Prove the following using a direct proof:
The sum of the squares of 4 consecutive integers is an even integer

Answer:

Show your solution ( 3. ) C + 18 = 29

Step-by-step explanation:

To solve the equation C + 18 = 29, we want to isolate the variable C on one side of the equation.

We can start by subtracting 18 from both sides of the equation:

C + 18 - 18 = 29 - 18

Simplifying the left side of the equation:

C = 29 - 18

C = 11

Therefore, the solution to the equation C + 18 = 29 is C = 11.

(a) The function x as a function of t is t = 250ln(499x/998)

(b) Half the population will have heard the rumor approximately 109.86 units of time after it was started.

(a) To solve the differential equation dt/dx = kx(N−x), we can separate the variables and integrate

dt/dx = kx(N−x)

dt/(N-x) = kx dx

Integrating both sides, we get

t = -1/k × ln(N-x) - 1/k × ln(x) + C

where C is the constant of integration.

To find C, we can use the initial condition that two people start the rumor at t=0, so x=2:

0 = -1/k * ln(N-2) - 1/k * ln(2) + C

C = 1/k * ln(N-2) + 1/k * ln(2)

Substituting C back into the equation, we get:

t = -1/k * ln(N-x) - 1/k * ln(x) + 1/k * ln(N-2) + 1/k * ln(2)

Simplifying, we get

t = 1/k * [ln((N-2)x/(2(N-x)))]

Substituting k=1/250 and N=1000, we get:

t = 250ln(499x/998)

(b) We want to find the time t when half the population has heard the rumor, so x = N/2 = 500. Substituting this into the equation we obtained in part (a), we get

t = 250ln(499(500)/998) = 250ln(249/499)

t ≈ 109.86

Therefore, half the population will have heard the rumor approximately 109.86 units of time after it was started.

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2 gallons of blue paint are used for the walls, which cover 700 square feet.

What is surface area?

The space occupied by a two-dimensional flat surface is called the area. It is measured in square units. The area occupied by a three-dimensional object by its outer surface is called the surface area.

Let's call the surface area of the trim "T" and the surface area of the walls "W". We know that the ratio of T to W is 1:10, which means that:

T = (1/11) * W

We also know that 2 gallons of blue paint are used for the walls. Let's call the amount of white paint needed for the trim "P" (in pints).

We can use the fact that the total surface area of the room is equal to the surface area of the walls plus the surface area of the trim:

W + T = total surface area

Since T = (1/11) * W, we can substitute and simplify:

W + (1/11) * W = total surface area

(12/11) * W = total surface area

Now we can use the fact that 2 gallons of blue paint are used for the walls to find the surface area of the walls:

2 gallons = 16 pints

2 gallons = W / 350 (since 1 gallon covers 350 square feet)

W = 700 square feet

Now we can use the formula above to find the total surface area of the room:

total surface area = (12/11) * W

total surface area = (12/11) * 700

total surface area = 763.64 square feet

We know that the blue paint covers the walls, so we don't need to worry about that. We only need to find the amount of white paint needed for the trim. Let's call the amount of white paint needed per square foot of trim "p" (in pints). Then the total amount of white paint needed is:

P = p * T

We know that the ratio of the surface area between the trim and the walls is 1:10, so we can use that to find the surface area of the trim:

T = (1/11) * W

T = (1/11) * 700

T = 63.64 square feet

Now we just need to find the amount of white paint needed per square foot of trim. Since the trim is white, we don't need to worry about coverage, so we just need to find the surface area of the trim in square pints:

P = p * T

P = p * 63.64

Finally, we know that 1 gallon of paint is equal to 8 pints, so we can convert the total amount of white paint needed from pints to gallons:

P = p * 63.64

P / 8 = gallons of white paint needed

Putting it all together, we get:

2 gallons of blue paint are used for the walls, which cover 700 square feet.

The total surface area of the room is (12/11) * 700 = 763.64 square feet.

The surface area of the trim is (1/11) * 700 = 63.64 square feet.

