find the linear approximation of the function fsx, y, zd − sx 2 1 y 2 1 z 2 at s3, 2, 6d and use it to approximate the number ss3.02d 2 1 s1.97d 2 1 s5.99d 2 . quizlet

The factored function is given as follows:

[tex]x^4 + 2x^3 - 20x^2 + 64x - 32 = (x^2 + 6x - 4)(x^2 - 4x + 8)[/tex]

The function for this problem is defined as follows:

[tex]y = x^4 + 2x^3 - 20x^2 + 64x - 32[/tex]

The zeros are given as follows:

Hence the function is factored as follows:

[tex]x^4 + 2x^3 - 20x^2 + 64x - 32 = (ax^2 + bx + c)(x - 2 + 2i)(x - 2 - 2i)[/tex]

[tex]x^4 + 2x^3 - 20x^2 + 64x - 32 = (ax^2 + bx + c)(x^2 - 4x + 8)[/tex]

[tex]x^4 + 2x^3 - 20x^2 + 64x - 32 = ax^4 + (-4 + b)x^3 + \cdots + 8c[/tex]

Hence the value of a is given as follows:

a = 1.

The value of b is given as follows:

-4 + b = 2

b = 6.

The value of c is given as follows:

8c = -32

c = -4.

Hence the factored expression is of:

[tex]x^4 + 2x^3 - 20x^2 + 64x - 32 = (x^2 + 6x - 4)(x^2 - 4x + 8)[/tex]

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A couple would probably budget for the wedding if they make $70,000 together, you can use the regression equation obtained from the analysis. The regression equation will provide you with the relationship between the couple's income and the wedding cost.

The excel regression tool using the cost as the dependent variable and the couple's income as the independent variable, only those weddings paid for by the bride and groom. A. Interpret all key regression results, hypothesis tests, and confidence intervals in the output. B. Analyze the residuals to determine if the assumptions underlying the regression analysis are valid. C. Use the standard residuals to determine if any possible outliers exist. D. If a couple makes $70,000 together, how much would they probably budget for the wedding?

Weddings Couple's Income Bride's age Payor Wedding cost Attendance Value Rating

$130,000 22 Bride's Parents $60,700. 00 300 3

$157,000 23 Bride's Parents $52,000. 00 350 1

$98,000 27 Bride & Groom $47,000. 00 150 3

$72,000 29 Bride & Groom $42,000. 00 200 5

$86,000 25 Bride's Parents $34,000. 00 250 3

$90,000 28 Bride & Groom $30,500. 00 150 3

$43,000 19 Bride & Groom $30,000. 00 250 3

$100,000 30 Bride & Groom $30,000. 00 300 3

$65,000 24 Bride's Parents $28,000. 00 250 3

$78,000 35 Bride & Groom $26,000. 00 200 5

$73,000 25 Bride's Parents $25,000. 00 150 5

$75,000 27 Bride & Groom $24,000. 00 200 5

$64,000 25 Bride's Parents $24,000. 00 200 1

$67,000 27 Groom's Parents $22,000. 00 200 5

$75,000 25 Bride's Parents $20,000. 00 200 5

$67,000 30 Bride's Parents $20,000. 00 200 5

$62,000 21 Groom's Parents $20,000. 00 100 1

$75,000 19 Bride's Parents $19,000. 00 150 3

$52,000 23 Bride's Parents $19,000. 00 200 1

$64,000 22 Bride's Parents $18,000. 00 150 1

$55,000 28 Bride's Parents $16,000. 00 100 5

$53,000 31 Bride & Groom $14,000. 00 100 1

$62,000 24 Bride's Parents $13,000. 00 150 1

$40,000 26 Bride's Parents $7,000. 00 50 3

$45,000 32 Bride & Groom $5,000. 00 50 5

Multiple R: The multiple correlation coefficient is 0.819, which indicates a strong positive relationship between couple's income and wedding cost.

R Square: The coefficient of determination is 0.671, which means that approximately 67% of the variation in wedding cost can be explained by couple's income.

Adjusted R Square: The adjusted R square is 0.650, which takes into account the number of independent variables in the model and is slightly lower than the R square.

Standard Error: The standard error of the estimate is 10,101.431, which represents the average deviation of the observed values from the predicted values.

The intercept coefficient is 24525.824, which represents the estimated wedding cost when the couple's income is zero. The coefficient for couple's income is 0.397, indicating that a one-unit increase in couple's income is associated with a $397 increase in wedding cost.

