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In mathematics, differentiation refers to the process of finding the derivative of a function. The basic means of differentiation include:

Limit definition: The derivative of a function is defined as the limit of the difference quotient as the interval over which the quotient is taken approaches zero.

Power rule: The derivative of a function of the form f(x) = xⁿ is given by f'(x) = nx⁽ⁿ⁻¹⁾

Product rule: The derivative of a product of two functions u(x) and v(x) is given by (u × v)'(x) = u'(x) × v(x) + u(x) × v'(x).

Quotient rule: The derivative of a quotient of two functions u(x) and v(x) is given by (u/v)'(x) = [u'(x) × v(x) - u(x) × v'(x)] / v(x)².

Chain rule: The derivative of a composition of functions f(g(x)) is given by f'(g(x)) × g'(x).

These basic means of differentiation are essential tools for analyzing and modeling functions in calculus and related fields.

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The equation of the semicircle upper side of the x-axis will be x = √[r² - (y - k)²] + h.

It is the close curve of an equidistant point drawn from the center. The radius of a circle is the distance between the center and the circumference.

Let r be the radius of the circle and the location of the center of the circle be (h, k). Then the equation of the circle is given as,

(x - h)² + (y - k)² = r²

The equation of the semicircle upper side of the x-axis will be given as,

(x - h)² + (y - k)² = r²

x - h = √[r² - (y - k)²]

x = √[r² - (y - k)²] + h

The equation of the semicircle upper side of the x-axis will be x = √[r² - (y - k)²] + h.

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The bearing are as follows:

(a) W from O = 70 degree south by east

(b) O from CA = 17 degree south by west

(c) CA from W = 21 degree west by north

What is angle ?

In geometry, an angle is the measure of the amount of rotation between two intersecting lines or surfaces, typically measured in degrees or radians.

An angle is formed when two lines or surfaces intersect at a point, which is called the vertex of the angle. The two lines or surfaces that form the angle are called the arms or sides of the angle. Angles are usually denoted using the vertex point, with a small angle symbol (such as ∠ABC).

Corresponding angles are a pair of angles that are in the same relative position at two different intersections of a line and a pair of parallel lines. In other words, if two parallel lines are intersected by a third line (called a transversal), then any pair of angles that are in the same position relative to the parallel lines are corresponding angles.

a. 70 degree south by east

b. 17 degree south by west

c. 21 degree west by north

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AThe bearing are as follows:

(a) W from O = 70 degree south by east

(b) O from CA = 17 degree south by west

(c) CA from W = 21 degree west by north

What is angle ?

In geometry, an angle is the measure of the amount of rotation between two intersecting lines or surfaces, typically measured in degrees or radians.

An angle is formed when two lines or surfaces intersect at a point, which is called the vertex of the angle. The two lines or surfaces that form the angle are called the arms or sides of the angle. Angles are usually denoted using the vertex point, with a small angle symbol (such as ∠ABC).

What is corresponding angle ?

Corresponding angles are a pair of angles that are in the same relative position at two different intersections of a line and a pair of parallel lines. In other words, if two parallel lines are intersected by a third line (called a transversal), then any pair of angles that are in the same position relative to the parallel lines are corresponding angles.

According to given conditions :

a. 70 degree south by east

b. 17 degree south by west

c. 21 degree west by north

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Answer: Hello! I'm JK!.....

B. I Only. (inside the triangle)

Step-by-step explanation:

I really hope this helps you.  XoXoGoldenMaknae <3

It takes 50 seconds to travel the full length at the constant rate R.

Remember the relation between speed, distance and time:

distance = speed*time

Here we know that at a rate R (or speed S, these are equivalent) it moves (1/20) of the total distance D in (2 + 1/2) seconds, then we can write:

R*(2 + 1/2) = (1/20)*D

Now we only need to isolate D on the right side, to do so we need to multiply both sides by 20.

20*R*(2 + 1/2) = D

The time is:

20*(2 + 1/2) seconds

40 + 10 seconds

50 seconds

It takes 50 seconds to travel the full length,

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The y = mx + b formula for these points is y = 2x - 67.

