Therefore, the global minimum of f(x) on S is at [tex]x = 1/\sqrt{5[/tex]and the global maximum does not exist. And the global minimum of f(x) on S is at x = 2 and the global maximum is at x = 1.
What is function?A function is a relationship between a set of inputs (called the domain) and a set of outputs (called the range) with the property that each input is associated with exactly one output. In other words, a function maps each element in the domain to a unique element in the range.
Functions are commonly denoted by a symbol such as f(x), where x is an element of the domain and f(x) is the corresponding output value. The domain and range of a function can be specified explicitly or can be determined by the context in which the function is used.
(i) To find the global maximizers and minimizers of the function f(x) = 4x²+1/4x on the interval S=[1/√5,[infinity]], we first need to find the critical points of f(x) within S.
Taking the derivative of f(x) with respect to x, we get:
f'(x) = 8x - 1/4x²
Setting f'(x) = 0 to find critical points, we get:
8x - 1/4x² = 0
Multiplying both sides by 4x², we get:
32x³ - 1 = 0
Solving for x, we get:
x = 1/∛32 = 1/2∛2
Note that this critical point is not in the interval S, so we need to check the endpoints of S as well as any vertical asymptotes of f(x).
At x = 1/√5, we have:
f(1/√5) = 4(1/5) + 1/(4(1/√5)) = 4/5 + √5/4
At x → ∞, we have:
Lim x→∞ f(x) = ∞
Therefore, the global minimum of f(x) on S is at [tex]x = 1/\sqrt5[/tex] and the global maximum does not exist.
(ii) To find the global maximizers and minimizers of the function[tex]f(x) = x^5 - 8x^3[/tex] on the interval S=[1,2], we first need to find the critical points of f(x) within S.
Taking the derivative of f(x) with respect to x, we get:
[tex]f'(x) = 5x^4 - 24x^2[/tex]
Setting f'(x) = 0 to find critical points, we get:
[tex]5x^4 - 24x^2 = 0[/tex]
Factoring out [tex]x^2[/tex], we get:
[tex]x^2(5x^2 - 24) = 0[/tex]
Solving for x, we get:
[tex]x = 0 or x =±\sqrt(24/5)[/tex]
Note that none of these critical points are in the interval S, so we need to check the endpoints of S.
At x = 1, we have:
[tex]f(1) = 1 - 8 = -7[/tex]
At x = 2, we have:
[tex]f(2) = 32 - 64 = -32[/tex]
Therefore, the global minimum of f(x) on S is at x = 2 and the global maximum is at x = 1.
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For an M/G/1 system with λ = 20, μ = 35, and σ = 0.005. Find the average length of the queue.
A. Lq = 0.6095
B. Lq = 0.3926
C. Lq = 0.4286
D. Lq = 0.964
The average length of the queue (Lq) for an M/G/1 system with λ = 20, μ = 35, and σ = 0.005 is Lq = 0.3926 (option B).
To find the average length of the queue (Lq) in an M/G/1 system, we can use the Pollaczek-Khintchine formula:
Lq = (λ² * σ² + (λ/μ)²) / (2 * (1 - (λ/μ)))
Given λ = 20 (arrival rate), μ = 35 (service rate), and σ = 0.005 (standard deviation of service time):
1. Calculate λ/μ: 20/35 = 0.5714
2. Calculate 1 - (λ/μ): 1 - 0.5714 = 0.4286
3. Calculate λ² * σ²: (20²) * (0.005²) = 0.01
4. Calculate (λ/μ)²: (0.5714²) = 0.3265
5. Plug these values into the Pollaczek-Khintchine formula:
Lq = (0.01 + 0.3265) / (2 * 0.4286) = 0.3926 . (B)
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is 132 divisible by 3
Answer:
yes clearly
Step-by-step explanation:
using rules of divisibility it is divisible
using your calculator it is divisible
the answer is 44
if you want to be 99% confident of estimating the population mean to within a sampling error of ± 6 and the standard deviation is assumed to be , what sample size is required
Sample size of at least 23 is required to be 99% confident that our estimate of the population mean is within ±6.
How to calculate the sample size?To calculate the required sample size, we can use the formula:
n = (Zα/2 * σ / E)²
Where:
n = sample size
Zα/2 = the Z-score for the desired confidence level, which is 2.58 for 99%
σ = the population standard deviation (assumed to be given)
E = the desired margin of error, which is 6 in this case.
