For the following exercises, find the average rate of change of each function on the interval specified for real numbers b or h in simplest form. f(x)=4x²-7 on (1,b)

To show that the improper integral g(x)dx is divergent, but the average value of g on the interval 4, infinity is finite, we can consider a function g(x) = 1/x, which is mostly used as an example for a divergent integral.

The improper integral of g(x) from 4 to infinity is given by:

[tex]\int\limits {g(x)} \, dx = \int\limits {(1/x)} \, dx[/tex]  from 4 to infinity

= lim (upper limit -> infinity) [ln|x|] from 4 to x

= [ln|x|] from 4 to infinity

= ln(infinity) - ln(4) = infinity - ln(4) = infinity

As the limit of the integral is infinity, the integral is divergent.

However, the average value of g on the interval [4, ∞) is given by:

average value = (1/∞ - 4) * ∫(1/x)dx from 4 to infinity

= (1/∞ - 4) * (infinity - ln(4))

= -4 + ln(4)

= ln(4) - 4

which is a finite value when the integral is divergent

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A population of bacteria growing exponentially can be modeled as P(t)= P0e kt, where is the time in hours and P0 is the initial population. if the population has a doubling time of 3 hours, the growth constant  is k = ln(2) / 3 = 0.231049.

The growth constant (k) can be calculated by taking the natural log of 2 (ln(2)) and dividing it by the doubling time (3 hours). Therefore, the growth constant is 0.231049.  This equation models exponential growth, which is a type of growth where the rate of increase is proportional to the population size. In this case, the population doubles every 3 hours.

This exponential growth is modeled using the equation P(t)= P0e^kt, where P(t) is the population size at a given time, P0 is the initial population size, and k is the growth constant.  By finding the growth constant, we can predict the population size at any given time. The growth constant is found by taking the natural log of 2 (ln(2)) and dividing it by the doubling time (3 hours).

This equation is used to find the amount of growth per unit time and is the same regardless of the population size. This equation can be used to predict the population size at any given time, allowing us to estimate the population size in the future.

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Another point on the line is  (2, 3)

To find another point on the line the equation of the line will be solved as used to get other points

The slope, m of the lines is calculated the formula

m = (y₀ - y₁) / (x₀ - x₁)

where

m = slope

x₂ and x₁ = points in x coordinates

y₂ and y₁ = points in y coordinates

The slope, m of the linear function is calculated using the points (0, 5), (4, 1)

m = (y₀ - y₁) / (x₀ - x₁)

m = (5 - 1) / (0 - 4)

m = (4) / (-4)

m = -1

equation passing through point (0, 5)

(y - y₁) = m  (x - x₁)

y - 5 = -1 (x - 0)

y - 5 = -x

y = -x + 5 equation of the line

for x = 2

y = -x + 5 = -2 + 5 = 3

the point is (2, 3)

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The piggy bank has 200 dimes and 50 quarters.

To find this, we can set up the following equation:

(number of dimes x $0.10) + (number of quarters x $0.25) = $39.25

We know that the piggy bank contains 250 coins, so we can substitute that in:

(x x $0.10) + (250-x x $0.25) = $39.25

Now we can solve for x, the number of dimes:

x0.10 + (250-x)0.25 = 39.25

x0.10 + 62.5 - x0.25 = 39.25

0.1x + 62.5 - 0.25x = 39.25

-0.15x = -23

x= 153

So there are 153 dimes in the piggy bank, and 250-153 = 97 quarters.

But we know that the piggy bank contains 250 coins, so we can check that 153 dimes + 97 quarters = 250 coins.

Answer: 3

Step-by-step explanation: 60 x 5/100 = 3

There were three first-year teachers in the meeting.

f(x) = 7x - 1, which is a linear equation with a positive leading coefficient, making f(x) > 0 for all x.

Set 1: {f(x) = x + 1, f(x) = x - 1}

Function f(x) = x^2 + 2 can be found in the span of the two given functions. This can be seen by expanding the equation:

f(x) = x^2 + 2 = (x + 1) + (x - 1) = 2x;

thus, f(x) = 2x + 2, which is a quadratic equation with a positive leading coefficient, making f(x) > 0 for all x.