Mathematically, the point-slope form of a straight line can be calculated by using this mathematical expression:

y - y₁ = m(x - x₁) or y - y₁ = (y₂ - y₁)/(x₂ - x₁)(x - x₁)

Where:

Based on the information provided, we can logically deduce the following data points on the line:

At point (67, 67), a linear equation of this line can be calculated in slope-intercept form (y = mx + b) as follows:

y - 67 = (65 - 67)/(65 - 67)(x - 67)

y - 67 = 2(x - 67)

y = 2x - 134 + 67

y = 2x - 67

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The image of (7, 3) after a translation of T(3, - 5) would be at  (10, - 2).

Two-dimensional figures can be transformed mathematically in order to travel about a plane or coordinate system.

Dilation: The preimage is scaled up or down to create the image.

Reflection: The picture is a preimage that has been reversed.

Rotation: Around a given point, the preimage is rotated to create the final image.

Translation: The image is translated and moved a fixed amount from the preimage.

Given, A point has a coordinate of (7, 3).

We know, Translation shifts coordinate into the x-y coordinate.

Therefore, The image of (7, 3) after a translation of (3, - 5) would be,

(7 + 3, 3 - 5) which is now at (10, - 2).

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Answer:

To write the radical ³√32m³ using rational exponents:

1. Rewrite the radical using exponential notation: 32m^3 ≈ 8m^3.

2. Convert the exponent into its equivalent fractional form: 8m^3 ≈ 2^3m^3/2^3.

3. Apply a rational exponent to the numerator: (2^3m^3)^(1/3)/2^3 = 2m.

4. The radical can now be written using rational exponents: (2m)^3/2^3.

Answer:

$1508.54

Step-by-step explanation:

A = p(1 + [tex]\frac{r}{n}[/tex])^nt

A = 7000(1+ .05)^4

A = 7000(1.05)^4

A = 8508.54  This is the total amount (principle and interest

Subtract out the principle

8508.54 - 7000 = 1508.54

Answer:

Step-by-step explanation:

Since there is a 30% chance of rain on any given day, there is a 70% chance that it will not rain on a given day.

To calculate the expected number of days it will rain during your visit, we can use the formula:

Expected number of rainy days = total number of days * probability of rain

Expected number of rainy days = 110 days * 0.3 = 33 days

Therefore, we can expect it to rain on approximately 33 days during a 110-day visit to Narnia.

Answer: x <- -8

x is less than negative eight.

Step-by-step explanation:

First, subtract 11 from both sides. Then divide both sides by 11. The arrow should be open and pointing to the left.

Answer:

x <- -8

Step-by-step explanation:

The distance from the boat to the dock will be 26 miles. Then the correct option is D.

The Pythagoras theorem states that the sum of two squares equals the squared of the longest side.

The Pythagoras theorem formula is given as

H² = P² + B²

A boat leaves the dock and travels 10 miles due south, then 24 miles due west.

Then the distance from the boat to the dock is given as,

d² = 10² + 24²

d² = 100 + 576

d² = 676

d = 26 miles

The distance from the boat to the dock will be 26 miles. Then the correct option is D.

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Answer:

True

Step-by-step explanation:

They Are all Equal and The Same By the Degrees and Angles and Numbers

The selling price of a shirt should be $8.97.

The profit or gain is equal to the selling price minus the cost price. Loss is equal to the cost price minus the selling price.

Given that, a store bought shirts for $5.75 each and profit percent = 65%.

We know that, Selling price = (100+Profit%)/100 ×CP

Here, Selling price =(100+65)/100 ×5.75

= 1.56×5.75

= $8.97

Therefore, the selling price of a shirt should be $8.97.

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The absolute value of y, and the square root of (xy to the power of z) is [tex]xy^{z}[/tex]

Floats are a type of number representation that allows us to store very large or very small numbers with a high degree of precision. In computer programming, it is often used to represent real numbers, such as decimals.

In this problem, we are given three float numbers x, y, and z and asked to perform various mathematical operations with them.

Let's start with the first operation: x to the power of z. This means that we want to find x raised to the power of z. This can be written mathematically as xᵃ.

The next operation is x to the power of (y to the power of z). This means that we want to find x raised to the power of y raised to the power of z. This can be written mathematically as x(yᵃ).

The third operation is the absolute value of y. The absolute value of a number is its magnitude, or distance from zero, regardless of its sign. For example, the absolute value of -5 is 5 and the absolute value of 5 is 5. Mathematically, it can be written as |y|.

Finally, we have to find the square root of (xy to the power of a). The square root of a number is the value that, when multiplied by itself, gives the original number.

The square root of 25 is 5 because 5 * 5 = 25. This can be written mathematically as

=> √(xyᵃ).

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Answer:

Pour savoir si un graphique est proportionnel, il faut vérifier si les valeurs de la variable indépendante (l'axe des x) sont proportionnelles aux valeurs de la variable dépendante (l'axe des y). Cela signifie que si on multiplie les valeurs de la variable indépendante par un facteur constant, les valeurs de la variable dépendante seront également multipliées par ce même facteur constant.

Un moyen de vérifier si un graphique est proportionnel est de tracer une ligne droite à travers les points de données. Si tous les points se trouvent sur cette ligne droite, alors le graphique est proportionnel. Si les points ne se trouvent pas sur une ligne droite, alors le graphique n'est pas proportionnel.

Il est également possible de calculer le coefficient de proportionnalité en divisant chaque valeur de la variable dépendante par la valeur correspondante de la variable indépendante. Si le résultat est constant pour toutes les paires de valeurs, alors le graphique est proportionnel.

Enfin, il convient de noter que la proportionnalité n'est qu'un type de relation entre deux variables et que d'autres types de relations peuvent exister, tels que les relations exponentielles, logarithmiques ou quadratiques.

Step-by-step explanation:

The area is 5.627 square units and the perimeter is 12.167 units of the given figure.

Triangle is defined as a basic polygonal shape of a triangle that has three sides and three interior angles. It is one of the fundamental shapes in geometry and is represented by the symbol Δ.

perimeter of the given figure = 3 5/6 + 4 4/3 + 3

perimeter of the given figure = 12 1/6 = 73/6 = 12.167 units

Area of triangle = √[s(s – a)(s – b)(s – c)],

where s is the semi-perimeter of the triangle, and a, b, and c are the lengths of the three sides of the triangle.

According to the given question, we have

a = 3 5/6

b = 4 4/3

c = 3

s = (3 5/6 + 4 4/3 + 3)/2 = 12.167/2 = 6.083

Substitute the values in the area formula, and we get

Area of triangle = √[6.083(6.083 – 3 5/6)(6.083 – 4 4/3)(6.083 – 3)]

Area of triangle = 5.627 square units

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The question seems incomplete, the missing figure has been attached below.

The table for the parabola is

x y

-2 24

0 0

2 8

4 0

6     -24

To create a table for a parabola with equation y = -2x^2 + 8x, we can simply plug in values of x and solve for y.

with x-values from -2 to 6,:

For x = -2, y = -2(-2)^2 + 8(-2) = -24

For x = 0, y = -2(0)^2 + 8(0) = 0

For x = 2, y = -2(2)^2 + 8(2) = 8

For x = 4, y = -2(4)^2 + 8(4) = 0

For x = 6, y = -2(6)^2 + 8(6) = -24

To plot the parabola on a graph, we can plot the points from the table and draw a smooth curve through them. The graph of y = -2x^2 + 8x looks like a downward-facing parabola with its vertex at the point (2, 8).

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The volume of the solid shown can be found to be 105 cm ³ .

The volume of the solid can be found by the formula :

= Length x Width x Height

This is because this is a rectangular prism and their volumes require multiplying the sides of the rectangle in the above manner .

The volume of the solid is therefore:

= 35 x 1 x 3

= 35 x 3

= 105 cm ³

In conclusion, the volume of the solid would be 105 cm ³ .

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Answer: The slope of a line can be found using the formula:

m = (y2 - y1) / (x2 - x1)

We can choose any two points from the table to find the slope. Let's use the points (1, 10) and (5, -5):

m = (-5 - 10) / (5 - 1) = -15 / 4 = -3.75

So, the slope of the line represented by the table (x = 1, 3, 5, y = 10, 5, 0, -5) is -3.75.

Step-by-step explanation:

Your friend's mistake when wanting to make a new mixture of pink paint is that she is only going to add 2 parts of white paint, the correct formula would be 3.

Error is, the difference between a true value and an estimate, or approximation, of that value.

Given that, yesterday your friend made needs to make 3 more gallons of the same shade. She 1 gallon of dark pink paint using the information in the table, (attached in solution)

The calculation of how much paint you need to make 3 gallons of Pink paint :-

To mix 3 gallons of pink paint we must take into account the following information:

1 gallon of pink paint is equal to 3 parts red paint and 1 part white paint.

To find out how many parts we need to prepare 3 gallons of pink paint we must multiply the mix values for a gallon by 3.

Red paint:

3 x 3 = 9

White paint:

1 x 3 = 3

We need to use 3 parts white paint and 9 parts red paint to make 3 gallons of pink paint.

Hence, She needs to add 3 parts of white paint and 9 parts of red paint. The error is she added 2 parts of white paint.

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The question is incomplete so, I attached the graph.

Answer:

Opposite rays point in opposite directions and therefore make a 180-degree angle with each other.

Step-by-step explanation:

The test score of 12 is at the 40th percentile.

A Percentile divides a data set into the 100 equal parts. A percentile is a measurement that tells us what percent of the total frequency of a data set was at or below that measure.

From the question,

The test score of 12 is less than or equal to 7, 9, 10, 12, so 4 out of the 10 scores are less than or equal to 12.

So, the percentile of the test score 12 can be calculated as:

Percentile = (number of scores less than or equal to 12) / (total number of scores) x 100

percentile = (4) / (10) x 100

percentile = 40

Hence, the test score of 12 is at the 40th percentile.

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The three consecutive integers are 46, 47, 48.

A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words.

here, we have,

Let the three consecutive integers be represented as x,x+1,x+2

As per the requirement, largest integer is 44 less than twice the smallest.

=> x+ 2 = 2x -44

=> 2x -x = 2+44

=> x= 46

So the three integers are 46, 47, 48

Verification:

Largest integer = 48

Smallest integer = 46

2 * smallest integer - 44

= 2* 46 - 44

= 48

Largest integer is also 48.

So the required condition in the question is satisfied by these three integers.

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They are related as u^T transposes of each other. v and v^T u are scalar values that represent the dot product of u and v. uv^T and vu^T are matrices of size n x n that represent the outer product of u and v.

If u and v are vectors in R^n, then:

u^T v and v^T u are related as transposes of each other. Specifically, if

u = [u_1, u_2, ..., u_n]^T

and v = [v_1, v_2, ..., v_n]^T

then:

u^T v = v^T u

= u_1 * v_1 + u_2 * v_2 + ... + u_n * v_n

So, u^T v and v^T u are scalar values that represent the dot product of u and v.

uv^T and vu^T are related as transposes of each other. Specifically, if u = [u_1, u_2, ..., u_n]^T and v = [v_1, v_2, ..., v_n]^T

then:

uv^T = [u_1 * v_1, u_1 * v_2, ..., u_1 * v_n;

u_2 * v_1, u_2 * v_2, ..., u_2 * v_n;

...;

u_n * v_1, u_n * v_2, ..., u_n * v_n]

vu^T = [v_1 * u_1, v_2 * u_1, ..., v_n * u_1;

v_1 * u_2, v_2 * u_2, ..., v_n * u_2;

...;

v_1 * u_n, v_2 * u_n, ..., v_n * u_n]

So, uv^T and vu^T are matrices of size n x n that represent the outer product of u and v.

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-8a + 32b using the distributive property as 8(-a + 4b).

An expression is combination of variables, numbers and operators.

To express -8a + 32b using the distributive property, we need to factor out the greatest common factor that the terms share.

The greatest common factor of -8a and 32b is 8.

-8a + 32b

= 8(-a + 4b)

Using the distributive property, we can check that this expression is equivalent to the original:

8(-a + 4b) = 8(-a) + 8(4b) = -8a + 32b

Therefore, we can express -8a + 32b using the distributive property as 8(-a + 4b).

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Answer:

[tex]x>-5[/tex]

Step-by-step explanation:

[tex]2x+10>0 \implies 2x>-10 \implies x>-5[/tex]

Answer: 0.36

Step-by-step explanation:

A=L*W

Since all sides of a square is the same you just do 3/5(3/5) which is 0.36 or 9/25

The cost of a folder is: $20

The cost of a package of sheets is $50.

The system of equations with variables:

Juan paid 120 pesos for a folder and 2 packages of sheets:

C + 2P = 120

Clearing C:

C = 120 - 2P

Manuel paid 190 pesos for 3 packages of sheets and 2 folders:

2C + 3P = 190

Substituting the "C":

2 * (120 - 2P) + 3P = 190

240 - 4P + 3P = 190

-P = 190 - 240

-P = -50

P = $50

The price for a folder is:

C = 120 - 2 * 50

C = $20

Translated question "Juan paid 120 pesos for a folder and 2 packages of sheets while Manuel paid 190 pesos for 3 packages of sheets and 2 folders.Calculate the cost of a folder and a package of sheets."

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