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The truth table for a selector will have 2^n rows and (n + 2^n) columns.

The number of rows in a truth table for a selector depends on the number of inputs the selector has. If the selector has n inputs, then the truth table will have 2^n rows. As for the columns, a selector typically has one output column and n input columns. So, for example, if the selector has 3 inputs, then the truth table will have 2^3 = 8 rows and 4 columns (3 input columns and 1 output column).

To construct a truth table for a selector, you need to first determine the number of input lines, which we will represent as 'n'. A selector is used to select one output line from multiple input lines based on the binary value of the selector inputs.

The number of rows in the truth table is determined by the possible combinations of input values, which is 2^n.

For the columns, you'll have n input columns for the selector inputs, and an additional 2^n output columns to represent each of the input lines.

So, to summarize, the truth table for a selector will have 2^n rows and (n + 2^n) columns (input and output columns combined).

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The value of unit tangent vector , unit normal vector , binormal vector and curvature at point is  T = (-2i + 3j) / √13 , N =(3/5)i + (2/5)j  , B = 12i + 8j and k = 4 / 13 respectively.

Vector function r(t) = sin(2t)i + 3tj + 2 sin²(t)k

To find the unit tangent vector, unit normal vector, and binormal vector of the given function, follow the following steps,

Find the derivative of r(t) with respect to t to obtain the velocity vector.

Evaluate the velocity vector at the given point to get the tangent vector.

Compute the magnitude of the tangent vector to obtain the unit tangent vector.

Find the second derivative of r(t) with respect to t to obtain the acceleration vector.

Evaluate the acceleration vector at the given point.

Compute the cross product of the tangent vector and the acceleration vector to obtain the binormal vector.

Compute the magnitude of the acceleration vector and divide it by the magnitude of the tangent vector squared to obtain the curvature.

Simplify it using all steps,

Differentiating r(t) = sin(2t)i + 3tj + 2 sin²(t)k, we get,

r'(t) = 2cos(2t)i + 3j + 4sin(t)cos(t)k

Evaluating r'(t) at t = 3π/2,

r'(3π/2)

= 2cos(3π) i + 3j + 4sin(3π/2)cos(3π/2)k

= -2i + 3j

Calculating the magnitude of the tangent vector,

|T| = √((-2)² + 3²)

= √(4 + 9)

= √13

The unit tangent vector, T, is obtained by dividing the tangent vector by its magnitude,

T = (-2i + 3j) / √13

Taking the second derivative of r(t),

r''(t)

= -4sin(2t)i + 0j + 4(cos²(t) - sin²(t))k

= -4sin(2t)i + 4cos(2t)k

Evaluating r''(t) at t = 3π/2,

r''(3π/2)

= -4sin(3π) i + 4cos(3π) k

= 4k

Taking the cross product of the tangent vector and the acceleration vector,

B = T x r''

= (-2i + 3j) x (0i + 0j + 4k)

= 12i + 8j

Calculating the magnitude of the acceleration vector,

|A| = |r''(3π/2)| = |4k| = 4

The curvature, κ, at the given point is given by the formula,

κ = |A| / |T|²

= 4 / (√13)²

= 4 / 13

Therefore, the unit tangent vector is T = (-2i + 3j) / √13, the unit normal vector is N = B / |B| = (12i + 8j) / 20 = (3/5)i + (2/5)j, and the binormal vector is B = 12i + 8j.

The curvature at the point (0, 3π/2, 2) is k = 4 / 13.

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A graph that has an edge between each pair of its vertices is a complete graph. The number of Hamilton circuits in such a graph with "n" vertices is given by the factorial expression.


A graph that has an edge between each pair of its vertices is called a complete graph. In a complete graph, every vertex is directly connected to every other vertex by an edge.

To calculate the number of Hamilton circuits in a complete graph with "n" vertices, we can use the factorial expression (n-1)!. This is because in a Hamilton circuit, each vertex is visited exactly once, except for the starting and ending vertex.

Starting from any vertex, there are (n-1) choices for the next vertex, (n-2) choices for the third vertex, and so on, until only one choice is left for the last vertex. Therefore, the number of Hamilton circuits is given by (n-1)!.

For example, in a complete graph with 4 vertices (n = 4), the number of Hamilton circuits would be (4-1)! = 3! = 6.

