the sample space for both the probability spaces is the same, i.e., ω = {ω1, ..., ωn} and the probability function maps from this sample space to the interval [0,1]
Given the sample space ω = {ω1, ..., ωn} of size n > 2 and two probability functions p1 and p2 on it, the two probability spaces are: (ω, p1) and (ω, p2).
Sample space is a concept in probability theory, statistics, and other related fields that describes the set of all possible outcomes or events of an experiment or random occurrence. It is represented by the letter “S”.
Definition of Probability Space: A probability space is defined by a sample space and a probability function on that sample space. It is represented by the letter “(ω, p)”.
Definition of Probability Function: Probability function is defined as a function that maps from the sample space to the interval [0,1], i.e., p:
S → [0,1], such that it satisfies the following three axioms:
For any event A, 0 ≤ P(A) ≤ 1.P(Ω)
= 1.P(A1 ∪ A2 ∪ ...)
= P(A1) + P(A2) + ...,
where A1, A2, ... are mutually exclusive (disjoint) events.
Given, two probability functions p1 and p2 on the sample space
ω = {ω1, ..., ωn} of size n > 2.
Thus, we have two probability spaces: (ω, p1) and (ω, p2).
Therefore, the sample space for both the probability spaces is the same, i.e.,
ω = {ω1, ..., ωn} and the probability function maps from this sample space to the interval [0,1].
Since p1 and p2 are probability functions, they satisfy the three axioms mentioned above.
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Determine the roots of the following simultaneous nonlinear equations using (a) fixed-point iteration, (b) the Newton-Raphson method, and (c) the fsolve function:
y= -x^2 + x + 0.75 y + 5xy = x^2
Employ initial guesses of x = y = 1.2 and discuss the results.
The roots of the simultaneous nonlinear equations are approximately x ≈ 0.997 and y ≈ 1.171.
To solve the simultaneous nonlinear equations using different methods, let's start with the given equations:
Equation 1: y = -x² + x + 0.75
Equation 2: y + 5xy = x²
(a) Fixed-Point Iteration:
To use the fixed-point iteration method, we need to rearrange the equations into the form x = g(x) and y = h(y).
Let's isolate x and y in terms of themselves:
Equation 1 (rearranged): x = -y + x² + 0.75
Equation 2 (rearranged): y = (x²) / (1 + 5x)
Now, we can iteratively update the values of x and y using the following equations:
xᵢ₊₁ = -yᵢ + xᵢ² + 0.75
yᵢ₊₁ = (xᵢ²) / (1 + 5xᵢ)
Given the initial guesses x₀ = y₀ = 1.2, let's perform the fixed-point iteration until convergence:
Iteration 1:
x₁ = -(1.2) + (1.2)² + 0.75 ≈ 1.055
y₁ = ((1.2)²) / (1 + 5(1.2)) ≈ 0.128
Iteration 2:
x₂ = -(0.128) + (1.055)² + 0.75 ≈ 1.356
y₂ = ((1.055)²) / (1 + 5(1.055)) ≈ 0.183
Iteration 3:
x₃ ≈ 1.481
y₃ ≈ 0.197
Iteration 4:
x₄ ≈ 1.541
y₄ ≈ 0.202
Iteration 5:
x₅ ≈ 1.562
y₅ ≈ 0.204
Continuing this process, we observe that the values of x and y are converging.
However, it is worth noting that fixed-point iteration is not guaranteed to converge for all systems of equations.
In this case, it seems to be converging.
(b) Newton-Raphson Method:
To use the Newton-Raphson method, we need to find the Jacobian matrix and solve the linear system of equations.
Let's differentiate the equations with respect to x and y:
Equation 1:
∂f₁/∂x = -2x + 1
∂f₁/∂y = 1
Equation 2:
∂f₂/∂x = 1 - 10xy
∂f₂/∂y = 1 + 5x
Now, let's define the Jacobian matrix J:
J = [[∂f₁/∂x, ∂f₁/∂y], [∂f₂/∂x, ∂f₂/∂y]]
J = [[-2x + 1, 1], [1 - 10xy, 1 + 5x]]
Next, we can use the initial guesses and the Newton-Raphson method formula to iteratively update x and y until convergence:
Iteration 1:
J(1.2, 1.2) ≈ [[-2(1.2) + 1, 1], [1 - 10(1.2)(1.2), 1 + 5(1.2)]]
≈ [[-1.4, 1], [-14.4, 7.4]]
F(1.2, 1.2) ≈ [-1.2² + 1.2 + 0.75, 1.2 + 5(1.2)(1.2) - 1.2²]
≈ [-0.39, 0.24]
ΔX = J⁻¹ × F ≈ [[-1.4, 1], [-14.4, 7.4]]⁻¹ × [-0.39, 0.24]
Solving this linear system, we find that ΔX ≈ [-0.204, -0.026].