Set 2: {f(x) = 4x - 2, f(x) = 3x + 1}

Function f(x) = 7x - 1 can be found in the span of the two given functions. This can be seen by expanding the equation:

f(x) = 7x - 1 = (4x - 2) + (3x + 1) = 7x;

thus, f(x) = 7x - 1, which is a linear equation with a positive leading coefficient, making f(x) > 0 for all x.

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Complete question:Find a function f(x) in the span of the functions f1(x) = x and f2(x) = -x such that f(x) > 0 for all x.

The image of quadratic function f(x) = x² is equal to g(x) = 6 · x².

In this question we see the representation of quadratic equation f(x) = x², from which we need to generate the image of this function, this can be done by a rigid transformation known as vertical dilation:

g(x) = k · f(x), k > 1

Where:

If we know that f(x) = x² and k = 6, then the image of function is:

g(x) = 6 · x²

Whose representation on Cartesian plane is shown in the image attached below.

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It is proven that the other nth roots of z can be found by multiplying the initial one by the nth roots of unity.

A complex number satisfying the equation [tex]z^{n-1} = 0[/tex] is known as the nth root of unity. There are about n nth roots of a unit, according to the fundamental theorem of algebra. We know that,[tex]1=e^{2\pi i}[/tex]. Then, the nth root is written as [tex]\omega=e^{2\pi i/n}[/tex].

The integer of powers of this root between 0 and n-1 provides the other nth roots. Then, the roots are [tex]1, \omega, \omega^2 ,\cdots \cdots, \omega^{(n-1)}[/tex]. Now, factorize the equation [tex]z^{(n-1)} = 0[/tex] we get the required identity as, [tex]z^{(n-1)} = (z- 1)(z^{(n-1)}+ z^{(n-2)} + \cdots \cdots+ z + 1) = 0[/tex].  

This indicates that the equation [tex]1+\omega+ \omega^2+\cdots \cdots+ \omega^{(n-1)}[/tex] is true for all nth roots of unity other than 1.

Roots of unity also have the great quality of being periodic, which results from the fact that n = 1. It is equal to a power of that power mod n if the power is bigger than n.

When ω is a third root of unity, [tex]\omega^5 + \omega^4 + 1 = \omega^2 + \omega + 1 = 0[/tex].

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The solution to the equation 10+4y=-4y+2 is -1 because the variable y represents the same value.

An equation is a formula in mathematics that expresses the equality of two expressions by connecting them with the equals sign =. In its most basic form, an equation is a mathematical statement that shows that two mathematical expressions are equal. 3x + 5 = 14, for example, is an equation in which 3x + 5 and 14 are two expressions separated by a 'equal' sign. A mathematical equation that depicts the relationship between two expressions on opposite sides of the sign. It mostly consists of one variable and one equal to symbol. 2x - 4 = 2 is an example.

Here,

10+4y=-4y+2

8y=-8

y=-1

The solution of equation 10+4y=-4y+2 is -1 as the variable y represents the same value.

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Answer:

x = 12

4 races

Step-by-step explanation:

2/3 = 8/x

Cross multiply.

2x = 8 × 3

x = 4 × 3

x = 12

Answer: x = 12

0.75 × 45 = 33.75

Since the number of races must be an integer, they must win a total of 34 races to have at least a 75% success rate.

They already won 30 races.

34 - 30 = 4

Answer: 4 races

Quadrilateral DEFG is similar to quadrilateral HIJK The measure of side JK is 8.

Since quadrilateral DEFG is similar to quadrilateral HIJK, we know that the ratio of corresponding side lengths is equal. This means that if we know the length of one side of one of the quadrilaterals, we can use that ratio to find the length of the corresponding side in the other quadrilateral.

For example, if we know that side DE is 8 units long, we can use the ratio of side JK to side DE (x/8) to find the length of side JK. We can also use the same ratio of side IJ to side DF which is x/4.

