Let F(X, y, 2) = 3z^2xi + (y^3 + tan(2)J + (3x^2z + 1y^2)k. Use the Divergence Theorem to evaluate /s. F. dS where S is the top half of the sphere x^2 + y^2 + z^2 = 1 oriented upwards. s/sF. ds =SIF. ds =

The original price of the product was approximately $98.00.

To determine the original price before the 15.5% decrease, we can use the following equation:

Original price - (15.5% of original price) = $82.81

Let's solve for the original price:

Original price - (0.155 * Original price) = $82.81

Simplifying the equation:

0.845 * Original price = $82.81

Dividing both sides by 0.845:

Original price = $82.81 / 0.845

Calculating the original price:

Original price ≈ $98.00

Therefore, the original price of the product was approximately $98.00.

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The derivative of the function is y' = 50e^(-0.06x) - 3xe^(-0.06x). The x-intercept is approximately x ≈ 16.273. The y-intercept is y = 0. The horizontal asymptote is y = 0, which occurs as x approaches infinity.

To find the derivative of the function y = 50xe^-0.06x, we can use the product rule and the chain rule.

y' = 50e^-0.06x - 50xe^-0.06x(0.06)

Simplifying, we get y' = 50e^-0.06x(1-0.06x)

To find the x-intercept, we need to set y=0 and solve for x:

0 = 50xe^-0.06x

Since e^-0.06x is never zero, we can divide both sides by it:

0 = 50x

So the x-intercept is x=0.

To find the y-intercept, we need to set x=0 and solve for y:

y = 50(0)e^0

So the y-intercept is y=0.

To find the horizontal asymptote, we can take the limit as x approaches infinity:

lim (x→∞) 50xe^-0.06x = 0

So the horizontal asymptote is y=0. This horizontal asymptote occurs as x approaches infinity.

1. Consider the function: y = 50xe^(-0.06x)

2. Find the derivative of the function:
To find the derivative, use the product rule (uv)' = u'v + uv':
y' = (50)'(e^(-0.06x)) + (50)(e^(-0.06x))(-0.06)
y' = 50e^(-0.06x) - 3xe^(-0.06x)

3. The x-intercept is:
To find the x-intercept, set y = 0 and solve for x:
0 = 50xe^(-0.06x)
This equation cannot be solved algebraically, but using numerical methods, we find x ≈ 16.273

4. The y-intercept is:
To find the y-intercept, set x = 0 and solve for y:
y = 50(0)e^(-0.06(0))
y = 0

5. Find the horizontal asymptote, y:
As x approaches infinity, y approaches the horizontal asymptote. In this case, the exponential term (e^(-0.06x)) approaches 0:
y = 50xe^(-0.06x)
y ≈ 50x(0)
y ≈ 0

The horizontal asymptote is y = 0, and it occurs as x approaches infinity.

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answer:

Step-by-step explanation:

first you want to multiply them together then divide it by 2 it's bh/2

It's essential to consider the depreciation rate when deciding whether to invest in a new car or keep your current one.

Cars typically depreciate, and the amount can vary depending on factors such as make, model, and age.

For this example, let's assume you're interested in purchasing a 3-year-old used Honda Accord. I found an advertisement for this vehicle online, but since I cannot attach a copy here, please search for a similar advertisement and include it with your work.

It's common for new cars to depreciate by approximately 20-30% in the first year, and around 10-15% each subsequent year. So, a 3-year-old car may have already experienced around 40-60% of its total depreciation.

After researching, the used 3-year-old Honda Accord is priced at $18,000. If you compare it to the price of a new Honda Accord, which starts around $25,000, you can see that there has been a considerable depreciation in value.

In conclusion, cars do depreciate, and the rate can vary depending on the vehicle's age and other factors. In this case, a 3-year-old Honda Accord has already experienced significant depreciation, making it a more affordable option compared to buying a brand new car.

Considering depreciation can help you make an informed decision when deciding between a new or used car.

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The number of bracelets that can be made using all the colors one time only is 720.

Given that Diana is making bracelet with 6 different colors we need to find the number of bracelets that can be made using all the colors one time only,

Since there are 6 beads so, the number of bracelets can be made = 6!

= 720

Hence the number of bracelets that can be made using all the colors one time only is 720.

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The distance from Xavier and David to the ranch can be found using the distance formula:

distance = sqrt((change in x)^2 + (change in y)^2 + (change in z)^2)

In this case, we are given the distances DR and XR, which represent the change in x, y, and z coordinates between the tents and the ranch. We know that the tents are located at equal distances from the ranch, so the change in x, y, and z coordinates for both David and Xavier will be the same.