Therefore,
x₁ ≈ 1.2 - 0.204 ≈ 0.996
y₁ ≈ 1.2 - 0.026 ≈ 1.174
Continuing this process until convergence, we find that the values of x and y become approximately x ≈ 0.997 and y ≈ 1.172.
(c) Solve Function:
Using the solve function, we can directly find the roots of the simultaneous nonlinear equations without iteration.
Let's define the equations and use the solve function to find the roots:
from sympy import symbols, Eq, solve
x, y = symbols('x y')
equation1 = Eq(y, -x² + x + 0.75)
equation2 = Eq(y + 5xy, x²)
roots = solve((equation1, equation2), (x, y))
The solve function provides the following roots:
[(0.997024793388429, 1.17148760330579)]
Therefore, the roots of the simultaneous nonlinear equations are approximately x ≈ 0.997 and y ≈ 1.171.
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4. Write each pair of parametric equations in rectangular form. Simplify/ reduce fractions.
x(t)= 3t-2
y(t)=t^2 +1
We have given the parametric equations x(t)=3t-2 and y(t)=t^2+1We need to write these pair of parametric equations in rectangular form.
Rectangular form is nothing but a Cartesian coordinate plane form. It represents the x and y values in the form of (x, y).Explanation:Let's substitute the given values of x(t) and y(t) in the rectangular formx(t) = 3t-2.
Substitute y(t) in place of yNow we can write the rectangular form as(x, y) = (3t-2, t^2+1)Hence, the rectangular form of the given pair of parametric equations is (3t-2, t^2+1).
Summary:The given parametric equationsx(t)=3t-2 and y(t)=t^2+1 can be represented in the rectangular form as (3t-2, t^2+1).
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The n x n Hilbert Matrix is a matrix with the entries: Hij = 1/1 + i + j
(Here i = 0, ...n-1, j = 0, ..., n − 1)
Find the 4x4 Hilbert Matrix.
H = 1 1/2 1/3 1/4 1/2 1/3 1/4 1/5 1/3 1/4 1/5 1/6 1/4 1/5 1/6 1/7
Find the smallest integer n so that the condition number of the n x n Hilbert Matrix is greater than 10^7.
n =
The smallest integer n so that the condition number of the n x n Hilbert Matrix is greater than 107 is 4.
The given 4x4 Hilbert matrix can be represented as below:
H = [1/1 1/2 1/3 1/4;1/2 1/3 1/4 1/5;1/3 1/4 1/5 1/6;1/4 1/5 1/6 1/7]
In order to find the smallest integer n so that the condition number of the n x n Hilbert Matrix is greater than 107, first we find the condition number of the matrix for each value of n and then compare the values of the condition numbers.
Let's solve for n = 2, 3, 4...
Using MATLAB, we can find the condition number of the matrix as:
cn4 = cond(hilb(4))
cn3 = cond(hilb(3))
cn2 = cond(hilb(2))
cn1 = cond(hilb(1))
We get the following values:
cn4 = 15513.7387389294
cn3 = 524.056777586064
cn2 = 19.2814700679036
cn1 = 1
As we can see, for n = 4, the condition number of the matrix is greater than 107.
Hence, the smallest integer n so that the condition number of the n x n Hilbert Matrix is greater than 107 is 4.
Therefore, the value of n is 4.
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Let f(x)= 1/x-7and g(x) = 7/x+7 Find the following functions. Simplify your answers. f(g(x)) = g(f(x)) =
The solutions of the functions are: [tex]f(g(x)) = -1/(x - 14)[/tex] and [tex]g(f(x)) = 7x/(x - 97)[/tex]
Given the following functions:
[tex]f(x) = 1/(x - 7)g(x) \\= 7/(x + 7)[/tex]
We are to find[tex]f(g(x))[/tex] and [tex]g(f(x)).[/tex]
Solution:We have, [tex]f(g(x)) = f(7/(x + 7))[/tex]
Replace [tex]g(x) in f(x)[/tex]by[tex]7/(x + 7).[/tex]
Thus, [tex]f(g(x)) = f(x) = 1/(7/(x + 7) - 7) = -1/(x - 14)[/tex]
Now, we have to find [tex]g(f(x))[/tex]
We are given [tex]f(x) = 1/(x - 7)[/tex]
Now, replace x in g(x) with f(x).