In summary, a graph that has an edge between each pair of its vertices is a complete graph, and the number of Hamilton circuits in such a graph with "n" vertices is given by the factorial expression (n-1)!.

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To find the point on the line -3x + 5y - 4 = 0 which is closest to the point (-4, 2), we can use the formula for the distance from a point to a line.First, we need to find the equation of a line perpendicular to -3x + 5y - 4 = 0 that passes through (-4, 2). The slope of -3x + 5y - 4 = 0 is 3/5, so the slope of the perpendicular line is -5/3.

The equation of the perpendicular line passing through (-4, 2) can be found using the point-slope form:

y - 2 = (-5/3)(x + 4)

y = (-5/3)x - 22/3

Now we can find the intersection of the two lines by solving the system of equations:

-3x + 5y - 4 = 0

y = (-5/3)x - 22/3

Substituting y from the second equation into the first, we get:

-3x + 5((-5/3)x - 22/3) - 4 = 0

-18x - 74 = 0

x = -37/9

Substituting x back into the second equation, we get:

y = (-5/3)(-37/9) - 22/3 = -7/3

So the point on the line -3x + 5y - 4 = 0 that is closest to (-4, 2) is (-37/9, -7/3).

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The best measures of center for the data set in the dot plot is mean

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

The dot plot

Where we have the properties to be

When a dataset has an outlier i.e. extreme values, the best measure of center to use is the median

Otherwise, we use the mean

Hence, the best measure is the mean

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The researcher should conclude that there is strong evidence which practice improves performance.

The researcher's hypothesis is that practice will improve performance, so they are conducting a one-tailed test.

The critical t-value for this test with a significance level of 0.05 and degrees of freedom is equal to the sample size minus one would be 1.96.

The observed t-value is -7.42 which means that the difference between the pre-practice and post-practice scores is much larger than what would be expected by chance.

This concept suggests that there is strong evidence that practice did indeed improve performance.

For make a decision about the hypothesis,

the researcher needs to compare the observed t-value to the critical t-value.

Since the observed t-value (-7.42) is much smaller in magnitude than the critical t-value (1.96).

The researcher can reject the null hypothesis that there is no difference between pre-practice and post-practice performance.

Therefore, the researcher should conclude that there is strong evidence which practice improves performance.

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Total number of edges new polyhedron have after attaching a regular tetrahedron to each face of a regular icosahedron is equal to 150 edges.

A regular tetrahedron has 4 faces and 6 edges, and a regular icosahedron has 20 faces and 30 edges.

When a regular tetrahedron is attached to each face of a regular icosahedron,.

Add 4 tetrahedral faces and 4 tetrahedral vertices to each of the 20 triangular faces of the icosahedron.

This means that the new polyhedron has 20 × 4 = 80 additional faces, and 4 × 20 = 80 additional vertices.

Each of these new vertices is connected to 3 other vertices, one from the original icosahedron and two from the added tetrahedra.

The number of new edges added is 80 × 3/2 = 120.

The original icosahedron had 30 edges, so the total number of edges in the new polyhedron is 30 + 120 = 150 edges.

Therefore, the new polyhedron formed by attaching a regular tetrahedron to each face of a regular icosahedron has 150 edges.

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In order to find a 99% confidence interval for the proportion of students who take notes, the student would need to follow certain steps. First, they would need to obtain a random sample of students and record whether or not each student takes notes. Based on this data, they would calculate the sample proportion, which is the number of students who take notes divided by the total number of students in the sample.

Next, they would use a statistical formula to calculate the margin of error, which is the amount by which the sample proportion could vary from the true proportion in the population. They would also use a table or calculator to find the critical value for a 99% confidence level.

Finally, the student would use these values to construct the confidence interval, which is the range of values that is likely to contain the true proportion of students who take notes in the population with 99% confidence. This interval would be expressed as a range of values, such as "between 0.55 and 0.75," and would indicate the level of uncertainty in the estimate based on the sample data.

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Careful consideration must be given to the design and implementation of the tax to ensure that it achieves its intended goals without creating additional problems.

The level of the tax will depend on a variety of factors, including the external costs associated with pig production and the elasticity of demand for pork products.

In general, the Pigouvian tax should be set equal to the marginal external cost of pig production, which is the amount by which the production of one additional pig imposes costs on society that are not reflected in the market price.

