Convert the following to exponential form 5⋅5⋅5⋅5⋅5⋅5⋅5⋅5

The proof of "if a | b and b | a, where a and b are integers, then either a = b or a = -b" is shown below .

The representation "a|b and b|a ", means that "a" is a divisor of "b" and "b" is a divisor of "a".

which means , there exists an integer "c" ; such that b=ac    ...equation(1) and an integer "d" such that a = bd    .....equation(2)

Substituting the first equation into the equation(2) , we get:

⇒ a = bd = (ac)d = adc ;

Since a|a, it must be that either a = adc or adc = -a.

But  "c" and "d" are integers, So , "adc" must also be an integer.

Hence, it must be that adc = a.

Thus , a = adc = a. This implies that d = c = 1, and

So ⇒ a = b .....Equation(3) .

Similarly , if adc = -a, then we have: a = -a

Which implies that a = 0, but since "a" and "b" are integers, they cannot both be 0. So , it must be that a = b   ...equation(4) .

Therefore , On combining equation(3) and equation(4)  , we proved that "if a|b and b|a, where a and b are integers, then either a = b or a = -b" .

The given question is incomplete , the complete question is

Prove/show that if a|b and b|a, where a and b are integers, then either a = b or a = -b.

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Answer: 1 7/10

Step-by-step explanation:

The solution is Option B.

The system of inequalities is x² - 2y ≥ 7000 and 2x + 5y < 1000

Inequalities are the mathematical expressions in which both sides are not equal. In inequality, unlike in equations, we compare two values. The equal sign in between is replaced by less than (or less than or equal to), greater than (or greater than or equal to), or not equal to sign.

In an inequality, the two expressions are not necessarily equal which is indicated by the symbols: >, <, ≤ or ≥.

Given data ,

Let the first inequality equation be represented as A

Let the second inequality equation be represented as B

Now , value of a company’s stock is represented by the expression x² - 2y

And , company’s goal is to maintain a stock value of at least $ 7,000

So , the inequality is x² - 2y ≥ 7000

And , the company’s purchases are modeled by 2x + 5y

Also , the company's purchases should be below $ 1,000

So , the inequality is 2x + 5y < 1000

Hence , the inequalities are solved

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The function that represents the amount of air required to inflate the balloon from a radius of r inches to a radius of r+3 inches is f(r) = 12π(r² + 3r + 3).

In this question, we are asked to determine a function for a spherical balloon that represents the amount of air required to inflate the balloon from a radius of r inches to a radius of r+3 inches.

Volume of spherical balloon = V(r) = 4/3 π r³

If the radius of balloon is r+3 inches = V(r+3) = 4/3 π (r+3)³

The amount of air required is the difference between V(r) and V(r+3):

So, f(r) = V(r+3) - V(r) = 4/3 π (r+3)³ - 4/3 π (r)³

                                  = 4/3 π ((r+3)³ -r³)

                                  = 4/3 π (r+3 -r)((r+3)² + r(r+3) + r²)

                                  = 4π(r² + 9 + 6r + r²+3r + r²)

                                  = 4π(3r² + 9r + 9)

                             f(r) = 12π(r² + 3r + 3)

Hence, the function representing is f(r) = 12π(r² + 3r + 3).

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The K-Means algorithm terminates when. a user-defined minimum value for the summation of squared error differences between instances and their corresponding cluster center is seen.

The K-Means algorithm terminates when

A) a user-defined minimum value for the summation of squared error differences between instances and their corresponding cluster center is seen.

B) the cluster centers for the current iteration are identical to the cluster centers for the previous iteration.

C) the number of instances in each cluster for the current iteration is identical to the number of instances in each cluster of the previous iteration.

D) the number of clusters formed for the current iteration is identical to the number of clusters formed in the previous iteration.

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The solution of dy/dt = (t*y^-.5)/3 with y(0)=1 is y(1) = 1.

The solution to the given differential equation is given by the following formula:

y(t) = [tex](3/t^2)^2/3 + C[/tex]

where C is the constant of integration.