Substituting these values, we get:
n = (2.58 * σ / 6)²
Since the population standard deviation is not given, we cannot find the exact sample size. However, we can use an estimated value of σ based on prior knowledge or a pilot study.
For example, if we assume σ = 10, then the sample size required would be:
n = (2.58 * 10 / 6)² = 22.25 ≈ 23
Therefore, we would need a sample size of at least 23 to be 99% confident that our estimate of the population mean is within ±6.
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Find the perimeter of the triangle:
The perimeter is 58.9
How to find the perimeter?Here we have an isosceles triangle.
To find the length of the sides that aren't the base we can use a trigonometric equation.
sin(60°) = 4*√15/hypotenuse
hypotenuse = 4*√15/sin(60°) = 18.8
The side in the left also measures that.
Now we need the base, we can define the base as:
(b/2)² + (4√15)² = 18.8²
b²/4 + 16*15 = 18.8²
b = √((18.8² - 16*15)*4)
b = 21.3
Then the perimeter is:
21.3 + 18.8 + 18.8 = 58.9
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Trapezium: Parallel side 1 is 7cm. Parallel side 2 is 11cm. Height is 6cm. What will be the area? Please show your working.
Answer:
54 square centimeters
Step-by-step explanation:
The area of a trapezium can be calculated by taking the average of the parallel sides and multiplying by the height. So, the area of this trapezium is:
(7 + 11) / 2 * 6 = 9 * 6 = 54 cm^2
Therefore, the area of the trapezium is 54 square centimeters.
Hope this helps!
Answer:
the area of the trapezium is 54 square centimeters.
Step-by-step explanation:
Given:
Parallel side 1 = 7cm
Parallel side 2 = 11cm
Height = 6cm
We can use the formula for the area of a trapezium, which is:
Area = (Sum of parallel sides / 2) × Height
Plugging in the values we have:
Area = ((7 + 11) / 2) × 6
Now, let's simplify the equation:
Area = (18 / 2) × 6
Area = 9 × 6
Area = 54
So, the area of the trapezium is 54 square centimeters.
complete the table to find the derivative of the function without using the quotient rule. function rewrite differentiate simplify y = (9x3⁄2)/x ____ x _____ ______
To complete the table and find the derivative of the function y = (9x^(3/2))/x without using the quotient rule, we'll rewrite, differentiate, and simplify the function and get dy/dx = 27/2x^(1/2) - 9x^(1/2) .
Step 1: Rewrite the function
y = 9x^(3/2) * x^(-1) (multiply the x term in the denominator by -1 to rewrite the division as multiplication)
Step 2: Differentiate the function using the power rule (dy/dx = nx^(n-1))
dy/dx = 9(3/2)x^(3/2 - 1) - 9x^(3/2 - 1)
Step 3: Simplify the expression
dy/dx = 27/2x^(1/2) - 9x^(1/2)
Your answer: To find the derivative of the function y = (9x^(3/2))/x without using the quotient rule, we rewrote the function, differentiated it, and simplified the result to obtain the derivative dy/dx = 27/2x^(1/2) - 9x^(1/2).
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If his company is worth $15 million, what is normally the maximum amount of funds that Entrepreneur Bill should raise
a)$1.0 m
b)$3.75 m
c)$1.5 m
d)none of the above
The maximum amount of funds that Entrepreneur Bill should raise typically depends on various factors such as the growth potential of the business, the market demand, and the financial needs of the company.
However, a general rule of thumb is that entrepreneurs should not raise more than 25% to 30% of the company's worth in a single fundraising round.
So, if his company is worth $15 million, the maximum amount of funds that Entrepreneur Bill should raise is around $3.75 million. This will help him maintain a fair ownership stake in the company while also ensuring that he has enough funds to achieve his business goals.
It is important to note that this is just a rough estimate and every business is unique. Entrepreneur Bill should seek the advice of experienced investors or financial advisors to determine the appropriate amount of funds to raise for his specific business needs.
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Priscilla can make 3 bracelets in 15 minutes. At this rate, how many bracelets can she make in 45 minutes?