Let's assume that the length of side DE is 8 units. We can use the ratio of side JK to side DE (x/8) to find the length of side JK.

[tex]x/8 = x/4[/tex]

Cross-multiplying and simplifying the equation we get

[tex]4x = 8x[/tex]

[tex]x = 8[/tex]

Therefore, the measure of side JK is 8 units.

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The solution of the given equation is X = B(C - A)⁻¹.

To solve for X in the equation AX + B = CX, we can rearrange the equation as follows:

AX - CX = -B

Factor out X on the left side:

(X(A - C)) = -B

To isolate X, we can multiply both sides of the equation by the inverse of (A - C), assuming it exists:

X = -(A - C)⁻¹ × B

X = (C - A)⁻¹ × B

Therefore, the solution for X is:

X = B(C - A)⁻¹

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In the problem, it is solved that the number of dogs John has is 14

The problem is solved by expression the word problem into a simultaneous equation as follows

let d be the number of dogs and c d number of chickens

d + c = 25 (since all must have a head)

4d + 2c = 78 (2 legs for chicken and 4 legs for dog)

using calculator to solve gives

d = 14 and c = 11

hence the number of dogs is 14

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1) the rate change of function f(x) in given interval =4(b+1)

2)  the rate change of function g(x) in given interval =1

3) the rate change of function j(x) in given interval =[tex]3h^2+9h+9[/tex]

To find out the average rate change of function on given interval is -

Use the graph of f to determine f (a) and f (b) for the provided range [a,b]. Remember that the graph of f's point with the y-coordinate of f (a) and the x-coordinate of an is called f. Similar to that, f ( b ) is the y-coordinate of the point on the f graph whose x-coordinate is b.

Step 2: Calculate the average rate of change of  f over the range [a, b] using the average rate of change formula.

[tex]\frac{f(b)-f(a)}{b-a}[/tex]

1)

[tex]f(x)=4x^2-7 \ on\ [1,b]\\\\f(1)=4-7=-3\\\\f(b)=4b^2-7\\\\the \ rate \ change =\frac{4b^2-7+3}{b-1}\\\\=4(b+1)[/tex]

2)

[tex]p(x)=3x+4 \ on\ [2,2+h]\\\\p(2)=10\\\\p(2+h)=10+h\\\\The\ average\ rate \ change= \frac{10+h-10}{2+h-2}\\\\=1[/tex]

3)

[tex]j(x)=3x^3 \ on \ [1,1+h]\\\\j(1)=3\\\\j(1+h)= 3(1+h)^3\\\\\\so, the\ rate\ change = 3h^2+9h+9[/tex]

The complete Question is: -

For the following exercises, find the average rate of change of each function on the interval specified for real numbers b or h in simplest form.

1. f(x)= 4x^2 - 7 on [1, b]

2. p(x)= 3x+4 on [2,2+h]

3. j(x)= 3x3 on [1, 1+h]

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The equivalent expression of i⁴¹ is i.

Expressions that are equivalent to the same thing, even when they have distinct appearances. When we enter the same value(s) for the variable, two algebraic expressions that are equivalent have the same value (s).

Given:

An expression i⁴¹.

Simplifying,

i⁴¹ = {(i⁴)¹⁰} i

i⁴¹ = {(1)¹⁰} i   {i⁴ = 1}

i⁴¹ = i

Therefore, the equivalent expression is i.

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Answer:

To find the result of 2.4 divided by -0.06, we need to use the following steps:

Step 1: Change the division problem into a multiplication problem by flipping the divisor and multiplying by its reciprocal.

-0.06 ÷ 2.4 = -0.06 x 1/(2.4)

Step 2: To find the reciprocal of 2.4, we need to divide 1 by 2.4

1/(2.4) = 0.416666666666667

Step 3: Multiply the original dividend (2.4) by the reciprocal of the divisor (0.416666666666667)

2.4 x 0.416666666666667 = -0.1

So, 2.4 divided by -0.06 is equal to -0.1

It's worth noting that -0.1 is a negative number which makes sense since the divisor is negative, hence the result is the opposite of the result when dividing by a positive number.

