How is the product of a complex number and a real number represented on the complex plane?



Consider the product of 2−4i and 3.



Drag a value or phrase into each box to correctly complete the statements

The derivative of f(x) is: f'(x) = [tan⁻¹(2)/2] - [tan⁻¹(1/2)/2]

The Fundamental Theorem of Calculus is a pair of theorems that link the concept of differentiation and integration. It states that if a function f(x) is continuous on an interval [a, b] and F(x) is the antiderivative of f(x) on the same interval, then:

Part I: The derivative of the integral of f(x) from a to x is equal to f(x):

d/dx ∫a to x[tex]f(t) dt = f(x)[/tex]

Part II: The integral of the derivative of a function f(x) on an interval [a, b] is equal to the difference between the values of the function at the endpoints of the interval:

∫a to b [tex]f'(x) dx = f(b) - f(a)[/tex]

Using Part I of the Fundamental Theorem of Calculus, we have:

f(x) = x∫4 1/(1+4t⁴) dt

Then, by the Chain Rule, we have:

f'(x) = d/dx [x∫4 1/(1+4t⁴) dt] = ∫4 d/dx [x(1/(1+4t⁴))] dt

= ∫4 (1/(1+4t⁴)) dt

= [tan⁻¹(2t)/2]₄¹

= [tan⁻¹(2)/2] - [tan⁻¹(1/2)/2]

Therefore, the derivative of f(x) is:

f'(x) = [tan⁻¹(2)/2] - [tan⁻¹(1/2)/2]

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Kimo should make 5 loads in the small truck and 4 loads in the large truck per day to minimize the total cost while meeting the delivery and union requirements.

What is the equivalent expression?

Equivalent expressions are expressions that perform the same function despite their appearance. If two algebraic expressions are equivalent, they have the same value when we use the same variable value.

the total cost will be minimized when the loads are distributed in the following way:

5 loads in the small truck (total capacity of 5 * 50 = 250 cubic yards)

4 loads in the large truck (total capacity of 4 * 70 = 280 cubic yards)

This will result in a total of 9 loads and a total capacity of 530 cubic yards, which meets both the daily minimum delivery requirement of 350 cubic yards and the union contract requirement of 9 loads per day.

The total cost can be calculated as follows:

Cost of 5 loads in small truck = 5 * $87 = $435

Cost of 4 loads in large truck = 4 * $73 = $292

Total cost per day = $435 + $292 = $727

Therefore, Kimo should make 5 loads in the small truck and 4 loads in the large truck per day to minimize the total cost while meeting the delivery and union requirements.

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Therefore , the solution of the given problem of expressions comes out to be  7n.

Instead of using random estimates, it is preferable to use shifting numbers that may also prove increasing, reducing, variable or blocking. They could only help one another by trading tools, information, or solutions to issues. The justifications, components, or quantitative comments for tactics like further disagreement, production, and blending may be included in the assertion of truth equation.

Here,

By dividing the number of rows by the number of trees in each row, one can calculate the total number of trees in Mr. Lee's apple orchard. Here are two expressions that demonstrate various approaches to determining the overall number of trees:

There are 7n =  trees in all.

=> Total number of trees = (Number of rows) x (Number of trees in each row) = 7n

The total number of trees in the orchard is the outcome of both expressions.

While the second statement more directly depicts the multiplication, the first expression merely merges the two elements into a single term.

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The arc measure of major arc BDC in degrees is 240 degrees.

To find the arc measure of major arc BDC in degrees, you'll need to provide more information about the given circle or angles within it. However, I can guide you on how to find the arc measure once you have the necessary information.

1. Determine the measure of the central angle corresponding to the major arc BDC. This can be done by subtracting the measure of the minor arc from 360 degrees.

2. Use the central angle measure to find the arc measure of major arc BDC. Since the arc measure is equal to the measure of the central angle in degrees, the arc measure of major arc BDC will be the same as the central angle measure you found in step 1.

Please provide more information or details about the given circle or angles to help you find the arc measure of major arc BDC.

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The combination of these side lengths uniquely determines the shape of the triangle.

Hi! To draw a triangle with side lengths 3 inches, 5 inches, and 6 inches, make sure that the sum of any two sides is greater than the third side. In this case, 3 + 5 > 6, 3 + 6 > 5, and 5 + 6 > 3, so a triangle can be formed.

