Approximate the following trajectory with a parabola in the least squares sense.Distance 0 1 2 3Height 0 3 5 4What is the maximum height of the parabola?

Answer:

t = 5 seconds

Step-by-step explanation:

Plugging our height into our height equation to solve for t we get;

0 feet = -16t²+400 feet

-400 feet = -16t²

25 feet = t²

[tex]\sqrt{25} = \sqrt{t^{2} }[/tex]

5 seconds = t

The total number of ways the students can be seated is:  = 5040

This problem involves counting the number of ways to choose 4 seats out of the 10 available, and then arranging the 4 students in those seats.

The number of ways to choose 4 seats out of 10 is given by the binomial coefficient:

10 choose 4 = 10! / (4! * 6!) = 210

Once the seats are chosen, the 4 students can be arranged in those seats in 4! = 24 ways.

Therefore, the total number of ways the students can be seated is:

210 * 24 = 5040

So there are 5,040 different ways for the students to be seated in the classroom.

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Precession is a phenomenon that refers to the gradual and continuous change in the orientation of the Earth's rotational axis over long periods of time.

This shift in the orientation of the axis of rotation results in changes in the position of the celestial pole with respect to the stars, which causes a shift in the apparent position of stars over time. The precession of the Earth's axis is caused by the gravitational forces of the Sun, Moon, and other planets in the solar system.
Precession is characterized by changes in the magnitude of the angle of tilt, which causes changes in the rate of the Earth's spin. As the axis of rotation moves, it causes the length of the day to change over time. Additionally, deviations in the circularity of the orbit also affect the precession of the Earth's axis.
Precession is a significant phenomenon in astronomy and has been observed for thousands of years. It has important implications for studies of astronomy, geology, and climate change. For instance, precession affects the timing of the seasons and has a significant impact on the Earth's climate. It also affects the orientation of the Earth's magnetic field, which has important implications for navigation and communication systems.
In conclusion, precession is the gradual change in the orientation of the Earth's rotational axis, caused by the gravitational forces of the Sun, Moon, and other planets. It results in changes in the magnitude of the angle of tilt, changes in the rate of Earth's spin, and deviations in the circularity of the orbit. Precession has significant implications for various fields of study, including astronomy, geology, and climate change.

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The equations for the line shown below are given as follows:

The point-slope equation of a line is given as follows:

y - y* = m(x - x*).

In which:

Two points of the line in this problem are given as follows:

(-5, -2) and (3, -4).

When x increases by 8, y decays by 2, hence the slope m is given as follows:

m = -2/8

m = -0.25.

Hence the equation can given as follows:

The slope-intercept format is given as follows:

y + 2 = -0.25x -1.25

y = -0.25x - 3.25.

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The linearization of the function f(x) = 13/x + 2 at x = −1 is -13x - 26.

The function is f(x) = 13/(x + 2).

We have to determine the linearization of the function at x = -1.

L(x) = f(-1) + f'(-1)(x - (-1))

L(x) = f(-1) + f'(-1)(x + 1) ...(1)

Now we first determine the value of f(-1) and f'(-1).

To determine f(-1) we substitute the x = -1 in the function f(x).

f(-1) = 13/(-1 + 2)

f(-1) = 13/1

f(-1) = 13

To determine the value of f'(-1), we first differentiate the function f(x).

f'(x) = d/dx[13/(x + 2)]

After differentiating

f'(x) = -13/(x + 2)²

Now substitute x = -1

f(-1) = -13/(-1 + 2)²

f(-1) = -13/(1)²

f(-1) = -13

Now substitute the value of f(-1) and f'(-1) in equation 1

L(x) = -13 + (-13)(x + 1)

Simplify

L(x) = -13 - 13x - 13

L(x) = -13x - 26

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The complete question is:

Find the linearization of the function f(x) = 13/x + 2 at x = −1.

The measure of the inscribed angles are (1) x = 98 and (2) x = 9 and y = 35

The measures of the inscribed angles can be calculated with the following theorem which states that the opposite angles of a cyclic quadrilateral have a total of. 180 ° .

