Use the form |x-b| c to write an absolute value inequality that has the solution set 5

Answer:

Step-by-step explanation:

Answer:

Step-by-step explanation:

Answer: [tex]y=(x-5)^{2} -23[/tex]

Step-by-step explanation:

Step 1: Subtract 2 from both sides to get [tex]y-2=x^{2} -10x[/tex].

Step 2: Divide B (-10) by 2 to get -5
Step 3: Square your answer to Step 2 to get 25.
Step 4: Use the answer you got to Step 3 as your C value. We get [tex]y-2=x^{2} -10x+25[/tex].
Step 5: Since we added 25 to the right side of the equal sign, we have to add 25 to the left side of the equal sign. We get [tex]y+23=x^{2} -10x+25[/tex].
Step 6: Complete the square, to do this keep the left side of the equal sign the same and change the right side to (x + or - B/2 [depending on if its positive or negative]) squared. In this case it's [tex]y+23=(x-5)^{2}[/tex].

Step 7: We still have to get our K value because our vertex formula is [tex]y=a(x-h)^{2} +k[/tex], but in this case our A value is just 1, so it doesn't have to be replaced. So, to get K we subtract 23 from both sides to get our final answer of [tex]y=(x-5)^{2} -23[/tex].

The exact length of the curve is 33/16

What is an equation?

An equation is a mathematical statement that proves two mathematical expressions are equal in algebra, and this is how it is most commonly used. In the equation 3x + 5 = 14, for instance, the two expressions 3x + 5 and 14 are separated.

Given:

[tex]x = \frac{y^4}{8} +\frac{ 1}{4y^2}[/tex]---------------------(1)

Arc length formula:

[tex]L=\int_{c}^d\sqrt{1+(\frac{dx}{dy})^2} ~~~dy[/tex]--------------(2)

Intervals c=1. d=2

differentiate (1) with respect to y

[tex]\frac{dx}{dy}=\frac{4y^3}{8}+\frac{-2}{4y^3}=\frac{y^3}{2}-\frac{1}{2y^3}[/tex]

Now,

(2)=>  [tex]L=\int_{1}^2\sqrt{1+(\frac{y^3}{2}-\frac{1}{2y^3})^2} ~~~dy[/tex]

Using the identity (a-b)² = a²-2ab+b²  and simplifying, we get

[tex]L=\int_{1}^2(\frac{y^3}{2}+\frac{1}{2y^3})^2 ~~~dy[/tex]

Integrate with respect to y

[tex]L= [(\frac{y^4}{8}-\frac{1}{4y^2})^2]_{1}^2[/tex]

Apply the limits and simplifying, we get

L= 33/16

The exact length of the curve is 33/16

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The exact length of the curve is 33/16

An equation is a mathematical statement that proves two mathematical expressions are equal in algebra, and this is how it is most commonly used. In the equation 3x + 5 = 14, for instance, the two expressions 3x + 5 and 14 are separated.

To find the length of the curve, we need to use the arc length formula:

L = ∫ [1, 2] √[1 + (dx/dy)²] dy

First, we need to find dx/dy:

dx/dy = 1/2 y³ + 1/2 y

Now we can substitute this into the arc length formula and simplify:

L = ∫ [1, 2] √[1 + (1/2 y^3 + 1/2 y)²] dy

L = ∫ [1, 2] √[1 + 1/4 y⁶ + y⁴ + 1/4 y²] dy

L = ∫ [1, 2] √[1/4 y⁶ + y⁴ + 1/4 y² + 1] dy

We can now use a trigonometric substitution, letting y² = tanθ:

y² = tanθ

2y dy = dθ

When y = 1, θ = π/4 and when y = 2, θ = π/3. So we can rewrite the integral as:

L = 2∫ [π/4, π/3] √[1/4 tan⁴θ + tan²θ + 1] dθ

We can then use a second substitution, letting u = tanθ:

u = tanθ

du/dθ = sec²θ

dθ = du/u²

Substituting this into the integral, we get:

L = 2∫ [1, √3] √[1/4 u⁴ + u² + 1] du/u²

We can simplify the integrand by multiplying both the numerator and the denominator by u²:

L = 2∫ [1, √3] √[u⁴/4 + u⁴ + u²] du/u⁴

L = 2∫ [1, √3] √[5/4 u⁴ + u²] du/u⁴

Now we can use a substitution, letting v = u²:

v = u²

du = dv/2√v

Substituting this into the integral, we get:

L = 4∫ [1, 3] √[5/4 v² + v] dv/v³

L = 4∫ [1, 3] √[5v² + 4v] dv/v³

At this point, we can use a partial fraction decomposition to evaluate the integral:

√[5v² + 4v]/v³ = A/v + B/v² + C/√[5v² + 4v]

Multiplying both sides by v³ and simplifying, we get:

√[5v² + 4v] = Av²√[5v² + 4v] + Bv + Cv³√[5v² + 4v]

We can solve for A, B, and C by equating coefficients:

A = 0

B = 1/2

C = √(5)/2

Now we can substitute these values back into the partial fraction decomposition:

√[5v² + 4v]/v³ = 1/2v + 1/2v² + √(5)/2 sqrt[5v² + 4v]

Substituting this back into the integral and evaluating, we get:

L = 4[1/2lnv + 1/2v - 1/√(5)ln(√(5)v + 2

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The required answer is  Sin u * Cos u * Sec u * Cot u = 1

In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities potentially involving angles but also involving side lengths or other lengths of a triangle.

Trigonometry' is a branch of mathematics concerned with relationships between angles and ratios of lengths. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies.The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions) such as sine.

The trigonometric identity is:

Sin u COS u = (1/2)Sin(2u)

Sec u = 1/Cos u

Cot u = Cos u/Sin u

To help you complete the trigonometric identity using the given terms, we will work step-by-step.

1. Sin u * Cos u: This is the given product of sine and cosine functions for angle u.
trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities potentially involving angles but also involving side lengths or other lengths of a triangle.These identities are useful whenever expressions involving trigonometric functions need to be simplified.

An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.
2. Sec u: The secant function is the reciprocal of the cosine function, so Sec u = 1/Cos u.

3. Cot u: The cotangent function is the reciprocal of the tangent function, which is the ratio of sine and cosine functions. So Cot u = Cos u / Sin u.

Now, let's combine these terms to complete the trigonometric identity:

Sin u * Cos u * Sec u * Cot u

Since Sec u = 1/Cos u and Cot u = Cos u / Sin u, we can substitute these values:

Sin u * Cos u * (1/Cos u) * (Cos u / Sin u)

When we multiply these terms, the Cos u and Sin u cancel out:

(Sin u * Cos u) / (Sin u * Cos u) = 1

Thus, the completed trigonometric identity is:

Sin u * Cos u * Sec u * Cot u = 1

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The required answer is y(x) = c1 e^x + c2 cos(√5 x) + c3 sin(√5 x)

To find the general solution to y′′′−y′′ 5y′−5y=0, we first write the characteristic equation:
An arbitrary constant is a symbol used to represent an object which is neither a specific number nor a variable. It is used to represent a general object (usually a number, but not necessarily) whose value can be assigned when the expression is instantiated.

the word constant conveys multiple meanings. As an adjective, it refers to non-variance (i.e. unchanging with respect to some other value); as a noun, it has two different meanings:

r^3 - r^2 + 5r - 5 = 0

This can be factored as:

(r-1)(r^2 + 5) = 0

Thus, the roots are r=1, r=i√5, and r=-i√5.
A constant may be used to define a constant function that ignores its arguments and always gives the same value.

A symbol that stands for an arbitrary input is called an independent variable, while a symbol that stands for an arbitrary output is called a dependent variable.

The general solution is then given by:

y(x) = c1 e^x + c2 cos(√5 x) + c3 sin(√5 x)

where c1, c2, and c3 are arbitrary constants.