Answer:
9
Step-by-step explanation:
because if she can make 3 in 15minutes then 45 minutes is triple the time so triple the bracelets please give brainliest bye have a good day :D
Solve the system using substitution. Check your solution
4x-y=62
2y=x
Answer:
(x, y) (124/7, 62/7)
Step-by-step explanation:
substitute x with 2y we get
4(2y)-y=62
8y-y=62
7y=62
y=62/7
substitute the value of y into the second eqaution
2(62/7)=x
124/7=x
substitute your values of x and y into the first eqaution to check if they are solutions
4(124/7)-62/7=62
you will get
62=62
which means they are solutions
find an equation of the slant asymptote. do not sketch the curve. y = x2 2 x 2y=?
The required answer is 2y = x / (x + 2)
To find the equation of the slant asymptote for y = (x^2)/(2x + 2), we can perform long division or synthetic division to divide x^2 by 2x + 2. The result is y = (1/2)x - 1. Therefore, the equation of the slant asymptote is y = (1/2)x - 1.
The asymptotes most commonly encountered in the study of calculus are of curves of the form y = ƒ(x). These can be computed using limits and classified into horizontal, vertical and oblique asymptotes depending on their orientation. Horizontal asymptotes are horizontal lines that the graph of the function approaches as x tends to +∞ or −∞. As the name indicates they are parallel to the x-axis. Vertical asymptotes are vertical lines (perpendicular to the x-axis) near which the function grows without bound.
It seems there might be some typos in the given function. I believe you meant the function to be written as y = (x^2 + 2x) / 2y. To find the equation of the slant asymptote, follow these steps:
Step 1: Rewrite the given function with proper notation:
y = (x^2 + 2x) / (2y)
Step 2: Solve for x in terms of y:
2y = x^2 + 2x
2yx = x^2 + 2x
Step 3: Factor out x on the right side:
2yx = x(x + 2)
Step 4: Divide both sides by (x + 2):
2y = x / (x + 2)
This equation represents the slant asymptote of the given function.
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En un triángulo rectángulo el cateto mayor excede en 2 cm al menor y la hipotenusa supera en 2cm al cateto mayor. Calcular la medida de cada lado
To indirectly measure the distance across a lake, Nachelle makes use of a couple landmarks at points D and E. She measures CF, FD, and FG as marked. Find the distance across the lake (DE), rounding your answer to the nearest hundredth of a meter
The distance across the lake (DE) is 207.68 m.
How to find the distance across the lake (DE)?The corresponding side lengths of two triangles that are similar are always proportional to each other.
Thus,
ΔCDE and ΔCFG are similar to each other
FG = 142.1 m
FC = 130 m
DF = 60 m
DC = 130 + 60 = 190 m
Thus, DE/FG = DC/FC
Substituting
DE/142.1 = 190/130
DE = (190*142.1)/130
DE = 207.68 m (nearest hundredth).
Therefore, the distance across the lake (DE) to the nearest hundredth is 207.68 m.
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suppose z has a standard normal distribution with a mean of 0 and standard deviation of 1. the probability that z is between -2.33 and 2.33 is
The probability that z is between -2.33 and 2.33 for a standard normal distribution with a mean of 0 and standard deviation of 1 is approximately 0.9802, or 98.02%. Here, he probability that z is between -2.33 and 2.33 for a standard normal distribution with a mean of 0 and a standard deviation of 1, you'll need to use a standard normal distribution table or a calculator with a built-in z-table function.
Step-by-step explanation:
Step:1. Identify the given values: Mean (µ) = 0, Standard Deviation (σ) = 1, and the range of z-scores is between -2.33 and 2.33.
Step:2. Use a standard normal distribution table or a calculator with a built-in z-table function to find the probabilities associated with z = -2.33 and z = 2.33.
Step:3. Look up the probability of z = -2.33 in the table, which should be approximately 0.0099.
Step:4. Look up the probability of z = 2.33 in the table, which should be approximately 0.9901.
Step:5. Subtract the probability for z = -2.33 from the probability for z = 2.33 to find the probability of z being between these two values: P(-2.33 < z < 2.33) = P(z = 2.33) - P(z = -2.33) = 0.9901 - 0.0099 = 0.9802
The probability that z is between -2.33 and 2.33 for a standard normal distribution with a mean of 0 and standard deviation of 1 is approximately 0.9802, or 98.02%.