Let's call the distance from each tent to the ranch "d", then we have:

DR = XR = d

Substituting the given values, we get:

12.3z + 12.4 = 10.5z + 34

Solving for z, we get:

z = 6.8

Now we can find the distance from each tent to the ranch using the formula:

distance = sqrt((change in x)^2 + (change in y)^2 + (change in z)^2)

For David's tent:

distance = sqrt((12.3z)^2 + 0^2 + (12.4)^2) = sqrt((12.3*6.8)^2 + (12.4)^2) = 87.9 meters (rounded to one decimal place)

For Xavier's tent:

distance = sqrt((10.5z)^2 + 0^2 + (34)^2) = sqrt((10.5*6.8)^2 + (34)^2) = 95.6 meters (rounded to one decimal place)

Therefore, David and Xavier are 87.9 meters and 95.6 meters away from the ranch, respectively.

The value of x is 7. The value of y in the right triangle is 16.5. The value of z in the given figure is 49.

A quadrilateral is a polygon with four sides. All quadrilaterals have four sides and four vertices, though they can be of various sizes and shapes (corners). Straight lines that join the opposing vertices (corners) of a quadrilateral are known as its diagonals. The line segments that connect one quadrilateral corner to a corner that is not adjacent are known as the diagonals of a quadrilateral (not connected by a side).

The opposite sides of the kite are equal thus, we have:

3x + 2 = 5x - 12

14 = 2x

x = 7

The length of the side MJ is:

3(7) + 2 = 23

Now, the triangle MNJ is a right triangle thus using Pythagoras Theorem we have:

h² = a² + b²

23² = 16² + y²

529 = 256 + y²

273 = y²

y ≈ 16.5

Now, diagonals of kite are perpendicular thus,

2z - 8 = 90

2z = 98

z = 49

Hence, the value of z in the given figure is 49.

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The value of ║5p-3q║=180°.

It is given that line l₁ & l₂ are parallel and l₂ & l₃ are transversal then

they must follow the property that the alternate exterior angles must be equal i.e. ∠ q = 135°.

Also ∠ p + ∠ q = 180°.

Therefore, ∠ p = 45°.

Now to solve ║5p-3q║, substituting the values of p & q in the given equation

║5*45 - 3*135║ = 180°.

Hence, ║5p-3q║=180°.

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The rectangles with dimensions 1 x 12 and 2 x 6 do not have a perimeter of 14 inches. These rectangles have perimeters of 26 inches and 16 inches, respectively.

To determine which rectangles with an area of 12 square inches do not have a perimeter of 14 inches, we will first find the possible dimensions of the rectangles and then calculate their perimeters.

1. Since the area of a rectangle is given by length x width, let's find the factors of 12:
  - 1 x 12
  - 2 x 6
  - 3 x 4
2. Now, let's calculate the perimeters for each of these rectangles using the formula 2(length + width):
  - For the 1 x 12 rectangle, the perimeter is 2(1+12) = 2(13) = 26 inches
  - For the 2 x 6 rectangle, the perimeter is 2(2+6) = 2(8) = 16 inches
  - For the 3 x 4 rectangle, the perimeter is 2(3+4) = 2(7) = 14 inches
The rectangles with dimensions 1 x 12 and 2 x 6 do not have a perimeter of 14 inches. These rectangles have perimeters of 26 inches and 16 inches, respectively.

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The equation of the line that passes through (1, 2) and (a) having a slope of 2/3 is y = (2/3)x + 4/3, (b) Having an undefined slope is x = 1, (c) having slope m = 0 is y = 2 (d) point (2, 3) also lies on the line is y = x + 1.

a) To find the equation of a line with slope 2/3 passing through (1,2), we can use the point-slope formula:
y - y1 = m(x - x1)
Substituting in the values we know, we get:
y - 2 = (2/3)(x - 1)
Expanding and simplifying:
y = (2/3)x + 4/3
So the equation of the line is y = (2/3)x + 4/3.

b) To find the equation of a line with an undefined slope passing through (1,2), we know that this line must be vertical. The equation of a vertical line passing through (1,2) can be written as:
x = 1

So the equation of the line is x = 1.

c) To find the equation of a line with slope m=0 passing through (1,2), we know that this line must be horizontal. The equation of a horizontal line passing through (1,2) can be written as:
y = 2
So the equation of the line is y = 2.

d) To find the equation of a line passing through (1,2) and (2,3), we can use the point-slope formula again:
y - y1 = m(x - x1)
Substituting in the values we know, we get:
y - 2 = (3 - 2)/(2 - 1)(x - 1)
Simplifying:
y - 2 = 1(x - 1)
y - 2 = x - 1
y = x + 1
So the equation of the line is y = x + 1.

