Find the average rate of change of the function over the given intervals.
f(x)=12x3 + 12: a) [5,7], b)[-4,4]
a) The average rate of change of the function f(x)= 12X +12 over the interval [5,7] is (Simplify your answer.) b) The average rate of change of the function f(x)=12x3 + 12 over the interval [-4A] is (Simplify your answer.)

S is not a basis for R3 because: c. S is linearly dependent and does not span R3.

A basis for a vector space is a set of vectors that are linearly independent and span the space.

Let's check if S = {(1, 1, 1), (0,1,1), (1,0,1), (0, 0, 0)} satisfies this definition of the basis or not.

Here, we have to find if the set is linearly dependent or not. If the set is linearly dependent, then it can't be a basis. To check the linearity of the set, we will set the equation for the linear combination of vectors of set S.

Then α1(1, 1, 1) + α2(0,1,1) + α3(1,0,1) + α4(0, 0, 0) = 0

For the above equation to hold true, α1 + α3 = 0α1 + α2 = 0α1 + α3 = 0

By solving the above equations, we get

α1 = -α3α2 = -α1α4 = 0

The above equation shows that α4 = 0 means there is always a trivial solution which makes it linearly dependent.

Therefore, it is not a basis.

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The answer of the given question based on Algebric notation is when x=4 and y=2, the value of 3x + 5y is 22.

An expression is combination of numbers, variables, and mathematical operations like addition, subtraction, multiplication, division, exponentiation, and others. Expressions can be simple or complex, and they can be used to represent a wide range of mathematical concepts, such as equations, inequalities, functions, and formulas. For example, "3x + 5y" is an expression that involves the variables x and y, and the operations of multiplication and addition.

In algebra, expressions are often used to represent relationships between variables, and to solve problems by manipulating the expressions to obtain a desired result.

To find the value of 3x + 5y when x=4 and y=2, we simply substitute these values into the expression and evaluate:

3x + 5y = 3(4) + 5(2) = 12 + 10 = 22

Therefore, when x=4 and y=2, the value of 3x + 5y is 22.

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Find 3x + 5y when x=4 and y=2 (algebric notation)?

The information makes sense in different scenarios like manufacturing, baking, researching survey, etc.

A sample is a portion of a population that is used to represent the full population in statistics. Researchers frequently utilise samples to draw conclusions about the population since it might be difficult or impossible to collect data from the complete population. In order for the conclusions made from the sample to be applicable to the entire population, it is important to pick a sample that is representative of the wider population. Many techniques, such as cluster sampling, stratified sampling, and random sampling, can be used to choose samples.

The knowledge of biscuit preferences may come in handy in a variety of situations. Here are three potential instances:

A biscuit manufacturer wants to launch a brand-new taste. They may opt to create a chocolate chip or peanut butter flavor as those are the most popular ones based on the preferences data.

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Complete question -

A national sampling of cookie preferences showed that 75% of people like chocolate chip, 50% like peanut butter, and only 3% like coconut. The use of this information makes sense in which of the following scenarios? A. The school cafeteria decides to make 3 coconut cookies for each 100 students who buy lunch, even though it puts them over budget for desserts. B. A national cookie company is thinking of changing the recipe in their peanut butter cookies. C. A national cookie company decides to spend $5 million in advertising to convince people to eat coconut cookies. D. Tom is about to open a small bakery and is using the result of the sampling to decide what kind of cookies to offer.

For expression a(x+2) = b(x-1), x will be (-b - 2a)/(a - b).

What exactly are expressions?

In mathematics, an expression is a combination of numbers, variables, operators, and/or functions that can be evaluated to produce a numerical value. Expressions can range from simple calculations involving only arithmetic operations such as addition, subtraction, multiplication, and division, to more complex combinations of algebraic expressions and functions.

Now,

To solve for x in the equation a(x+2) = b(x-1), we can use algebraic manipulation:

a(x+2) = b(x-1)

ax + 2a = bx - b [distribute the terms]

ax - bx = -b - 2a [subtract bx and 2a from both sides]

x(a - b) = -b - 2a [factor out x]

x = (-b - 2a)/(a - b)

Therefore,

                 x = (-b - 2a)/(a - b).

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Answer:

C

Step-by-step explanation:

the nth term of a geometric sequence is

[tex]a_{n}[/tex] = a₁ [tex]r^{n-1}[/tex]

where a₁ is the first term and r the common ratio

here a₁ = - 1 and r = [tex]\frac{a_{2} }{a_{1} }[/tex] = [tex]\frac{-5}{-1}[/tex] = 5

then nth term is

[tex]a_{n}[/tex] = - 1 [tex](5)^{n-1}[/tex]

The passage and Aylmer's dialogue describe his obsession with perfection and his attempts to remove a birthmark from his wife's face.