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find the average rate of change of the function between the given values of x. y = 6 3x 0.5x2 between x = 4 and x = 6.
The average rate of change of the function between the given values of x. y = 6 3x 0.5x2 between x = 4 and x = 6 is 8
To find the average rate of change of the function y = 6 + 3x + 0.5x^2 between x = 4 and x = 6, we need to find the difference between the y-values at x = 6 and x = 4, and divide by the difference between the x-values.
When x = 4, y = 6 + 3(4) + 0.5(4)^2 = 22
When x = 6, y = 6 + 3(6) + 0.5(6)^2 = 36
The difference in y-values is 36 - 22 = 14.
The difference in x-values is 6 - 4 = 2.
Therefore, the average rate of change of the function between x = 4 and x = 6 is 14/2 = 7.
So, the average rate of change of the function is 7 units per 1 unit change in x between the given values of x.
To find the average rate of change of the function y = 6 + 3x + 0.5x^2 between x = 4 and x = 6, follow these steps:
1. Evaluate the function at x = 4 and x = 6:
y(4) = 6 + 3(4) + 0.5(4^2) = 6 + 12 + 8 = 26
y(6) = 6 + 3(6) + 0.5(6^2) = 6 + 18 + 18 = 42
2. Calculate the average rate of change:
Average rate of change = (y(6) - y(4)) / (6 - 4) = (42 - 26) / 2 = 16 / 2 = 8
So, the average rate of change of the function between x = 4 and x = 6 is 8.
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Given the 4 points below, identify what shape is formed and how you found your answer. A(-1, 0), B(0, 2), C(4, 0), and D(3, -2)
Answer:
The shape formed is a quadrilateral.
Step-by-step explanation:
The four points A(-1,0), B(0,2), C(4,0), and D(3,-2) can be used to form a quadrilateral. To identify the shape formed by these points, we can use the distance formula to find the length of each side of the quadrilateral, and then compare the side lengths.
AB: Distance between A(-1,0) and B(0,2)
= sqrt((0 - (-1))^2 + (2 - 0)^2)
= sqrt(1 + 4)
= sqrt(5)
BC: Distance between B(0,2) and C(4,0)
= sqrt((4 - 0)^2 + (0 - 2)^2)
= sqrt(16 + 4)
= sqrt(20)
= 2 sqrt(5)
CD: Distance between C(4,0) and D(3,-2)
= sqrt((3 - 4)^2 + (-2 - 0)^2)
= sqrt(1 + 4)
= sqrt(5)
DA: Distance between D(3,-2) and A(-1,0)
= sqrt((-1 - 3)^2 + (0 - (-2))^2)
= sqrt(16 + 4)
= 2 sqrt(5)
Since the length of AB is not equal to the length of CD, and the length of BC is not equal to the length of DA, we can conclude that the quadrilateral formed by these four points is not a parallelogram or a rhombus. Additionally, since the length of AB is not equal to the length of CD, we can conclude that the quadrilateral is not a kite.
By comparing the angles formed by the line segments AB, BC, CD, and DA, we can see that the angle at B is a right angle, while the other three angles are all acute angles. This indicates that the quadrilateral is a trapezoid. Specifically, it is a right trapezoid, since it has one right angle.
give the laplace transofrm of -6 0<=x and x
The Laplace transform of -6 for the given conditions is -6/s + 1/s^2
The Laplace transform is a mathematical operation that transforms a function of time into a function of a complex variable s, commonly used in solving linear ordinary differential equations. The Laplace transform of a function f(t), denoted by L{f(t)}, is defined as:
L{f(t)} = F(s) = ∫[f(t)e^(-st)]dt, where s is a complex variable.
In this case, the given function is -6 for 0<=x and x. Since the function is constant, it can be represented as a step function, where the value of the function changes abruptly at x=0. The step function is denoted as u(x), where u(x) = 0 for x<0, and u(x) = 1 for x>=0.
So, the given function can be written as -6u(x), where u(x) is the step function.