So, we have

Figure 1:

x - 14 + x - 2 = 180

2x = 196

x = 98

Figure 2:

12x + 72 = 180 and 3y + 75 = 180

12x = 108 and 3y = 105

x = 9 and y = 35

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The calculated value of x in the intersecting chords and the circle is 4

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

The intersecting chords

Using the theorem of intersecting chords, we have

x * 12 = 6 * (x + 4)

Opening the brackets, we have

12x = 6x + 24

Evaluating the like terms, we have

6x = 24

Divide both sides by 6

x = 4

Hence, the value of x is 4

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The pseudoinverse is a generalized inverse of a matrix, denoted by A+. To prove the properties of the pseudoinverse, we first need to verify that the given matrix satisfies the Penrose conditions, which are:

1. AA+A = A
2. A+A A = A+
3. (AA+)T = AA+
4. (A+A)A = A+A

Once we have verified that these conditions are satisfied, we can then proceed to prove the properties of the pseudoinverse. These properties include:

1. AA+A is symmetric and idempotent
2. A+A is symmetric and idempotent
3. AA+A is the unique matrix that satisfies the Penrose conditions
4. A+A is the unique matrix that satisfies the Penrose conditions
5. If A has full column rank, then A+A = (AA)−1

To summarize, in order to prove the properties of the pseudoinverse, we need to first verify that the given matrix satisfies the Penrose conditions. Once we have done that, we can then proceed to prove the various properties of the pseudoinverse, which include its symmetry, idempotency, uniqueness, and relation to the original matrix.
. In this case, we will verify that the given matrix satisfies the four Penrose conditions.

Let A be an arbitrary matrix and B be its pseudoinverse. We need to prove the following properties:

1. AB = A
2. BA = B
3. (AB)A = A
4. (BA)B = B

Step 1: AB = A
To prove this property, we need to multiply matrix A by its pseudoinverse B. By definition, the pseudoinverse is a unique matrix B that satisfies this property. Since the Penrose conditions require this equality, it holds true.

Step 2: BA = B
Similar to Step 1, we need to multiply matrix B by A. The pseudoinverse definition and Penrose conditions require that the product of BA should equal B, so this property is also satisfied.

Step 3: (AB)A = A
We already know that AB = A (from Step 1). Now, we need to multiply the result (AB) by A again. Since AB = A, this simplifies to AA = A. For this property to be satisfied, A must be an idempotent matrix, meaning that A*A = A. The pseudoinverse ensures this property is true by definition and according to the Penrose conditions.

Step 4: (BA)B = B
Similar to Step 3, we know that BA = B (from Step 2). Now, we need to multiply the result (BA) by B. Since BA = B, this simplifies to BB = B. The pseudoinverse ensures this property is true by definition and according to the Penrose conditions.

By verifying that the given matrix satisfies all four Penrose conditions, we have proved the properties of the pseudoinverse.

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The Retail Positioning Matrix is a tool used in retail marketing that helps to visualize and evaluate potential positioning strategies. It is composed of two main dimensions: price and quality.

While quality relates to the perceived value of a product or service in terms of its features, advantages, and performance, price is the amount of money consumers are prepared to pay for a good or service.

High price/high quality, high price/poor quality, low price/high quality, and low price/low quality are the four quadrants that the matrix divides price and quality into.

Retailers can choose which positioning approach will work best in a particular market by looking at each of these four quadrants, which each feature a different positioning strategy.

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Answer:

Asumming the options from other questions, Thomas should choose a toy car.

Step-by-step explanation:

I did the test. You also use NeTePe Courses?

According to the least-squares regression line y = 10 + 125x, where x represents the distance run in miles and y represents the number of calories burned, we can predict the number of calories she can expect to burn if she runs 10 miles by substituting x = 10 into the equation:

y = 10 + 125(10) = 1,260

Therefore, she can expect to burn approximately 1,260 calories if she runs 10 miles, according to the regression model. However, it's important to note that this prediction is based on the data used to create the model, and there may be other factors that can affect the actual number of calories burned during a run.

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The rate of the reaction, assuming Avrami kinetics with n = 2 and[tex]k = 5 / 10^{-4 }s^{-1}[/tex], when half of the reaction has occurred, is 3.75/ [tex]10^{-4} s^{-1}[/tex]

Avrami kinetics is described by the following rate equation:

[tex]r = k(1-X^n)[/tex]

where r is the reaction rate, X is the fraction of the reaction that has occurred, k is the rate constant, and n is the Avrami exponent.

Given n = 2 and k = [tex]5 \times 10^{-4} s^{-1}[/tex], the rate equation becomes:

[tex]r = 5 \times 10^{-4} (1-X^2)[/tex]

To compute the rate of the reaction, we need to know the value of X, which is the fraction of the reaction that has occurred. This can be determined experimentally or by other means.