Aylmer's language is formal and scientific, using technical terms and precise language to describe his work. This enhances the meaning of the passage by highlighting Aylmer's belief in the power of science and his desire to control nature.

For example, Aylmer says, “Georgiana, I have spent much thought upon the subject,” and “I feel myself fully competent to render this dear cheek as faultless as its fellow.”

This language conveys Aylmer's confidence in his abilities and his belief that he can achieve perfection through scientific means. The precise language also reinforces the idea that Aylmer sees his wife's birthmark as a flaw that needs to be eradicated.

Therefore, the diction in Aylmer's dialogue enhances the meaning of Passage 2 by highlighting the theme of obsession with perfection and the power of science to control nature.

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The rate of change of y when x = 1 is 0.005 unit/sec.

To find the rate of change of y when x = 1 and given that x increases at a rate of 0.02 unit/sec, we'll first find the derivative of the equation y = (2x - 5)/(x + 1) with respect to x, and then multiply it by the rate of change of x (0.02 unit/sec). Differentiate the equation with respect to x:

y = (2x - 5)/(x + 1)

To differentiate this function, we'll use the quotient rule: [tex](u'/v) - (uv')/v^2[/tex], where u = (2x - 5) and v = (x + 1).

u' = derivative of (2x - 5) = 2

v' = derivative of (x + 1) = 1

Now, apply the quotient rule:

dy/dx =[tex](2\times(x + 1) - (2x - 5)\times1) / (x + 1)^2[/tex]

Plug in x = 1:

dy/dx =[tex](2\times(1 + 1) - (2\times1 - 5)\times1) / (1 + 1)^2[/tex]

dy/dx = (4 - 3) / 4

dy/dx = 1/4

Multiply the derivative by the rate of change of x:

The rate of change of x is 0.02 unit/sec, so the rate of change of y when x = 1 is:

Rate of change of y = (dy/dx) × (rate of change of x)

Rate of change of y = (1/4) × 0.02 = 0.005 unit/sec

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mAD is 74 ° .we can get this answer by using properties of chord and also using fact that  the sum of angles in a quadrilateral is 360 degrees

what is properties of chord?

A chord is a straight line segment that joins two points on the circumference of a circle. Here are some properties of chords

1)The length of a chord is less than the diameter of the circle.

2) A chord that passes through the center of the circle is a diameter, which is the longest chord in a circle.

In the given question,

angle CAD = 180 - angle ADC - angle DCA

= 180 - 102 - 62

= 16 degrees

Now, we can use the fact that angles that subtend the same arc are equal. In particular, angles CAD and ABD subtend the same arc AD, so we have:

angle ABD = angle CAD = 16 degrees

Similarly, angles ADB and ACB subtend the same arc AB, so we have:

angle ACB = angle ADB

Now, we can use the fact that the sum of angles in a quadrilateral is 360 degrees. In particular, we have:

angle ABD + angle BAC + angle CAD + angle ACB = 360

Substituting the values we know, we get:

16 + angle BAC + 16 + angle ADB = 360

Simplifying, we get:

angle BAC + angle ADB = 328

But we also know that angle BAC + angle ADB + angle BAD = 180 degrees (since they form a triangle). Substituting the value we just found, we get:

328 + angle BAD = 180

Solving for angle BAD, we get:

angle BAD = 180 - 328 = -148 degrees

Wait, a negative angle? What's going on here? Well, it turns out that angles are usually measured in a counterclockwise direction, so if we go clockwise instead, we can end up with negative angles. In this case, we can think of angle BAD as a clockwise rotation from DB to BA, which is why it's negative.

But we can fix this by adding 360 degrees to angle BAD, which will bring it back into the range of 0 to 360 degrees. So:

angle BAD = -148 + 360 = 212 degrees

Now, we can use the fact that angles that subtend the same arc are equal again. In particular, angles ABD and ACD subtend the same arc AD, so we have:

angle ABD = angle ACD

Substituting the values we know, we get:

16 = angle ACD

Finally, we can use the fact that the sum of angles in a triangle is 180 degrees again. In particular, we have:

angle ADC + angle ACD + angle CDA = 180

Substituting the values we know, we get:

102 + 16 + angle CDA = 180

Simplifying, we get:

angle CDA = 62 degrees

But we also know that angles ACD and ADB subtend the same arc AD, so they're equal. Therefore:

angle ADB = angle ACD = 16 degrees

Now we can use the fact that the sum of angles in a triangle is 180 degrees again to find mAD:

angle ADB + angle ABD + angle BAD = 180

Substituting the values we know, we get:

