Given f(x) = x³ + kx + 9, and the remainder when f(x) is divided by x − 2 is 7,
then what is the value of k?

The negative of the gradient (-grad f(x)) points in the direction of the maximum decrease of f from x.

The gradient of a scalar-valued function is a vector that points in the direction of the maximum increase of the function.

In other words, if we consider a point in the domain of the function and take the gradient at that point, the direction of the gradient vector indicates the order in which the function increases the most from that point. Conversely, the negative gradient points in the direction of the maximum decrease of the function.

Specifically, let f be a scalar-valued function of n variables [tex](f: R^n - > R),[/tex]and let x be a point in the domain of f. The gradient of f at x is defined as the vector:

[tex]grad f(x) = (∂f/∂x1, ∂f/∂x2, ..., ∂f/∂xn)[/tex]

where ∂f/∂xi denotes the partial derivative of f with respect to xi evaluated at x, the direction of the gradient vector grad f(x) at x is the direction in which f increases the most from x, and the magnitude of the gradient vector is the rate of change of f at x in that direction.

The negative of the gradient (-grad f(x)) points in the direction of the maximum decrease of f from x.

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Answer:

0.600 or 600, 0.500 or 500, select option 2

Step-by-step explanation:

Answer: y = 3x-2 (for first question)

Step-by-step explanation:

Equation of the line: y = mx+c. For A, it passes through (0, -2) and has a gradient of 3. So, you substitute the values inside the equation.

x = 0

y = -2

m = 3

-2 = 0+c

c= -2

ans: y = 3x-2

Use the same method to complete the rest of the questions. GL

Note: I am just a student, if I get this wrong I hope someone corrects me, thanks

Answer:

Step-by-step explanation:

Для решения задачи мы можем использовать свойства треугольников и окружностей.

Первое, что мы можем заметить, это что треугольник ABD является равносторонним, так как все его стороны имеют одинаковую длину 13. Это означает, что угол ABD также равен 60 градусам.

Также мы можем заметить, что точка O является центром окружности, вписанной в треугольник ABD, так как все ее стороны касаются окружности в точке D. Из свойств вписанных углов, мы знаем, что угол AOD равен половине угла ABD, то есть 30 градусам.

Далее, мы можем заметить, что треугольник AOC является равнобедренным, так как угол ACO равен углу OCA (они оба равны 75 - 30 = 45 градусов), а сторона AC имеет одинаковую длину с стороной AB.

Таким образом, мы можем найти длину стороны AC, используя теорему косинусов для треугольника AOC:

AC^2 = AO^2 + OC^2 - 2 * AO * OC * cos(45)

Заметим, что AO = DO, так как точка O является центром вписанной окружности, а DO является радиусом этой окружности. Из прямоугольного треугольника ADO мы можем выразить DO как DO = AD/2 = 6.5.

Также, мы можем выразить OC, используя равенство углов в треугольнике ACO (ACO и AOD являются вертикальными углами):

ACO = AOD = 30 градусов

Тогда, угол OCA равен 180 - 2 * 45 = 90 градусам, что означает, что треугольник OCA является прямоугольным, и мы можем использовать теорему Пифагора:

OC^2 + AC^2 = OA^2

OC^2 + AC^2 = DO^2

AC^2 = DO^2 - OC^2

Теперь мы можем подставить выражения для DO и OC, и получить:

AC^2 = 6.5^2 - (6.5/sqrt(2))^2

AC^2 = 42.25 - 22.5625

AC^2 = 19.6875

AC = sqrt(19.6875)

AC = 4.43 (с точностью до сотых)

Таким образом, длина стороны AC равна пр

Answer:

The width of the frame is 10 inches.

5x7 = 35

80-35 = 45

45/7 = 6.4

5 + 6.4 = 11.4

11.4 rounded down = 11

11 - 5 = 10

Width of the frame = 10 inches

Answer:

We can substitute the given values of p, m, and n into the formula for h:

h = p√(m^2 + n)

h = 3√(6^2 + 20)

h = 3√(36 + 20)

h = 3√56

We can simplify this by factoring 56 into its prime factors:

h = 3√(2^3 × 7)

h = 3 × √(2^2 × 7) × √2

h = 3 × 2√7

Therefore, when p = 3, n = 20, and m = 6, the value of h is 6√7 or approximately 13.42.