The total amount of white paint needed is P = p * 63.64.

The amount of white paint needed in gallons is P / 8.

We don't know the value of p, so we can't solve for P directly. However, we do know that the ratio of the surface area between the trim and the walls is 1:10. This means that the surface area of the trim is 1/11 of the total surface area of the room.

Therefore, we can solve for p as follows:

T = (1/11) * W

63.64 = (1/11) * 700

p = P / T

p = P / 63.64

Hence, 2 gallons of blue paint are used for the walls, which cover 700 square feet.

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(a) Negation: C) This vertex is connected to at least one other vertex in the graph. (b) Negation: D) This number is related to at least one even number.

(a)

A) Some vertex is connected to some other vertex in the graph.

B) At least one vertex is not connected to any other vertex in the graph.

C) This vertex is connected to at least two other vertices in the graph.

D) There exists at least one vertex that is not connected to at least one other vertex in the graph.

E) This vertex is connected to some other vertices in the graph, but not necessarily to all of them.

(b)

A) This number is related to at least one odd number.

B) There exists at least one number that is not related to any even number.

C) All numbers are related to at least one even number.

D) This number is not related to at least one even number.

E) All numbers are related to at least one even number.

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Complete question:

Find a negation for each of the statements in (a) and (b).

(a) This vertex is not connected to any other vertex in the graph.

A) No vertex is connected to any other vertex in the graph.

B) All vertices are connected to all other vertices in the graph.

C) This vertex is connected to at least one other vertex in the graph.

D) All vertices are connected to at least one other vertex in the graph.

E) This vertex is connected to all other vertices in the graph.

(b) This number is not related to any even number.

A) This number is not related to any odd number.

B) All numbers are related to at least one even number.

C) All numbers are not related to any even number.

D) This number is related to at least one even number.

E) No number is related to any even number.

The value of the quadratic equation in the standard form is y = -x² -6x + 2.

y = ax² + bx + c, where a, b, and c are constants and an is not equal to 0, is a quadratic equation in standard form. A parabolic function's vertex, axis of symmetry, and intercepts with the x- and y-axes are all expressed by the quadratic equation in standard form. While the positions of the vertex and intercepts are determined by the factors b and c, the direction and form of the parabola are determined by the coefficient a. Every quadratic equation may be changed into standard form by applying the quadratic formula or the square method, which simplifies the analysis and comparison of various functions.

The standard form of the quadratic equation is given by:

y = ax² + bx + c

Substituting the values of x and y from the table we have:

For (-4, 10):

10 = a(-4)² + b(-4) + c

10 = 16a - 4b + c......(1)

For (-3, 11):

11 = a(-3)² + b(-3) + c

11 = 9a -3b + c......(2)

For (-2, 10):

10 = 4a - 2b + c .........(3)

Equation 1 can be written as follows:

10 = 16a - 4b + c

c = 10 - 16a + 4b

Substitute the value of c in equation 2 and 3:

11 = 9a -3b + c

11 = 9a - 3b + 10 - 16a + 4b

1 = - 7a + b .........(4)

And,

10 = 4a - 2b + c

10 = 4a - 2b + 10 - 16a + 4b

0 = -12a + 2b

12a = 2b

b = 6a .......(5)

Substitute the value of b in equation 4:

1 = - 7a + 6a

1 = -a

a = -1

Substitute the value of a in equation 5:

b = -6

Now, substitute the value of a and b in equation 1:

10 = 16a - 4b + c

10 = 16(-1) - 4(-6) + c

10 = -16 + 24 + c

10 = 8 + c

c = 2

Substituting the value in the quadratic equation we have:

y = -x² -6x + 2

Hence, the value of the quadratic equation in the standard form is y = -x² -6x + 2.

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The p-values for the observed values of the test statistic are 0.0094, 0.0628, 0.0228, 0.032 and 0.4404.