By setting the tax equal to this amount, the market price of pigs will increase to reflect the full social cost of production, which will lead to a reduction in the quantity of pigs produced and consumed, bringing it in line with the socially optimal level.

In practice, determining the appropriate level of a Pigouvian tax can be challenging. It requires an accurate estimate of the external costs associated with pig production, which can be difficult to quantify.

Additionally, setting the tax too high can lead to unintended consequences, such as creating black markets or reducing the welfare of small-scale pig producers.

Therefore, careful consideration must be given to the design and implementation of the tax to ensure that it achieves its intended goals without creating additional problems.

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The matrix a is diagonalizable if and only if it has n linearly independent eigenvectors, where n is the dimension of the matrix. In this case, a is a 3x3 matrix with diagonal elements 4, 3, and k, and therefore, has three eigenvectors.

To find the eigenvalues, we need to solve the characteristic equation det(a - λI) = 0, where I is the 3x3 identity matrix and λ is the eigenvalue. This yields:

det(a - λI) = (4 - λ)(3 - λ)k - 90 = 0

Expanding and simplifying, we get:

kλ^2 - 7λ^2 + 12λ - 90 = 0

We can factor this quadratic as:

(k - 10)(λ - 6)(λ - 3) = 0

Therefore, the eigenvalues of a are λ = 6, 3, and k - 10. Since a has three linearly independent eigenvectors, it is diagonalizable if and only if all three eigenvalues are distinct. Thus, we need to find the values of k that make the eigenvalues distinct.

If k = 6 or k = 3, then a has repeated eigenvalues and is not diagonalizable. Therefore, the only values of k for which a is diagonalizable are those that make k - 10 ≠ 6 and k - 10 ≠ 3, or k ≠ 16 and k ≠ 13. Thus, the answer is k ≠ 16, 13.

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(a) We find that t ≈ 5.7 years. So the car will be worth $9000 when it is approximately 5.7 years old. (b) we find that t ≈ 5.0 years. So the car will be worth $8000 when it is approximately 5.0 years old. (c) we find that t ≈ 13.4 years. So the car will be worth $1000 when it is approximately 13.4 years old.

To find when the car will be worth a certain amount, we can set V(t) equal to that amount and solve for t.
(a) To find when the car will be worth $9000:
$9000 = [tex]14,704(0.84)^t[/tex]
Divide both sides by 14,704:
0.612 = [tex]0.84^t[/tex]
Take the logarithm of both sides (using either base 10 or natural logarithm):
log(0.612) = t log(0.84)
Solve for t:
t = log(0.612) / log(0.84)
Using a calculator, we find that t ≈ 5.7 years. So the car will be worth $9000 when it is approximately 5.7 years old.
(b) To find when the car will be worth $8000:
$8000 = [tex]14,704(0.84)^t[/tex]
Following the same steps as before, we find that t ≈ 5.0 years. So the car will be worth $8000 when it is approximately 5.0 years old.
(c) To find when the car will be worth $1000:
$1000 = [tex]14,704(0.84)^t[/tex]
Following the same steps as before, we find that t ≈ 13.4 years. So the car will be worth $1000 when it is approximately 13.4 years old.

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To find the divergence of the vector field F(x, y) = −y i + x j / x2 + y2, we need to calculate the partial derivatives of its components with respect to x and y and then take their sum:

∂F/∂x = ∂/∂x (−y) i + ∂/∂x (x) j / x2 + y2

= 0 + 1/(x2 + y2) j

∂F/∂y = ∂/∂y (−y) i + ∂/∂y (x) j / x2 + y2

= −1/(x2 + y2) i + 0

Therefore, the divergence of F is:

div F = ∇ · F = ∂F/∂x + ∂F/∂y

= 1/(x2 + y2) j − 1/(x2 + y2) i

= (1/(x2 + y2), −1/(x2 + y2))

Thus, the divergence of the vector field F(x, y) = −y i + x j / x2 + y2 is the vector (1/(x2 + y2), −1/(x2 + y2)).

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By answering the presented question, we may conclude that As a result, the slope of  y3= -4(x - 5) is -4.

Here, we have,

The slope of a line indicates how steep it is. The term "gradient overflow" refers to a mathematical equation for the gradient (the change in y divided by the change in x).

The slope is defined as the ratio of the vertical change (rise) between two places to the horizontal change (run). The slope-intercept form of an equation is used to express a straight line's equation, which is written as y = mx + b.