Yes, this is the only triangle you can draw using these side lengths.

The reason is that the side lengths are fixed, and according to the triangle inequality theorem, the combination of these side lengths uniquely determines the shape of the triangle.

The combination of these side lengths uniquely determines the shape of the triangle.

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The exact tax owed cannot be determined without knowing the specific income bracket for Alex's taxable income.

We know that,

Based on the provided information, Alex's investment income consists of

$500 of qualified dividends and $2,000 of short-term capital gains.

Here, we have to calculate the taxes owed on his investment income, we

need to determine the applicable tax rate based on his taxable income.

As the specific income bracket for Alex's taxable income is not mentioned, it is not possible to provide an exact amount of taxes owed.

The tax rate and corresponding income brackets should be referenced to calculate the taxes accurately.

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The minimum value of the function is -13 and the minimum value of the function occurs at the point (4, -13).

The function F(x) is a quadratic function with a negative coefficient of the squared term.

Therefore, the function has a maximum value.

To find the maximum value, we need to find the vertex of the parabola.

The x-coordinate of the vertex is given by x = -b/2a, where a and b are the coefficients of the x² and x terms respectively.

In this case, a = -1 and b = -8, so x = -(-8)/(2(-1)) = 4.

To find the minimum value, we substitute this x-value into the function to get F(4) = -(4²) - 8(4) + 19 = -13.

Therefore, the minimum value of the function is -13.

We found in part (1) that the x-coordinate of the vertex is x = 4.

To find the y-coordinate, we substitute this x-value into the function to get F(4) = -13.

Therefore, the minimum value of the function occurs at the point (4, -13).

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The future value of the investment when Jack is ready to purchase the vacation home is $44,924.

To solve this problem, we can use the formula for future value of an annuity:

FV = Pmt x [(1 + r)^n - 1] / r

Where:

Pmt = $1000 (quarterly contribution)
r = 0.021 (annual interest rate)
n = 40 (number of quarters until Jack turns 50)

Plugging in the numbers, we get:

FV = $1000 x [(1 + 0.021)^40 - 1] / 0.021
FV = $44,924.38

Therefore, the future value of Jack's investment, rounded to the nearest dollar, is $44,924. So the correct answer is $44,924.

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The inverse functions:

f'(2) = 4.

[tex]f^{-1}(x)[/tex] = sqrt(x - 4).

f'(s1(x)) = sqrt(x - 4).

(a) To find f'(2), we need to take the derivative of f(x) with respect to x and then substitute x = 2.
[tex]f(x) = x^2 + 4[/tex]
f'(x) = 2x
f'(2) = 2(2) = 4
Therefore, f'(2) = 4.
(b) To find the inverse function [tex]f^{-1}(x)[/tex], we need to first solve for x in terms of f(x) and then switch the roles of x and f(x).
[tex]f(x) = x^2 + 4[/tex]
[tex]x^2[/tex] = f(x) - 4
x = sqrt(f(x) - 4)
Switching x and f(x), we get:
[tex]f^{-1}(x)[/tex] = sqrt(x - 4)
Therefore, the inverse function is [tex]f^{-1}(x)[/tex] = sqrt(x - 4).
(c) To find the compound function f'(s1(x)),

we need to first find s1(x) and then take the derivative of f(x) with respect to s1(x) and then multiply by the derivative of s1(x) with respect to x.
s1(x) = sqrt(x - 4)
f(s1(x)) = (sqrt(x - 4)[tex])^2[/tex] + 4 = x
Taking the derivative of f(x) with respect to s1(x), we get:
f'(s1(x)) = 2s1(x)
Taking the derivative of s1(x) with respect to x, we get:
s1'(x) = 1/(2sqrt(x - 4))
Multiplying these two derivatives, we get:
f'(s1(x))s1'(x) = 2s1(x) * 1/(2sqrt(x - 4))
f'(s1(x))s1'(x) = sqrt(x - 4)
Therefore, the compound function is f'(s1(x)) = sqrt(x - 4).
(d) The given expression "derivative mets-() de 1-12" does not make sense and seems incomplete. Please provide more information or context so that I can help you with this part of the question.

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The distance between the two cruise liners is approximately 3.6 km.

