Maths equation need help please

The first two iterations of the Method of Steepest Descent algorithm starting from the initial point Xo = (2, 1) for the function f(x1, x2) = 3x1 + 2x2 are as follows:

Iteration 1:

1. Compute the gradient at the current point Xo: ∇f(Xo) = [∂f/∂x1, ∂f/∂x2] = [3, 2].

2. Choose a step size (learning rate) α.

3. Update the current point Xo using the gradient and step size: X1 = Xo - α * ∇f(Xo).

Iteration 2:

1. Compute the gradient at the current point X1: ∇f(X1) = [∂f/∂x1, ∂f/∂x2].

2. Choose a step size (learning rate) α.

3. Update the current point X1 using the gradient and step size: X2 = X1 - α * ∇f(X1).

In the given function f(x1, x2) = 3x1 + 2x2, the partial derivatives with respect to x1 and x2 are 3 and 2, respectively. These represent the gradients in the x1 and x2 directions at any given point (x1, x2).

The Method of Steepest Descent is an iterative optimization algorithm that aims to minimize a function by moving in the direction of the steepest descent (negative gradient) at each iteration.

It starts from an initial point Xo and updates the current point by taking steps in the opposite direction of the gradient, multiplied by a step size or learning rate α.

In the first iteration, we compute the gradient at the initial point Xo = (2, 1), which is ∇f(Xo) = [∂f/∂x1, ∂f/∂x2] = [3, 2]. Let's assume we choose a learning rate α of 0.1.

Using the gradient and learning rate, we update Xo to X1:

X1 = Xo - α * ∇f(Xo) = (2, 1) - 0.1 * [3, 2] = (2, 1) - [0.3, 0.2] = (1.7, 0.8).

In the second iteration, we compute the gradient at the current point X1 = (1.7, 0.8), which is ∇f(X1) = [∂f/∂x1, ∂f/∂x2]. Let's assume we again choose a learning rate α of 0.1.

Using the gradient and learning rate, we update X1 to X2:

X2 = X1 - α * ∇f(X1) = (1.7, 0.8) - 0.1 * [∂f/∂x1, ∂f/∂x2] = (1.7, 0.8) - [0.1 * ∂f/∂x1, 0.1 * ∂f/∂x2].

The above calculations provide the values of X1 and X2 after the first two iterations of the Method of Steepest Descent algorithm for the given function.

Now, let's move on to the second part of your question.

If the initial start point Xo is changed to a different position, it can significantly affect the operation of the algorithm. The Method of Stee

pest Descent aims to find a local optimum of the function, and the starting point plays a crucial role in determining the convergence behavior.

If the new initial point is closer to a local optimum, the algorithm may converge faster as it takes smaller steps towards the optimal point. However, if the new initial point is far from any local optima, the algorithm may take longer to converge or even converge to a different suboptimal point.

The choice of learning rate α also affects the algorithm's performance. A larger learning rate may lead to faster convergence but can also cause overshooting and instability. On the other hand, a smaller learning rate may lead to slower convergence but better stability.

In summary, changing the initial start point xo can affect the convergence behavior and the final solution obtained by the Method of Steepest Descent algorithm. It is crucial to choose an appropriate initial point and learning rate to achieve the desired optimization outcome.

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Answer:

I think it is 30

Step-by-step explanation:

Answer:

x = 30

Step-by-step explanation:

The angles shown are vertical angles

If you didn't know vertical angles are congruent

That being said we create an equation to solve for x

(because vertical angles are congruent) 104 = 3x + 14

now that we have created an equation we want to solve for x

Step 1 subtract 14 from each side

104 - 14 = 90

14 - 14 cancels out

now we have 90 = 3x

step 2 divide each side by 3

90/3=30

3x/3=x

we're left with x = 30

The Sample, we estimate that there are approximately 40 small T-shirts in the box.

The number of small T-shirts in the box, sampling and assume that the proportion of small T-shirts in the sample is representative of the proportion in the entire box.

In the given sample of 10 shirts, we have the following sizes: large, small, extra large, medium, small, extra large, large, small, medium, small.

Out of the 10 shirts, 4 of them are small. To estimate the number of small T-shirts in the entire box, we can set up a proportion:

Small shirts in sample / Total shirts in sample = Small shirts in box / Total shirts in box

Plugging in the values we have:

4 / 10 = x / 100

Cross-multiplying:

4 * 100 = 10 * x

400 = 10x

Dividing both sides by 10:

x = 400 / 10

x = 40

Based on the sample, we estimate that there are approximately 40 small T-shirts in the box.