16 + 16 + 212 = 180

Simplifying, we get:

mAD = (180 - 32) / 2 = 74 degrees

Therefore, mAD is 74 degrees

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Answer:

- 5 , - 3 , 3

Step-by-step explanation:

to find the first three terms, substitute n = 1, 2, 3 into the explicit formula

[tex]T_{n}[/tex] = 2n² - 4n - 3

T₁ = 2(1)² - 4(1) - 3

    = 2(1) - 4 - 3

    = 2 - 7

    = - 5

T₂ = 2(2)² - 4(2) - 3

    = 2(4) - 8 - 3

    = 8 - 11

    = - 3

T₃ = 2(3)² - 4(3) - 3

    = 2(9) - 12 - 3

    = 18 - 15

    = 3

the first three terms are - 5, - 3 , 3

The probability of selecting a pen from the first box is 1/8 (since there is only 1 pen out of 8 items in the box). The probability of selecting a crayon from the second box is 3/6 (since there are 3 crayons out of 6 items in the box).

To find the probability of both events happening together, we multiply the probabilities:

P(pen and crayon) = P(pen) x P(crayon)

P(pen and crayon) = (1/8) x (3/6)

Simplifying the fraction 3/6 to 1/2:

P(pen and crayon) = (1/8) x (1/2)

Multiplying the numerators and denominators:

P(pen and crayon) = 1/16

Therefore, the probability of selecting a pen from the first box and a crayon from the second box is 1/16.

580.017 is equivalent to 580 tenths and 17 thousandths.

What is Algebraic expression ?

An algebraic expression is a mathematical phrase that can contain variables, constants, and mathematical operations such as addition, subtraction, multiplication, division, and exponentiation. Algebraic expressions are used to represent and solve problems in many areas of mathematics, science, engineering, and finance.

The number that is equal to 580 tenths and 17 thousandths is 580.017.

To understand this, we need to understand the place value system used in decimal notation. Each digit in a decimal number has a place value that is a power of 10. The rightmost digit represents the ones place, the next digit to the left represents the tens place, and so on.

In the number 580.017, the digit in the ones place is 0, the digit in the tens place is 1, the digit in the hundreds place is 7, the digit in the thousands place is 0, the digit in the ten-thousands place is 8, and the digit in the hundred-thousands place is 5.

Thus, the number 580.017 can be written as the sum of the digits multiplied by their corresponding place values:

580.017 = 5 x 100,000 + 8 x 10,000 + 0 x 1,000 + 0 x 100 + 1 x 0.1 + 7 x 0.01

Simplifying this expression, we get:

580.017 = 500,000 + 80,000 + 0 + 0 + 0.1 + 0.17

580.017 = 580,000 + 0.017

Therefore, 580.017 is equivalent to 580 tenths and 17 thousandths.

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To solve the inequality -6x < 36, we need to isolate x on one side of the inequality sign. We can do this by dividing both sides of the inequality by -6, remembering to reverse the inequality sign because we are dividing by a negative number:

-6x < 36

x > -6

Therefore, the solution set for the inequality is {x | x > -6}.

A sentence to explain how I know this is not a perpendicular bisector is that line AB was not divided into two equal halves.

A possible mistake which Aaron made in his construction is: C. he did not use a pair of compasses to draw his arcs.

In Mathematics and Geometry, a perpendicular bisector can be defined as a type of line that bisects or divides a line segment exactly into two (2) equal halves and forms an angle that has a magnitude of 90 degrees at the point of intersection.

This ultimately implies that, the steps taken by Aaron for the construction of the perpendicular bisector for line AB is incorrect because it was not divided into two equal halves, even though it formed an angle that has a magnitude of 90 degrees at the point of intersection.

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the rate of discount is double the rate of VAT. The selling price of the article without VAT is Rs. [tex]7,680.[/tex]

Let's assume that the original selling price of the article is Rs. [tex]8,000.[/tex]

Let the rate of discount be x and the rate of VAT be y.