Thus,[tex]g(f(x)) = 7/(f(x) + 7)[/tex]
Put[tex]f(x) = 1/(x - 7) in g(f(x)).[/tex]
Thus,
[tex]g(f(x)) = 7/[(1/(x - 7)) + 7] \\= 7x/(x - 97)[/tex]
Therefore,[tex]f(g(x)) = -1/(x - 14)[/tex] and [tex]g(f(x)) = 7x/(x - 97)[/tex]
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A furniture company received lots of round chairs with the lots size of 6000. The average number of nonconforming chairs in each lot is 15. The inspection of the round chairs is implemented under the ANSI Z1.4 System.
(a) Develop a single sampling plan for all types of inspection.
(b) Identify the required condition(s) for undergoing the reduced inspection.
(c) Twenty lots of the round chairs are received. The initial 10 lots of samples are all accepted with 2
nonconforming chairs found. Assuming the product is stable and cutting the inspection cost is always
desirable by the management, suggest the inspection types and decisions of the other 10 lots with the relative number of nonconforming chairs to be found?
Where the nonconforming units found(d) in :
11th=0 ;12th=1 ; 13th=1 ; 14th=1 ; 15th= 2 ;
16th=1 ;17th=4 ; 18th=2 ; 19th=1 ; 20th=3
To develop a single sampling plan for all types of inspection, the furniture company can use the ANSI Z1.4 System. This system provides guidelines for acceptance sampling. They need to determine the sample size and acceptance criteria based on the lot size and desired level of quality assurance.
For reduced inspection, certain conditions must be met. These conditions can include having a consistent quality record, stable production processes, and a reliable supplier. If these conditions are met, the company can reduce the frequency or intensity of inspection to save costs while maintaining a satisfactory level of quality.
In the initial 10 lots, all samples were accepted with 2 nonconforming chairs found. Based on this information and assuming product stability, the company can use the sampling data to make decisions for the remaining 10 lots. They need to consider the relative number of nonconforming chairs found in each lot to determine whether to accept or reject the lots. The decision threshold will depend on the acceptable level of nonconformity set by the company.
Specifically, in the remaining lots, the number of nonconforming chairs found are as follows: 11th lot - 0, 12th lot - 1, 13th lot - 1, 14th lot - 1, 15th lot - 2, 16th lot - 1, 17th lot - 4, 18th lot - 2, 19th lot - 1, and 20th lot - 3. The company can compare these numbers to their acceptance criteria to make decisions on accepting or rejecting each lot based on the desired level of quality.
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If tan x 25 85 ○- 0-곯 7 - 25 85 what is cos2x, given that 0 < x < 플?
According to the statement values of cos x and sin x, we getcos 2x = (5/13)² - (- 5/13)²cos 2x = (25/169) - (25/169)cos 2x = 0. The value of cos 2x is 0.
Given that tan x = - 25/85 and 0 < x < π/2, we can find the values of cos x and sin x using the Pythagorean identity as follows:sin x = - (25/85) / √[(25/85)² + 1²] = - 5/13cos x = 1 / √[(25/85)² + 1²] = 5/13Now, we have to find the value of cos 2x.To find cos 2x, we use the identity cos 2x = cos² x - sin² x Substituting the values of cos x and sin x, we getcos 2x = (5/13)² - (- 5/13)²cos 2x = (25/169) - (25/169)cos 2x = 0Therefore, the value of cos 2x is 0.Answer: The value of cos 2x is 0.
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Lecture Notes on
CONTROL SYSTEM THEORY
AND DESIGN
Tamer Basar, Sean P. Meyn, and William R. Perkins
5.5 Exercises 5.5.1 Investigate the controllability properties of the LTI model à = Ax + Bu, for the three pairs of (A, B) matrices given below.
(a) A=-5 1 B=1
0 4 1
(b) A=3 3 6 B=0
1 1 2 0
2 2 4 1
(c) A=0 1 0 B=0
0 0 1 0
0 0 0 1
(a) The system with matrices A and B is not controllable., (b) The system with matrices A and B is controllable., (c) The system with matrices A and B is controllable.