To find the value of y(1), we need to substitute t = 1 in the above formula.

y(1) =[tex](3/1^2)^2/3 + C[/tex]

y(1) = 9 + C

Since y(0) = 1, we can find the value of C by substituting t = 0 in the above formula.

y(0) = ([tex]3/0^2)^2/3 + C[/tex]

y(0) = 9 + C

1 = 9 + C

C = -8

Substituting C = -8 in the formula for y(1),

y(1) = 9 + (-8) = 1

Therefore, y(1) = 1.

Thus, the solution of [tex]dy/dt = (t*y^-.5)/3 , y(0)=1[/tex] is y(1) = 1.

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Answer:

Option 1 is correct

Step-by-step explanation:

When two linear equations in two variables have no solution, it means that the equations are not compatible. This means that the equations cannot be solved simultaneously, as there is no set of values that can satisfy both equations. This can be seen by graphing the equations on a coordinate plane. If the two lines are parallel, then they will never intersect, and thus, there is no solution. If the two lines are the same line, then they will intersect at every point, and thus, any point on the line is a solution.

In the case of two linear equations in two variables, if the equations are not parallel and not the same line, then the equations will intersect at one point. This point is the solution to the two equations. If the equations are not parallel and not the same line, but do not intersect, then there is no solution. This means that the equations are incompatible and cannot be solved simultaneously.

In summary, two linear equations in two variables have no solution when the equations are incompatible. This means that the equations are either parallel or the same line, but do not intersect. In this case, there is no set of values that can satisfy both equations, and thus, there is no solution.

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Two linear equations in two variables have no solution if and only if the lines represented by the equations are parallel and do not intersect.

In other words, the lines do not have a common point, which means that there is no pair of values for the variables that satisfy both equations simultaneously.

Here's an example:

y = 2x + 1

y = 2x + 3

In this case, the lines represented by the equations are parallel and do not intersect, so there is no solution for the system of equations.

You can see this graphically by plotting the lines on a coordinate plane: the lines are close to each other, but they never intersect, so there is no common point that satisfies both equations.

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Since, the formula for variance is Variance = (σ)², where σ denotes standard deviation, the statement "the standard deviation is the positive square root of the variance" is false.

What is standard deviation?

The standard deviation is a metric that reveals how much variance from the mean there is, including spread, dispersion, and spread. A "typical" variation from the mean is shown by the standard deviation. Because it uses the data set's original units of measurement, it is a well-liked measure of variability.

The dispersion of the set of data values is measured by the standard deviation. The sigma symbol (σ) serves as its symbol.

This indicates that the data values tend to be more dispersed over a greater range for a higher standard deviation. While a lower number of standard deviation results from data being closer to its mean.

Variance, which is the expectation of the squared standard deviation, is a measure of how widely distributed the data are with respect to their mean.

As a result, the relationship between standard deviation and variance can be expressed numerically as, Variance = (σ)².

Therefore, standard deviation is not square root of variance.

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The perimeter of the entire shape that is made up of a semi circle and an equilateral triangle would be = 116.66in

The perimeter of a shape is defined as the total area of all the sides surrounding that shape.

The perimeter of the object can be calculated by the addition of the perimeter of the triangle and the circle.

The perimeter of the triangle = a+b+c = 19+19+19 = 57in

The perimeter of the circle = 2πr

where π = 3.14

r = 19/2 = 9.5 in

perimeter = 2×3.14×9.5 = 59.66in

The perimeter of the whole object = 57+59.66 = 116.66in.

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If the expression is the square of a binomial, then it must have the form [tex]$(ax + b)^2 = a^2x^2 + 2abx + b^2$[/tex].

Comparing the coefficients of like terms, we have:

[tex]a^2 = x^2$, so $a = x[/tex]

[tex]2ab = 18x$, so $b = 9[/tex]

Therefore, the missing number is [tex]$\boxed{9}$[/tex].

The missing number was found by using the fact that the expression must have the form [tex]$(ax + b)^2 = a^2x^2 + 2abx + b^2$[/tex] if it is the square of a binomial.