To calculate the p-value for each observed value of the test statistic, we need to find the area under the standard normal distribution curve to the right of each z-score. This is because the alternative hypothesis is one-tailed, with the inequality sign pointing to the right (i.e., H1: µ > µ0). Here are the steps to calculate the p-value for each observed value:

For z0 = 2.35, the area to the right of the z-score can be found using a standard normal distribution table or calculator. The area is 0.0094, which is the p-value.

For z0 = 1.53, the area to the right of the z-score is 0.0628, which is the p-value.

For z0 = 2.00, the area to the right of the z-score is 0.0228, which is the p-value.

For z0 = 1.85, the area to the right of the z-score is 0.0322, which is the p-value.

For z0 = -0.15, the area to the right of the z-score is 0.5596. However, since the alternative hypothesis is one-tailed with the inequality sign pointing to the right, we need to subtract this area from 1 to get the p-value. Therefore, the p-value is 1 - 0.5596 = 0.4404.

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Answer: 2s²(39 - 74s - 280s)(s - 2)(s + 7/2)

Step-by-step explanation:

To factor the polynomial 78s - 148s³ - 560s² completely, we can first factor out a common factor of 2s²:

2s²(39 - 74s - 280s)

Then, we can factor the quadratic expression inside the parentheses using the quadratic formula:

s = [-(-74) ± √((-74)² - 4(39)(-280))] / 2(39)

s = [74 ± √(54724)] / 78

s = [74 ± 2√13681] / 78

s = [74 ± 2×117] / 78

Therefore, the roots of the quadratic expression are:

s = 2 or s = -7/2

Substituting these values back into the factored expression, we get:

2s²(39 - 74s - 280s) = 2s²(39 - 74(2) - 280(2)) = -1240s²

2s²(39 - 74s - 280s) = 2s²(39 - 74(-7/2) - 280(-7/2)) = 2450s²

So the completely factored form of the polynomial is:

2s²(39 - 74s - 280s)(s - 2)(s + 7/2)

We'll presume that the cash in question were initially split between two accounts since we don't know the answer to question #9: the amount that has been sitting in a savings account for 60 days is $78.00.

Because the FDIC for savings accounts and the NCUA for community bank accounts guarantee all deposit made by consumers, savings are a secure location to put your money.

Savings accounts might assist you avoid overspending by keeping the money away from your spending account. You should save emergency cash in your bank account for easy access. Savings accounts keep money secure because the Deposit Insurance Corporation of the United States insures them for up to $250,000.

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Answer:

[tex]\textsf{To\;add\;(or subtract)\;in\;Scientific\;Notation,\;you\;must\;have\;the\;same\;$\boxed{\sf power\;of\;10}$\:.}\\\textsf{Then\;you\;can\;$\boxed{\sf add\;or\;subtract}$\;the\;coefficients\;and\;$\boxed{\sf keep}$\;the\;power\;of\;10.}[/tex]

[tex]\textsf{To\;multiply\;in\;Scientific\;Notation,\; you\;must\;$\boxed{\sf multiply}$\;the\;coefficients}\\\textsf{and\;$\boxed{\sf add}$\;the\;powers\;of\;10.}[/tex]

[tex]\textsf{To\;divide\;in\;Scientific\;notation,\;you\;must\;$\boxed{\sf divide}$\;the\;coefficients}\\\textsf{and\;$\boxed{\sf subtract}$\;the\;powers\;of\;10.}[/tex]

Step-by-step explanation:

To add (or subtract) in Scientific Notation, you must have the same power of 10. Then you can add or subtract the coefficients and keep the power of 10.

Example expression:

[tex]2.3 \times 10^3 +3.2 \times 10^3[/tex]

Factor out the common term 10³:

[tex]\implies (2.3 +3.2) \times 10^3[/tex]

Add the numbers:

[tex]\implies (5.5) \times 10^3[/tex]

[tex]\implies 5.5\times 10^3[/tex]

Therefore, we have added the coefficients and kept the power of 10.

[tex]\hrulefill[/tex]

To multiply in Scientific Notation, you must multiply the coefficients and add the powers of 10.