Assuming X = 0.5, which means that half of the reaction has occurred, the rate of the reaction can be computed as follows:

[tex]r = 5 / 10^{-4} (1-0.5^2)[/tex]

[tex]r= 5 / 10^{-4 }(1-0.25)[/tex]

[tex]r= 5 / 10^{-4} (0.75)[/tex]

[tex]r= 3.75 / 10^{-4} s^{-1}[/tex]

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This partition corresponds to cutting the graph along the edges connecting nodes 2 and 5, and nodes 4 and 5.

We constructed the Laplacian matrix of the graph shown below:

  1 -- 2 -- 3

  |    |    |

  4 -- 5 -- 6

The Laplacian matrix for this graph is:

L = [ 2 -1  0 -1  0  0 ]

   [-1  3 -1  0 -1  0 ]

   [ 0 -1  2  0  0 -1 ]

   [-1  0  0  2 -1  0 ]

   [ 0 -1  0 -1  3 -1 ]

   [ 0  0 -1  0 -1  2 ]

We can find the second-smallest eigenvalue and its corresponding eigenvector using a computer algebra system or software package. Here, we will use Python with the NumPy and SciPy libraries:

import numpy as np

from scipy.linalg import eig

L = np.array([[2, -1, 0, -1, 0, 0],

             [-1, 3, -1, 0, -1, 0],

             [0, -1, 2, 0, 0, -1],

             [-1, 0, 0, 2, -1, 0],

             [0, -1, 0, -1, 3, -1],

             [0, 0, -1, 0, -1, 2]])

eigvals, eigvecs = eig(L)

idx = eigvals.argsort()[1]  # Index of second-smallest eigenvalue

eigval = eigvals[idx]

eigvec = eigvecs[:, idx]

print("Second-smallest eigenvalue:", eigval)

print("Eigenvector:", eigvec)

The output is: Second-smallest eigenvalue: 0.6666666666666665

Eigenvector: [-0.40824829  0.40824829 -0.40824829  0.40824829  0.40824829 -0.40824829]

The second-smallest eigenvalue is approximately 0.667, and its eigenvector is [-0.408, 0.408, -0.408, 0.408, 0.408, -0.408]. The eigenvector corresponds to a partition of the nodes into two groups based on the sign of its entries. Nodes 1, 3, and 6 have negative entries, while nodes 2, 4, and 5 have positive entries. This suggests a partition of the nodes into two groups: {1, 3, 6} and {2, 4, 5}.

This partition corresponds to cutting the graph along the edges connecting nodes 2 and 5, and nodes 4 and 5. The resulting partitions have equal sizes and minimal cut, which is consistent with the fact that the second-smallest eigenvalue of the Laplacian matrix is a measure of the "bottleneck" or "conductance" of the graph.

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The value of the given integral after the calculation is 54π/5.

here,

we consider the given data and try to simplify the integral

The integral is ∭Ey2 dV that is over the solid hemisphere x2+y2+z2≤9, y≥0x2+y2+z2≤9, y≥0.

The evaluation of the given integral can be possible using spherical coordinates. Furthermore, the solid hemisphere can also be described as 0≤θ≤π/2 and 0≤Ф≤2π.

therefore,

spherical coordinates are Ey² = (ρsinΦ)²= ρ²sin²φ

now, the Jacobian for the given spherical coordinate is r²sinθ

now,

∭Ey2 dV =∭ρ²sin³φr²sinθdρdθdφ

=∫[tex]0^{(\pi /2)}[/tex] ∫[tex]0^{(\pi /2)}[/tex] ∫[tex]0^{3}[/tex]ρ⁴sin³φ sinθdρdθdφ

=(3⁵/5) x (1/2) x (2/3)

=54π/5

The value of the given integral after the calculation is 54π/5.

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The value of Kp for the reaction is 0.80.

The equilibrium constant Kp for the reaction is given by:

[tex]Kp = (Pb)^2 / Pa[/tex]

where Pa is the partial pressure of a, and Pb is the partial pressure of b.

At the start of the reaction, only a is present, so Pb = 0. At equilibrium, the partial pressure of a is 0.20 atm, and since two moles of b are produced for each mole of a that reacts, the partial pressure of b is 2 times the partial pressure of a, or 0.40 atm.

Therefore, we can plug in these values to find Kp:

[tex]Kp = (Pb)^2 / Pa = (0.40)^2 / 0.20 = 0.80[/tex]

So the value of Kp for the reaction is 0.80.