We can use the Pythagorean theorem to find the distances between Liam and the two cruise liners, and then use the distance formula to find the distance between the two cruise liners. Let's call the distance between Liam and the first cruise liner "d1" and the distance between Liam and the second cruise liner "d2". Then:

d1 = sqrt(5² - 2²) = sqrt(21) km

d2 = sqrt(6.8² - 2²) = sqrt(44.44) km

To find the distance between the two cruise liners, we can use the distance formula:

distance = sqrt((d2 - d1)² + (6.8 - 5)²) km

Plugging in the values, we get:

distance = sqrt((sqrt(44.44) - sqrt(21))² + 1.8²) km

Simplifying this expression gives:

distance = sqrt(44.44) - sqrt(21) km

So the distance between the two cruise liners is approximately 3.9 km.

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Answer:

Step-by-step explanation:

You're going to want to break up the shape into three parts, two triangles, and the rectangle.

Starting with the left-most triangle: A=(L*W)/2

The length is 4ft and the width is 3ft, multiply and divide by 2 to get:  A=6 square feet.

Do the same with the second triangle on the bottom left (L=2ft, W=2ft) to get A=2 square feet.

Now the rectangle, A=L*W and total length is 10ft (8ft+2ft) and the width is 3ft. Multiply these values to get A=30 square feet.

Last step: add up all three areas for the total area of the entire shape, 6+2+30=38.

Area= 38 square feet.

The prediction for the number of times the event of landing on an even number in 550 rolls is 275

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

The number of times = 550

The sample space of a fair 6-sided die is

S = {1, 2, 3, 4, 5, 6}

And as such the even numbers are

Even = {2, 4, 6}

This means that in a fair 6-sided die, we have

P(Even) = 3/6

When evaluated, we have

P(Even) = 1/2

So, when the die is rolled 550 times, we have

Expected value = 1/2 * 550

Evaluate

Expected value = 275

Hence, the number of times is 275

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The fraction is equivalent to a whole number  are 9/3, -16/8, 7/0, 0/5

A fraction can simply be described as the part of a whole variable, a whole numbers, or a whole element.

In mathematics, there are different types of fractions. These fractions are listed thus;

From the information given, we have that;

Equivalent expressions or fractions are fractions with the same solutions

Then, we have;

9/3

Divide the values

3

-16/8

-2

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Answer:

(d)

Step-by-step explanation:

The positive value of z indicates option e) the value z is to the right of the mean.

A standard normal random variable has a mean of 0 and a standard deviation of 1. Positive values of z represent values above the mean, while negative values of z represent values below the mean.

The probability associated with a value of z is always positive since it represents the likelihood of observing a certain value. The area between zero and z is also always positive since it represents the probability of observing a value between 0 and z.

Therefore, option e) is the correct answer as it reflects the relationship between positive values of z and their location relative to the mean.

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Given the function y = 4x² + 3x, we will find dy/dt by differentiating y with respect to t. Therefore, the value of dy/dt is -15.

Using the chain rule, we have:

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

Differentiating y with respect to x, we get:

dy/dx = 8x + 3

Now, we are given that x = -1 and dx/dt = 3. We can substitute these values into our equation:

dy/dt = (8(-1) + 3)(3)

dy/dt = (-5)(3)

dy/dt = -15

So, when x = -1 and dx/dt = 3, the value of dy/dt is -15.

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The cross-product of u and v:
w = u × v = (5, -4, 8) × (-8, -4, 5) = (-20, -60, 32)

Thus, w is orthogonal to u. Since we need two vectors in opposite directions, we can negate w:
-w = (20, 60, -32)
Therefore, the two orthogonal vectors in opposite directions are w = (-20, -60, 32) and -w = (20, 60, -32).

To find two vectors that are orthogonal to u, we can use the cross-product. Let v = (4,5,0) and w = (-8,0,5). Then v x u = (40,40,45) and w x u = (20,-40,20). So two vectors orthogonal to u are (40,40,45) and (20,-40,20).

To determine whether two planes are orthogonal, parallel, or neither, we can look at the normal vectors of each plane. Let the first plane be defined by the equation 2x + 3y - z = 4 and the second plane being defined by the equation :

4x + 6y - 2z = 8.

The normal vector of the first plane is (2,3,-1) and the normal vector of the second plane is (4,6,-2).