By comparing the coefficients of like terms in this equation and in the expression given, [tex]$x^2 18x \boxed{\phantom{00}}$[/tex], we were able to solve for the values of [tex]$a$[/tex] and [tex]$b$[/tex] as follows:

[tex]$a^2 = x^2 \Rightarrow a = x$[/tex]

[tex]$2ab = 18x \Rightarrow b = 9$[/tex]

Finally, we see that the coefficient of the [tex]$x$[/tex] term in the expression is [tex]$2ab = 2 \cdot x \cdot 9 = 18x$[/tex], which matches the given expression. Hence, the missing number is [tex]$\boxed{9}$[/tex].

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The value of k is "i".

What is cube and square?

Any number multiplied by itself is a square number. 0 through 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100 are the square numbers. A number multiplied by itself three times is a cube number. Up to 100, the cube numbers are 1, 8, 27, and 64. Cubes in three dimensions can be used to represent cube numbers.

Given that the value of a cube and a square, A, is zero when A = -1, we can write the equation:

a^3 + A^2 = 0

Substituting A = -1 into this equation gives us:

a^3 + (-1)^2 = 0

Simplifying the right-hand side of this equation:

a^3 + 1 = 0

So,

a^3 = -1

Taking the cube root of both sides:

a = -1^(1/3)

The value of k is the cube root of -1, which is a complex number and is represented as an imaginary number, i.e., "i".

Therefore, the value of k is "i".

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The complex number -1, which is represented as an imaginary number, or I has the value of k, which is the cube root of that number.

Any number multiplied by itself is a square number. 0 through 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100 are the square numbers. A number multiplied by itself three times is a cube number. Up to 100, the cube numbers are 1, 8, 27, and 64. Cubes in three dimensions can be used to represent cube numbers.

Given that the value of a cube and a square, A, is zero when A = -1, we can write the equation:

[tex]a^3+A^2=0[/tex]

Substituting A = -1 into this equation gives us:

[tex]a^3+(-1)^2=0[/tex]

Simplifying the right-hand side of this equation:

[tex]a^3+1=0[/tex]

So,

[tex]a^=-1[/tex]

Taking the cube root of both sides:

[tex]a=(-1)^{\frac{1}{3} }[/tex]

The value of k is the cube root of -1, which is a complex number and is represented as an imaginary number, i.e., "i".

Therefore, the value of k is "i".

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The expression that is equivalent to x^(2/5) is

Information given in the problem

the expression x^(2/5)

This equation represents exponents in fraction form and the basics for the calculation is from

a^(b/c) =

[tex]\sqrt[c]{a^{b} }[/tex]

Rewriting the expression is done as below

x^(2/5)

= [tex]\sqrt[5]{x^{2} }[/tex]

comparing with [tex]\sqrt[c]{a^{b} }[/tex]

gives the following values

a will be equal to x

b will be equal to 2

c will be equal to 5

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We get to the conclusion that the response is 10x after examining the two sides.

The above equation's roots are 5 + 3i and 5 - 3i.

This indicates that x - (5+3i) and x - are the two factors of the provided polynomial (5 - 3i). The original statement must be the sum of the elements. We can thus write:

x² - (answer) + 34 = (x - 5 -3i)(x - 5 + 3i).........(1)

Simplifying the right hand side:

[tex]& (x-5-3 i)(x-5+3 i) \\[/tex]

[tex]& =x^2-5 x+3 x i-5 x+25-15 i-3 x i+15 i-9 i^2 \\[/tex]

[tex]& =x^2-10 x+25-9 i^2 \\[/tex]

[tex]& =x^2-10 x+25-9(-1) \\[/tex]

[tex]& =x^2-10 x+25+9 \\[/tex]

[tex]& =x^2-10 x+34[/tex]

Thus, we can write the Equation 1 as:

[tex]x^2-\left(\text { answer) }+34=x^2-10 x+34\right.[/tex]

Our analysis of the two sides leads us to the conclusion that the response is 10x.