Therefore, the solution to y′′′−y′′ 5y′−5y=0, using c1 as c1, c2 as c2, and c3 as c3, is:
y(x) = c1 e^x + c2 cos(√5 x) + c3 sin(√5 x)

To find the general solution to the given differential equation, y''' - y'' + 5y' - 5y = 0, follow these steps:
A symbol that stands for an arbitrary input is called an independent variable, while a symbol that stands for an arbitrary output is called a dependent variable. The most common symbol for the input is x, and the most common symbol for the output is y; the function itself is commonly written y = f(x).

it is possible to have multiple independent variables or multiple dependent variables. For instance, in multivariable calculus, one often encounters functions of the form z = f(x ,y), where z is a dependent variable and x and y are independent variables

Step 1: Identify the characteristic equation for the given differential equation.

For the given differential equation, the characteristic equation is:
r^3 - r^2 + 5r - 5 = 0

Step 2: Solve the characteristic equation for r.
This cubic equation is difficult to solve by hand, but using a numerical method or software, we find the roots to be approximately:
r1 ≈ 0.201
r2 ≈ 1.159
r3 ≈ 2.640

Step 3: Construct the general solution using the roots and the arbitrary constants c1, c2, and c3.
The general solution to the differential equation is given by:
y(x) = c1 * e^(r1 * x) + c2 * e^(r2 * x) + c3 * e^(r3 * x)

So, the general solution to y''' - y'' + 5y' - 5y = 0 is:
y(x) = c1 * e^(0.201 * x) + c2 * e^(1.159 * x) + c3 * e^(2.640 * x)

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Let A be the number of adults and C be the number of children. We know that A + C = 500 and 2A + C = 780. Solving for A, we get A = 260.

To solve this problem, we use a system of equations with two variables: A and C. From the problem, we know that the total number of people who attended the play was 500.

We also know that the admission price for adults was $2 and for children was $1. Finally, we know that the total admission receipts were $780.

Using this information, we can set up two equations: A + C = 500 (equation 1) and 2A + C = 780 (equation 2). We can then solve for A by eliminating C. Subtracting equation 1 from equation 2, we get A = 260. Therefore, there were 260 adults who attended the play.

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Based on the given information, this is not a realistic idea

The most a cat can have in 2 months is typically 6 kittens.

18months / 2 months is 9

So she can have 9 litters in a year, if she's absolutely pumping them out; however, the average number of litters a female can have is 3 litters.

So, let's try 3 x 6 = 18 kittens in a year. Okay, that's much less than 2000.

Let's try the other one then, the 9 time litter.

9 x 6 = 54 Still a lot less than 2000.

If only that one female cat was breeding, there is no way she could make 2000 descendants oh her own within 18 months.

If her kittens were added into the equation, it'd be possible, but otherwise, absolutely not.

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To convert categorical variables to dummy variables, follow these steps:

1. Identify the categorical variable(s) in your dataset that you wish to convert.

2. For each categorical variable, determine the number of unique categories (levels).

3. Create new binary variables (dummy variables) equal to the number of unique categories minus one for each categorical variable.

4. Assign a unique combination of 0s and 1s to represent each category within the new dummy variables. Typically, 1 indicates the presence of a category, while 0 indicates its absence.

5. Replace the original categorical variable(s) with the corresponding dummy variables in your dataset.

By converting categorical variables to dummy variables, you can use them in statistical analyses that require numerical data, such as regression models.

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What is the equation for line that is perpendicular to the line 3x + 6y = 24 and goes through the point (1,-5) is y = 2x - 7.

The formula for equation of line is expressed as;

y = mx + b

Where m is slope and b is y-intercept.

Given the equation of the original line: 3x + 6y = 24

Rewritten in slope-intercept form as:

6y = -3x + 24

y = (-1/2)x + 4

The slope of the given line is -1/2.

To find the equation of a line that is perpendicular to this line, we need to find a line with a slope that is the negative reciprocal of -1/2, which is 2.

