Let f be a polynomial with integer coefficients, of degree n>1. What is the maximal number of consecutive integers belonging to the sequence f(1),f(2),f(3),...

a) The average rate of change of a function over an interval is the change in the function's output (f(b) - f(a)) divided by the change in the function's input (b - a). So, to find the average rate of change of f over the interval -4 <= x <= 3, we can use the formula:

(f(3) - f(-4)) / (3 - (-4)) = (sqrt(25 - 9) - sqrt(25 - 16)) / (3 - (-4)) = (-2sqrt(7) + 4sqrt(3)) / 7

b) To find the derivative of f(x), we need to use the power rule and the chain rule. The power rule states that the derivative of x^n is nx^(n-1), and the chain rule states that the derivative of f(g(x)) is f'(g(x)) * g'(x).

So, the derivative of f(x) = sqrt 25 - x^2 is:

-2x

c) To find the equation of the line tangent to the graph of f at x = -3, we need to use the point-slope form of a line, which is:

y - y1 = m(x - x1)

where m is the slope of the line (which is f'(-3) = -6), (x1, y1) is the point on the graph where the line touches (which is (-3, f(-3)) = (-3, sqrt(25 - 9) = 2sqrt(7))

so we have: y - 2sqrt(7) = -6(x + 3)

d) To check if g(x) is continuous at x = -3, we need to check if the limit of g(x) as x approaches -3 exists and is equal to g(-3).

Since g(x) = f(x) for -5 <= x <= -3 and g(x) = x + 7 for -3 <= x <= 5, we have:

g(-3) = f(-3) = sqrt(25 - 9) = 2sqrt(7)

and the limit of g(x) as x approaches -3 is:

lim x->-3 g(x) = lim x->-3 (x + 7) = -3 + 7 = 4

Since g(-3) = 2sqrt(7) and lim x->-3 g(x) = 4, g(x) is not continuous at x = -3.

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The correct representations of the inequality expression are (a) 6x – 15 > -10 + 5x and (c) 3(2x – 5) > -5(2 – x)

From the question, we have the following expression that can be used in our computation:

–3(2x – 5) < 5(2 – x)

Divide bot sides of the inequality by -1

So, we have the following representation

3(2x – 5) > -5(2 – x)

Next, we open the brackets

This gives

6x – 15 > -10 + 5x

Hence, the representation is 6x – 15 > -10 + 5x

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Complete question

Which are correct representations of the inequality –3(2x – 5) < 5(2 – x)? Select two options.

(a) 6x – 15 > -10 + 5x

(b) 6x – 15 < -10 + 5x

(c) 3(2x – 5) > -5(2 – x)

(d) 3(2x – 5) < -5(2 – x)

Step-by-step explanation:

y = 2x + 1 -------- (I)

y = 4x + 6 --------(ii)

from eqn (I)

y = 2x + 1

sub y = 2x + 1 in eqn (ii)

2x + 1 = 4x + 6

2x - 4x = 6 - 1

-2x = 5

x = -5/2

sub x = -5/2 in eqn (I)

y = 2(-5/2) + 1

y = -5 + 1

y = -4

Therefore, (x, y) = (-5/2, -4)

Answer: The Sale price is $196

Step-by-step explanation:

We can start by converting 30% to a decimal 0.3, then we will multiply the $280 by the decimal (280*0.3 = $84) This means it is marked down by $84, so to get the sale price we subtract the mark down by the original price, ($280-$84 = $196).

Answer:

The suit's sale price is $196.

Step-by-step explanation:

⭐What is "mark down"?

To find the sale price, we need to form an equation representing the given situation.

MARK DOWN:

original price - (mark down percentage)(original price) = sale price

Substitute the known values.

280 - (0.3)(280) = sale price

Compute.

280 - 84 = sale price

∴ $196 = sale price

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The lengths of a,b,c, and d have been derived and expressed in terms of trigonometric ratios below.

As we can see from the figure itself, the value of a = sin θ

This can be written as such because = sin θ = Perpendicular / Hypotenuse

Here, the perpendicular = a and hypotenuse = radius = 1

= sin θ = a

The value of b = tan θ

This is because tan θ = perpendicular/base

Here, the base = radius = 1

= tan θ = b  

The value of c = sec θ

This is because sec θ = Hypotenuse / Base

= sec θ = c

The value of d = cos θ

This is because cos θ = Base / Hypotenuse

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The probability that the product has a shelf life of at least 1 week and at most 2 weeks is 0.30.

Potential is described by probability. This branch of mathematics deals with the occurrence of a random event. The value's range is 0 to 1. Probability is a concept in mathematics that helps predict the likelihood of certain events. Probability generally refers to the degree to which something is likely to occur.