The relative order of the metric prefixes Milli- < Centi- < Micro- < Kilo-

The relative order of the metric prefixes of kilo-, micro-, centi-, and milli- is milli-, centi-, micro-, and kilo-. This order is determined by the exponential values of each prefix.

Milli- is equal to 10-3, centi- is equal to 10-2, micro- is equal to 10-6 and kilo- is equal to 103. The exponential values are used to determine the relative order of the prefixes. The lowest exponential value is milli-, making it the smallest metric prefix and the highest exponential value is kilo-, making it the largest metric prefix.

When the exponential values are compared, it is clear that the order of the prefixes is milli- < centi- < micro- < kilo-. This order is used to determine the relative size of the metric prefixes when measuring the same base unit.

For example, if the base unit is meter, then millimeter is the smallest measure and kilometer is the largest measure. The exponential values of the metric prefixes determine the relative order of the base unit.

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Center of the sphere would be (-4.5, 1, -6)  and radius be 6.086.

Write the equation of the sphere?

A sphere's general equation is (x - a)2 + (y - b)2 + (z - c) 2 = r2, where (a, b, c) represents the sphere's center, r represents the radius, and x, y, and z are the coordinates of points on the sphere's surface.

To complete the square, we need to add and subtract the square of half of the x coefficient and y coefficient, and the square of half of the z coefficient.

The equation will be : [tex](x + 9/2)^2 + (y - 1)^2 + (z + 6)^2 = (9/2)^2 + 1^2 + 6^2[/tex]

So the standard form of the equation of the sphere is,    

          [tex](x + 4.5)^2 + (y - 1)^2 + (z + 6)^2 = 37[/tex]

The center of the sphere is (x, y, z) = (-4.5, 1, -6) and the radius is √37 ≈ 6.086

Therefore, Center of the sphere would be (-4.5, 1, -6) and radius be 6.086.

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The problems with Alisha's statistical study are that the difficulty levels of the sudoku puzzles might be different, self timing can lead to inaccurate measures of time and also the experience and skill level of her friends will be different.

In her statistical study, Alisha asked 10 of her friends to try 10 different sudoku puzzles from the internet and time themselves while doing it. In this study there are a few errors. The first one being that the difficulty level of the puzzles that her friends will be solving are going to be different.

Another issue is that since her friends are self-timing, there can be inaccurate measures of time and lastly, the experience and skill of her friends for sudoku is going to create a bias in her study.

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Python lines to get the output for the probability P(x<0.325) are -

a) import scipy.stats as

st print(st.norm.cdf(0.325, 0, 1))

Hence, option a is the correct answer.

The Python code -

        import scipy.stats as

        st print(st.norm.cdf(0.325, 0, 1))

This line of code imports the scipy.stats module as "st", and then uses the norm.cdf function to calculate the cumulative distribution function (CDF) of a normal distribution with mean 0 and standard deviation 1, evaluated at x=0.325.

The CDF gives the probability that a random variable from the normal distribution is less than or equal to a certain value, so this line of code is outputting the probability P(x<0.325).

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Answer: (-4t - 7) + (3t + 8) cannot be simplified further as the terms have different variables.

(-4r and -7) and (3r and 8) cannot be combined as these terms have different variables.

(-41 and 3r) cannot be combined as these terms have different variables.

Step-by-step explanation:

When evaluating the likelihood of an outcome, the claim or hypothesis statement is assumed. The required answer is a hypothesis claim or statement.

It is possible to test one's assumptions regarding a population parameter through the statistical analysis known as hypothesis testing. The association between two statistical variables can then be estimated by this hypothesis.

We start by writing the claim mathematically and determining if it is the null or alternative hypothesis. The null hypothesis is the assertion that there is no change or no relationship between the two variables; otherwise, the alternative hypothesis is the assertion that tells that there is a change or relationship.