Let the equation of the perpendicular line be:

y = 2x + b

where b is the y-intercept.

To find the value of b, we can use the fact that the line passes through the point (1,-5).

Substituting these values into the equation of the line, we get:

-5 = 2(1) + b

-5 = 2 + b

b = -7

Hence, the equation of the line that is perpendicular to 3x + 6y = 24 and passes through the point (1,-5) is y = 2x - 7.

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a) The due date of the loan is April 15th of the following year.

b) The interest due on October 12th is $200 and the balance of the loan after the October 12th payment is $6,700.

The percentage amount a lender charges a borrower for using money or the amount a saver earns for depositing money in a bank or other financial institution is known as an interest rate.

a) Let's assume that the loan term is 12 months.

The loan is taken out on April 15th, so the due date will be 12 months later, which is:

April 15th + 12 months = April 15th of the following year.

Therefore, the due date of the loan is April 15th of the following year.

b) The interest for the 6 months between April 15th and October 12th is:

Interest = Principal x Rate x Time

= $10,000 x 0.04 x (6/12)

= $200

Therefore, the interest due on October 12th is $200.

The payment made on October 12th is $3,500, so the remaining balance of the loan after that payment is:

Balance = Principal + Interest - Payment

= $10,000 + $200 - $3,500

= $6,700

So, the balance of the loan after the October 12th payment is $6,700.

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The volume of his barn that comprises a triangular prism and a rectangular prism is calculated as: 40,000 ft³.

The barn of Mr. Fowler as shown in the image attached below is a composite solid which is made up of a rectangular prism and a triangular prism.

To find the volume of his barn, we would apply the formula below:

Volume of the barn = (volume of triangular prism) + (volume of rectangular prism)

Volume of triangular prism = 1/2 * b * h * L

base of triangular face = 40 ft

height of triangular face = 10 ft

Length of prism = 50 ft

Plug in the values:

Volume of triangular prism = 1/2(40 * 10) * 50 = 10,000 ft³.

Volume of the rectangular prism = length * width * height

Length = 50 ft

Width = 40 ft

Height = 15 ft

Plug in the values:

Volume of the rectangular prism = 50 * 40 * 15 = 30,000 ft³.

Volume of his barn = 10,000 + 30,000 = 40,000 ft³.

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Answer:

0

Step-by-step explanation:

Answer:

7.1

Step-by-step explanation:

b = 2A / h

7.1 = 2(11.36) / 3.2

Answer: -2

Step-by-step explanation:

-2(x-1)-3x=12(x+3)

-2x+2-3x=12x+36

-5x-12x=36-2

-17x=34

x=-2

A box plot shows the minimum, maximum, median, and any outliers of a data set. It does not show individual values in the data set. Therefore, options A, C, D, and E are the correct answers.

A box plot is a graphical representation of a data set that displays the median, individual values, outliers, minimum, and maximum of the data set.

The box plot is created by drawing a box from the lower quartile, or the 25th percentile, to the upper quartile, or the 75th percentile, with a line in the middle of the box representing the median of the data set.

The individual values in the data set are represented by dots, marks, or lines outside of the box. Outliers, or values that are significantly different from the rest of the set, are also represented outside of the box. T

he minimum and maximum of the data set are typically represented by either a line or a dot outside of the box.

Therefore, options A, C, D, and E are the correct answers.

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The drone's battery will last 16 minutes minutes if no weight is added. The battery life will decrease by 0.0333 of weight added.

Given data ,

Let the first point be A ( 0 , 16 )

Let the second point be B ( 60 , 14 )

Now , the slope of the line is

m = ( 16 - 14 ) / ( 0 - 60 )

m = - 2 / 60

m = - 0.0333

The y-intercept of the line is when x = 0

So , when x = 0 , y = 16

Now , The drone's battery will last 16 minutes if no weight is added.

Hence , the equation of line is solved

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The answer is: A. r = 0.3, indicating a weak positive correlation between percent body fat and preferred amount of salt.