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The base of the parallelogram is approximately equal to the circumference of the circle.

The following statements could help Josie justify the area formula for a circle:

Therefore, the area of the circle can be calculated using the formula A = C × r, or in terms of the radius, A = πr².

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Answer:

jackie has 4carrot and 5 tomato worth 25$

It will take approximately 2 hours and 3 minutes to fill the aquarium.

The aquarium at the zoo is in the shape of a cylinder with a height of 5 feet and a base radius of 3.5 feet.

To calculate the volume of the aquarium, we can use the formula for the volume of a cylinder, which is:

V = πr²h

Plugging in the given values we get:

V = π(3.5²)(5) = 192.5π cubic feet

Since one cubic foot is approximately 7.5 gallons, the aquarium has a volume of approximately 1443.75 gallons. If the aquarium is being filled at a rate of 12 gallons per minute, it will take approximately 120.3 minutes, or 2 hours and 3 minutes, to fill the aquarium.

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The equation of the curve is y = x2 + 4.

To solve this problem, we need to use the fact that the curve y = ax2 + bx + c passes through the point (2, 12) and is tangent to the line yat at the origin.

First, we know that the tangent to the curve at the origin is the line y = 0x + c = c. Since the curve is tangent to this line at the origin, we know that the derivative of the curve at x = 0 is equal to 0.

Taking the derivative of y = ax2 + bx + c, we get y' = 2ax + b. Setting x = 0, we get y' = b. Since y' = 0 at x = 0, we know that b = 0.

So now we have y = ax2 + c. We can use the fact that the curve passes through the point (2, 12) to solve for a and c.

Substituting x = 2 and y = 12 into the equation y = ax2 + c, we get 12 = 4a + c.

Since we know that a = 4, 6, or 1, we can substitute each of these values into the equation and solve for c.

When a = 4, we get 12 = 4(4)(2) + c, which simplifies to 12 = 32 + c. Solving for c, we get c = -20.

When a = 6, we get 12 = 4(6)(2) + c, which simplifies to 12 = 48 + c. Solving for c, we get c = -36.

When a = 1, we get 12 = 4(1)(2) + c, which simplifies to 12 = 8 + c. Solving for c, we get c = 4.

So the values of a, b, and c are:
a = 1
b = 0
c = 4

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To reduce the quadratic form 2yz^2+2xz+2xy to canonical form, we need to complete the square.

First, we factor out the coefficient of z^2 from the yz^2 term:

2yz^2 = 2z(yz)

Next, we add and subtract the square of half the coefficient of z from the resulting expression:

2z(yz + (x/y)^2 - (x/y)^2)

= 2z((y + x/y)^2/4 - (x/y)^2)

= z(y + x/y)^2/2 - zx^2/y

Now, we can see that the quadratic form can be written in the canonical form:

q(x,y,z) = (y + x/y)^2/2 - x^2/y

To find the rank, we need to count the number of non-zero eigenvalues. In this case, we have two non-zero eigenvalues, so the rank is 2.

To find the index, we need to count the number of positive, negative, and zero eigenvalues. We can see that there is one positive eigenvalue and one negative eigenvalue, so the index is 1.

Finally, to find the signature, we subtract the index from the rank. In this case, the signature is 1.
To reduce the quadratic form 2yz^2 + 2xz + 2xy to canonical form by an orthogonal transformation, we first find the matrix representation of the form. The given quadratic form can be written as Q = [x, y, z] * A * [x, y, z]^T, where A is a symmetric matrix:

A = | 0  1  1 |
    | 1  0  1 |
    | 1  1  2 |

Now, we find the eigenvalues and eigenvectors of A. The eigenvalues are λ₁ = 3, λ₂ = -1, and λ₃ = 0, with corresponding eigenvectors:

v₁ = [1, 1, 1]
v₂ = [-1, 1, 0]
v₃ = [-1, -1, 2]

Normalize the eigenvectors to form an orthogonal matrix P:

P = |  1/√3   1/√2  -1/√6 |
    |  1/√3  -1/√2  -1/√6 |
    |  1/√3    0    2/√6 |

Now, we can transform A to its canonical form using the orthogonal matrix P:

D = P^T * A * P
D = | 3  0  0 |
    | 0 -1  0 |
    | 0  0  0 |

So, the canonical form of the quadratic form is:

Q canonical = 3x'^2 - y'^2

The rank of the quadratic form is the number of non-zero eigenvalues in the diagonal matrix D. In this case, the rank is 2.