Now, applying the definition of the Laplace transform, we get:
L{-6u(x)} = ∫[-6u(x)e^(-sx)]dx
Since u(x) = 0 for x<0, the integral becomes:
L{-6u(x)} = ∫[0*e^(-sx)]dx = 0
Since u(x) = 1 for x>=0, the integral becomes:
L{-6u(x)} = ∫[-6*e^(-sx)]dx = -6∫[e^(-sx)]dx
Integrating e^(-sx) with respect to x, we get:
L{-6u(x)} = -6 * (-1/s) * e^(-sx) + C, where C is the constant of integration.
Finally, substituting back u(x) = 1, we get:
L{-6u(x)} = -6 * (-1/s) * e^(-sx) + C = 6/s * e^(-sx) + C
So, the Laplace transform of -6 for the given conditions is -6/s * e^(-sx) + C, which can also be written as -6/s + C/s, where C is a constant.
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suppose v1,v2,v3 is an orthogonal set of vectors in r5. let w be a vector in span(v1,v2,v3) such that v1⋅v1=6,v2⋅v2=18,v3⋅v3=25, w⋅v1=−6,w⋅v2=−90,w⋅v3=−75,
According to the information, we can express the vector w as a linear combination of v1, v2, and v3 like this: w = -v1 - 5v2 - 3v3
How to express the vector w as a linear combination?We can express the vector w as a linear combination of v1, v2, and v3. Let's say:
w = c1 v1 + c2 v2 + c3 v3
We can find the values of c1, c2, and c3 using the dot product properties of orthogonal vectors. Since v1, v2, and v3 are orthogonal:
w ⋅ v1 = (c1 v1 + c2 v2 + c3 v3) ⋅ v1 = c1 (v1 ⋅ v1) = 6c1
w ⋅ v2 = (c1 v1 + c2 v2 + c3 v3) ⋅ v2 = c2 (v2 ⋅ v2) = 18c2
w ⋅ v3 = (c1 v1 + c2 v2 + c3 v3) ⋅ v3 = c3 (v3 ⋅ v3) = 25c3
Using the given values, we can set up a system of equations:
-6 = 6c1 + 0c2 + 0c3
-90 = 0c1 + 18c2 + 0c3
-75 = 0c1 + 0c2 + 25c3
Solving for c1, c2, and c3, we get:
c1 = -1
c2 = -5
c3 = -3
Therefore, we have:
w = -v1 - 5v2 - 3v3
Note: The solution is not unique, as any linear combination of v1, v2, and v3 that satisfies the given dot product conditions would work.
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This question has several parts that must be completed sequentially. If you skip able to come back to the skipped part. Tutorial Exercise Find the dimensions of a rectangle with perimeter 120 m whose area is as large as possible. Step 1 If a rectangle has dimensions x and y, then we must maximize the area A= xy. Since the perimeter is 2x +2y = 120, then y= __ - x. Step 2 We must maximize the area A= xy x=(60-x)=60x- x^2,where 0
The dimensions of the rectangle with the largest possible area and a perimeter of 120 meters are 30 meters by 30 meters.
Explanation: -
To find the dimensions of a rectangle with a perimeter of 120 meters and the largest possible area, we need to follow these steps:
Step 1: Given the dimensions x and y, we have the area A = xy. then the perimeter of the rectangle is 2x + 2y = 120. Solving for y, we get y = 60 - x.
Step 2: To maximize the area A = xy, we substitute y with the expression from step 1: A(x) = x(60 - x) = 60x - x^2, where 0 < x < 60.
To find the maximum area, we can use calculus to find the critical points.
Step 3: Find the derivative of the area function, use the formula
d/dx(x^n) =nx^n-1
so that derivative is A'(x) = 60 - 2x.
Step 4: Set A'(x) = 0 and solve for x. In this case, 60 - 2x = 0, so x = 30.
Step 5: Plug x = 30 back into the expression for y: y = 60 - x = 60 - 30 = 30.
The dimensions of the rectangle with the largest possible area and a perimeter of 120 meters are 30 meters by 30 meters.
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The areas of 10 cities are given in the table.
How much greater is the range for the cities in the south than the cities in the north?
Enter your answer in the box.
mi²
Northern Cities Southern Cities
Portland 145 mi² Orlando 111 mi²
Seattle 84 mi² New Orleans 350 mi²
New York City 305 mi² Los Angeles 503 mi²
Detroit 143 mi² San Diego 372 mi²
Minneapolis 58 mi² Atlanta 132 mi²
The range for the southern cities is 145 mi² greater than the range for the northern cities.