The y-intercept is found where the slope of the line is m, b is b, and (0, b). For example, the slope and y-intercept of the equation y = 3x - 7 (0, 7). The slope of the line is m. b is b at the y-intercept, and (0, b).

The above equation is in point-slope form: y -y1 = m(x -x1 ), where (x1,y1 ) is a point on the line and m is the slope.

We can see by comparing the provided equation to the point-slope form that (x1,y1) = (5, 3) and m = -4.

As a result, the slope of y3 = -4(x - 5) is -4.

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complete question;

Linear Functions

Quiz Active

1

2

3

-4

O-3

020

23

4

5

6

DI

What is the slope of the equation y-3 = -4(x - 5)?

Answer:

-2/7 is 3 and -6 is 2

Step-by-step explanation:

to make a slope perpendicular, you need to swap the numerator and denominator and multiply by negative so - becomes + and + becomes -

Answer:

-2/7 is 3 and -6 is 2

Step-by-step explanation:

Answer:

No 1:85

no2:95

no3:96

no4 :78

no5:98

no6:100

no7:10999

no8:200

no9:48

no10:56

The nearest integer 136,043 people. To find the total population within a 3-kilometer radius of the city center, we need to integrate the population density function (∂(x, y)) over the circular region with a radius of 3 kilometers.

Given that the population density (∂) is defined as 7000 (x^2 + y^2)^-0.2 people per square kilometer, we can express the population density as a function of the distance from the origin (r).

Let's perform the integration using polar coordinates, where x = rcos(θ) and y = rsin(θ):

∂(r) = 7000[tex](r^2)^-0.2[/tex]

∂(r) = 7000[tex]r^(-0.4)[/tex]

Now, we need to integrate this population density function (∂(r)) over the circular region with a radius of 3 kilometers.

To do this, we integrate from 0 to 2π for the angle (θ), and from 0 to 3 kilometers for the radius (r).

Total population = ∫∫R ∂(r) r dr dθ

Total population = ∫[0 to 2π] ∫[0 to 3] 7000 [tex]r^(-0.4)[/tex] r dr dθ

Simplifying the integral:

Total population = 7000 ∫[0 to 2π] ∫[0 to 3] [tex]r^(0.6)[/tex] dr dθ

Total population = 7000 ∫[0 to 2π] [([tex]r^(1.6)[/tex])/(1.6)]|[0 to 3] dθ

Total population = 7000 [tex](1.6)^(-1)[/tex]∫[0 to 2π] [([tex]3^(1.6))[/tex]/(1.6)] dθ

Total population = (7000/1.6) [tex](3^(1.6))[/tex] ∫[0 to 2π] dθ

Total population = (7000/1.6) [tex](3^(1.6)[/tex]) (θ)|[0 to 2π]

Total population = (7000/1.6) ([tex]3^(1.6)[/tex]) (2π)

Now, let's evaluate this expression:

Total population ≈ (7000/1.6) ([tex]3^(1.6)[/tex]) (2π)

Total population ≈ 136042.195 people

Rounding up to the nearest integer, the total population within a 3-km radius of the city center is approximately 136,043 people.

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The "Fundamental Statistics for the Behavioral Sciences" 9th edition provides a comprehensive guide to the statistical methods used in the behavioral sciences. The book covers important topics such as descriptive statistics, probability, hypothesis testing, and regression analysis.

It also includes numerous examples and exercises to help readers practice and apply these concepts. The answers to the exercises are provided at the end of the book, allowing readers to check their work and ensure that they understand the material. This edition is an essential resource for students and researchers in the behavioral sciences who want to develop a strong foundation in statistics and its application to their field.
"Fundamental Statistics for the Behavioral Sciences" is a textbook that teaches essential statistical concepts and techniques for students in behavioral science fields. The 9th edition covers topics like descriptive statistics, probability, hypothesis testing, and inferential statistics.

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The Fundamental Statistics for the Behavioral Sciences 9th edition is a college-level textbook focusing on different concepts in statistics, as applicable to behavioral sciences. The book helps students learn and apply statistical concepts through examples and exercises. Available on OpenStax, it supports learning using calculators and statistical software.