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The equation for the line that goes through (-10, 6) is given as y = -4x - 34

A parallel line has the same slope as the original line, but a different y-intercept. So, we want to find an equation of the form y = mx + b where m = -4 (the slope of the original line) and b is the y-intercept that gives the desired point (-10, 6).

We can find b by plugging in the coordinates of the point (-10, 6) into the equation y = mx + b:

6 = -4 * (-10) + b

6 = 40 + b

Subtracting 40 from both sides, we get:

-34 = b

So, the equation for the parallel line that goes through (-10, 6) is:

y = -4x - 34

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The height of the cone is - 3.4 centimeters.

Given data :

The total surface area of the cone is calculated by the formula -

Total surface area = πrl² + πr²

Keep the values in the formula to find the value of length or hypotenuse.

90π = πr(l² + r)

90 = 5(l² + 5)

18 = l² + 5

l² = 13

Now, find the height of the cone.

length² = perpendicular² + base²

13 = perpendicular² + 5²

Perpendicular² = 25 - 13

Perpendicular² = 12

Perpendicular = 3.4 centimeters.

Thus, the height of the cone is 3.4 centimeters

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The scale factor is equal to 4 which is calculated by determining the coordinates.

Let's begin by determining the coordinates.

These are the ones we begin with!

D (1,1)

E (2,2)

F (4,2)

G (3,1)

Now we must locate the best image cords.

D' (4,4)

E' (8,8)

F' (16,8)

G' (12,4)

We're looking for the scale factor, so we'll see what number is multiplied by those coordinates to get the prime coordinates.

D is (1,1) and D' is (4,4), but we know that 1x4 equals 4, so 4 is the scale factor by which we multiply all of the numbers.

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Using Priya's method to calculate (0.0015) x (0.024) gives 0.0000360

To multiply decimals using Priya's method, first, multiply as if there is no decimal. Next, count the number of digits after the decimal in each factor. Finally, put the same number of digits behind the decimal in the product.

Now, 15 × 24 = 360

0.0015 has 4 numbers after the decimal and  0.024 has 3 numbers after the decimal. The total is 7.

Moving the decimal point 7 spaces to the left, we get 0.0000360.

Thus,  (0.0015) x (0.024) = 0.0000360

Note: 0.0000360 = 0.000036

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The probability that the system works under these conditions is 0.81 (0.90 x 0.90 = 0.81).

This question uses the concept of independent probabilities, Independent probabilities refer to the likelihood of two or more events occurring independently of each other.

In other words, the probability of one event occurring does not influence the probability of another event occurring.

This means that if the probability of one event occurring is known, the probability of the other event occurring can be calculated independently.

given below

Step 1: Determine the probability that both components A and B will work.

Step 2: Multiply the probability that component A will work by the probability that component B will work.

Step 3: The result of Step 2 is the probability that the system will work under these conditions.

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The range of possible values for the third side of the triangle is d)6 < x < 26.

The third side of an acute triangle must be shorter than the sum of the other two sides and longer than the difference between the other two sides. By using the Pythagorean Theorem, we can find the range of possible values for the third side.

The Pythagorean Theorem states that the sum of the squares of the two sides of a right triangle equals the square of the hypotenuse. So, for this triangle, we have 10^2 + 16^2 = x^2.

Solve for x: x^2 = 256, so x = √256 = 16.

The third side must be shorter than the sum of the other two sides. So, the maximum value for x is 16 + 10 = 26.

The third side must be longer than the difference of the other two sides. So, the minimum value for x is 16 - 10 = 6.

Therefore, the range of possible values for the third side of the triangle is 6 < x < 26.

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An acute triangle has sides measuring 10 cm and 16 cm. The length of the third side is unknown.

Which best describes the range of possible values for the third side of the triangle?

x < 12.5, x > 18.9

12.5 < x < 18.9

x < 6, x > 26

6 < x < 26

KL and MN are chords of circle Q. They intersect at point J. the value of x is 5.9.

A circle's chord is the line segment that connects any two locations on the circle's circumference. It should be noted that a circle's diameter is defined as the lengthiest chord that runs through the centre of the circle.