How to find the range of the cities?We must determine the difference between the largest and smallest areas in each group in order to determine the range of the cities.
Minneapolis has the smallest area (58 miles2) and the largest (305 miles2) of the northern cities. Therefore, the northern cities' range is:
305 mi² - 58 mi² = 247 mi²
For the southern cities, the smallest area is 111 mi² (Orlando) and the largest area is 503 mi² (Los Angeles). So the range for the southern cities is:
503 mi² - 111 mi² = 392 mi²
To find how much greater the range is for the southern cities, we subtract the range for the northern cities from the range for the southern cities:
392 mi² - 247 mi² = 145 mi²
As a result, the cities in the south have a range that is 145 miles longer than the cities in the north.
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A 16 ounce box of pasta costs $1.12. A 32 ounce box cost 1.92. A 5 pound box cost $4.00. Which box is the best deal?
Answer:
To find the best deal, we must get the value of 1 oz for each of the boxes.
Box 1: 1.12 divided by 16 equals 0.07 / oz
Box 2: 1.92/32 equals 0.06 / oz
5 LB Box = 80 OZ
Box 3: 4/80 = 0.05 / oz
The third box is the best deal.
Answer the question below in the picture. Thank you! Have a wonderful day.
Answer:
1) Statistical
2)Not statistical
3)Not statistical
Step-by-step explanation:
If your not sure just visit it will tell you the answers.
Hope it helps!!
Find the measures of angle A and B. Round to the nearest degree.
Answer:
∠ A = 60° , ∠ B = 30°
Step-by-step explanation:
using the cosine ratio in the right triangle
cosA = [tex]\frac{adjacent}{hypotenuse}[/tex] = [tex]\frac{AC}{AB}[/tex] = [tex]\frac{7}{14}[/tex] = [tex]\frac{1}{2}[/tex] , then
∠ A = [tex]cos^{-1}[/tex] ( [tex]\frac{1}{2}[/tex] ) = 60°
the sum of the 3 angles in Δ ABC = 180°
∠ A + ∠ B + ∠ C = 180°
60° + ∠ B + 90° = 180°
∠ B + 150° = 180° ( subtract 150° from both sides )
∠ B = 30°
Widely known kite ABCD
35cm^2
. Gerrard made a kite
with the length of each diagonal
each twice the length of the diagonal of the kite
ABCD kite. Calculate the area of the kite
the new one !
The area of the kite the new one is 70 cm^2.
Calculating the area of the kite the new oneThe area of a kite is given by half the product of its diagonals. Let's call the diagonals of kite ABCD d1 and d2, and the diagonals of the new kite d1' and d2'.
We know that d1' = 2d1 and d2' = 2d2, so we can write:
Area of new kite = 1/2 * d1' * d2'
= 1/2 * (2d1) * (2d2)
= 2 * (1/2 * d1 * d2)
= 2 * Area of kite ABCD
= 2 * 35 cm^2
= 70 cm^2
Therefore, the area of the new kite is 70 cm^2.
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WILL GIVE BRAINLIEST + 100 PTS
The mean of four positive integers is 5. The median of the four integers is 6.
What is the mean of the largest and smallest of the integers?
Answer:
4
Step-by-step explanation:
(b + c)/2 = 6
b + c = 12
(a + b + c + d)/4 = 5
(a + 12 + d) = 20
a + d = 8
Hence,
the sum of the largest and smallest is 8. The mean has to be 8/2 = 4.
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Zach bought 200 shares of Goshen stock years ago for $21.35 per share. He sold all 200 shares today for $43 per share. What was his gross capital gain?
Answer: 8,600
Zach's gross capital gain can be calculated as follows:
Total proceeds from selling the stock = 200 shares x $43/share = $8,600
Total cost of buying the stock = 200 shares x $21.35/share = $4,270
Gross capital gain = Total proceeds - Total cost = $8,600 - $4,270 = $4,330
Therefore, Zach's gross capital gain from selling 200 shares of Goshen stock is $4,330.
apply the convolution theorem to find the inverse laplace transform of the given function. 1/s(s2+ 36)
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l-1{1/s(s2+36}
The inverse Laplace transform of 1/s(s^2 + 36) using the convolution theorem is (1/6)sin(6t) + (1/6)cos(6t).