Example expression:

[tex]2.3 \times 10^3 \times 3.2 \times 10^3[/tex]

Collect like terms:

[tex]\implies 2.3 \times 3.2 \times 10^3 \times 10^3[/tex]

Multiply the numbers (coefficients):

[tex]\implies 7.36 \times 10^3 \times 10^3[/tex]

[tex]\textsf{Apply the exponent rule:} \quad a^b \cdot a^c=a^{b+c}[/tex]

[tex]\implies 7.36 \times 10^{(3+3)}[/tex]

[tex]\implies 7.36 \times 10^{6}[/tex]

Therefore, we have multiplied the coefficients and added the powers of 10.

[tex]\hrulefill[/tex]

To divide in Scientific notation, you must divide the coefficients and subtract the powers of 10.

Example expression:

[tex]\dfrac{8.6 \times 10^6}{2.15 \times 10^2}[/tex]

Collect like terms:

[tex]\implies \dfrac{8.6}{2.15} \times \dfrac{10^6 }{10^2}[/tex]

Divide the numbers (coefficients):

[tex]\implies 4\times \dfrac{10^6 }{10^2}[/tex]

[tex]\textsf{Apply the exponent rule:} \quad \dfrac{a^b}{a^c}=a^{b-c}[/tex]

[tex]\implies 4\times 10^{(6-2)}[/tex]

[tex]\implies \implies 4\times 10^{4}[/tex]

Therefore, we have divided the coefficients and subtracted the powers of 10.

Answer: 39

39 is the only answer option greater than 17

The values of a continuous function f are given which are false is  By the Mean Value Theorem applied to f on the interval (2,5), there is a value c such that f' (c) = 10. So, the correct option is statement (B). Let f be the function defined by f (x) = r - 6x2 + 9x + 4 for 0 < 3 < 3 then f is decreasing on the interval (1,3) because f' (2) < 0 on the interval (1, 3). So, the correct option is D).

For continuous function f the statement (B) is false. Although the Mean Value Theorem guarantees the existence of a point c such that f'(c) = (f(5)-f(2))/(5-2) = 2, there is no guarantee that this value will be exactly 10.

When f (x) = r - 6x2 + 9x + 4 for 0 < 3 < 3 is statement (D) is true. We have f'(x) = -12x + 9, which is negative for x in the interval (1,3). Therefore, f is decreasing on this interval. Statement (A) is false, as f'(2) = 3 is positive, so f is increasing on the interval (0,1).

Statement (B) is also false, as f'(x) is not negative on the interval (0,1). Statement (C) is false, as f" (x) = -12 is negative everywhere, so f is concave down on the entire interval (0,3).

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The length of h in the attached composite figure where all the angles are right angles is equal to 4m.

In the attached diagram of composite figure,

All are right angles.

Composite figure consist two rectangles,

Upper and lower rectangles.

length of the upper rectangle is equal to 5m

Width of the upper rectangle is equal to 'h' m

Width of the lower rectangle is equal to 2m

Length of each dash '-' mark  is equals to 1m.

length of 'h'm  is equals

= 2 m + 2 dash marks

= 2m + 2m

= 4m

Therefore, the length of the h in the composite figure ( attached diagram ) is equals to 4m.

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The above question is incomplete, the complete question is:

What is the length of h in the following composite figure? All angles are right angles.

5 m

4 m

3 m

2 m

Diagram is attached.

Answer:

4 m is ur answer

Step-by-step explanation:

hope this helps

For part A the 90% confidence interval for the percentage of deer carrying ticks that test positive for Lyme disease is (0.106, 0.214), or 10.6% to 21.4% (rounded to one decimal place). And for part b cut the margin of error in half, we need to quadruple the sample size.