A standard normal variable with a mean of 0 and a variance of 1 can be created from a normally distributed random variable with a defined mean and variance.

Thus, The probability that the product has a shelf life of at least 1 week and at most 2 weeks is 0.30.

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Answer: Thirty-eight thousand two hundred.

Step-by-step explanation: You'd get Thirty-eight thousand two hundred because the standard form is writing out the number using letters. So, we should break this down:

38,000 + 200 is a good strategy to divide big numbers.

So, 38,000 = Thirty-eight thousand

And 200 = two hundred

So, combine it and you get Thirty-eight thousand two hundred or Thirty-eight thousand and two hundred. (it's the same thing)

The change in elevation would be -225/4 feet

A fraction represented with its quotient and remainder is a mixed fraction. For example, 2 1/3 is a mixed fraction, where 2 is the quotient, and 1 is the remainder. So, a mixed fraction is a combination of a whole number and a proper fraction.

Given here, Speed of scuba diver as -12¹/₂=-25/2 and time in minutes 4¹/2=9/2

Thus distance traversed -25/2 × 9/2 = -225/4

Hence, the change in elevation is -225/4 feet

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The function be f(x) = (6) then growth factor for the given equation (b) = 6.

Quantity grows exponentially over time. It happens when a quantity's instantaneous rate of change with regard to time is proportional to the quantity itself.

A function called exponential growth illustrates an expansion within a population that happens at the same pace across time. When a population's per capita growth rate remains constant across time, regardless of population size, exponential growth occurs, causing the population to grow exponentially as the population increases.

The general exponential growth function is given by :-

[tex]$f(t)=A b^t$[/tex] , where A is the initial amount , b is the growth factor and t is the time period.

The given function : [tex]$f(x)=(6)^x$[/tex]

When we compare it to the general exponential equation , we get

The growth factor for the given equation (b) = 6.

The complete question is:

Consider the function f(x) = (6)x. What is the value of the growth factor of the function?

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The function be f(x) = (6) then growth factor for the given equation (b) = 6.

Quantity grows exponentially over time. It happens when a quantity's instantaneous rate of change with regard to time is proportional to the quantity itself.

A function called exponential growth illustrates an expansion within a population that happens at the same pace across time. When a population's per capita growth rate remains constant across time, regardless of population size, exponential growth occurs, causing the population to grow exponentially as the population increases.

The general exponential growth function is given by :-

F(t) = AB^t, where A is the initial amount , b is the growth factor and t is the time period.

The given function : f(t) = (6)^x

When we compare it to the general exponential equation , we get

The growth factor for the given equation (b) = 6.

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The complete question is:

Consider the function f(x) = (6)x. What is the value of the growth factor of the function?

Answer: 2i+5

Step-by-step explanation: so (-2+14i)-(-7+12i)

Multiply (-1) to 7 and -12

Then: -2+14i+7-12i

Common numbers: (7-2) and 14i-12i

Then you get 2i+5

A polynomial in which the degrees of all the variables is 0 is called a zero degree polynomial .

The zeros of a polynomial p(x) are all the x-values that make the polynomial equal to zero.

For example- 6 x 0 , − 9 a 0.

Polynomial

A polynomial is defined as an expression which is composed of variables, constants and exponents, that are combined using mathematical operations such as addition, subtraction, multiplication and division .

Degree of polynomial

The degree of a polynomial is defined as the highest exponent of a monomial within a polynomial. Thus, a polynomial equation having one variable which has the largest exponent is called a degree of the polynomial .

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The standard way to solve 3 linear equations with 3 variables is to use elimination or substitution.

-Elimination: To solve the 3 equations using elimination, add or subtract equations together to eliminate one of the variables. Keep repeating this process until you are left with one equation with one variable. Solve the equation and then use the value of the variable to substitute back into the other equations to find the values of the other variables.

-Substitution: To solve the 3 equations using substitution, pick one of the equations and solve it for one of the variables. Then substitute the value of the variable into the other equations. Keep repeating this process with the other equations and variables until you have a system of equations with one variable that you can solve. Once you have the value of the variable, substitute it back into the other equations to find the value of the other variables.

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The value of x is 0

Now, According to the question:

Let's know:

How to find the value of 'x'

Generally, the algebraic expression should be any one of the forms such as addition, subtraction, multiplication and division. To find the value of x, bring the variable to the left side and bring all the remaining values to the right side. Simplify the values to find the result.