The textbook Fundamental Statistics for the Behavioral Sciences 9th edition covers diverse topics in statistics, with a focus on their application to the behavioral sciences. The chapters of the book include key statistics concepts, like Sampling and Data, Descriptive Statistics, Probability Topics, Discrete Random Variables, Continuous Random Variables, Normal Distribution, The Central Limit Theorem, Confidence Intervals, Hypothesis Testing with one and two samples, The Chi-Square Distribution, Linear Regression and Correlation, and F Distribution and One-Way ANOVA. The textbook helps students calculate degrees of freedom, test statistics, and p-values both manually and using calculators or statistical software. Through examples and exercises, students reinforce their understanding of statistics concepts applied to behavioral science data. This textbook is available for free in web view or PDF on OpenStax.

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If she follows the path shown by the dotted line in the graph, the distance from her house to the store would be = 128m. That is option B.

To calculate the distance between her house and the store the Pythagorean formula should be used which is given as follows;

C² = a² + b²

where;

a= 80

b= 100

c= ?

That is;

c²= 80²+100²

= 6400+10000

= 16,400

c = √16400

= 128.1m

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The limit lim t→∞ arctan(6t) = π/2, lim t→∞ [tex]e^{-4t}[/tex] = 0, lim t→∞ ln(t)/t = 0.

To find the limit as t approaches infinity for the given expressions, we will evaluate each limit separately:

lim t→∞ arctan(6t):

As t approaches infinity, the argument of the arctan function, 6t, also approaches infinity. The arctan function has a range of (-π/2, π/2), so as the argument grows larger, the arctan(6t) will approach π/2. Therefore, the limit of arctan(6t) as t approaches infinity is π/2.

lim t→∞ [tex]e^{-4t}[/tex] :

As t approaches infinity, the exponential function  [tex]e^{-4t}[/tex]  will approach zero. This is because the negative exponent leads to a rapidly decreasing function. Therefore, the limit of  [tex]e^{-4t}[/tex]  as t approaches infinity is 0.

lim t→∞ ln(t)/t:

To evaluate this limit, we can apply L'Hôpital's rule. Taking the derivative of the numerator and denominator gives:

lim t→∞ [d/dt ln(t)] / [d/dt t]

= lim t→∞ (1/t) / 1

= lim t→∞ 1/t

As t approaches infinity, 1/t approaches zero. Therefore, the limit of ln(t)/t as t approaches infinity is 0.

In summary:

lim t→∞ arctan(6t) = π/2

lim t→∞  [tex]e^{-4t}[/tex]  = 0

lim t→∞ ln(t)/t = 0

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The value of  r(s(-4)) is -32 and the value of composite function , hoh⁻¹(2) is -4/7.

Given that r and s are functions.

r(x)=-x-1

s(x)=2x²-1

We have to find the value of r(s(-4)):

s(-4)=2(-4)²-1

=32-1

=31

Now r(s(-4))=r(31)=-31-1=-32

So the value of  r(s(-4)) is -32.

The one to one functions g and h are defined as follows

g={(3, 8), (4, 7), (6, -5), (7, 4), (8, -9)}

To find g⁻¹(x), we switch the x and y values:

{(8, 3), (7, 4), (-5, 6), (4, 7), (-9, 8)}.

So, g⁻¹(x) = {(8, 3), (7, 4), (-5, 6), (4, 7), (-9, 8)}.

The function h(x) = -x + 10/7.

To find h⁻¹(x), we switch the x and y values and solve for y:

x = -y + 10/7

x - 10/7 = -y

y = -x + 10/7

So, h⁻¹(x) = -x + 10/7.

To find hoh⁻¹(2), we substitute h⁻¹(x) = 2 into h(x):

h(2) = -2 + 10/7

h(2) = -2 + 10/7 = -4/7

Therefore, hoh⁻¹(2) = -4/7.

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The requried solution to the equation is x = -3/5.

To solve the equation, we need to simplify and isolate the variable x on one side of the equation.

Starting with:

-3x + 8x - 5 = -8

Combining like terms on the left side, we get:

5x - 5 = -8

Adding 5 to both sides, we get:

5x - 5 + 5 = -8 + 5

Simplifying, we get:

5x = -3

Finally, dividing both sides by 5, we get:

x = -3/5

Therefore, the solution to the equation is x = -3/5.

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For the quadratic function:

(a) Using the information given, determine that the quadratic function has a vertex at (3, -8). Therefore, the output values at x = 0 and x = 6 are also -8.