How to solve?

a) To create a 90% confidence interval for the percentage of deer carrying ticks that test positive for Lyme disease, we can use the following formula:

CI = p ± z×(√(p×(1-p)/n))

where:

p is the sample proportion of deer carrying ticks that test positive for Lyme disease (p = 23/144 = 0.16)

z× is the critical value for a 90% confidence level, which is approximately 1.645 (from a standard normal distribution table)

n is the sample size (n = 144)

Substituting these values into the formula, we get:

CI = 0.16 ± 1.645×(√(0.16×(1-0.16)/144))

CI = 0.16 ± 0.054

Therefore, the 90% confidence interval for the percentage of deer carrying ticks that test positive for Lyme disease is (0.106, 0.214), or 10.6% to 21.4% (rounded to one decimal place).

b) To cut the margin of error in half, we need to quadruple the sample size. Since the original sample size was 144, we need to inspect 4×144 = 576 deer.

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The end balance is negative, so Mr. Manuel lost money.

We know that Mr. Manuel spended $800,000 in a product, and it can be sold with an extra 60% over the cost. Then the revenue here is:

$800,000*(1.6) = $1,280,000

We also know that there are costs of $75,680, $247.000 and $185.460.

Now we know that profit is defined as the difference between the revenue and the costs, so to get the profit we need to solve the equation below:

P = $1,280,000 - $800,000 - $75,680 - $247.000 - $185.460

P = -$28,140

So we can see that Mr. Manuel had a loss at the end.

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100 times the square root of 3 is also a possible value of a for this function.

In mathematics, the square root of a non-negative real number "a" is a non-negative real number that, when multiplied by itself, gives the original number "a". It is denoted by the symbol "√".

We can substitute each of the given values into the function f(a) = a² + 7 to determine if they are possible values of a.

Substituting the square root of 5:

f(√(5)) = (√(5))² + 7 = 5 + 7 = 12

So, the square root of 5 is not a possible value of a for this function.

Substituting the square root of 7:

f(√(7)) = (√(7))² + 7 = 7 + 7 = 14

So, the square root of 7 is a possible value of a for this function.

Substituting 100 times the square root of 3:

f(100√(3)) = (100√(3))² + 7 = 30000 + 7 = 30007

So, 100 times the square root of 3 is also a possible value of a for this function.

Therefore, the possible values of a for the given function are:

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0-24-  Tally (1)

25-49   Tally (4)

50-74  Tally (5)

75-99  Tally (2)

The taylor series (centered at c) for the function f(x) = 1/x, c = 1 is f(x) = 1 - (x-1) - (x-1)^2 + (x-1)^3 + ...

The Taylor series is a representation of a function as an infinite sum of terms that involve the function's derivatives evaluated at a particular point. The Taylor series centered at a point c for a function f(x) is given by:

f(x) = f(c) + f'(c)(x-c)/1! + f''(c)(x-c)^2/2! + f'''(c)(x-c)^3/3! + ...

In this case, we want to find the Taylor series centered at c=1 for the function f(x) = 1/x. We can start by finding the derivatives of f(x):

f'(x) = -1/x^2

f''(x) = 2/x^3

f'''(x) = -6/x^4

f''''(x) = 24/x^5

We can then evaluate these derivatives at c=1 to get:

f(1) = 1/1 = 1

f'(1) = -1/1^2 = -1

f''(1) = 2/1^3 = 2

f'''(1) = -6/1^4 = -6

f''''(1) = 24/1^5 = 24

Substituting these values into the Taylor series formula, we get:

f(x) = 1 - (x-1) - (x-1)^2 + (x-1)^3 + ...

This is the Taylor series centered at c=1 for the function f(x) = 1/x. It represents an approximation of the function in the neighborhood of x=1. By adding more terms to the series, we can improve the accuracy of the approximation.

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Complete question is:

Use the definition of Taylor series to find the taylor series (centered at c) for the function. f(x) = 1/x, c = 1.