Answer: 1/3 chance or 33.33333%

Step-by-step explanation: There are only 3 odd numbers on the die, 1,3,5 so there is a 1/3 chance it is 3

The dimensions of the rectangular frame is found as : 12 and 6 inches.

When a triangle is just a right triangle, the hypotenuse square is equal to the sum of the squares of the triangle's legs.

That's a picture frame, therefore pay attention that it must be rectangular.

Hence, the triangle is really a right triangle, and the Pythagorean theorem will eventually be applied.

You are aware that the square of the hypotenuse is 20 and equals 400.

hence, a²  + b²  = 400 and...

So because length is 4 times more than the breadth, a = b + 4.

This can be resolved if "b + 4" is substituted for "a":

(b + 4)² + b²  = 400,

(b + 4)(b + 4) + b²  = 400,

b² + 8b + 16 + b²  = 400,

2b² + 8b = 384

Further solving;

b² + 4b = 192

b² + 4b - 192 = 0

(b + 16)(b - 12) = 0

Due to the fact that a length cannot be negative, b must therefore be between b - 16 or 12 (negative value not taken)

The second leg is 12 + 4 = 6.

Thus,  the dimensions of the rectangular  frame is found as : 12 and 6 inches.

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Answer:

∠1 = 50

∠2 = 50

∠3 = 80

∠4 = 130

∠5 = 130

Step-by-step explanation:

∠1 = 180 - 130 = 50

∠2 = ∠1 = 50

∠3 = 180 - ∠1 - ∠2 = 180 - 50 - 50 = 80

∠4 = 180 - ∠2 = 180 - 50 = 130

∠5 = ∠4 = 130

Angle PLE is measured at 90°. The options supplied do not contain the solution.

The acute angle is defined as being less than 90 degrees. Right angles are 90 degrees in length. Angles that are obtuse have a greater angle than 90 degrees. Discover the various sorts of angles and examples of each.

As complementary angles, ZPLA and ZELA, we can infer that:

ZPLA + ZELA = 90°

Using the following expressions in place of ZPLA and ZELA, we obtain:

5x-2 + x+8 = 90

When we simplify the equation, we obtain:

6x + 6 = 90

6 is subtracted from both sides to yield:

6x = 84

Dividing by 6, we get:

x = 14

With the knowledge of x, we can determine the dimensions of ZPLA and ZELA:

ZPLA = 5x-2 = 5(14)-2 = 68°

ZELA = x+8 = 14+8 = 22°

Therefore, we can find the measure of angle PLE by subtracting the measures of ZPLA and ZELA from 180°:

PLE = 180 - ZPLA - ZELA

PLE = 180 - 68 - 22

PLE = 90

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To show that the properties of a probability distribution for a discrete random variable are satisfied, we need to verify that the probability distribution function (PDF) satisfies, The PDF must be non-negative for all values of the random variable and The sum of the probabilities for all possible values of the random variable must be equal to 1.

Let X be a discrete random variable taking values x1, x2, ..., xn, and let P(X) be the probability distribution function for X, such that P(X = xi) = pi for all i.

The PDF must be non-negative for all values of the random variable. This means that for any value xi of the random variable, the probability pi must be greater than or equal to zero. That is,

pi ≥ 0 for all i.

This property ensures that probabilities are never negative, which is a necessary condition for a valid probability distribution.

The sum of the probabilities for all possible values of the random variable must be equal to 1. That is,

∑ pi = 1 for all i.

This property ensures that the total probability of all possible outcomes is equal to 1, which is a necessary condition for a valid probability distribution.

Therefore, if the PDF satisfies these two properties, we can conclude that it represents a valid probability distribution for the given discrete random variable.

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The marginal profit at the following expenditures for $8000 is 0 for $6000 is 8 for $12,000 is -16 and for $20,000 is -48.

When one more unit is produced and sold, a company or individual will make a profit known as marginal profit. The additional expense or profit made while manufacturing the next unit is referred to be marginal. Whereas marginal cost is the additional expense for manufacturing one extra unit, marginal product is the additional income received.