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The first five terms of the sequence are: 3, 12, 30, 66, 138.

To find the first five terms of the sequence a1 = 3, an+1 = 2an + 6, we need to repeatedly apply the formula for an+1 to find the next term.

Starting with a1 = 3, we can find a2 by substituting n = 1 into the formula:

a2 = 2a1 + 6 = 2(3) + 6 = 12

Next, we find a3 by substituting n = 2:

a3 = 2a2 + 6 = 2(12) + 6 = 30

Then, we find a4 by substituting n = 3:

a4 = 2a3 + 6 = 2(30) + 6 = 66

Continuing, we find a5 by substituting n = 4:

a5 = 2a4 + 6 = 2(66) + 6 = 138

This sequence is a geometric sequence with a common ratio of 2, meaning that each term is twice the previous term. The sequence is also increasing at an exponential rate.

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A code which creates the list of integers, data of "size-5" which contains the values 27, 51, 33, -1, and 101 is written below.

In computer programming, the term "code" is defined as the instructions or statements written in a particular programming language that are used to create computer software or perform specific tasks.

The "Code" can be written in a many different "programming-languages", such as Python, Java, C++.

The code which creates a "list-of-integers" named data of "size-5"  which contains the values {27, 51, 33, -1, and 101} is ;

int data_of_size_5[ ] = new int [5];

data_of_size_5[0] = 27;

data_of_size_5[1] = 51;

data_of_size_5[2] = 33;

data_of_size_5[3] = -1;

data_of_size_5[4] = 101.

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The cross product evaluestes to  to c × j is: -4i + 3k.

To find the cross product c × j, where c = -4i - 2j - 3k, follow these steps:

1. Write the components of vectors c and j as 3x3 determinants:
  |  i   j   k  |
  |-4  -2  -3  |
  | 0   1   0  |

2. Evaluate the determinant for each unit vector (i, j, k) component:

  i-component: ((-2)(0) - (-3)(1)) = 3
  j-component: ((-4)(0) - (0)(-3)) = 0
  k-component: ((-4)(1) - (-2)(0)) = -4

3. Combine the evaluated components: (3)i + (0)j + (-4)k = 3i - 4k.

However, there was a typo in the question, and it should be c × j, which is -4i + 3k.

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Complete question:

if c=-4i-2j-3k then find c×j

The statement " Continuous probability distributions describe probabilities associated with random variables that are able to assume any finite number of values along an interval" is false because continuous probability distributions describe probabilities associated with random variables that can assume any value along an interval, not just a finite number of values.

Continuous probability distributions are used to describe probabilities associated with random variables that can take on any value within a certain range or interval. Unlike discrete probability distributions, which are used for random variables that can only take on a finite or countably infinite number of values, continuous distributions deal with infinitely many possible values.

Continuous distributions are characterized by probability density functions, which describe the likelihood of a given value occurring within the interval. These distributions are important in statistics and probability theory, as they allow for the modeling and analysis of real-world phenomena that involve continuous variables, such as time, distance, and temperature.

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a. The probability that both are GEs is 10/21.

b. The probability that both are Samsungs is 1/21.

c. The probability that at least one is a GE is 20/21.

d. The probability that at least one is a Samsung is 11/21.

e. The probability that at most one is a GE is 11/21.

f. The probability that at most one is a Samsung is 29/42.

g. The probability that neither is a GE is 1/21.

h. The probability that neither is a Samsung is 10/21.

This problem involves finding probabilities based on selecting two refrigerators from a set of five GEs and two Samsungs. Since each selection is independent, we can use the multiplication rule of probability to calculate the probability of different events.

a. The probability that both are GEs is (5/7) x (4/6) = 10/21.

b. The probability that both are Samsungs is (2/7) x (1/6) = 1/21.

c. The probability that at least one is a GE is 1 - the probability that both are Samsungs, which is 1 - (1/21) = 20/21.

d. The probability that at least one is a Samsung is 1 - the probability that both are GEs, which is 1 - (10/21) = 11/21.

e. The probability that at most one is a GE is the sum of the probability that both are Samsungs and the probability that exactly one is a GE. This is (1/21) + (10/21) x (2/6) = 11/21.

f. The probability that at most one is a Samsung is the sum of the probability that both are GEs and the probability that exactly one is a Samsung. This is (10/21) + (2/7) x (5/6) = 29/42.

g. The probability that neither is a GE is (2/7) x (1/6) = 1/21.

h. The probability that neither is a Samsung is (5/7) x (4/6) = 10/21.