The correct answer is A. r = 0.3.

Correlation coefficient (r) is a statistical measure that indicates the strength and direction of a linear relationship between two variables.

It ranges from -1 to 1, where 1 indicates a perfect positive correlation, -1 indicates a perfect negative correlation, and 0 indicates no correlation.

In this case, a correlation coefficient of 0.3 indicates a weak positive correlation between percent body fat and preferred amount of salt.

This means that as the preferred amount of salt increases, there is a

slight tendency for percent body fat to also increase, but the relationship is not very strong.

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The derivatives of the following functions are

1. Derivative of the f(x) = log8(x) is f'(x) = (1 / x) * (1 / ln(8)).

2. Derivative of the h(x) = log5(x + 9) is h'(x) = (1 / (x + 9)) * (1 / ln(5)).

3. Derivative of the h(x) = e^x^8 − x + 3 is h'(x) = e^(x^8 - x + 3) * (8x^7 - 1).

4. Derivative of the g(x) = 2^x is g'(x) = 2^x * ln(2).

1. For the function f(x) = log8(x), find its derivative:
To find the derivative of f(x) with respect to x, we can use the change of base formula for logarithms and the chain rule:
f(x) = log8(x) = ln(x) / ln(8)
f'(x) = (1 / x) * (1 / ln(8))

2. For the function h(x) = log5(x + 9), find its derivative:
Similar to the previous function, use the change of base formula and the chain rule:
h(x) = log5(x + 9) = ln(x + 9) / ln(5)
h'(x) = (1 / (x + 9)) * (1 / ln(5))

3. For the function h(x) = e^(x^8 − x + 3), find its derivative:
Apply the chain rule:
h'(x) = e^(x^8 - x + 3) * (8x^7 - 1)

4. For the function g(x) = 2^x, find its derivative:
Use the exponential rule and the chain rule:
g'(x) = 2^x * ln(2)

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Answer:

(0,-9)

Step-by-step explanation:

When you're on the y-axis, the x-coordinate is 0. In the point (0,-9), x=0 and y=9. Reflecting it across the y axis wont do anything because x is so it is (0,-9)

Answer:

5 9/16 or 5.5625

Step-by-step explanation:

To solve make the denominator the same by multiplying

4x4=16 and multiply the numerator by the same amount 3x4=12 so 12/16

Lastly, solve with subtraction.

Answer: The correct answer for this is 5 8/16 which is a mixed fraction.

Step-by-step explanation: Since it is a mixed fraction, we first convert both the terms into improper fractions and then carry out the operation.

on solving mixed fractions we get 31/4 - 35/16

Then we further solve this to get 189/ 16 which is an improper fraction.

Then we convert this into mixed fraction: 5 8/16 (answer)

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The third word in the series is an a x r², and this is the answer to the given question based on the sequence.

A progression in mathematics is a particular form of sequence where the distance between succeeding terms is constant. A collection of numbers or other mathematical elements arranged in a specific order is called a sequence.

Arithmetic progressions, geometric progressions, and harmonic progressions are only a few of the several forms of progressions. The formula for the nth term of the sequence varies depending on the type of progression.

By dividing the first term by the common ratio r, one may get the second term in the sequence:

Second term = a x r

The second term can also be multiplied by the common ratio r to find the third term:

Third term = (a x r) x r = a x r²

As a result, an a x r² is the third term in the series.

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The unit tangent vector t(t) at the point with the given value of the parameter t = 1 is t(1) = (1/√3)i + (1/√3)j + (1/√3)k.

To find the unit tangent vector t(t) at the point with the given value of the parameter t, we will follow these steps:

1. Find the derivative of the vector function r(t) with respect to t.
2. Evaluate the derivative at the given value of t.
3. Normalize the derivative to find the unit tangent vector.

Given r(t) = 4t i + [tex]2t^2[/tex] j + 4t k and t = 1.