Also, since the function is symmetric about the vertex, the outputs at x = 4 and x = 2 are the same as those at x = -1 and x = 1, respectively.

Thus, the missing output values are:

x | -1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7

g(x) | 24 | -8 | 0 | -6 | -8 | -6 | 0 | 10 | 48

(b) To find the zeroes of the function, solve for g(x) = 0. Since the vertex is at (3, -8), use the vertex form of the quadratic function:

g(x) = a(x - 3)² - 8

where a = coefficient of the squared term.

To find a, use any of the three points on the graph (0, -8), (1, 0), or (6, -6):

-8 = a(0 - 3)² - 8

a = 8/9

Therefore, the quadratic function is:

g(x) = (8/9)(x - 3)² - 8

To find the zeroes, set g(x) = 0 and solve:

0 = (8/9)(x - 3)² - 8

8 = (8/9)(x - 3)²

9 = (x - 3)²

x = 3 ± 3

Thus, the zeroes of the function are x = 0 and x = 6.

(c) To find the y-intercept, set x = 0 and solve for g(x):

g(0) = (8/9)(0 - 3)² - 8

g(0) = 17

Therefore, the y-intercept is (0, 17).

(d) The vertex of the quadratic function is at (3, -8) and the coefficient of the squared term is positive, which means that the parabola opens downwards. Therefore, the maximum value of the function occurs at the vertex and the range of the function is g(x) ≤ -8 for -1 ≤ x ≤ 7.

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The number of hours it will take Shamus to bike the same distance at a speed of 10 mph is 1 ⅓ hours.

Time taken for John = 2 hours

John speed = 15 mph

Time taken for Shamus = x

Shamus speed = 10 mph

Equate the time taken ratio speed of both

2 : 15 = x : 10

2/15 = x/10

cross product

2 × 10 = 15 × x

20 = 15x

divide both sides by 15

x = 20/15

x = 1 ⅓ hours

Therefore, it will take Shamus 1 ⅓ hours to bike the same distance as John.

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The correct answers are option C for when n is odd and option E for when n is even.

The number of loops in the polar curves given by r sin(n), where n is a positive integer, is related to the parity of n. If n is odd, then the curve will have n loops, and if n is even, the curve will have 2n loops. Therefore, options C and E are correct.

To understand why this is the case, we can consider how the sine function behaves. The sine function oscillates between -1 and 1 as its argument increases from 0 to 2π. When n is odd, the argument of sin(nθ) increases from 0 to 2π as θ goes from 0 to π, resulting in n oscillations of the sine function in this interval. When n is even, the argument of sin(nθ) increases from 0 to 4π as θ goes from 0 to π, resulting in 2n oscillations of the sine function in this interval. This behavior translates into the number of loops in the polar curve, where each oscillation of the sine function corresponds to one loop.

Therefore, the number of loops in the polar curve r sin(n) depends on the parity of n, with n loops for odd values of n and 2n loops for even values of n.

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The objective function q=[tex]x^{2}[/tex] y can be written as q=[tex]x^{2}[/tex](22-x) in terms of x, and as q=[tex](22-y)^2[/tex] y in terms of y, given that x+y=22.

To write the objective function first in terms of x, we need to solve the equation x+y=22 for y. Subtracting x from both sides, we get y=22-x. We can then substitute this expression for y into the objective function q=[tex]x^{2}[/tex] y, giving q=[tex]x^2(22-x)[/tex].
To write the objective function in terms of y, we need to solve the equation x+y=22 for x. Subtracting y from both sides, we get x=22-y. We can then substitute this expression for x into the objective function q=[tex]x^{2}[/tex] y, giving q=[tex](22-y)^2[/tex] y.

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The derivatives for x, y, and dy/dx are dx/dt = 15t² + 3, dy/dt = 4 - 12t, and dy/dx = (4 - 12t) / (15t² + 3), respectively.

To find dx/dt, we need to take the derivative of x with respect to t:

dx/dt = d/dt (5t³ + 3t) = 15t² + 3

To find dy/dt, we need to take the derivative of y with respect to t:

dy/dt = d/dt (4t - 6t²) = 4 - 12t

To find dy/dx, we need to take the derivative of y with respect to x:

dy/dx = (dy/dt) / (dx/dt)

Substituting the expressions we found above, we get:

dy/dx = (4 - 12t) / (15t² + 3)

Note that dy/dx is a function of t, so its value depends on the value of t at a particular point.

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