The given equation is:

9x - 7 = -7

To solve the value of 'x'

9x - 7 = -7

Add 7 on both sides:

9x - 7 + 7 = -7 + 7

9x = 0

x = 0/9

x = 0

Hence, The value of x = 0

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The complete question is this:

How do you solve 9x - 7 = -7

The amount luke made more than alexa is $3.91 .

What is simple interest ?

simple interest can be defined as the ratio of product of principal amount , rate and time and 100.

Luke investment,

This is compound interest so the future value formula will be used:

= Amount * ( 1 + rate) ^ no. of years

= 4,000 * ( 1 + 1.8%)³

= $4,219.91

Alexa investment:

Simple interest is calculated with the formula:

= Amount * ( 1 + rate * time)

= 4,000 * ( 1 + 1.8% * 3)

= $4,216

Amount Luke made more than Alexa:

= 4,219.91 - 4,216

= $3.91

Hence , The amount luke made more than alexa is $3.91 .

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Consequently, the solution of the given problem of perimeter comes out to be 24.5 inches will be adequate to build the bird house, which needs 24 inches.

The complete length of a shape's boundary is referred to as the perimeter in geometry. A shape's perimeter is calculated by adding the lengths of all of its sides and edges.

Here,

As stated, Hodi is making a birdhouse.

Four walls make up a birdhouse.

There must be 5.5 inches on each wall.

Board = 24.5 inches for Hodi.

Using comparable numbers, we must now estimate.

6 in. is a suitable size for 5.5 in.

The entire amount of wood needed to construct the bird home will be determined first.

The amount of board needed is equal to the number of walls in the house times the length of each wall.

Required total board:   4 x 6 = 24 inches for a wall count multiplied by the length of each wall.

It takes 24 inches of board in total.

His board is 24.5 inches long.

Consequently, the solution of the given problem of perimeter comes out to be 24.5 inches will be adequate to build the bird house, which needs 24 inches.

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The equation to find the root of the statement given in the question is , 12x = 2x - 24.

Let the number of tamp which Ritu have be x

Then the number of tamp which Raj have be 2x = 24

Now as Ritu has 6 times more than Raj , 6( 2x) = 24 x 6

Therefore, the required equation becomes,

12x = 2x - 24

=> 10x = 24

=> x = 2.4 answer.

An equation is a mathematical statement which is made up of constants , variables and operators. There are two parts of an equation which are equated using the equal sign . The right hand and the left hand side of the equation.

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The answer which we get after calculating from the data given is that  Larry has 15 dollars more than Lenny.

Let the amount which Lenny is having be x

then the amount which Larry is having will be (2/5)x

and on combining them it makes $35 .

Which means,  x + 2x/5 = 35

On simplifying this equation we will get ,

(7/5)x= 35   ,

multiplying both the sides by 5/7

x= (5/7)*35 = $25

Therefore , Lenny is having 4 25.

So Larry must be having ,  $(35 - 25) = $10

So , we can conclude that Lenny has more dollars. The difference is of 15 dollars.

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To calculate the amount of food to order for the graduation party with drop-in guests, you should multiply the number of "full" guests, which is 40 + x/2, by the recommended amounts listed in the table.

To calculate the amount of food to order for the graduation party, you need to take into account the number of guests and the recommended amounts listed on the table. Since the party is for 40 guests and there will also be "drop-in" guests, you need to convert these guests into an equivalent number of "full" guests. To do this, you can divide the number of drop-in guests by 2.

So if there are x drop-in guests, the number of "full" guests will be 40 + x/2.

Now you can use this number of "full" guests to calculate how much of each food item to order. For example, if the table recommends 1/4 lb of meat per guest, you would order (40 + x/2) × 1/4 lb = 10 + x/8 lb of meat.

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Check the picture below.

Answer:

The answer to your question is C, 15

Step-by-step explanation:

Answer is 15, C

Tim received 8 SMS on Monday, 4 emails on Thursday, and 0.52 of the communications were emails, which is the solution to (a), (b), and (c).

When two sets of data are joined and their frequencies are combined, a composite bar chart is utilized. Two pieces of data—emails and texts—are included in this inquiry, but they both fall under the same heading, "messages." With regard to question (a), it is clear from the diagram that Tim got 8 texts on Monday. In order to answer question (b), we must determine how many emails Tim received on Thursday. To do this, we must look at the bar for Thursday and divide the total number of texts by the total number of messages to determine the number of emails received. 15-11=4, indicating that Tim received 4 emails on Thursday.

given

Tim for five days, total the emails received, and then determine the percentage of emails relative to messages.

Total messages: 14+16+14+15+14=73.

Emails totaled: 6 + 10 + 10 + 4 + 8 = 38

Email to message ratio is equal to 0.52.