Since the dot product of these two normal vectors is -2(3) + 3(6) - 1(2) = 14, which is not equal to 0, the planes are not orthogonal.

To determine if they are parallel, we can check if the ratio of their normal vectors is constant. Dividing the second normal vector by the first, we get (4/2, 6/3, -2/-1) = (2,2,2). Since this is a constant ratio, the planes are parallel.

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The centroid of the given triangle (2, 8/3), the circumcenter of the triangle is (0,2), the orthocenter of the triangle is (2,8).

What is centroid?

In geometry, the centroid of a triangle is the point where the three medians of the triangle intersect.

To find the centroid of a triangle with vertices at (x1,y1), (x2,y2), and (x3,y3), we can use the formula:

(x1 + x2 + x3)/3 , (y1 + y2 + y3)/3

Using this formula, we get the centroid of the given triangle as:

((-4 + 2 + 8)/3 , (0 + 8 + 0)/3) = (2, 8/3)

To find the circumcenter, we first need to find the equations of the perpendicular bisectors of any two sides of the triangle. Let's choose the sides formed by the points (-4,0) and (2,8), and (2,8) and (8,0).

The midpoint of the first side is ((-4+2)/2, (0+8)/2) = (-1,4), and the slope of the line passing through (-4,0) and (2,8) is (8-0)/(2-(-4)) = 8/6 = 4/3. So the equation of the perpendicular bisector of this side is y-4 = -(3/4)(x+1), or 3x + 4y = 8.

Similarly, the midpoint of the second side is ((2+8)/2, (8+0)/2) = (5,4), and the slope of the line passing through (2,8) and (8,0) is (0-8)/(8-2) = -8/6 = -4/3. So the equation of the perpendicular bisector of this side is y-4 = (3/4)(x-5), or 3x - 4y = -8.

The intersection of these two lines gives us the circumcenter of the triangle. Solving the system of equations:

3x + 4y = 8

3x - 4y = -8

We get x = 0, y = 2. So the circumcenter of the triangle is (0,2).

To find the orthocenter, we first need to find the equations of the altitudes from any two vertices of the triangle. Let's choose the vertices (2,8) and (8,0).

The altitude from (2,8) is perpendicular to the side formed by the points (-4,0) and (8,0), so its slope is 0. Therefore, its equation is y = 8.

The altitude from (8,0) is perpendicular to the side formed by the points (-4,0) and (2,8), so its slope is the negative reciprocal of the slope of that side, which is -4/3. Using the point-slope form, we get the equation:

y - 0 = (-4/3)(x - 8)

y = -4x/3 + 32/3

To find the intersection of these two lines, we can substitute y = 8 into the second equation:

8 = -4x/3 + 32/3

-8/3 = -4x/3

x = 2

Substituting x = 2 into either equation gives us y = 8, so the orthocenter of the triangle is (2,8).

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Answer:

[tex]x = \frac{ log(125) }{ log(25) } = \frac{ log( {5}^{3} ) }{ log( {5}^{2} ) } = \frac{3 log(5) }{2 log(5) } = \frac{3}{2} = 1 \frac{1}{2} [/tex]

Answer: 81

Step-by-step explanation: this would be 81 because [tex]3^{3}[/tex] is basically 3 x 3 x 3 and 3 x 3 x 3=27 then multiply 27 by 3, and you get 81

Answer:

3^4. my answer needs to be 20+character sooooooooo

The simplified expression is 4.

To simplify the expression (-3)(-2) -2), we need to follow the order of operations, which are parentheses first, then multiplication and division from left to right, and finally addition and subtraction from left to right.

First, we need to simplify (-3)(-2) to get:

(-3)(-2) = 6

Now we can substitute this value into the expression to get:

6 - 2) = 4

Therefore, the simplified expression is 4.

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Answer:

x = -9

Step-by-step explanation:

As this is an absolute value function, the line of symmetry is the x-value of the maximum/minimum point. An absolute value function can be denoted as y = |x - h| + k, where (h, k) is the maximum/minimum point. We only need the x-value of the maximum/minimum point, so we only have to look at "h". Now, we can use y = |x - h| + k and turn it into g(x):

y = |x - h| + k

g(x) = |x - -9| + -11   --> this means h = -9, and the line of symmetry is at x = -9

g(x) = |x + 9| - 11

Answer:

I think x equals --9

To move triangle KLM so that point K lies on the Y-axis, you need to apply a translation that shifts the entire triangle horizontally. By translating triangle KLM using the vector (-4, 0), point K now lies on the Y-axis.