First, we need to find the Laplace transform of the given function 1/s(s^2 + 36). We can use the table of Laplace transforms to find that L{1/s(s^2 + 36)} = (1/6)sin(6t).
Next, we need to find the Laplace transform of the function f(t) = cos(6t)u(t), where u(t) is the unit step function. Using the table of Laplace transforms, we find that L{cos(6t)u(t)} = (s)/(s^2 + 36).
Now, we can apply the convolution theorem, which states that the inverse Laplace transform of the product of two functions in the frequency domain is equal to the convolution of their inverse Laplace transforms in the time domain.
The convolution of (1/6)sin(6t) and (s)/(s^2 + 36) is given by the integral of (1/6)sin(6(t - τ)) * (s)/(s^2 + 36) dτ from 0 to t.
To solve the integral, we can use partial fraction decomposition. We can express (s)/(s^2 + 36) as (A/s) + (B(s)/(s^2 + 36)), where A and B are constants to be determined.
Solving for A and B, we get A = 1/6 and B(s) = -s/6.
Substituting A and B(s) back into the integral and evaluating the integral, we get (1/6)sin(6t) + (1/6)cos(6t).
Therefore, the inverse Laplace transform of 1/s(s^2 + 36) using the convolution theorem is (1/6)sin(6t) + (1/6)cos(6t).
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In a right-skewed distribution the median is greater than the mean. a. the median equals the mean. b. the median is less than the mean. c. none of the above. d. Dravious Skip
In a right-skewed distribution, the correct answer is b. the median is less than the mean. In a right-skewed distribution, the data has a longer tail on the right side, indicating that there are more values greater than the mean. This causes the mean to be greater than the median.
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Consider a linear model of the form:
y(x,theta)=theta0+∑=1thetaxy(xn,θ)=θ0+∑d=1Dθdxnd
where x=(x1,...,x)xn=(xn1,...,xnD) and weights theta=(theta0,...,theta)θ=(θ0,...,θD). Given the the D-dimension input sample set x={x1,...,x}x={x1,...,xn} with corresponding target value y={y1,...,y}y={y1,...,yn}, the sum-of-squares error function is:
(theta)=12∑=1{y(x,theta)−y}2ED(θ)=12∑n=1N{y(xn,θ)−yn}2
Now, suppose that Gaussian noise ϵn with zero mean and variance 2σ2 is added independently to each of the input sample xxn to generate a new sample set x′={x1+1,...,x+}x′={x1+ϵ1,...,xn+ϵn}. For each sample xxn, x′=(x1+1,...,x+)xn′=(xn1+ϵn1,...,xnD+ϵnd), where n and d is independent across both n and d indices.
(3pts) Show that y(x′,theta)=y(x,theta)+∑=1thetay(xn′,θ)=y(xn,θ)+∑d=1Dθdϵnd
Assume the sum-of-squares error function of the noise sample set x′={x1+1,...,x+}x′={x1+ϵ1,...,xn+ϵn} is (theta)′ED(θ)′. Prove the expectation of (theta)′ED(θ)′ is equivalent to the sum-of-squares error (theta)ED(θ) for noise-free input samples with the addition of a weight-decay regularization term (e.g. 2L2 norm) , in which the bias parameter theta0θ0 is omitted from the regularizer. In other words, show that
[(theta)′]=(theta)+z
Step-by-step explanation:
Part 1:
We know that y(x,θ) = θ0 + ∑d=1Dθdxnd and x′n = xn + ϵn.
So,
y(x′,θ) = θ0 + ∑d=1Dθd(xnd+ϵnd)
= θ0 + ∑d=1Dθdxnd + ∑d=1Dθdϵnd
Since ϵn is independent of the weights θ, we can take it outside the summation:
y(x′,θ) = y(x,θ) + ∑d=1Dθdϵnd
Therefore, we have shown that y(x′,θ) = y(x,θ) + ∑d=1Dθdϵnd.