The difference between marginal cost and marginal product is known as marginal profit (also known as marginal revenue). Managers can utilise margin analysis to help them decide whether to increase output, reduce production, or halt production completely, a decision known as the shutdown point.

we have,

P(x)= 1000+32x - 2x²

Formula for derivative of function:

[tex]f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}[/tex]

Finding the derivative,

P(x+h) = 1000 + 32(x+h) - 2(x+h)²

P(x+h) = -2x² -4xh - 2h² + 32x + 32h + 1000

P(x+h) - P(x) = h(-4x - 2h + 32)

P'(x) = -4x + 32

Finding derivatives for given values of x is,

P'(8) = -4(8) + 32 = 0

P'(6) = -4(6) + 32 = 8

P'(12) = -4(12) + 32 = -16

P'(20) = -4(20) + 32 = -48

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Complete question:

The profit (in thousands of dollars) from the expenditure of x thousand dollars on advertising is given by P(x)= 1000+32x - 2x². Find the marginal profit at the following expenditures.

a. $8000

b. $6000

c. $12,000

d. $20,000

Answer:

Step-by-step explanation:

An equation of the line that goes through the point (-1, -3) and (3, 5) is y = 2x - 1.

An equation of the line in slope-intercept form that is perpendicular to the equation for obstacle 1 is y = -x/2 + 3.

In Mathematics, the point-slope form of a straight line can be calculated by using the following mathematical expression:

y - y₁ = m(x - x₁) or [tex]y - y_1 = \frac{(y_2- y_1)}{(x_2 - x_1)}(x - x_1)[/tex]

Where:

At data point (-1, -3), a linear equation in slope-intercept form for this line can be calculated by using the point-slope form as follows:

[tex]y - y_1 = \frac{(y_2- y_1)}{(x_2 - x_1)}(x - x_1)\\\\y - (-3) = \frac{(5- (-3))}{(3-(-1))}(x -(-1))\\\\y +3 = \frac{(5+3)}{(3+1)}(x +1)[/tex]

y + 3 = 2(x + 1)

y = 2x + 2 - 3

y = 2x - 1

In Mathematics, a condition that must be met for two lines to be perpendicular is given by:

m₁ × m₂ = -1

2 × m₂ = -1

m₂ = -1/2.

At point (-4, 5), an equation of the line can be calculated by using the point-slope form as follows:

y - y₁ = m(x - x₁)

y - 5 = -1/2(x + 4)

y = -x/2 - 2 + 5

y = -x/2 + 3

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The common factors of sixteen and twenty are 1, 2, and 4.

The factors of sixteen are 1, 2, 4, 8, and 16. The factors of twenty are 1, 2, 4, 5, 10, and 20. The factors that are common to both numbers are 1, 2, and 4.

To calculate the factors of sixteen, we can start by dividing sixteen by two until we cannot divide any further. We can start with sixteen divided by two, which equals eight. Eight divided by two equals four. Four divided by two equals two. Two divided by two equals one. As we can see, the factors of sixteen are 1, 2, 4, 8, and 16

To calculate the factors of twenty, we can start by dividing twenty by two until we cannot divide any further. We can start with twenty divided by two, which equals ten. Ten divided by two equals five. Five divided by two equals two. Two divided by two equals one. As we can see, the factors of twenty are 1, 2, 4, 5, 10, and 20.The common factors of sixteen and twenty are 1, 2, and 4. This can be determined by comparing the two lists of factors. As we can see, 1, 2, and 4 are present in both lists.

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Complete question

What are the common factors of sixteen and twenty ?

The graph repeats itself every 32 units hοrizοntally. [ The graph is attached belοw ]

An interactive widget in a graph is a tοοl that allοws the user tο manipulate different elements οf the graph using their mοuse οr tοuch screen.

Interactive widgets can be used tο explοre the behaviοr οf a functiοn οr dataset in a mοre dynamic and engaging way than a static graph.

Sοme cοmmοn examples οf interactive widgets in graphs include sliders, buttοns, checkbοxes, and drοpdοwn menus.

Here we have

y = -5 cοs (π/16 x) - 4    

Represents a periοdic functiοn with a cοsine wave shape that is shifted dοwnwards by 4 units.

The cοsine functiοn οscillates between -1 and 1, sο multiplying it by -5 stretches the wave vertically by a factοr οf 5 and reflects it acrοss the x-axis.

The cοefficient οf x, π/16, cοntrοls the periοd οf the functiοn, which is the distance between twο cοnsecutive peaks οr trοughs οf the wave.