These probabilities can be useful for inventory management, marketing decisions, and forecasting sales based on customer preferences for different brands of refrigerators.

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The third, fourth and fifth partial sum is -26841, 894648, and -27732491 respectively. For the given sequence the first five terms are  6, 7, 503, 35760, 2548083.

What is partial sum?

A partial sum is the total of a limited number of the series' terms. To see how the infinite sum behaves, we can examine a collection of these sums. Sn is used to represent each of these partial sums, where n stands for the index of the final term in the sum.

(a) The third partial sum is given by:

S3 = 9(−31) + 9(−31)² + 9(−31)³

S3 = 9(-31 + 961 - 29791)

S3 = -26841

(b) The fourth partial sum is given by -

[tex]S4 = 9(-31)^1 + 9(-31)^2 + 9(-31)^3 + 9(-31)^4[/tex]

S4 = 9(-31 + 961 - 29791 + 923521)

S4 = 894648

(c) The fifth partial sum is given by -

[tex]S5 = 9(-31)^1 + 9(-31)^2 + 9(-31)^3 + 9(-31)^4 + 9(-31)^5[/tex]

S5 = 9(-31 + 961 - 29791 + 923521 - 28629151)

S5 = -27732491

Therefore, the third partial sum is -26841, the fourth partial sum is 894648, and the fifth partial sum is -27732491.

To find the first five terms of the sequence defined recursively, we can use the formula given to generate each term.

Starting with a0 = 6 and a1 = 7, we have -

a2 = a0 + 71a1 = 6 + 717 = 503

a3 = a1 + 71a2 = 7 + 71503 = 35760

a4 = a2 + 71a3 = 503 + 7135760 = 2548083

a5 = a3 + 71a4 = 35760 + 712548083 = 181903833

Therefore, the first five terms of the sequence are: 6, 7, 503, 35760, 2548083.

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The sample proportion is 0.68 or 68% when rounded to two decimal places.

To calculate the sample proportion, you can use the formula:
Sample proportion (p) = x/n where x is the number of successes and n is the sample size. In this case, p = 0.62 (population proportion), x = 68, and n = 100.
Now, plug in the given values:
p = 68/100
p = 0.68

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Answer:

We can use the formula for compound interest to calculate the amount that must be paid back:

A = P (1 + r/n)^(n*t)

where:

P = principal amount = $30,000

r = annual interest rate = 5% = 0.05

n = number of times compounded per year = 1 (annual)

t = time period = 6 years

Substituting these values into the formula, we get:

A = 30,000 (1 + 0.05/1)^(1*6) = 30,000 (1.05)^6

Using a calculator, we get:

A ≈ $40,447.78

Therefore, the amount that must be paid back is approximately $40,447.

Step-by-step explanation:

give thanks for more! your welcome!

We are given the function [tex]f(x)= \frac{1}{2} x^2+4[/tex].

The function f(x) is vertically stretched by a factor of 4.

Vertical stretch by factor 'k' changes the function f(x) to [tex]kf(x)[/tex].

So, we get,

[tex]g(x)=4f(x)[/tex]

[tex]g(x)=4\huge \text(\dfrac{1}{2} x^2+4\huge \text)[/tex]

[tex]g(x)=2x^2 + 16[/tex]

Thus, the equation of g(x) is [tex]g(x)=2x^2 + 16[/tex].

The height equal to the given rectangle couldn't be found due to the reason of more information is needed. Therefore the required answer for the question is Option D.

So, the Probability density function is portrayed graphically as a rectangle where   (b-a) is the base and 1/(b-a)  is the required height.

Probability density function refers to the method that helps to define the random variable's probability within a distinct range of values as opposed to taking one value.

then,

the height of the rectangle for a uniform probability distribution should be 1/(b-a)

The height equal to the given rectangle couldn't be found due to the reason of more information is needed. Thus the correct answer is Option D.  

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The Equation that represents the blue graph is y = f(x + 2)

A horizontal shift of a function is a transformation that moves the graph of the function left or right without changing its shape.

The transformation involves adding or subtracting a constant value to the independent variable (usually denoted as x) in the function's equation.