Part 2:
The sum-of-squares error function for the noise sample set x′ is given by:
ED'(θ) = 1/2 ∑n=1N [y(x′n,θ) - yn]^2
Using the expression for y(x′,θ) derived in part 1, we have:
ED'(θ) = 1/2 ∑n=1N [y(xn,θ) + ∑d=1Dθdϵnd - yn]^2
Expanding the square term and taking the expectation with respect to the noise ϵ, we get:
E[ED'(θ)] = E[1/2 ∑n=1N [(y(xn,θ) - yn)^2 + 2(y(xn,θ) - yn)∑d=1Dθdϵnd + (∑d=1Dθdϵnd)^2]]
Now, since ϵ is a zero-mean Gaussian noise with variance 2σ^2, we have:
E[ϵnd] = 0
E[ϵnd^2] = σ^2
Using these properties, we can simplify the above expression:
E[ED'(θ)] = E[1/2 ∑n=1N [(y(xn,θ) - yn)^2 + 2(y(xn,θ) - yn)∑d=1DθdE[ϵnd] + (∑d=1Dθd^2E[ϵnd^2])]]
= E[1/2 ∑n=1N (y(xn,θ) - yn)^2] + E[θ]^T E[Z] E[θ]
where Z is a (D-1) x (D-1) matrix with (i,j)-th element being E[ϵiϵj], and E[Z] is the matrix obtained by adding σ^2 to the diagonal elements of Z. The terms involving the cross-product of ϵ are ignored as they are zero.
The first term in the above expression is just the sum-of-squares error for the noise-free input samples. The second term is the weight-decay regularization term, which is proportional to the L2 norm of the weights θ, with the bias parameter θ0 omitted.
Therefore, we have shown that:
E[ED'(θ)] = (theta)^T(theta) + z
where z is the weight-decay regularization term.
construct an arrow diagram to show the relation is the square of from ×=(1,4,9) TO y=(3,2,1,-1,-2,-3)
The arrow symbolizes the directional connection from each member in set x to its corresponding member in set y. The members of set y are evident squares of their respective counterparts in set x.
How to solveHere is an arrow diagram to show the relation between the sets x and y, where y is the set of all elements in x squared:
(1, 4, 9)
/ \
/ \
/ \
1, 4, 9 --> 1, 4, 9, 16, 25, 36
\ /
\ /
\ /
(3, 2, 1, -1, -2, -3)
The arrow symbolizes the directional connection from each member in set x to its corresponding member in set y. The members of set y are evident squares of their respective counterparts in set x.
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6) One hundred tickets numbered 1,2,3,...,100, are sold to 100 different people for a drawing. Three different prizes are awarded, including the grand prize (trip to Serbia), and no person can get more than one prize. a) How many ways are there to award prizes? b) In how many ways can you do this if person holding the ticket 23 must get a prize? c) In how many ways can you do this if the person holding ticket 8 and the person holding ticket 11 must win prizes? d) In how many ways can you do this if the grand prize winner is a person holding ticket 8,11 or 23?
The following parts can be answered by the concept of Combination.
a. The total number of ways to award prizes is 100 x 99 x 98 = 970,200.
b. The total number of ways to award prizes in this scenario is 99 x 98 x 97 = 941,094.
c. The total number of ways to award prizes in this scenario is 98 x 97 x 96 = 912,192.
d. The grand prize winner is a person holding ticket 8,11 or 23 is 3 x 99 x 98 = 29,178.
a) There are a total of 100 choices for the first prize, 99 choices for the second prize (since one person already won a prize), and 98 choices for the third prize (since two people already won prizes). So, the total number of ways to award prizes is 100 x 99 x 98 = 970,200.
b) If person holding the ticket 23 must get a prize, then there are only 99 choices for the first prize (since ticket 23 is already taken), 98 choices for the second prize, and 97 choices for the third prize. So, the total number of ways to award prizes in this scenario is 99 x 98 x 97 = 941,094.
c) If the person holding ticket 8 and the person holding ticket 11 must win prizes, then there are only 98 choices for the first prize (since tickets 8 and 11 are already taken), 97 choices for the second prize, and 96 choices for the third prize. So, the total number of ways to award prizes in this scenario is 98 x 97 x 96 = 912,192.
d) If the grand prize winner is a person holding ticket 8,11 or 23, then there are only 3 choices for the first prize (since only these three tickets are eligible for the grand prize), 99 choices for the second prize (since one person already won a prize and the grand prize winner is not eligible for the second prize), and 98 choices for the third prize (since two people already won prizes). So, the total number of ways to award prizes in this scenario is 3 x 99 x 98 = 29,178.
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