KL and MN are chords of circle Q. They intersect at point J .MKN = (10x + 2 )° , mLM = (3x–4)°, and m∠KJM=103°.

(10x + 2 )° +  (3x–4)° +103° = 180 degree

13x + 2 + 103 = 180 degree

13x = 180 - 103 = 77

x = 77 / 13 = 5.92 .

KL and MN are chords of circle Q. They intersect at point J. the value of x is 5.9.

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"Owning a home provides a partial tax shelter for housing-related expenses" the following is an advantage of purchasing and owning a home.

Hence, option (B) is correct choice.

The Benefits of Owning a Home:

Interest rates are low.

Creating equity: The difference between what you can sell the house for and what you owe is your equity. As you pay down your mortgage, your equity rises. Over time, more of your monthly payments will be applied to the loan balance rather than the interest, resulting in more equity.

Federal tax breaks: Mortgage interest is deductible on the first $750,000 of the house's purchase price, as are home equity loan interest, property taxes up to $10,000 if married ($5,000 if married filing separately), and various closing fees at the time of purchase.

Greater privacy: Because you own the property, you may renovate it to your desire, something tenants do not have.

Home office: The work-at-home craze may persist after the epidemic has passed, implying that more of us will require a home office. The appropriate configuration affects both comfort and productivity.

Monthly payments that are predictable: A fixed-rate mortgage requires you to pay the same amount each month for principle and interest until the loan is paid off. Rents are subject to rise with each annual lease renewal. Changes in property taxes or homeowner's insurance can affect monthly payments, although not as often as rent hikes.

Stability: People tend to stay in a house they own for a longer period of time, if only because purchasing, selling, and relocating is difficult.

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Answer:

B. Owning a home provides a partial tax shelter for housing related expenses

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The required percent error of the measurement is 0.8%.

Error bar, are the line through a point on a graph, axes, which emphasizes the uncertainty or variation of the corresponding coordinate of the point.

The percent error of Adrian's measurement can be calculated as follows:

(|Measured value - Actual value| / Actual value) * 100

= (|12.8 - 12.7| / 12.7) * 100

= (0.1 / 12.7) * 100

= 0.786%

Rounded to the nearest tenth of a percent, the error is 0.8%.

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Answer:6.6%

Step-by-step explanation:

Part a) A. The population must be normally distributed

Part b) P(overbar x < 68.2​) = 0.7967

Part c)P(overbar x ≥ 65.6​)=  0.3745

a) Population is normally distributed -

b) P(overbar x < 68.2​)

Estimate the z score (z).

z = (x -  μ) / s

z = (68.2 - 64) / 19/√14

z = 0.83

Use z table,

P(overbar x < 68.2​) = P(z < 0.83)

P(overbar x < 68.2​) = 0.7967

c)  P(overbar X  ≥  65.6)

Estimate the z score (z).

z = (x -  μ) / s

z = (65.6 - 64) / 19/√14

z = 0.32

Use z table,

P(overbar x ≥  65.6)  =  P(z  ≥  0.32)

                                  = 1 -  P(z  <  0.32)

                                 = 1 - 0.6255

                                 = 0.3745

P(overbar x ≥  65.6)  = 0.3745

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The day she runs faster is Wednesday.

It is the quantity of an amount of something at a rate of one of another quantity.

In 2 hours, a man can walk for 6 miles

In 1 hour, a man will walk for 3 miles.

We have,

On Wednesday:

She ran 6 miles in 50 minutes.

Unit rate:

= 6/50

= 0.12 miles per minute _____(1)

On Thursday:

She ran 4 miles in 32 minutes.

Unit rate:

= 4/32

= 0.125 miles per minute _______(2)

From (1) and (2),

0.12 > 0.125

Thus,

On Wednesday she runs faster.

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The vector orthogonal for both vectors is [tex](20, -10, -4) / \sqrt{444}[/tex] = ([tex]20 / \sqrt{444}[/tex], [tex]-10 / \sqrt{444}[/tex], [tex]-4 / \sqrt{444}[/tex])

What is an orthogonal vector?

