is k = 49/41.

If a function's limit at a given position exists and is the same as the function's value there, the function is said to be continuous at that location. If a function is continuous over its whole domain, then it is continuous everywhere. As the name implies, a continuous function is one whose graph is continuous throughout without any pauses or leaps. To put it another way, we say that a function is continuous if we can draw the curve (graph) of the function without ever picking up the pencil.

For f(x) to be continuous everywhere, it must be continuous at x = 7.

Using the left limit we have:

k(7)² = 49k

Using the right limit we have:

(7x + k) = 49 + k

Setting the limits equal we have:

49k = 49 + k = 8k

49k = 8k + 49

41k = 49

k = 49/41

Hence, the constant k that makes f(x) continuous everywhere is k = 49/41.

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The marginal profit at the following expenditures for $8000 is 0 for $6000 is 8 for $12,000 is -16 and for $20,000 is -48.

When one more unit is produced and sold, a company or individual will make a profit known as marginal profit. The additional expense or profit made while manufacturing the next unit is referred to be marginal. Whereas marginal cost is the additional expense for manufacturing one extra unit, marginal product is the additional income received.

The difference between marginal cost and marginal product is known as marginal profit (also known as marginal revenue). Managers can utilise margin analysis to help them decide whether to increase output, reduce production, or halt production completely, a decision known as the shutdown point.

we have,

P(x)= 1000+32x - 2x²

Formula for derivative of function:

[tex]f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}[/tex]

Finding the derivative,

P(x+h) = 1000 + 32(x+h) - 2(x+h)²

P(x+h) = -2x² -4xh - 2h² + 32x + 32h + 1000

P(x+h) - P(x) = h(-4x - 2h + 32)

P'(x) = -4x + 32

Finding derivatives for given values of x is,

P'(8) = -4(8) + 32 = 0

P'(6) = -4(6) + 32 = 8

P'(12) = -4(12) + 32 = -16

P'(20) = -4(20) + 32 = -48

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Complete question:

The profit (in thousands of dollars) from the expenditure of x thousand dollars on advertising is given by P(x)= 1000+32x - 2x². Find the marginal profit at the following expenditures.

a. $8000

b. $6000

c. $12,000

d. $20,000

Roberto needs 4 square yards of fabric to make his costume.

A fraction that has the numerator higher than or equal to the denominator is said to be inappropriate. For instance, the fraction 7/3 is incorrect since 7 is bigger than 3. Mixed numbers, which combine a whole number and a correct fraction, can be created from improper fractions.

Given that, piece of fabric that is 2 2/3 yards long and 1 1/2 yard wide.

Convert the length from a mixed number to an improper fraction:

2 2/3 = (2 x 3 + 2)/3 = 8/3

1 1/2 = 3/2

The area of the rectangle is:

Area = Length x Width

Substituting the values we have:

Area = (8/3) x (3/2) = 4

Hence, Roberto needs 4 square yards of fabric to make his costume.

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The value of y in the given isosceles triangle is 140°.

A triangle with two equal edges is said to be isosceles. Also equal are the two angles that face the two identical sides. In other terms, an isosceles triangle is a triangle with two sides that are the same length.

If the sides AB and AC of a triangle ABC are equivalent, then the triangle ABC is an isosceles triangle with B = C. "If the two sides of a triangle are congruent, then the angle opposite to them is also congruent," states the theorem that explains the isosceles triangle.

An angle is a shape created by two lines that share an endpoint and are referred to as the angle's sides and vertex, respectively. Angles created by two beams are in the plane where the rays are located. The meeting of two surfaces also creates angles.

In the given question,

∠P=∠Q

2x+10=x+15

x=5

Therefore,

∠P=2(5)+10=20°

∠Q=5+15= 20°

∠P+∠Q+∠R= 180°

20+20+∠R= 180°

∠R= 140°

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