Answer:

[tex]f(7.3)\approx9.9[/tex]

Step-by-step explanation:

Use point-slope form

[tex]y-y_1=m(x-x_1)\\y-9=3(x-7)\\y-9=3x-21\\y=3x-12[/tex]

[tex]f(7.3)=3(7.3)-12=21.9-12=9.9[/tex]

Answer:

{y,x}={-1,1}

to leave and take

Solving a system of equations we the length and width of the playground is length = 9 yards, width = 12 yards

Remember that the area of a rectangle of length L and width W is:

Area = L*W

Here we know that the area is 108 square yards, and we know that he width is 3 yards longer than the length, then we can write a system of equations:

W =L + 3

108 = L*W

Replacing the first equation into the second one we will get:

108 = (L + 3)*L

108 = L² + 3L

Then we have the quadratic equation:

L² + 3L - 108 = 0

Using the quadratic formula we get the solutions:

[tex]L = \frac{-3 \pm \sqrt{3^2 - 4*1*-18} }{2}[/tex]

We only care for the positive solution, which is:

L = 9

Then the width is:

W = L + 3 = 9 + 3 = 12

Then the correct option is:

•length = 9 yards, width = 12 yards

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Answer:

Step-by-step explanation:

subtract 434-46

Step-by-step explanation:

Original job = x men * 40 days = 40x man days  to complete

  now add 8 men     =    x+8  men

                                           man days now is   (x+8) (30)  to complete job

so     40x = (x+8)(30)

        40x = 30x + 240

          10 x = 240

              x = 24 men originally

Answer: In the table x part, it increases from -2 all the way to 4. In the table y part, it decreases from 17 to -1, but then increases back from -1 to 17.

The state in question is using a five-character license plate system, with each character having 36 possible combinations. Multiplying the possible combinations of each character gives us a total of 60,466,176 possible license plates.

Multiplication is an iterative process of addition where the multiplier is the quantity of times the multiplicand is added to itself. When a number is multiplied, it is multiplied by itself a predetermined amount of times.

This implies that each license plate will have five distinct characters, each of which can be any of the following: the 26 characters of the alphabet, the digits 1, 2, 3, or 4, or the numbers 0 through 9. It provides us with a total of 5 characters, each of which has 36 different potential combinations (4 digits + 26 letters + 10 digits).

The number of character combinations is multiplied to determine the total number of potential license plates. In this instance, the result is 36 times itself.

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270,400 license plates are possible from the combinations of each character given in the question.

Combinations are used to calculate the number of ways a certain number of items can be selected from a given set of items.

To calculate the total possible license plates in the state, we need to consider the total number of possible combinations of the five characters.

For the first character, there are four possible digits (1, 2, 3, or 4).

For the second character, there are 26 letters of the alphabet. (A-Z)

For the third character, there are again 26 letters of the alphabet.

For the fourth character, there are 10 possible digits (0 through 9).

For the fifth character, there are again 10 possible digits.

We can calculate the number of possible license plates by multiplying the number of possibilities for each character.

4 x 26 x 26 x 10 x 10 = 270,400 possible license plates.

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Answer:

x = 37.5

Step-by-step explanation:

the top triangle has 2 congruent sides and is therefore isosceles with base angles being congruent, then

base angles = (180 - 75) ÷ 2 = 105 ÷ 2 = 52.5

the angle on the left of the outer triangle is right , then

x + 52.5 = 90 ( subtract 52.5 from both sides )

x = 37.5

this is just a quick addition to the superb reply above by "jimrgrant1"

Check the picture below.

Answer:

The value of the derivative at (2, 3) is zero.

Step-by-step explanation:

Given function:

[tex]g(x)=x+\dfrac{4}{x^2}[/tex]

To differentiate the given function, use the power rule of differentiation.

[tex]\boxed{\begin{minipage}{5.4 cm}\underline{Power Rule of Differentiation}\\\\If $y=x^n$, then $\dfrac{\text{d}y}{\text{d}x}=nx^{n-1}$\\\end{minipage}}[/tex]

[tex]\textsf{Rewrite\;the\;function\;using\;the\;exponent\;rule\;\;$a^{-n}=\dfrac{1}{a^n}$}:[/tex]

[tex]\implies g(x)=x+4x^{-2}[/tex]

Apply the power rule:

[tex]\implies g'(x)=1+(-2) \cdot 4x^{-2-1}[/tex]

[tex]\implies g'(x)=1-8x^{-3}[/tex]

[tex]\implies g'(x)=1-\dfrac{8}{x^3}[/tex]

An extremum is a point where a function has a maximum or minimum value. From inspection of the given graph, the minimum point of the function is (2, 3).