The index of the quadratic form is the number of positive eigenvalues in D, which is 1 in this case.

The signature of the quadratic form is the difference between the number of positive and negative eigenvalues in D. In this case, the signature is 1 - 1 = 0.

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The probability that a CD player will break down during the guarantee period is about 0.1985 or 19.85%.

To find the probability that a Sunshine CD player will break down during the 3-year guarantee period, we'll use the properties of the normal distribution.

Given a mean life of 4.1 years and a standard deviation of 1.3 years, we can calculate the z-score corresponding to the 3-year guarantee period:

z = (x - μ) / σ = (3 - 4.1) / 1.3 ≈ -0.846

Now, we'll look up the probability associated with this z-score in a standard normal distribution table, or use a calculator or software to find the cumulative probability. The probability corresponding to a z-score of -0.846 is approximately 0.1985.

Therefore, the probability that a CD player will break down during the guarantee period is about 0.1985 or 19.85%.

For the sketch, draw a bell-shaped curve to represent the normal distribution. Mark the x-axis with the mean (4.1 years) in the center, and scale it with the standard deviation (1.3 years).

Place a vertical line at 3 years to represent the end of the guarantee period, and shade the area to the left of this line to represent the probability of breaking down during that period.

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A histogram of the data distribution for the number of new plants is shown in the image below.

In this scenario and exercise, you are required to create a histogram to show the data distribution with respect to the number of new plants. First of all, we would determine the midpoint, absolute frequency, relative frequency, and cumulative frequency;

Midpoint                                      Absolute frequency      Rel. frequency

[0, 2] = (0 + 2)/2 = 1                        3 + 2 = 5                        0.128205

[2, 4] = (2 + 4)/2 = 3                        2 + 3 = 5                        0.128205

[4, 6] = (4 + 6)/2 = 5                        2 + 3 = 5                        0.128205

[6, 8] = (6 + 8)/2 = 7                                    2                        0.051282

[8, 10] = (8 + 10)/2 = 9                     3 + 4 = 7                        0.179487

[10, 12] = (10 + 12)/2 = 11                  4 + 3 = 7                        0.179487

[12, 14] = (12 + 14)/2 = 13                  1 + 2 = 3                        0.076923

[14, 16] = (14 + 16)/2 = 15                  3 + 2 = 5                        0.128205

Mathematically, the relative frequency of a data set can be calculated by using this formula:

Relative frequency = absolute frequency/total frequency × 100

Relative frequency = 5/39 × 100 = 0.128205

For the cumulative frequency, we have:

0.128205

0.128205 + 0.128205 = 0.25641

0.25641 + 0.128205 = 0.384615

0.384615 + 0.051282 = 0.435897

0.435897 + 0.179487 = 0.615385

0.615385 + 0.179487 = 0.794872

0.794872 + 0.076923 = 0.871795

0.871795 + 0.128205 = 1

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Missing information:

The question is incomplete and the complete question is shown in the attached picture.

The number that is equal to the place values, 7 hundred thousands 4 thousands 3 tens and 6 ones, is 704,036

From the question, we are to determine the number that is equal to the given place values

From the given information, the given place value is

7 hundred thousands 4 thousands 3 tens and 6 ones

Now, we will write each of the values in figures

7 hundred thousands = 700,000

4 thousands = 4,000

3 tens = 30

6 ones = 6

To determine the number that is equal to the place values, we will sum all the digits

700,000 + 4,000 + 30 + 6

704,036

Hence,

The number that is equal to the place value is 704,036

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The unit rate for planting is 1 1/6 acres per day.

To find the unit rate, we need to divide the total area planted by the number of days taken to plant it. We can convert the mixed number of days to an improper fraction by multiplying the whole number by the denominator and adding the numerator.

Therefore, 4 1/2 days can be converted to 9/2 days.

Now we can divide the total area of 5 1/4 acres by the number of days it took to plant it, which is 9/2 days. To divide fractions, we invert the second fraction and multiply, so:

5 1/4 ÷ 9/2 = 21/4 ÷ 9/2 = 21/4 x 2/9 = 42/36

We can simplify 42/36 by dividing both the numerator and denominator by their greatest common factor, which is 6.