Emails made up 0.52 of the messages, therefore.

Tim got 8 texts on Monday, 4 emails on Thursday, and 0.52 of the messages were emails, according to an analysis of the composite bar chart we used to determine the answers to the three questions.

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The complete question is :- The composite bar chart Messages received shows the number of messages Tim received each day for five days.

a) How many texts did Tim receive on Monday?

b) How many emails did Tim receive on Thursday?

c) In total, what fraction of the messages were emails?

Give your answer in its simplest form.

Yes, it is a perfect square trinomial.

Perfect squares are numbers that are the product of a number multiplied by itself. Examples of perfect squares include 4 (2 x 2), 9 (3 x 3), 16 (4 x 4), 25 (5 x 5), 36 (6 x 6), 49 (7 x 7), and 64 (8 x 8). Perfect squares are also referred to as "square numbers" or "whole-number squares."

The equation X² + 10X + 25 can be factored into (X + 5)²,

which is a perfect square trinomial. In general, a perfect square trinomial is any polynomial of the form A² + 2AB + B² where A and B are constants or variables.

This is because such a polynomial can be factored into (A + B)².

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answer:10

Step-by-step explanation:

the graph show it is 1 hour and shows 10 miles and it continues

Answer:

it should be that the miles are 10 times the amount of hours

Step-by-step explanation:

Considering that to bike 10 miles it took 1 hour and to bike 20 miles it took 2 hours it will continue going like that so the miles will always be 10 times the amount of hours

Yes, [tex]9x^2-30x +25[/tex]  is a perfect square trinomial.

A perfect square trinomial expression can be created by taking the binomial equation's square. If and only if a trinomial of the form [tex]ax^2 + bx + c[/tex] meets the requirement [tex]b^2 = 4ac[/tex], it is said to be a perfect square.

Two of your terms will be perfect squares in a perfect square trinomial. This trinomial is not a perfect square trinomial if it lacks two perfect square terms. The square root of each perfect square word should now be determined, combine the two square roots, then multiply the result by two.

This trinomial is a perfect square. The only thing left to determine is whether the middle term matches when you square (3x-5) because both [tex]9x^2=(3x)^2\ and\ 25=52[/tex] are perfect squares. The minus sign was used to at least produce a negative coefficient for the middle term.

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The algebraic expression to show Jill's age will be;

Jill = 1/3 x b

Algebraic expressions are the mathematical statements that are derived when operations such as addition, subtraction, multiplication, division, etc. are operated upon on variables and constants.

There are different parts of an algebraic expression and they are; Variables - a symbol that doesn't have a fixed value is called a variable. It can take any value assigned to it.

Constants - a symbol that has a fixed numerical value is called a constant. All numbers are constants.

Terms - A term is a variable alone (or) a constant alone (or) it can be a combination of variables and constants and the operation of multiplication or division with it.

Coefficients - the numbers that are multiplying the variables are called coefficients.

Beth's age = b

Jill's age = 1/3 of Beth's age

Jill's age = 1/3 x b

Jill's age = 1/3b

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The prove of given limit lim θ → 0 ( sin θ / θ ) = 1 is shown in below.

Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

We have to given that;

⇒ lim θ → 0 ( sin θ / θ ) = 1

Here, The form of given limit is 0/0, so we can apply the L - Hospital Rule.

⇒ lim θ → 0 ( sin θ / θ ) = 1

Take LHS;

⇒ lim θ → 0 ( d/dθ (sin θ)/ dθ/dθ )

⇒ lim θ → 0 ( cos θ / 1 )

⇒ ( cos 0 )

⇒ 1

⇒ RHS

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The required money to be invested in order to save $5000 is $4677.26.

The compound interest is a form of interest where the rate of interest is applied on the amount obtained after every interval of time.

The principal for each interval is the sum of interest And the principal for the previous interval.

Suppose the amount to be invested be P.

The rate of interest for monthly compounding is given as 4.5/12 = 0.375%.

Time is given as 18 months.

This implies number of time period is 18.

Plug n = 18, r = 0.375% and A = 5000 in A = P(1 + r/100)ⁿ as,

⇒ 5000 = P(1 + 0.375/100)¹⁸

⇒ P = 4677.26

Hence, the amount of money that should be invested is obtained as $4677.26.

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Answer:

multiply top and bottom of 2/6 to get 6/18, then subtract the answer by 6/18 , you get 5/18

Step-by-step explanation:

Answer:

Step-by-step explanation:

2/6 + x = 11/18

x = 11/18 - 2/6

x = 5/18

----------------------------

2/6 + 5/18 = 11/18

11/18 = 11/18