Specifically, the periοd οf this functiοn is 32 units, since the cοsine functiοn has a periοd οf 2π and 2π/(π/16) = 32.

Therefοre,

The graph repeats itself every 32 units hοrizοntally. [ The graph is attached belοw ]

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Answer:

Dont understand. Repeat question?

Step-by-step explanation:

The expected value of playing this game is -$0.251. This means that, on average, you can expect to lose 25.1 cents every time you play this game.

The term "outcome" generally refers to the result or effect of an action, process, or event. It refers to what happens as a consequence of a particular decision, activity, or circumstance.

To calculate the expected value, we multiply the possible outcomes by their probabilities and then sum them up.

Let's start with the possible outcomes:

If you win, you make a profit of $749, so the outcome is $749.

If you lose, you lose $1, so the outcome is -$1.

The probabilities are:

Probability of winning is 0.001.

Now we can calculate the expected value:

Expected value = (0.001 * $749) + (0.999 * -$1)

Expected value = $0.748 - $0.999

Expected value = -$0.251

Therefore, the expected value of playing this game is -$0.251. This means that, on average, you can expect to lose 25.1 cents every time you play this game.

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The correlation between X and Y can be calculated using the formula r = SQRT(R-squared).The correlation coefficient is a measure of the strength of the linear relationship between two variables and can range from -1 to 1

In this case, the R-squared value is 0.81, so the correlation between X and Y is r = SQRT(0.81) = 0.9 (or -0.9 depending on the direction of the relationship).The correlation between X and Y can be calculated using the formula r = SQRT(R-squared). The correlation coefficient is a measure of the strength of the linear relationship between two variables and can range from -1 to 1, where -1 is a perfectly negative linear relationship, 0 is no linear relationship, and 1 is a perfectly positive linear relationship. In this case, the correlation between X and Y was 0.9, indicating a strong linear relationship between the two variables.

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Step-by-step explanation:

this is solution which is given above in photo

The mean and standard deviation for 30 multiple-choice questions for the number of correct answers is Mean(μ) = 6 and Standard deviation(σ) = 2.19.

What is mean?

A collection of values are averaged to form the mean. The total points were divided by the total scores. When population samples are tiny, the mean is sensitive to extreme scores. For example, if two students in a class of 20 earned significantly higher than the rest, the mean will be skewed higher than the other students' scores might suggest. When using means, larger sample sizes are preferable.

Define standard deviation.

In statistics, a measure of variability known as the standard deviation ( standard deviation ) is frequently used. It demonstrates how different things are from the norm. (mean). When the SD is low, the data tend to be close to the mean, while when it is high, the data are dispersed over a wide variety of values.

Given: number of questions(n) = 30, p = [tex]\frac{1}{5}[/tex], q = [tex]\frac{4}{5}[/tex]

Mean μ = n*p

            = 30  *[tex]\frac{1}{5}[/tex]

         μ = 6

Standard deviation σ = [tex]\sqrt{n*p*q}[/tex]

                                   = [tex]\sqrt{30*\frac{1}{5}*\frac{4}{5}}[/tex]

                                   = [tex]\sqrt{4.8}[/tex]

                                σ = 2.19

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Answer:

To find the maximum number of bags that Thomas can pack, we need to find the greatest common divisor (GCD) of 120, 168, and 192.

Prime factorizing the three numbers:

120 = 2^3 x 3 x 5

168 = 2^3 x 3 x 7

192 = 2^6 x 3

The GCD is the product of the common prime factors with the lowest exponents, which is 2^3 x 3 = 24.

So, Thomas can pack the items into 24 bags, each containing an equal number of whistles, yo-yos, and tops.

Answer:

Step-by-step explanation:

To find the maximum number of bags that Thomas can pack, we need to find the greatest common divisor (GCD) of 120, 168, and 192.

We can start by finding the prime factorization of each number:

120 = 2^3 × 3 × 5

168 = 2^3 × 3 × 7

192 = 2^6 × 3

Then we can find the GCD by taking the product of the smallest power of each common prime factor:

GCD = 2^3 × 3 = 24

Therefore, Thomas can pack a maximum of 24 bags.

The total number of miles from Sharon's house to her mother's house would be = 60 miles.