An orthogonal vector is a vector that is perpendicular to another vector in a given space. In other words, two vectors are orthogonal if their dot product is equal to zero.

To find a vector orthogonal to both 2, 2, 0 and 0, 2, 5 of the form 1, we can take the cross product of the two vectors, which results in a vector that is orthogonal to both. The cross product of 2, 2, 0 and 0, 2, 5 is given by:

(2, 2, 0) x (0, 2, 5) = (20, -10, -4)

So, a vector orthogonal to both 2, 2, 0 and 0, 2, 5 is given by (20, -10, -4). We can normalize this vector by dividing it by its magnitude, which is equal to sqrt(20^2 + (-10)^2 + (-4)^2) = sqrt(444). The normalized vector will be of the form 1:

[tex](20, -10, -4) / \sqrt{444}[/tex] = ([tex]20 / \sqrt{444}[/tex], [tex]-10 / \sqrt{444}[/tex], [tex]-4 / \sqrt{444}[/tex])

Hence, the vector orthogonal for both vectors is [tex](20, -10, -4) / \sqrt{444}[/tex] = ([tex]20 / \sqrt{444}[/tex], [tex]-10 / \sqrt{444}[/tex], [tex]-4 / \sqrt{444}[/tex]).

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Protons are a type of subatomic particle with a positive charge. With more electrons than protons, the particle is negatively charged.

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The difference quotient of f; that is, [tex]\frac{f(x+h)-f(x)}{h}[/tex], h ≠ 0 for the function f(x) = x^2 - 9x + 6 is 2x - 9.

The function is f(x) = x^2 - 9x + 6.

We have to determine the differential equation; that is [tex]\frac{f(x+h)-f(x)}{h}[/tex].

To find the value of f(x+h) substitute x+h in place of x the the function.

f(x+h) = (x+h)^2 - 9(x+h) + 6

Solving

F(x+h) = x^2 + 2xh + h^2 - 9x - 9h + 6

Now putting the value

[tex]\frac{f(x+h)-f(x)}{h} = \frac{(x^2 + 2xh + h^2 - 9x - 9h + 6)-(x^2 - 9x + 6)}{h}[/tex]

Simplify

[tex]\frac{f(x+h)-f(x)}{h} = \frac{x^2 + 2xh + h^2 - 9x - 9h + 6-x^2 + 9x - 6}{h}[/tex]

[tex]\frac{f(x+h)-f(x)}{h} = \frac{2xh + h^2 - 9h}{h}[/tex]

Taking common h on both side, we get

[tex]\frac{f(x+h)-f(x)}{h} = \frac{h(2x + h - 9)}{h}[/tex]

[tex]\frac{f(x+h)-f(x)}{h}[/tex] = (2x + h - 9)

Now setting the limit h tends to 0

[tex]\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}[/tex] = [tex]\lim_{h\rightarrow 0}[/tex](2x + h - 9)

Now putting h = 0, we get

[tex]\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}[/tex] = 2x - 9

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The complete question is:

Find the difference quotient of f; that is, find [tex]\frac{f(x+h)-f(x)}{h}[/tex], h ≠ 0 for the following function. Be sure to simplify.

f(x) = x^2 - 9x + 6

It will take Naomi approximately 4 quarters and 8 months, or 4.67 quarters to reach her goal.

To find the number of quarters required to reach $100,000 with a quarterly payment of $1500 and an interest rate of 4.4% compounded quarterly, we will  use the  formula below:

N = log((A / P) + 1) / log(1 + r)

Where:

N = number of quarters

A = target amount ($100,000)

P = payment per quarter ($1500)

r = interest rate per quarter (4.4%/4 = 1.1%)

Substituting the values, we have that

N = log((100,000 / 1500) + 1) / log(1 + 0.011)

N = log(66.67) / log(1.011)

N = 4.67 quarters

In conclusion, It will take Naomi approximately 4 quarters and 8 months, or 4.67 quarters to reach her goal.

Learn more about interest rate per quarter at: https://brainly.com/question/25793394

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