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Answer:

The triangles are congruent because they are both exactly the same.

Answer:

How to Find the Mean. The mean is the average of the numbers. It is easy to calculate: add up all the numbers, then divide by how many numbers there are. In other words it is the sum divided by the count.

Answer:

x = 8[tex]\sqrt{3}[/tex] , y = 8

Step-by-step explanation:

Using the sine and cosine ratios in the right triangle and the exact values

sin60° = [tex]\frac{\sqrt{3} }{2}[/tex] , cos60° = [tex]\frac{1}{2}[/tex]

sin60° = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{x}{16}[/tex] = [tex]\frac{\sqrt{3} }{2}[/tex] ( cross- multiply )

2x = 16[tex]\sqrt{3}[/tex] ( divide both sides by 2 )

x = 8[tex]\sqrt{3}[/tex]

----------------------------------------------------------

cos60° = [tex]\frac{adjacent}{hypotenuse}[/tex] = [tex]\frac{y}{16}[/tex] = [tex]\frac{1}{2}[/tex] ( cross- multiply )

2y = 16 ( divide both sides by 2 )

y = 8

Answer:

117+x=180°(sum of straight line)

Step-by-step explanation:

x=180-117

x=63

Step-by-step explanation:

[tex]\pi \times {7}^{2} = 3.14 \times 49 = 153.86 \\ 15 \times 25 = 375 \\ 153.86 + 375 = 523.86 \: {ft}^{2} [/tex]

Answer:

the answer is the first one

Step-by-step explanation:

Explanation: be im smart

Function transformation involves changing the form of a function

A function that could represent the transformed function is function (c) [tex]f(x) = -\sqrt{\frac 12 x}[/tex]

The equation of the function is given as:

[tex]f(x) = \sqrt x[/tex]

The rule of horizontal stretch is:

[tex](x,y) \to (ax,y)[/tex]

Where:

[tex]0 < a < 1[/tex]

Take for instance:

[tex]a = \frac 12[/tex]

So, we have:

[tex]f(x) = \sqrt{\frac 12 x}[/tex]

Next, the function is reflected in across the x-axis.

The rule of this transformation is:

[tex](x,y) \to (x,-y)[/tex]

So, we have:

[tex]f(x) = -\sqrt{\frac 12 x}[/tex]

Hence, a function that could represent the transformed function is function (c)

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Answer: About 4

Step-by-step explanation:

16 divided by 3 2/3 is 4. decimal so round that and about 4 is you’re answer.

Answer:

4 dress will be the max

Step-by-step explanation:

it is a division question

16 yards on a roll ÷ 3⅔ yards of fabric

16÷3⅔

16÷ 11/3

16*3/11

48/11

4.3636.

Answer:

the answer is option (b)

good day mate

Answer:

384 cubic inches (in^3)

Step-by-step explanation:

volume = length * width * height

So find the volume of the large rectangle of the prism, then find the volume of the small rectangle of the prism. Add the two volumes together and the sum is your final answer.

I would appreciate Brainliest, but no worries.

Answer:

180 degrees

Step-by-step explanation:

i think im not sure???????                                                                                                                      i hope im right

you put it in the question

Answer:

I think it's the second I e at the top

The exact values of A and B that satisfy the equation are A = -65/22 and B = 65/11.

To find the exact values of A and B in the equation 16Acos(x) - Bsin(x) - 2Asin(x) + 19Bcos(x) = 65cos(x), we need to equate the coefficients of the corresponding trigonometric functions on both sides of the equation.

Comparing the coefficients of cos(x) on both sides:

16A + 19B = 65 (Equation 1)

Comparing the coefficients of sin(x) on both sides:

-2A - B = 0 (Equation 2)

We now have a system of two equations with two unknowns (A and B). We can solve this system to find the values of A and B.

Let's solve the system of equations:

From Equation 2, we can express B in terms of A:

B = -2A

Substituting this expression for B in Equation 1:

16A + 19(-2A) = 65

16A - 38A = 65

-22A = 65

A = -65/22

Substituting the value of A back into the expression for B:

B = -2A

B = -2(-65/22)

B = 65/11

Therefore, the exact values of A and B that satisfy the equation are:

A = -65/22

B = 65/11

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Dividing 3x^2 - 11x - 4 by x - 4 results in the quotient of 3x + 1 and a remainder of -23.

To divide 3x^2 - 11x - 4 by x - 4, we can use long division.

First, we divide the highest degree term of the dividend by the divisor. In this case, (3x^2) / (x) gives us 3x as the first term of the quotient.

Next, we multiply the divisor (x - 4) by the first term of the quotient (3x) to obtain (3x)(x - 4) = 3x^2 - 12x.

We subtract this result from the dividend (3x^2 - 11x - 4) to get a new polynomial: (3x^2 - 11x - 4) - (3x^2 - 12x) = x + 8x - 4.

Now, we repeat the process with the new polynomial (x + 8x - 4). We divide the highest degree term (x) by the divisor (x - 4), which gives us the second term of the quotient, 8.

Multiplying the divisor (x - 4) by the second term of the quotient (8) gives us (8)(x - 4) = 8x - 32.

Subtracting this from the new polynomial (x + 8x - 4) - (8x - 32) = 40, we obtain the remainder.

Therefore, the division of 3x^2 - 11x - 4 by x - 4 gives the quotient 3x + 1 and the remainder -23.

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Given:

An archer hits a target 50% of the time.

To find:

The experimental probability that the archer hits the target exactly four of the next five times.

Solution:

It is given that an archer hits a target 50% of the time. It means the probability of hitting the target is

[tex]p=\dfrac{50}{100}[/tex]

[tex]p=0.5[/tex]

The probability of not hitting the target is

[tex]q=1-p[/tex]

[tex]q=1-0.5[/tex]

[tex]q=0.5[/tex]

Binomial distribution formula:

[tex]P(x=r)=^nC_rp^rq^{n-r}[/tex]

We need to find the probability that the archer hits the target exactly four of the next five times. So, [tex]n=5,r=4,p=0.5,q=0.5[/tex].

[tex]P(x=4)=^5C_4(0.5)^4(0.5)^{5-4}[/tex]

[tex]P(x=4)=\dfrac{5!}{4!(5-4)!}(0.5)^4(0.5)^{1}[/tex]

[tex]P(x=4)=5(0.5)^{5}[/tex]

[tex]P(x=4)=0.15625[/tex]

Therefore, the experimental probability that the archer hits the target exactly four of the next five times is 0.15625.

The sample variance for the given data set is 3.6.

To calculate the sample variance, we follow a series of steps. First, we need to find the mean (average) of the data set. Adding up all the numbers and dividing by the total count gives us the mean, which in this case is (4+3+4+7+4+8)/6 = 30/6 = 5.

Next, we calculate the deviations of each data point from the mean. We subtract the mean from each data point to get the deviations: (4-5), (3-5), (4-5), (7-5), (4-5), and (8-5), which simplify to -1, -2, -1, 2, -1, and 3, respectively.

Then, we square each deviation to eliminate negative values:[tex](-1)^2[/tex], [tex](-2)^2[/tex], [tex](-1)^2[/tex], [tex]2^2[/tex], [tex](-1)^2[/tex], and [tex]3^2[/tex], which simplify to 1, 4, 1, 4, 1, and 9, respectively.

The next step is to find the sum of the squared deviations. Adding up all the squared deviations gives us 20.

Finally, we divide the sum of squared deviations by the total count minus 1 (n-1) to calculate the sample variance: 20/(6-1) = 20/5 = 4.

Rounding the sample variance to one decimal place, we get 3.6 as the final result.

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Answer:

20 blocks

Step-by-step explanation:

The closest answer for each area is B. 0.052, D. 0.015, and C. 0.936, respectively.

(a) The area to the left of 32 on a N(45, 8) distribution. The area to the left of 32 on a N(45, 8) distribution is given by: P(Z < (32 - 45)/8)P(Z < -1.625)= 0.052, approximately. So, the closest answer is B. 0.052.

(b) The area to the right of 12 on a N(9.4, 1.2) distribution. The area to the right of 12 on a N(9.4, 1.2) distribution is given by: P(Z > (12 - 9.4)/1.2)P(Z > 2.166)= 1 - P(Z < 2.166)= 1 - 0.985= 0.015. So, the closest answer is D. 0.015.

(c) The area between 43 and 100 on a N(75, 15) distribution. The area between 43 and 100 on a N(75, 15) distribution is given by: P((43 - 75)/15 < Z < (100 - 75)/15)P(-1.5333 < Z < 1.6666)= P(Z < 1.6666) - P(Z < -1.5333)= 0.9525 - 0.0624= 0.8901. So, the closest answer is C. 0.936.