The total amount of time it takes Sharon to reach the mother's house at constant speed = 180 minutes

The speed she used after the storm = 20 mile/hr

Therefore the miles she needs to cover before she reaches the mother's house = ?

That is;

20 miles = 1hr or 60 mins

X miles = 180 minutes

make X the subject of formula;

X miles = 20×180/60

= 3600/60

= 60 miles.

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Step-by-step explanation:

[tex]\textsf{For this problem, we are asked to draw an angle that is 150}^{\circ}.[/tex]

[tex]\textsf{We will need a Protractor to draw this angle.}[/tex]

[tex]\large\underline{\textsf{To Draw an Angle with a Protractor;}}[/tex]

[tex]\textsf{First, notice a point at the bottom of the Protractor. This is where we will start.}[/tex]

[tex]\textsf{Draw a Straight Line from the point to the 0}^{\circ} \ \textsf{mark.}[/tex]

[tex]\textsf{Secondly, use a ruler to carefully draw a straight line towards the 150}^{\circ} \ \textsf{mark.}[/tex]

[tex]\textsf{Note that some Protractors may be different from others. Similar steps should apply.}[/tex]

[tex]\textsf{Refer to the picture. It represents what the angle should look like afterwards.}[/tex]

To do :-

Instruments required:-

Steps of construction:-

Hence angle ECB represents 150° .

Precaution:- do not change the arm length of the compass.

and we are done!

Answer: 32

Step-by-step explanation: got you broski

Answer:

The critique of the sample method and survey used by the guild is as follows:

In summary, the survey used by the guild suffers from various types of bias, which may affect the validity and generalizability of the results. To improve the validity and reliability of the survey, the guild should use random sampling and address issues such as non-response and response bias.

Answer: At most 18 acute angles

Step-by-step explanation:

The sum of the interior angles of an n-sided polygon is (n-2) × 180 degrees. In a convex polygon, each interior angle is less than 180 degrees.

Let a₁, a₂, ..., a₁₁ be the interior angles of the 11-sided polygon. Then the sum of the interior angles is:

a₁ + a₂ + ... + a₁₁ = (11-2) × 180 = 1620 degrees

Since each angle is acute, we know that each angle is less than 90 degrees. Let A be the number of acute angles. Then the sum of the acute angles is at most:

A × 90

So we have:

a₁ + a₂ + ... + a₁₁ ≤ A × 90

Substituting the sum of the interior angles, we get:

1620 ≤ A × 90

Solving for A, we get:

A ≤ 18

Therefore, the polygon can have at most 18 acute angles.

[tex]T(11) = -1[/tex] represents the initial negative term inside the series.

A phrase that acknowledges a characteristic or trait in anything is said to be positive. Negative terms are those that downplay some aspect or characteristic of something. The positive or negativizes of a phrase is termed its quality.

Positive simply denotes excellent or the polar opposite of negative. For instance, you're less likely to receive favorable feedback on your scorecard if you've a good mindset toward your assignments. It might be difficult to keep track of all the different definitions of the word positive.

T(n) is negative when [tex]32 - 3n < 0[/tex], which can be rearranged as:

[tex]3n > 32[/tex]

[tex]n > 32/3[/tex]

Since [tex]n[/tex] is an integer, the smallest value of n that satisfies this inequality is [tex]n = 11[/tex].

Therefore, the [tex]11th[/tex] term of the sequence is:

[tex]T(11) = 32 - 3(11) = -1[/tex]

So the first negative term in the sequence is [tex]T(11) = -1.[/tex]

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Answer: Let's denote the two unknown angles as x and y.

We know that the sum of the angles in any quadrilateral is 360°, so we can set up an equation using this fact:

235° + 40° + x + y = 360°

Simplifying this equation, we get:

x + y = 85° (equation 1)

We also know that the other two angles are in a ratio of 5:12. This means that:

x/y = 5/12

Multiplying both sides by y, we get:

x = (5/12)y (equation 2)

Now we can substitute equation 2 into equation 1 and solve for y:

(5/12)y + y = 85°

(17/12)y = 85°

y = (12/17) * 85°

y = 60°

Substituting y = 60° into equation 2, we can solve for x:

x = (5/12) * 60°

x = 25°

Therefore, the two angles that are in a ratio of 5:12 measure 25° and 60°, respectively.

Step-by-step explanation: