example of writing a linear approximation and approximating the value of a functionIf y = 2x2 + x - 4, find the equation of the tangent when x = 1.

A function can have zero to many parameters, and it can return only one (option c).

In mathematics, a function is a rule that maps each element from one set, called the domain, to a unique element in another set, called the range.

The number of parameters a function can take determines the number of arguments required to call the function.

For instance, a function f(x) takes one parameter, and it requires one argument to be called.

On the other hand, a function g(x,y,z) takes three parameters, and it requires three arguments to be called. However, a function h() takes no parameters and does not require any argument to be called.

Hence the correct option is (c).

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From the given dimensions, area of the figure is approximately equal to 75.8 cm².

A regular polygon is a closed shape with straight sides and equal-length edges, as well as equal angles between those sides. Examples of regular polygons include equilateral triangles, squares, and hexagons. The number of sides a regular polygon has is referred to as its order, while the measure of each interior angle of a regular polygon can be calculated using the formula (n-2) x 180/n, where n represents the number of sides.

From the image we can see that, ABHI is a square with sides 6 cm, BCDGH is a regular pentagon with sides 6 cm (BH = AI = 6 cm) and DEFG is a rectangle with length 11 cm and breadth 6 cm ( DG = AI = 6 cm).

To find the area of the figure we have to find the area of each shape and add it together.

Area of square = side × side = 6 × 6 = 36 cm²

Area of regular pentagon = [tex]\frac{1}{4} \sqrt{5(5+25)a^{2} }[/tex]

= [tex]\frac{1}{4} \sqrt{5(5+25)6^{2} }[/tex]

= [tex]\frac{1}{4} \sqrt{25 + 10*1.41*36}[/tex]

= [tex]\frac{1}{4} \sqrt{532.6 }[/tex]

≈ 5.77 cm²

Area of rectangle = 2(l + b) = 2(11 + 6) = 34 cm²

Therefore area of the figure = 36 cm² + 5.77 cm² + 34 cm² ≈ 75.8 cm².

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The new vertices of the triangle after the translation are A'(-3, 7), B(-2, 0), and C(1, 7), which form the resultant matrix:

|-3 -2 1 |

| 7 0 7 |

| 1 1 1 |

In mathematics, a translation refers to a geometric transformation that moves every point in a figure or a space by the same distance in a given direction. In other words, it involves sliding a figure or a point to a new location without changing its size, shape, or orientation.

To translate the triangle by a vector (x, y), we use the following transformation matrix:

|1 0 x|

|0 1 y|

|0 0 1|

For the given triangle with vertices at A(-1, 3), B(0, -4), and C(3, 3), let's assume we want to translate it by a vector (−2, 4). Then the transformation matrix for translation is:

|1 0 -2|

|0 1 4 |

|0 0 1 |

To apply this transformation matrix to each vertex of the triangle, we represent the vertices as column vectors and multiply them by the matrix:

| -1 | | 1 0 -2 | | -3 |

| 3 | -> | 0 1 4 | = | 7 |

| 1 | | 0 0 1 | | 1 |

| 0 | | 1 0 -2 | | -2 |

| -4 | -> | 0 1 4 | = | 0 |

| 1 | | 0 0 1 | | 1 |

| 3 | | 1 0 -2 | | 1 |

| 3 | -> | 0 1 4 | = | 7 |

| 1 | | 0 0 1 | | 1 |

So, the new vertices of the triangle after the translation are A'(-3, 7), B(-2, 0), and C(1, 7), which form the resultant matrix:

|-3 -2 1 |

| 7 0 7 |

| 1 1 1 |

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a.) The position of the train moving from station A to station B after 10 minutes 20 km east

b.)  The position of the train moving from station A to station B after 10 minutes, if station B is assumed to be the origin is 50 km west

c.) The position of the train moving from station A to station B after 10 minutes, if station C is assumed to be the origin is 80 km west

we have that:

Railway station -     A            B              C

Distance(km) -       0            30             60

Starts from A , train reaches B be in 15 minutes

Starts from A , train reaches C be in 30 minutes

a.)

If A is origin

As train covers 30km = 15 minutes

⇒ 15 minutes = 30 km

   1 minute =  km

⇒ 10 minutes = 2×10 = 20 km

The position of the train moving from station A to station B after 10 minutes = 20 km east

b.)

If B is origin then , the position of  the train moving from station A to station B after 10 minutes = 30 + 20  = 50 km west

c.)

If C is the origin then , the position of  the train moving from station A to station B after 10 minutes = 60 + 20  = 80 km west

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#possible complete question:

Three railway stations, A, B, and C, are situated on a straight railway route. Station B is at 30 km in the east from station A and station C is at 60 km in east from station A. A train, with a constant average velocity, starts from A and reaches B in 15 minutes, and reaches C in 30 minutes. Assuming that station A is the origin, find the following:

a. The position of the train moving from station A to station B after 10 minutes.

b. The position of the train moving from station A to station B after 10 minutes, if station B is assumed to be the origin.

c. The position of the train moving from station A to station B after 10 minutes, if station C is assumed to be the origin.

Answer:

3 cm

Step-by-step explanation:

because the correct answer is 3cm

Answer:

2) The smallest amount of fabric Amy can buy is 78 inches = 2 1/6 yards, so she will need 2 1/4 yards of fabric.

3) $2.25(5) + $0.35(2) + $0.25(3) + $0.60 = $13.30 before sales tax

Sales tax is $13.30 × .08 = $1.06

Total is $13.30 + $1.06 = $14.36

The type of error that could have occurred in this scenario is a Type II error. A Type II error occurs when the null hypothesis is not rejected, even though it is false.

In this case, the null hypothesis is that the proportion of defective bulbs in the shipment is greater than or equal to 5%. By not rejecting this null hypothesis, the company is accepting the possibility that the shipment has a high proportion of defective bulbs, when in reality it does not. This can lead to significant costs for the company, such as the cost of returning the shipment or the cost of replacing defective bulbs in the future.

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For a 10,000 linear feet of fencing and wants to enclose a rectangular field, the maximum area of each pasture is equals to the 2,083,333.33 square feet .

A rancher wants to enclose about 10,000 linear feet of fencing in a rectangular field. There are two equal pastures. Let, W the Width and L the Length with 3 pieces. The total length will be equal to 2W + 3L. As we know, the Area of rectangle, A = L×W ---(1)

Perimeter of rectangle = 10,000 feet, so sum of all sides of rectangle, 2W + 3L = 10000

Solve for determining value of L, 3L

= 10000 - 2W

[tex]L = \frac{ 10000 - 2W }{3}[/tex]

Substitutes for L into the first equation, A = L×W

[tex]A = W(\frac{ 10000 - 2W }{3})[/tex]

For maximum area, set the 1st derivative = 0.

differentiating above Area equation w.r.t W,[tex] A = \frac{(10000 \: W - 2W^2)}{3}[/tex]

[tex] \frac{dA}{dW} = \frac{d(\frac{10000 \: W - 2W^2}{3})}{dW}[/tex]

=> [tex]\frac { 1}{3}(10000 - 4W) = 0 [/tex]

=> W = 2500 meters

Now, using above relation, 2W + 3L= 10000

=> 5000 + 3L = 10000

=> 3L = 5000

=> L = 1667 meters

Area = 2500× 5000/3 = 4,166,666.67 sq feet. Now, maximum area of each pasture

= 4,166,666.67/2 = 2,083,333.33333 sq. feet. Hence, required value is 2,083,333.33 sq. feet.

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The GCF of 84 and 36 is 12.

The GCF of two non-zero integers, x(36) and y(84), is the greatest positive integer m(12) that divides both x(36) and y(84) without any remainder.

GCF of 36 and 84 by Listing Common Factors

Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84

There are 6 common factors of 36 and 84, that are 1, 2, 3, 4, 6, and 12. Therefore, the greatest common factor of 36 and 84 is 12.

Rewrite in the form of a(b + c), a is the greatest common factor of 84 and 36

84/ 12 = 7

36/12 = 3

So, 12(84 + 36)

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We can predict that out of 200 repetitions of the experiment, approximately 75 of them will result in exactly 2 of the 3 flips landing on tails.

Probability is a branch of mathematics that deals with the study of random events and the likelihood of their occurrence. It is used to quantify the uncertainty associated with a particular event or outcome. In simpler terms, probability is a measure of the likelihood that a certain event will occur, expressed as a number between 0 and 1, where 0 indicates that the event is impossible and 1 indicates that the event is certain.

There are different types of probability, including classical probability, empirical probability, and subjective probability. Classical probability is based on the assumption of equally likely outcomes, while empirical probability is based on actual observations or experiments. Subjective probability, on the other hand, is based on personal judgment or belief.

The probability of getting tails on any single flip of a fair coin is 1/2. Since we are flipping the coin three times, the probability of getting exactly two tails is given by the binomial probability formula:

P(exactly 2 tails) = (number of ways to get 2 tails out of 3) * (probability of getting tails)² * (probability of getting heads)¹

The number of ways to get 2 tails out of 3 is given by the binomial coefficient "3 choose 2", which is 3. So we have:

P(exactly 2 tails) = 3 * (1/2)² * (1/2)¹ = 3/8

This means that the probability of getting exactly 2 tails in one trial of flipping the coin 3 times is 3/8.

To predict the number of repetitions out of 200 that will result in exactly 2 tails, we can multiply the probability of getting exactly 2 tails in one trial by the total number of trials:

Number of repetitions with exactly 2 tails = P(exactly 2 tails) * total number of trials

Number of repetitions with exactly 2 tails = (3/8) * 200

Number of repetitions with exactly 2 tails = 75

Therefore, we can predict that out of 200 repetitions of the experiment, approximately 75 of them will result in exactly 2 of the 3 flips landing on tails.

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A. a prism with a height of x cm and a square base with side length of x cm = option 2

B. a cylinder with a radius of x and height of cm= option 6

C. a pyramid with a height of x cm and a square base with a side length of x cm = option 3.

For statement A=

The formula for a square based prism= a²h

where a= X

h = X

Vol = x³

For statement B;

The formula for a cylinder=πr²h

where r= X, h= X

Vol= π×x³

For statement C;

The formula for a square based pyramid= 1/3a²h

a = X

H = X

Vol = 1/3 * x³

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Answer:

1) f(-2) = -2 + 3

2) f(-2) = 1

The equation of the circle is (x - 2)²+ (y - 3)² = 169

A circle is a two-dimensional form that may be described as a collection of points that are equally spaced apart from a central fixed point. The radius of a circle is the separation between its centre and any other point on the circle. Another way to think of a circle is as the collection of all points at a certain radius from the centre.

The following is the equation for a circle with centres (a, b) and radius r:

(x - a)² + (y - b)² = r²

In this case, the center of the circle is A(2,3), and a point on the circle is B(7,15). The radius of the circle may be calculated using the distance formula:

r = √((7 - 2)² + (15 - 3)²) = √(5² + 12²) = 13

So the equation of the circle is:

(x - 2)² + (y - 3)² = 13²

(x - 2)²+ (y - 3)² = 169

Therefore, the equation of the circle is (x - 2)²+ (y - 3)² = 169

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The two solutions to the system of equations are (x,y) = (7/5,-3) and (x,y) = (144/65,-12/13).

What is quadratic equation?

it's a second-degree quadratic equation which is an algebraic equation in x.

We can use substitution to solve this system of equations.

First, we solve the second equation for one of the variables, say x:

5x + 12y + 13 = 0

5x = -12y - 13

x = (-12/5)y - (13/5)

Next, we substitute this expression for x into the first equation:

x² + y² = 1

[(-12/5)y - (13/5)]² + y² = 1

Expanding the square and simplifying, we get:

y² + [(144/25)y² + (312/25)y + (169/25)] + y² = 1

(26/5)y² + (312/25)y + (144/25) = 0

Multiplying both sides by 25 to eliminate the fractions:

26y² + 312y + 144 = 0

Dividing by 4 to simplify:

6.5y² + 78y + 36 = 0

Using the quadratic formula:

y = (-b ± sqrt(b² - 4ac)) / 2a

where a = 6.5, b = 78, and c = 36.

Plugging in these values:

y = (-78 ± sqrt(78² - 4(6.5)(36))) / 2(6.5)

y = (-78 ± sqrt(4761)) / 13

y = (-78 ± 69) / 13

y = -3 or -12/13

If y = -3, then substituting into the equation for x gives:

x = (-12/5)(-3) - (13/5) = 7/5

So one solution is (x,y) = (7/5,-3).

If y = -12/13, then substituting into the equation for x gives:

x = (-12/5)(-12/13) - (13/5) = 144/65

So another solution is (x,y) = (144/65,-12/13).

Therefore, the two solutions to the system of equations are (x,y) = (7/5,-3) and (x,y) = (144/65,-12/13).

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Answer:

Step-by-step explanation:

36

The trend line for the y = 0.5x + 2 is attached accordingly. Note that the the predicted gas price for month 13 would be $8.50.

a. To draw a trend line on a scatter plot, you need to perform a linear regression analysis. This involves finding the line of best fit that passes through the data points. The equation for a linear regression line is typically of the form y = mx + b, where y is the dependent variable (gas price in this case), x is the independent variable (month), m is the slope of the line, and b is the y-intercept.

Once you have the equation for the line of best fit, you can plot it on the scatter plot to visualize the trend. The slope of the line will tell you the direction and steepness of the trend (i.e., whether prices are increasing or decreasing, and how quickly).

b. To predict future values using the trend line, you can simply plug in the value of the independent variable (month) for the month you want to predict, and solve for the dependent variable (gas price). For example, if the equation for the trend line is y = 0.5x + 2, and you want to predict the gas price for month 13, you would plug in x = 13 and solve for y:

y = 0.5(13) + 2

y = 6.5 + 2

y = 8.5

So the predicted gas price for month 13 would be $8.50.

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Answer:

20 3/5

Step-by-step explanation:

this is the correct result when 30 9/10 is multipled by 2/3

Understanding the meaning of addition and subtraction will allows us to make predictions about how changing the value of a variable will affect the value of an expression.

In mathematics, integers are a set of whole numbers that include both positive and negative numbers, as well as zero. The set of integers is denoted by the symbol Z, and it includes all the natural numbers (1, 2, 3,...), their negatives (-1, -2, -3,...), and zero (0). Integers are used in a variety of mathematical contexts, including algebra, number theory, and geometry. Integers can be used to represent many real-world situations, such as temperatures above or below zero, changes in altitude, gains or losses in money, and more. They are also used extensively in computer programming and other fields that rely heavily on mathematics.

Knowing what addition and subtraction mean allows you to understand how changing the value of a variable can affect the value of an expression.

For example, consider the expression x + 2. If x has a value of 3, then the value of the expression is 5. However, if x increases to 4, then the value of the expression increases to 6. This is because we are adding 2 to the value of x.

On the other hand, if we have the expression x - 2 and x has a value of 3, then the value of the expression is 1. If x increases to 4, then the value of the expression decreases to 2. This is because we are subtracting 2 from the value of x, so as x increases, the value of the expression decreases.

Understanding the meaning of addition and subtraction is essential to be able to manipulate expressions and variables, and it allows us to make predictions about how changing the value of a variable will affect the value of an expression.

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Answer:c. F(x)=2x

Step-by-step explanation:

Answer: 80

Step-by-step explanation:

The sum of the NPV and false omission rate for any test will always be 100%.

The NPV (Negative Predictive Value) is the proportion of people who test negative for a condition and actually do not have it, while the false omission rate is the proportion of people who have the condition but test negative for it.

When we add together the NPV and the false omission rate for any test, we are essentially considering all the cases where the test result is negative.

The NPV represents the proportion of people who truly do not have the condition and test negative for it, while the false omission rate represents the proportion of people who actually have the condition but test negative for it.

Together, these two values cover all possible cases where the test result is negative, which means that they represent the entirety of the negative results for the test.

Since the sum of all probabilities for an event must always equal 100%, the sum of the NPV and false omission rate must also equal 100%.

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Based on the information, the y-intercept is (0, 6.5).

The function is increasing.

Based on the information, we can use the variable x. This will be:

0 = 6.5 + 5.25%x

-6.5 = 5.25%x

x = -6.5 / 5.25% ≈ -123.81

There's no x intercept due to the negative value which doesn't make sense in this scenario.

In order to find the y-intercept, we set x = 0 and evaluate b(x):

b(0) = 6.5 + 5.25% × 0 = 6.5

Therefore, the y-intercept is (0, 6.5).

The function b(x) is increasing since the coefficient of x is positive (5.25%). This means that as the number of engines produced (x) increases, the time it takes to manufacture the engines also increases.

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Answer:

f(x) = 2(x + 1)² + 1

Step-by-step explanation:

the equation of a parabola in vertex form is

f(x) = a(x - h)² + k

where (h, k ) are the coordinates of the vertex and a is a multiplier

given

f(x) = 2x² + 4x + 3 ← factor out 2 from the first two terms

    = 2(x² + 2x) + 3

using the method of completing the square

add/subtract ( half the coefficient of the x- term )² to x² + 2x

f(x) = 2(x² + 2(1)x + 1 - 1) + 3

     = 2(x + 1)² - 2(1) + 3

     = 2(x + 1)² - 2 + 3

    = 2(x + 1)² + 1 ← in vertex form

The probability that Ann picked a ball from the second box given that she selected an orange ball is 0.56 or 56% (rounded to two decimal places).

We can use Bayes' theorem to calculate the probability that Ann picked a ball from the second box given that she selected an orange ball.

Let A be the event that Ann selected an orange ball, and B be the event that Ann picked a ball from the second box. We want to find P(B|A), the probability that Ann picked a ball from the second box given that she selected an orange ball.

We know that there are two boxes, each with a probability of 1/2 of being selected. The probability of selecting an orange ball from the first box is 3/7, and the probability of selecting an orange ball from the second box is 7/11. Therefore, the probability of selecting an orange ball overall is:

P(A) = P(A|B)P(B) + P(A|B')P(B')

= (7/11)(1/2) + (3/7)(1/2)

= 25/42

Now we can use Bayes' theorem:

P(B|A) = P(A|B)P(B)/P(A)

= (7/11)(1/2)/(25/42)

= 14/25

Therefore, the probability that Ann picked a ball from the second box given that she selected an orange ball is 0.56 or 56% (rounded to two decimal places).

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Answer:

When 2.7 is multiplied by 10 to the 2nd power, it becomes 270. The digit 7 is in the ones place, so the value of the digit 7 is 7.

Step-by-step explanation:

The correct list of functions ordered from 0 ≤ x ≤ 3 is:

h(x) = sin(x) < f(x) = x^2 < g(x) = 2x + 1 < k(x) = e^x

Line connecting the interval's endpoints in order to compute the average rate of change for each function over range 0 x 3.

For f(x) = x^2, the slope between x = 0 and x = 3 is:

[tex](f(3) - f(0)) / (3 - 0) = (9 - 0) / 3 = 3[/tex]

For g(x) = 2x + 1:

[tex](g(3) - g(0)) / (3 - 0) = (7 - 1) / 3 = 2[/tex]

For h(x) = sin(x):

[tex](h(3) - h(0)) / (3 - 0) = (sin(3) - sin(0)) / 3[/tex] ≈ 0.279

For k(x) = e^x, the slope between x = 0 and x = 3 is:

[tex](k(3) - k(0)) / (3 - 0) = (e^3 - 1) / 3[/tex] ≈ 6.076

Therefore, correct list of functions ordered from least to greatest by average rate of change over the interval 0 ≤ x ≤ 3 is:

[tex]h(x) = sin(x) < f(x) = x^2 < g(x) = 2x + 1 < k(x) = e^x[/tex]

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--The complete Question is, Consider the following four functions:

f(x) = x^2

g(x) = 2x + 1

h(x) = sin(x)

k(x) = e^x

What is the correct list of functions ordered from least to greatest by average rate of change over the interval 0 ≤ x ≤ 3? --

Answer:

32m

Step-by-step explanation:

area= L×B

L= 10m ×2 =20m

B= 6m×2 =12m

Area= 20m + 12m = 32m

Answer:

A

Step-by-step explanation:

It is not a function because both x-values -4 and 2 have multiple y-values. Try the vertical line test if you don't understand.

Answer:

Less

Step-by-step explanation:

Look at the hundreds number in set L the numbers are greater in Set L than Set K.

The missing angles are sides are;

We can see that we would have to first apply the cosine rule here;

[tex]a^2 = b^2 + c^2 - 2abCos A\\a^2 = (17)^2 + (19.6)^2 - 2(17 * 19.6)Cos 30\\a^2 = 673.16 - 666.4Cos 30[/tex]

a = 9.8

We can now apply the sine rule to obtain the angle B

[tex]a/Sin A = b/Sin B\\SinB = bSin A/a\\B = Sin-1 ( bSin A/a)\\B = Sin-1(17 * Sin 30/9.8)[/tex]

B = 60

Then;

C = 180 - (30 + 60)

C = 90 degrees

Thus the triangle in the problem have been solved.

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