In conclusion, the closest answer for each area is B. 0.052, D. 0.015, and C. 0.936, respectively.

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Answer:

they both = eachother

10x+9y=-17 = 10x-2y=16

Step-by-step explanation:

Answer:

No, Collin didn't reach his goal. He got a 81% on his test.

Step-by-step explanation:

The surface area of the part of the x2 + y2 + z2 = 8z that lies inside the paraboloid z = x2 + y2, using the cylindrical coordinates is given as below:

The integral to find the surface area in the cylindrical coordinates, we can write as,

∫∫ dS = ∫∫ r dθ dr The given surface is x2 + y2 + z2 = 8z and the paraboloid is z = x2 + y2

By substituting the value of z from the paraboloid to the first equation,

we get,x2 + y2 + (x2 + y2)2 = 8(x2 + y2) Simplify it by expanding the square term as,

x2 + y2 + x4 + 2x2y2 + y4 = 8x2 + 8y2Now,

re-write the equation as,

x2 + y2 - 8x2 - 8y2 + x4 + 2x2y2 + y4 = 0On

solving this equation, we get

x2 + y2 - 8x2 - 8y2 + x4 + 2x2y2 + y4 = (x2 - 4x + y2 - 4y + 8)(x2 + 4x + y2 + 4y - 8) = 0

The equation of the paraboloid is given as, z = x2 + y2Hence,

the integral to find the surface area of the given surface in cylindrical coordinates,

∫∫ dS = ∫∫ r dθ dr Bounds of the integral to find the surface area are 0 ≤ θ ≤ 2π and r1 ≤ r ≤ r2,

where r1 and r2 are the radii of the cylinder.

Solve this equation and get the values of r1 and r2,r2 = 2r1

On solving the quadratic equation of (x2 - 4x + y2 - 4y + 8)(x2 + 4x + y2 + 4y - 8) = 0,

we get,

x2 + y2 - 4x - 4y + 4 = 0

The equation of the circle is given as

,x2 + y2 = 4x + 4y - 4 Solve for x and y to get,

x = 2 + cos θ y = 2 + sin θ

The radius of the circle is given as,

√(42 + 42) = √32 Thus,

the limits of integration of r are r1 = 0 and r2 = √32.

Integrating over the limits,

∫0^2π ∫0^√32 r dr dθ= 1/2(32) (2π)= 16πTherefore,

the surface area of the given surface is 16π.

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Answer:

9x10^7 is your scienctific notation, which is also 90000000 in expanded form

Answer: i think $35

Step-by-step explanation: have a great day

Given that R is the region enclosed by y = x² and y = 9. The value of the double integral over the region is I =81.

We are given the region R that is enclosed by y = x² and y = 9.

The x values range from -3 to 3.

The y values range from x² to 9.

We thus evaluate the double integral as follows:

I = [tex]\int_{(-3)}^ {(3)} \int_{(x^2)}^{( 9)[/tex]  dA

I= [tex]\int_{(-3)}^ {(3)} \int_{(x^2)}^{( 9)[/tex]  dydx

We integrate the integral with respect to y from x² to 9, and then integrate that expression with respect to x from -3 to 3.

We get: I = [tex]\int_{(-3)}^ {(3)} \int_{(x^2)}^{( 9)[/tex]  dydx

I= [tex]\int_{(-3)}^ {(3)[/tex] (9 - x²) dx

= [tex]\int_{(-3)}^ {(3)} 9 dx - \int_{(-3)}^ {(3)[/tex] x² dx

= 18[tex]\int_{(0)}^ {(3)} x dx - \int_{(-3)}^ {(3)[/tex] x² dx

= 18[(3²/2) - (0²/2)] - [(3³/3) - (-3³/3)]

= 18(9/2) - 54

= 81

Answer: I = 81.

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Answer:

0.603

Step-by-step explanation:

Given the data:

X 57 58 59 59 60 61 62 64

Y 77 78 75 78 82 82 79 81

The Correlation Coefficient, R value gives a measure of the degree of correlation between two variables, the dependent and independent variable. The correlation Coefficient value ranges from - 1 to 1. With negative values depicting a negative relationship and positive values meaning a positive relationship. The closer the R value is to + or - 1, the higher the strength of the relationship. With a value of 0 meaning 'no correlation'.

The correlation Coefficient value of the data above is 0.603, this gives a fairly strong positive correlation