We know that the selling price of the article with discount and VAT is Rs. [tex]6,689.60.[/tex]

So, we can write the following equation:

Selling price = (Original price - Discount) + VAT

[tex]Rs. 6,689.60 = (Rs. 8,000 - Rs. 8,000 x (x/100)) + (Rs. 8,000 - Rs. 8,000 x (x/100)) x (y/100)[/tex]

Simplifying the equation, we get:

[tex]Rs. 6,689.60 = Rs. 8,000 (1 - x/100 + y/100 - xy/10000)[/tex]

Now, we are given that the rate of discount is double the rate of VAT. So, we can write:

[tex]x = 2y[/tex]

Substituting this value in the equation above, we get:

[tex]Rs. 6,689.60 = Rs. 8,000 (1 - 3y/100 + 2y^2/10000)[/tex]

Simplifying this equation further, we get a quadratic equation in y:

[tex]2y^2 - 3y + 0.41505 = 0[/tex]

Solving this quadratic equation using the quadratic formula, we get:

[tex]y = 10.5 or y = 0.01988[/tex]

Since the rate of VAT cannot be 10.5%, we will take y = 0.01988 (approx. 2%).

Now, we can calculate the rate of discount using x = 2y:

x = 4%

So, the selling price of the article without VAT can be calculated as follows:

Selling price without VAT = Original price - Discount

[tex]= Rs. 8,000 - Rs. 8,000 x\times (4/100)[/tex]

[tex]= Rs. 7,680[/tex]

Therefore, the selling price of the article without VAT is Rs. [tex]7,680.[/tex]

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Answer:

$1529

Step-by-step explanation:

The formula for continuous compounding is:

A = Pe^(rt)

Where:

A = the final amount (balance) in the account

P = the initial principal (deposit)

e = Euler's number (approximately 2.71828)

r = the annual interest rate (as a decimal)

t = the time period (in years)

Plugging in the given values, we get:

A = 1000e^(0.0855)

A = 1000*e^0.425

A = 1000*1.529

A = 1529

Therefore, the balance after 5 years with continuous compounding at 8.5% interest rate is $1529.

Answer: 21

In order to find the answer you have to multiply FJ and JH which is equal to 6x+24.

After, multiply GJ and JE which is equal to 90.

After doing all these steps equal your findings:

6x+24= 90

x= 11

The length of FH is asked so, you have found x, which is equal to 11

FH= 6+(11+4)= 21

The statement "a) the mean of the sampling distribution of sample mean is always the same as that of x, the distribution from which the sample is taken" is not necessarily true.

While the mean of the sampling distribution of the sample mean is an unbiased estimator of the population mean, it is not always equal to the mean of the original distribution from which the sample is taken. This is because the sample mean will vary from sample to sample, and the sampling distribution will reflect this variability.

For example, if the population distribution is skewed or has outliers, the sampling distribution of the sample mean may be more symmetrical and have a different mean. However, as the sample size increases, the sampling distribution of the sample mean will approach a normal distribution with a mean equal to the population mean. This is known as the Central Limit Theorem.

Therefore, the incorrect statement is a).

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Answer:

1

Step-by-step explanation:

One to any exponent is just 1 so this fraction simplifies to 1/1 or just simply 1

Answer:

1 or 1 / 1.

Step-by-step explanation:

Square the numbers:

1^22 / 1^7

Any number squared with 1 is always 1:

1 / 1

= 1 or 1 / 1

Answer: The variables h and k in the general formula for a cube root function affect the graph by causing a horizontal and vertical shift, respectively. The value of h determines the horizontal position of the graph by shifting it to the right or left along the x-axis. If h is positive, the graph will shift to the right, and if h is negative, the graph will shift to the left. On the other hand, the value of k determines the vertical position of the graph by shifting it up or down along the y-axis. If k is positive, the graph will shift upward, and if k is negative, the graph will shift downward. It is important to note that these shifts do not affect the shape of the graph, only its position on the coordinate plane. Therefore, changing the values of h and k in the general formula for a cube root function will cause the graph to shift, but the overall shape of the graph will remain the same.

Step-by-step explanation:

The probability that a randomly selected student studies Hindi or Maths is 0.60 or 60%.

We can use the formula P(A or B) = P(A) + P(B) - P(A and B) to find the probability that a randomly selected student studies Hindi or Maths is calculated by adding the probabilities of studying Hindi and Maths and subtracting the probability of studying both Hindi and Maths.

P(Hindi or Maths) = P(Hindi) + P(Maths) - P(Hindi and Maths)

P(Hindi or Maths) = 0.30 + 0.45 - 0.15

P(Hindi or Maths) = 0.60

Therefore, if a student is selected randomly, then the probability that he/she studies Maths or Hindi is 0.60 or 60%.

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Answer:

273/500

Step-by-step explanation:

54.6% can be changed into a fraction by making it over 100.
This gives us 54.6/100
However, you can change it by making it into an integer by multiplying both sides by 10.
This gives us 546/1000
You can then go on by simplifying this by dividing both sides by 2

273/500.

Hope this helped,
No problem

Approximately 40.4 defective items in a sample of 101 items from this manufacturing machine with a 40% defect rate. The correct symbol for this random variable is X.