To move the triangle so that point K lies on the Y axis, we need to perform a translation. First, we need to determine how far point K is from the Y axis. We can do this by finding the x-coordinate of point K, which is 4. This means that point K is 4 units away from the Y axis.

Next, we need to determine the direction of the translation. Since we want to move point K onto the Y axis, we need to move the triangle in the negative x direction. Therefore, the translation that will move the triangle so that point K lies on the Y axis is a horizontal translation of -4 units. We can express this translation as follows:

T(-4, 0)

This means that we need to move each point of the triangle 4 units to the left (negative x direction) to achieve the desired position of point K on the Y axis.

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Answer:

Let's solve this problem using algebra. Let x be the cost of one orange and y be the cost of one apple. Then we have the system of equations:

12x + 7y = 5.36

8x + 5y = 3.68

To solve for x and y, we can use elimination. Multiplying the second equation by 3 and subtracting it from the first equation multiplied by 5, we get:

(5*12 - 7*8)x + (5*7 - 3*5)y = 26.8 - 11.04

20x = 15.76

x = 0.788

Substituting x back into one of the equations, we can solve for y:

12(0.788) + 7y = 5.36

y = 0.308

Therefore, one orange costs $0.788 and one apple costs $0.308.

An equation of the line that passes through the pair of points (0,4) and (6, −3) in slope-intercept form is y = -7x/6 + 4.

In Mathematics and Geometry, the point-slope form of a straight line can be calculated by using the following mathematical expression:

y - y₁ = m(x - x₁)

Where:

First of all, we would determine the slope of this line;

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

Slope (m) = (-3 - 4)/(6 - 0)

Slope (m) = -7/6

At data point (0, 4) and a slope of -7/6, a linear equation for this line can be calculated by using the point-slope form as follows:

y - y₁ = m(x - x₁)

y - 4 = -7/6(x - 0)  

y = -7x/6 + 4

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Answer:

a) apple juice

b) whole milk

easy pagel

A. proj_v(u) = (586 / 68) * (8i + 2j) = (293 / 34) * (8i + 2j) ≈ 8.62i + 2.15j

B. u_orthogonal = (71 + 9j) - (8.62i + 2.15j) ≈ 62.38i + 6.85j

(a) To find the projection of vector u onto vector v, we use the formula:

proj_v(u) = (u·v / ||v||^2) * v

where u = 71 + 9j, v = 8i + 2j, "·" represents the dot product, and ||v|| represents the magnitude of v.

First, let's find the dot product u·v:

u·v = (71)(8) + (9)(2) = 568 + 18 = 586

Next, we find the magnitude of v:

||v|| = √((8)^2 + (2)^2) = √(64 + 4) = √68

Now, we find ||v||^2:

||v||^2 = 68

Finally, we can find the projection of u onto v:

proj_v(u) = (586 / 68) * (8i + 2j) = (293 / 34) * (8i + 2j) ≈ 8.62i + 2.15j

(b) To find the vector component of u orthogonal to v, we subtract the projection of u onto v from u:

u_orthogonal = u - proj_v(u)

u_orthogonal = (71 + 9j) - (8.62i + 2.15j) ≈ 62.38i + 6.85j

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Its translation is 5×3-3=12

Marsha would save $26,040 over the life of the loan if she purchases the 3 points.

First, let's calculate the monthly payment if Marsha doesn't purchase any points. We can use a mortgage calculator or the PMT function in Excel to find;

PMT = $1,987.13

Now, let's calculate the monthly payment if Marsha purchases 3 points;

Loan amount = $350,000

Points cost = 3 points × 1% × $350,000 = $10,500

Effective loan amount = $350,000 - $10,500 = $339,500

Interest rate = 4.5% / 12 = 0.375%

Number of payments=20 years × 12 = 240

Using the PMT function, we get;

PMT = $1,878.63

So, by purchasing 3 points, Marsha can save;

$1,987.13 - $1,878.63 = $108.50 per month

Over the life of the loan, which is 20 years or 240 months, the total savings would be;

$108.50 × 240 = $26,040

Therefore, Marsha would save $26,040 amount of money.

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