Step 1: Find the derivative of r(t) with respect to t.
r'(t) = (d(4t)/dt)i + (d([tex]2t^2[/tex])/dt)j + (d(4t)/dt)k
r'(t) = 4i + 4tj + 4k

Step 2: Evaluate r'(t) at t = 1.
r'(1) = 4i + 4(1)j + 4k
r'(1) = 4i + 4j + 4k

Step 3: Normalize r'(1) to find the unit tangent vector t(1).
Magnitude of r'(1) = sqrt[tex](4^2 + 4^2 + 4^2)[/tex] = sqrt(48) = 4√3
t(1) = (1/(4√3))(4i + 4j + 4k) = (1/√3)i + (1/√3)j + (1/√3)k

Your answer: The unit tangent vector t(t) at the point with the given value of the parameter t = 1 is t(1) = (1/√3)i + (1/√3)j + (1/√3)k.

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To find a point estimate of the mean from the given data set, we simply take the average of the sample values.

To find the mean of a data set, you need to add up all the values in the data set and then divide the total by the number of values in the data set.

The formula for the mean is:

Step 1: Add the sample values. 14 + 20 + 22 + 26 + 33 = 115

Step 2: Divide the sum of the sample values by the number of observations (n = 5).

115 ÷ 5 = 23

The point estimate of the mean for the simple random sample of 5 observations from the 400-element population is 23.

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Answer:

  A.  f(x) = -4|x +2| +3

Step-by-step explanation:

You want the function that matches the graph of the absolute value function shown. Its vertex is (-2, 3) and it opens downward.

The parent function must be reflected across the x-axis for its graph to open downward. That means the function must be multiplied by a negative number. (Eliminates choices B and D.)

The vertex of the function is translated up 3 units, so 3 will be added to the function value. (Eliminates choices C and D.)

The only remaining viable choice is A.

  A.  f(x) = -4|x +2| +3

__

Additional comment

The translation left 2 units replaces x in the function by (x -(-2)) = (x+2). This matches choice A and eliminates choice C.

  g(x) = a·f(x -h) +k

translates f(x) by (h, k). When a < 0, reflects f(x) across the x-axis. Here, (h, k) = (-2, 3).

The value of x on solving the given problems are

a. X+7x+10x = 20 x(0) = 5 (0) = 3 ; x= 0

b. 5x+20t + 20x = 28 x(0) = 5 (0) = 8; x = (28=20t)/25

c..f + 16x = 144 x() = 5X(0) = 12; x= (144-f)/16

d.X+6f+34x = 68 x(0) = 5x10) = 7; x= (68-6f)/35

a. To solve for x, we first need to combine like terms: a·X + 7x + 10x = 20x. Simplifying this equation gives us 18x = 20x - we subtracted 7x and 10x from both sides. To isolate x, we need to subtract 20x from both sides as well, giving us -2x = 0. Finally, we divide both sides by -2 to solve for x, which gives us x = 0.
b. Similar to part a, we need to combine like terms first: 5x + 20t + 20x = 28. Simplifying this equation gives us 25x + 20t = 28. To isolate x, we need to subtract 20t from both sides, giving us 25x = 28 - 20t. Finally, we divide both sides by 25 to solve for x, which gives us x = (28 - 20t)/25.
c. To solve for x, we need to isolate it by itself. We can start by subtracting f from both sides: 16x = 144 - f. Finally, we divide both sides by 16 to solve for x, which gives us x = (144 - f)/16.
d. Similar to parts a and b, we need to combine like terms first: x + 6f + 34x = 68. Simplifying this equation gives us 35x + 6f = 68. To isolate x, we need to subtract 6f from both sides, giving us 35x = 68 - 6f. Finally, we divide both sides by 35 to solve for x, which gives us x = (68 - 6f)/35.

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The finite collection of subsets A1, A2,..., An belongs to an algebra F0 if it is closed under finite unions, finite intersections, and complementation.