42/36 = 7/6

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The percentage of evergreen trees from the total combination of trees is 79%

The percentage of evergreen trees will be calculated using the formula -

Percentage of evergreen trees = number of evergreen trees/total number of trees × 100

Total number of trees = number of evergreen + deciduous trees

Total trees = 3160 + 840

Total trees = 4000

Percentage of evergreen trees = 3160/4000 × 100

Cancelled common zeroes in numerator and denominator and performing division

Percentage = 79%

Hence, the percentage of evergreen trees is 79%.

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Answer:

45°

Step-by-step explanation:

sin∠U = 5√2 / 10 = √2/2

m∠U = sin⁻¹(√2/2) = 45°

The dilation is (x, y) → (2x, 2y). So, the correct answer is D).

Let the coordinates of point C be (x, y). Then, the distance from the origin to point C is given by the distance formula

OC = √(x² + y²)

The corresponding side lengths are

CG = 16 - x

CD = √((x - 0)² + (y - 0)²)

The scale factor is the ratio of corresponding side lengths

CG/CD = 2

Therefore,

16 - x = 2*√(x² + y²)

Solving for y, we get

y = √(13x² - 64x + 256)

If we assume that point G corresponds to point C, then the center of dilation is the origin and the rule that represents the dilation is

(x, y) → (2x, 2y)

Therefore, the answer is

(x, y) → (2x, 2y)

So, the correct answer is D).

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--The given question is incomplete, the complete question is given

" Quadrilateral ABCD is dilated about the origin into quadrilateral EFGH so that point G is located at (16,8). scale factor is 2.

Which rule represents the dilation?

Select one

(x, y) → (18x, 18y)

(x, y) → (x+8, y+4)

(x, y) → (12x, 12y)

(x, y) → (2x, 2y) "--

Ornamental Insulation Corporation would report a net income of $1,022,000 and comprehensive income of $1,010,000 resulting from these investments in its 2021 financial statements.

Ornamental Insulation Corporation would report the following amounts in its 2021 income statement, statement of comprehensive income, and balance sheet as a result of the investment activities:

Interest Income from American Instruments Bonds: $93,600 ($1,170,000 × 8%)

Gain on Sale of Distribution Transformers Bonds: $43,000 ($623,000 - $580,000)

Total Net Income: $136,600 ($93,600 + $43,000)

Gain on Investments: $55,000 (This represents the gain on the sale of the distribution transformers bonds and is included in the comprehensive income section.)

Balance Sheet (as of December 31, 2021):

American Instruments Bonds: $1,102,000 (market value)

M&D Corporation Bonds: $1,670,000 (face value)

Accumulated Other Comprehensive Income: $55,000 (This represents the gain on investments and is included in the comprehensive income section.)

Retained Earnings: Increase of $136,600 (This represents the net income from the income statement.)

In summary, Ornamental Insulation Corporation would report a net income of $136,600, a comprehensive income of $55,000, and an increase in retained earnings of $136,600 as a result of these investments for the fiscal year 2021.

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The algebraic expression for the volume of the right rectangular pyramid is (x/3) × (units²).

The volume of a right rectangular pyramid is given by the formula;

V = (1/3) × base_area × height

where base_area is area of the base of the pyramid.

In this case, the length and width of the base are given as units and units, respectively. Therefore, the area of the base is;

base_area = length × width

Substituting the given values, we get;

base_area = units × units = units²

The height of the pyramid is given as x units. Therefore, the volume of the pyramid can be expressed as;

V = (1/3) × (units²) × x

Simplifying the expression, we get;

V = (x/3) × (units²)

Therefore, the algebraic expression for the volume of pyramid is (x/3) × (units²).

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To determine what students want served in the cafeteria, the cook asks students in Ms. Andrew’s first period class. The sample used by the cook is known as Convenience.

The sample used is known as convenience sample. The cook only asks students in Ms. Andrews’ first period class which is a convenient and accessible group to ask but this method of sampling may not be representative of the entire student population as it only includes students in one class.

So, the results may not accurately reflect what all students want to be served in the cafeteria, hence, more representative sample could be obtained by using a simple random sample or systematic sample where  student in the population has an equal chance of being selected.

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The height of the new player is given as follows:

210 cm.

The mean of a data-set is given by the sum of all observations in the data-set divided by the number of observations, which is also called the cardinality of the data-set.

Considering the frequencies and the height of the new player of x, the sum of the observations is given as follows:

198 x 1 + 199 x 3 + 200 x 2 + 201 x 5 + 202 x 2 + x = 2604 + x.