For this specific question, the random variable can be described as the number of defective items in a sample of 101

items taken from a manufacturing machine with a 40% defect rate. The correct wording for the random variable in this

case is "the number of defective items in a sample of 101 items."

The correct symbol for this random variable is X, which is often used to represent an unknown variable. Therefore, the

random variable can be written as X = the number of defective items in a sample of 101 items.

To find the expected number of defective items in this sample, we can use the formula for the expected value of a

discrete random variable: E(X) = np, where n is the sample size and p is the probability of the event of interest (in this

case, the probability of an item being defective).

Using this formula, we can calculate the expected number of defective items in the sample as:

E(X) = (101)(0.4) = 40.4

Therefore, we would expect to find approximately 40.4 defective items in a sample of 101 items from this manufacturing

machine with a 40% defect rate.

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The answer is 2.0625

Answer:

33/16=2.0625

Step-by-step explanation:

33/16=2 rounded to the nearest integer

33/16=2.1 rounded to 1 decimal place

33/16=2.06 rounded to 2 decimal places

33/16=2.063 rounded to 3 decimal places

33/16=2.0625 rounded to 4 decimal places

The coordinates of the image based on the transformation rule (x, y) → (x - 2, y) include the following:

A' (2, -4).

B' (3, -4).

C' (3, -1).

D' (2, -1).

In Mathematics, the translation of a graph to the left is a type of transformation that simply means subtracting a digit from the value on the x-coordinate of the pre-image while the translation of a graph to the right is a type of transformation that simply means adding a digit to the value on the x-coordinate of the pre-image.

By translating the pre-image of quadrilateral ABCD horizontally right by 4 units, the coordinates of quadrilateral ABCD include the following:

(x, y)                               →                  (x - 2, y)

A (4, -4)                        →                  (4 - 2, -4) = A' (2, -4).

B (5, -4)                        →                  (5 - 2, -4) = B' (3, -4).

C (5, -1)                        →                  (5 - 2, -1) = C' (3, -1).

D (4, -1)                        →                  (4 - 2, -1) = D' (2, -1).

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The angles that form a linear pair is 6 and 8.

option C.

Linear pair angles are two adjacent angles formed by two intersecting lines. These two angles are always supplementary, which means that their sum is equal to 180 degrees. In other words, if we add the measure of one angle to the measure of the other angle, we will get a total of 180 degrees.

A linear pair of angles can be easily identified by observing the angles formed when two straight lines intersect each other. For example, if line AB intersects line CD at point E, then the two pairs of adjacent angles formed are:

Each pair of adjacent angles in a linear pair is supplementary, which means that ∠AED + ∠DEB = 180 degrees and ∠CEB + ∠BED = 180 degrees.

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As a result, the monument's shadow is approximately 28.9 metres long as the monument's height is on the other side .

The length of an object or the separation of two points is a measurement of its size in one dimension. It is an important physical quantity that is employed in many disciplines, including physics, technology, and daily life. Depending on the situation, length is typically expressed in quantities like metres (m), centimetres (cm), feet (ft), inch (in), and miles (mi).

given

Assume that the monument's shadow extends "x" metres in length.

Because we are aware of the monument's height and the sun's elevation angle, we can use the tangent function to get "x":

tan(60) = adjacent/opposite

where the monument's height is on the other side, and the shadow's length is right next to it.

Adding the values we obtain:

tan(60) = 50/x

To solve for "x," we obtain:

x = 50/tan(60) (60)

(Using the value of the tangent of 60 degrees, which is 3 or 1.732) x = 50/1.732

28.9 metres x

As a result, the monument's shadow is approximately 28.9 metres long as the monument's height is on the other side .

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Answer:

The amplitude of this graph is 4.

Answer:

Step-by-step explanation:

75

Answer:$75

Step-by-step explanation:

The domain of the expression is (−∞,−5)∪(−5,7)∪(7,∞).

The range of values that we are permitted to enter into our function is known as the domain of a function. The x values for a function like f make up this set (x). A function's range is the collection of values it can take as input.

Here the given expression is,

=> [tex]\frac{3}{x+5}-\frac{1}{7-x}[/tex]

Now simplifying the expression then,

=> [tex]\frac{3(7-x)-1(x+5)}{(x+5)(7-x)}[/tex]

=> [tex]\frac{21-3x-x-5}{7x-x^2+35-5x}[/tex]

=> [tex]\frac{-4x-26}{-x^2+2x+35}[/tex]

Now to find domain then x≠-5 and x≠7.

The domain is (−∞,−5)∪(−5,7)∪(7,∞).

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