An algebra, F0, is a collection of subsets of a set S with three key properties:

1. S is in F0.
2. If A is in F0, then its complement, is also in F0.
3. If A1, A2,..., An are in F0, then their finite union, A1∪A2∪...∪An, and finite intersection, A1∩A2∩...∩An, are in F0.

For A1, A2,..., An to belong to the algebra F0, they must satisfy these properties. In other words, for each subset Ai (1 ≤ i ≤ n), Ai and its complement must be in F0, and any finite union or intersection of these subsets must also be in F0. By fulfilling these conditions, A1, A2,..., An form a finite collection of subsets in the algebra F0.

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The smallest positive integer k  is big-O of nk is k = 3

To find the smallest positive integer k such that the expression 12 + 22 + 32 + ... + n2 is big-O of nk .

we need to determine the growth rate of the given expression and compare it with the growth rate of nk.

The expression 12 + 22 + 32 + ... + n2 represents the sum of squares of integers from 1 to n. We can express this sum using the formula for the sum of squares:

1[tex]^2 + 2^2 + 3^2 + ... + n^2[/tex] = n(n + 1)(2n + 1)/6

Now, we can compare the given expression with nk:

n(n + 1)(2n + 1)/6 = O(nk)

We need to find the smallest positive integer k for which this expression is big-O of nk.

Let's simplify the expression on the left-hand side:

n(n + 1)(2n + 1)/6 = ([tex]n^3 + n^2 + n[/tex])/6

Now, we can compare the growth rates of ([tex]n^3 + n^2 + n[/tex])/6 and nk.

As n approaches infinity, the term n^3 dominates the other terms in the numerator (n^2 and n), and the constant coefficient 1/6 can be ignored for big-O notation. Therefore, the growth rate of ([tex]n^3 + n^2 + n[/tex])/6 is dominated by n^3.

So, we can conclude that [tex](n^3 + n^2 + n)/6 = O(n^3)[/tex].

Thus, the smallest positive integer k such that 12 + 22 + 32 + ... + n2 is big-O of nk is k = 3, as the expression ([tex]n^3 + n^2 + n[/tex])/6 has a growth rate of O([tex]n^3[/tex]).

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The slope of the tangent line at (x,f(x)) is given by the derivative f'(x). Thus, we have: The function f(x) is:
f(x) = -4/x - (7/5)/x^5 + 27/5

f'(x) = 4/x^2 - 7/x^6

To find the function f(x), we need to integrate f'(x) with respect to x. We have:

∫ f'(x) dx = ∫ (4/x^2 - 7/x^6) dx

Integrating each term separately, we get:

f(x) = -4/x - 7/(5x^5) + C

where C is the constant of integration. We can find the value of C by using the fact that the curve passes through the point (1,0):

0 = -4/1 - 7/(5*1^5) + C

C = 4/5

Therefore, the function f(x) is:

f(x) = -4/x - 7/(5x^5) + 4/5

Note that this function is defined for x > 0, as specified in the problem statement.

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The give differential dy . y = tan x dy is incorrect an the correct one is dy = [dy/dx * y - sec^2(x) * dy/dx] / (dy/dx - tan(x))

To find the correct differential, we need to use the product rule of differentiation.

Starting with the given equation:

dy/dx * y = tan(x) * dy/dx

Now, we can use the product rule:

d/dx [ y * dy/dx ] = d/dx [ tan(x) * dy/dx ]

Using the chain rule on the right side:

d/dx [ y * dy/dx ] = sec^2(x) * dy/dx + tan(x) * d^2y/dx^2

Simplifying:

dy/dx * d/dy [y] + d^2y/dx^2 = sec^2(x) * dy/dx + tan(x) * d^2y/dx^2

Rearranging and factoring out the common factor of d^2y/dx^2:

(dy/dx - tan(x)) * d^2y/dx^2 = dy/dx * y - sec^2(x) * dy/dx

Finally, solving for the differential dy:

dy = [dy/dx * y - sec^2(x) * dy/dx] / (dy/dx - tan(x))

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