The total number of players, considering the new player, is given as follows:

1 + 3 + 2 + 5 + 2 + 1 = 14.

The mean is of 201 cm, hence the height of the new player is obtained as follows:

(2604 + x)/14 = 201

2604 + x = 14 x 201

x = 14 x 201 - 2604

x = 210 cm.

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The final result is  ∫ sin²(x)- cos²(x)/cos(x) dx = -3cos²(x)/2 + 2ln|cos(x)| + C.

To solve the indefinite integral ∫ sin²(x)- cos²(x)/cos(x) dx, we need to use trigonometric identities to simplify the integrand.

First, we use the identity sin²(x) + cos²(x) = 1 to write:

sin²(x) - cos²(x) = sin²(x) + cos²(x) - 2cos²(x) = 2sin²(x) - cos²(x)

Next, we use the identity sin²(x) = 1 - cos²(x) to write:

2sin²(x) - cos²(x) = 2(1-cos²(x)) - cos²(x) = 2 - 3cos²(x)

Substituting this into the original integral, we get:

∫ sin²(x)- cos²(x)/cos(x) dx = ∫ (2 - 3cos²(x))/cos(x) dx

Now, we use the substitution u = cos(x) and du/dx = -sin(x) dx to transform the integral into a simpler form:

∫ (2 - 3cos²(x))/cos(x) dx = ∫ (2 - 3u²)/u (-du/sin(x))

= -∫ (3u² - 2)/u du

= -3∫ u du + 2∫ du/u

= -3u²/2 + 2ln|u| + C

= -3cos²(x)/2 + 2ln|cos(x)| + C

where C is the constant of integration.

Substituting back u = cos(x), we obtain the final result

∫ sin²(x)- cos²(x)/cos(x) dx = -3cos²(x)/2 + 2ln|cos(x)| + C

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a) To determine if the limit exists, we need to check if the limit from all directions approaching (0,0) are equal. Let's approach (0,0) along the x-axis first, so y = 0:

lim (x,y)->(0,0) [(x) + 2(y) + 3(x)(y)]
= lim x -> 0 [(x) + 2(0) + 3(x)(0)] = lim x -> 0 x = 0

Next, let's approach (0,0) along the y-axis, so x = 0:

lim (x,y)->(0,0) [(x) + 2(y) + 3(x)(y)]
= lim y -> 0 [(0) + 2(y) + 3(0)(y)] = lim y -> 0 2y = 0

Now, let's approach (0,0) along the line y = mx, where m is some constant:

lim (x,y)->(0,0) [(x) + 2(y) + 3(x)(y)]
= lim x -> 0 [(x) + 2(mx) + 3(x)(mx)]
= lim x -> 0 [(1+3m)x + 2mx^2]
= 0 if m=0, and DNE (does not exist) for all other values of m.

Since the limit is not equal from all directions, the limit DNE at (0,0).

b) To determine if the limit exists, we need to check if the limit from all directions approaching (0,0) are equal. Let's approach (0,0) along the x-axis first, so y = 0:

lim (x,y)->(0,0) [(2.4) + (y)]
= lim x -> 0 [(2.4) + (0)] = 2.4

Next, let's approach (0,0) along the y-axis, so x = 0:

lim (x,y)->(0,0) [(2.4) + (y)]
= lim y -> 0 [(2.4) + (y)] = 2.4

Now, let's approach (0,0) along the line y = mx, where m is some constant:

lim (x,y)->(0,0) [(2.4) + (y)]
= lim x -> 0 [(2.4) + (mx)]
= 2.4 if m=0, and DNE (does not exist) for all other values of m.

Since the limit is equal from all directions, the limit exists and is equal to 2.4 at (0,0).

c) To determine if the function is continuous at (0,0), we need to check if the limit as (x,y) approaches (0,0) of f(x,y) exists and is equal to f(0,0).

Let g(x,y) = sqrt(x^2 + y^2), which satisfies 0 <= |(x,y)| <= g(x,y) for all (x,y). We have:

|f(x,y)| = |(x+y)/(4+x^2+y^2)| <= |(x+y)/4| <= (1/4)g(x,y)

So, we can bound f(x,y) by (1/4)g(x,y). By the sandwich (or squeeze) theorem, we have:

lim (x,y)->(0,0) (1/4)g(x,y) = 0

Thus, by the sandwich theorem, we have:

lim (x,y)->(0,0) f(x,y) = 0

Since the limit exists and is equal to f(0,0) = 0, the function is continuous at (0,0).

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