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For the value of x = 4, the inequality equation that is true is option C: x/2 < 3.

What is an inequality?

In Algebra, an inequality is a mathematical statement that uses the inequality symbol to illustrate the relationship between two expressions. An inequality symbol has non-equal expressions on both sides. It indicates that the phrase on the left should be bigger or smaller than the expression on the right, or vice versa.

The first inequality equation is - x + 5 < 3

Substitute the value of x = 4 in the equation.

4 + 5 < 3

Now, simplify the equation -

9 < 3

It is known that 9 is not less than 3, but greater than 3.

Therefore, this inequality is false.

The second inequality equation is - 9x > 36

Substitute the value of x = 4 in the equation.

9(4) > 36

Now, simplify the equation -

36 < 36

It is known that 36 is not less than 36, but equal 36.

Therefore, this inequality is false.

The third inequality equation is - x/2 < 3

Substitute the value of x = 4 in the equation.

4/2 < 3

Now, simplify the equation -

2 < 3

It is known that 2 is less than 3.

Therefore, this inequality is true.

The fourth inequality equation is - 18 < x + 8

Substitute the value of x = 4 in the equation.

18 < 4 + 8

Now, simplify the equation -

18 < 12

It is known that 18 is not less than 12, but greater than 12.

Therefore, this inequality is false.

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Which inequality is true when x = 4?

A. x + 5 < 3

B. 9x > 36

C. ×/2 < 3

D. 18 < x + 8

One can infer some information about a line's slope just by looking at its graph, especially in comparison to other lines plotted on the same coordinate plane.

Rise / run = a slope.

The Complete Question is line's slope.

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You cannot use the Hypotenuse-Leg congruence theorem to prove the similarity of triangles JKM and LKM, as they are not right triangles.

The Hypotenuse-Leg congruence theorem states that if the hypotenuse and one leg of two right triangles have the same measure then these two triangles are said to be congruent to each other.

The triangles JKM and LKM are not right triangles, as they do not have an angle of 90º, hence the HL congruence theorem cannot be used to prove their similarity.

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Answer:

less than or equal to N7500

Step-by-step explanation:

N2x - N5000 =  N2500

N2x = N7500

The amount each boy will have to contribute is $56.23

How to calculate the amount that each boy will contribute?

Andrew and his two brother want to contribute money together to buy a new game system

The cost of the game system is $168.69

Each boy want to contribute the same amount

Therefore the amount that each boy will contribute can be calculated as follows

There are 3 boys in total, Andrew and his two brothers

= 168.69/3

= 56.23

Hence each boy will contribute an amount of $56.23

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Answer:

X = 11

Step-by-step explanation:

The total degrees of a line is 180

They gave us 39 and 13x-2

We can get rid of the 39 by subtracting it from 180

180-39=141

That means 13x-2 = 141

Now to find X. Adding 2 on both sides, gets rid of the 2 on the left side

13x = 143

Dividing 13 on both sides, we can get X alone

X = 11

Answer:

9.5

Step-by-step explanation:

Distance (d) = [tex]\sqrt{(2-(-7.5)^2)+(-11-(-11)^2)}\\ \sqrt{(9.5)^2+(0)^2}\\ \sqrt{90.25} \\9.5[/tex]

Therefore, the 3 patterns are
3,6,9,3,6,9            

Explanation:                      

Given that,

we will carry out the multiplications of succession  counting number by 3 and if there is more than a single digit answer .we add the digits.
we are looking for is a pattern for there single digit numerals.
carryout the plan                  

1x3=3→3
2x3=6→6
3x3=9→9
4x3=12→1+2=3
5x3=15→1+5=6
and so on
continue with some additional at terms.
6x3=18→1+8=9
7x3=21→2+1=3

∵ Therefore the 3 patterns are
3,6,9,3,6,9


A pattern is defined in mathematics as a sequence of repeating objects, shapes, or numbers. A pattern can be associated with any type of event or object. A pattern has a rule that tells us which objects belong to the pattern and which objects do not belong to the pattern.


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