To determine the value of the derivative at the minimum point, substitute x = 2 into the differentiated function.

[tex]\begin{aligned}\implies g'(2)&=1-\dfrac{8}{2^3}\\\\&=1-\dfrac{8}{8}\\\\&=1-1\\\\&=0\end{aligned}[/tex]

Therefore, the value of the derivative at (2, 3) is zero.

Answer:

(2u - 1) (3u + 10)

Step-by-step explanation:

Let's Check

(2u - 1) (3u + 10)

6u² + 20u - 3u + 10

6u² + 17u + 10

So, (2u - 1) (3u + 10) is the correct answer.

Paul practiced playing the piano for one hour on Friday and for the same amount of time on Saturday. Therefore, he practiced for 1 hour on both days.

Time is a concept used to measure the duration or sequence of events, actions or processes, and to organize them into a coherent and meaningful structure. It is a fundamental aspect of the physical universe and an essential element of human experience, enabling us to make sense of our environment and our lives.

The measurement of time is typically based on the movement of objects or the cycles of natural phenomena, such as the rotation of the Earth on its axis, the orbit of the Moon around the Earth, or the vibrations of an atomic oscillator. Time is commonly expressed in units such as seconds, minutes, hours, days, weeks, months, and years.

In addition to its scientific and practical applications, time also plays an important role in culture, language, and philosophy. It has been the subject of extensive debate and speculation throughout history, with questions about its nature, meaning, and relationship to other concepts such as causality, free will, and eternity.

Paul practiced playing the piano for one hour on Friday and for the same amount of time on Saturday. Therefore, he practiced for 1 hour on both days.

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Paul practiced playing the piano for one hour on Friday and for the same amount of time on Saturday. Therefore, he practiced for 1 hour on both days.

Time is a concept used to measure the duration or sequence of events, actions or processes, and to organize them into a coherent and meaningful structure. It is a fundamental aspect of the physical universe and an essential element of human experience, enabling us to make sense of our environment and our lives.

The measurement of time is typically based on the movement of objects or the cycles of natural phenomena, such as the rotation of the Earth on its axis, the orbit of the Moon around the Earth, or the vibrations of an atomic oscillator. Time is commonly expressed in units such as seconds, minutes, hours, days, weeks, months, and years.

In addition to its scientific and practical applications, time also plays an important role in culture, language, and philosophy. It has been the subject of extensive debate and speculation throughout history, with questions about its nature, meaning, and relationship to other concepts such as causality, free will, and eternity.

Paul practiced playing the piano for one hour on Friday and for the same amount of time on Saturday. Therefore, he practiced for 1 hour on both days.

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Using the properties of  absolute value function, proved that |x - y| > |x| - |y| is true for all x and y.

To prove that |x - y| > |x| - |y|, we can consider two cases

Case 1

x >= 0 and y >= 0

In this case, |x - y| = x - y and |x| - |y| = x - y. So we have

|x - y| = x - y

| x | - | y | = x - y

Substituting these expressions into the original inequality, we get:

x - y > x - y

This inequality is true for all x and y where x >= 0 and y >= 0, since the difference between x and y is always greater than or equal to zero.

Case 2

x < 0 and y < 0

In this case, |x - y| = -(x - y) and |x| - |y| = -x + y. So we have:

|x - y| = -(x - y)

| x | - | y | = -x + y

Substituting these expressions into the original inequality, we get

-(x - y) > -x + y

Simplifying both sides, we get

y - x > -x + y

Adding x to both sides, we get

y > 0

This inequality is true for all x and y where x < 0 and y < 0, since both x and y are negative and the difference between x and y is always less than or equal to zero.

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