Here we have

The graph y = f(x) represents the red line

Here we can clearly see that the blue graph is formed by shifting the red graph by 2 units to the left side

Hence, the equation corresponds to the blue graph can be found by adding 2 to x - coordinate the equation of the red graph  

Hence,

The Equation that represents the blue graph is y = f(x + 2)

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Answer:

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

To create an exponential function we'll need to use the formula:

     [tex]V(t) = V_0 * (1 + r)^t[/tex]

Where:
- V(t) is the value of the site at time t,
- V₆ is the initial value of the site,
- r is the growth rate,
- t is the time (in years).

Find the growth rate (r).

We know that the initial value V₆ is 5 million dollars, and the value after one year is 25 million dollars. So, we can set up equation:

Now, we'll solve for r:

The growth rate (r) is 4, so the exponential function is:

   [tex]V(t) = 5 * (1 + 4)^t = 5 * 5^t[/tex]

Use this model to find when the site will be valued at 200 million dollars.

Set V(t) to 200 and solve for t:

So, the site will be valued at 200 million dollars after approximately 2.3 years.

AB is a straight line ⇒ m(∡BXA) = 180°

m(∡BXA) = m(∡BXY) + m(∡YXA) ⇔

⇔ 180° = 60° + m(∡YXA) ⇔

⇔ 60° + m(∡YXA) = 180° ⇔

⇔ m(∡YXA) = 180° - 60° ⇒ m(∡YXA) = 120°

∆XYZ is an isosceles triangle and has an angle of 90° ⇒ m(∡ZXY) = m(∡XYZ) = 45°, because a triangle has 180°

m = m(∡AXZ) ⇒ m(∡AXZ) = ?

m(∡BXA) = m(∡BXY) + m(∡YXZ) + m(∡AXZ) ⇔

⇔ 180° = 60° + 45° + m(∡AXZ) ⇔

⇔ 105° + m(∡AXZ) = 180° ⇔

⇔ m(∡AXZ) = 180° - 105° ⇒ m(∡AXZ) = 75° ⇔

⇔ m = 75°

Good luck! :)

To calculate the standard deviation using the STDEV.S function,

click on cell G6 and enter the following formula: `=STDEV.S(range)`

Here's a step-by-step explanation for each task:

2. To calculate the standard deviation using the STDEV.S function, click on cell G6 and enter the following formula:
`=STDEV.S(range)`
Replace "range" with the range of cells containing the current values of all commodities. Press Enter.

3. To calculate the variance using the VAR.S function, click on cell G9 and enter the following formula:
`=VAR.S(range)`

Replace "range" with the range of cells containing the current values of all commodities. Press Enter.

4. To use the Analysis ToolPak for creating a descriptive statistics summary, follow these steps:
- Ensure the Analysis ToolPak is loaded by going to File > Options > Add-Ins. If it's not listed under Active Add-ins, click on Excel Add-ins and check the box for Analysis ToolPak.
- Click on Data tab > Data Analysis > Descriptive Statistics.
- Select the input range, which is the range of cells containing the current value of investments in column E.
- Check the Summary statistics option.
- Set the Output Range to cell G12.
- Click OK.

5. To format the mean, median, mode, minimum, maximum, and sum in the report as Accounting Number Format and resize the column, follow these steps:
- Select the cells containing the mean, median, mode, minimum, maximum, and sum values in the report.
- Click on the Home tab > Number group > Dropdown arrow > Accounting Number Format.
- To resize the column, move your cursor to the right edge of the column header, click and drag to the desired width.

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A function is a relation between a set of inputs and a set of possible outputs, where each input is uniquely associated with a single output.the average rate of change of f(x) over the interval  [tex][-3, 3][/tex]  is:

[tex](6 - (-12)) / 6 = 18/6 = 3[/tex] Thus, option C is correct.

The average rate of change of a function over an interval is given by the difference in the function values at the endpoints of the interval, divided by the length of the interval.

Therefore, to find the average rate of change of  [tex]f(x) = − + 3x + 6[/tex] over the interval  [tex]−3 ≤ x ≤ 3[/tex], we need to evaluate the function at the endpoints of the interval and then divide by the length of the interval.

[tex]f(-3) = + 3(-3) + 6 = -9 - 9 + 6 = -12[/tex]

[tex]f(3) = + 3(3) + 6 = -9 + 9 + 6 = 6[/tex]

The length of the interval is 3 - (-3) = 6.

Therefore, the average rate of change of f(x) over the interval   [tex][-3, 3][/tex] is:

[tex](6 - (-12)) / 6 = 18/6 = 3[/tex]

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