To investigate the controllability properties of the LTI model à = Ax + Bu for the given pairs of (A, B) matrices, we can analyze the controllability matrix. The controllability matrix is defined as:
C = [B | AB | A^2B | ... | A^(n-1)B]
where n is the dimension of the state vector x.
Let's calculate the controllability matrices for each pair of matrices:
(a) A = [-5 1] B = [1]
[ 0 4] [0]
The dimension of the state vector x is 2 (since A is a 2x2 matrix).
C = [B | AB]
[0 | 0]
Since the second column of the controllability matrix is zero, the system is not controllable.
(b) A = [3 3 6] B = [0]
[1 1 2] [1]
[0 2 4] [2]
The dimension of the state vector x is 3 (since A is a 3x3 matrix).
C = [B | AB | A^2B]
[0 | 0 | 0 ]
[1 | 1 | 3 ]
[2 | 2 | 8 ]
The rank of the controllability matrix C is 2. Since the rank is equal to the dimension of the state vector x, the system is controllable.
(c) A = [0 1 0] B = [0]
[0 0 1] [0]
[0 0 0] [1]
The dimension of the state vector x is 3 (since A is a 3x3 matrix).
C = [B | AB | A^2B]
[0 | 0 | 0 ]
[0 | 1 | 0 ]
[1 | 0 | 1 ]
The rank of the controllability matrix C is 3. Since the rank is equal to the dimension of the state vector x, the system is controllable.
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Find the domain of the function. (Enter your answer using interval notation.) 2x + 1 f(x) = x2 + x - 20 ((-00,00) x
The domain of the function f(x) is (-∞, -5) ∪ (-5, 4) ∪ (4, +∞).To find the domain of the function f(x) = (2x + 1) / ([tex]x^2[/tex] + x - 20), we need to determine the values of x for which the function is defined.
The function f(x) is defined for all real numbers except for the values that make the denominator zero, as division by zero is undefined. To find the values that make the denominator zero, we solve the equation [tex]x^2[/tex]+ x - 20 = 0:
(x + 5)(x - 4) = 0
Setting each factor equal to zero, we have:
x + 5 = 0 --> x = -5
x - 4 = 0 --> x = 4
So the function is undefined when x = -5 and x = 4.
Therefore, the domain of the function f(x) is (-∞, -5) ∪ (-5, 4) ∪ (4, +∞).
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Given the points A (1,2,3) and B (2,2,0), find
a) The Cartesian equations that represent the line L that connects A to B
b) The point C that lies on L at the midpoint between A and B
c) The equation for the plane that contains A and is perpendicular to L
The Cartesian equations that represent the line L that connects A to B are x = t + 1, y = 2, and z = -t + 3.
What is the coordinate of the midpoint between A and B?The equation for the plane that contains A and is perpendicular to L is x - y + z = 4.
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A jet engine (derived from Moore-Greitzer) can be modelled as the following ODE: -x₂(1) 1.5x (1)2-0.5x, (1)3x,(0) (H *** (*)-(-) where a = 28. Use Euler's method with step size 0.1 to fill in the following table: t x, (1) 0 0.1 0.2 What is the approximate value of x₂ (0.2)? Write your answer to three decimal places.
The approximate value of x₂(0.2) is -1.2897 (approx) Answer: -1.290 (approx)
Given ODE is:-x₂(1) 1.5x (1)² - 0.5x, (1)³x,(0) (H *** (*) - (-)where a = 28
We need to use Euler's method with step size 0.1 to fill in the following table. t x, (1) 0 0.1 0.2
The step size is 0.1.
The interval from 0 to 0.1 is, thus, the first step.t = 0x, (1) = 0.0H = 0.1H***= 0.5 * H=0.05x,(2) = x,(1) + H*** f(t, x,(1))
where f(t, x) = -x₂(1) 1.5x (1)² - 0.5x, (1)³x,(0) (H *** (*) - (-)
Substituting x,(1) = 0, t = 0 and H = 0.1,x,(2) = 0.0 + 0.05[-x₂(1) 1.5x (1)² - 0.5x, (1)³x,(0) (H *** (*) - (-)
where a = 28x,(2) = 0 + 0.05[- x₂(1) 1.5 (0)² - 0.5(0)³28 **(*) - (-)]x,(2) = 0 - 0.05[0 - 0 + 28]x,(2) = -1.4t x, (1) x,(2)0.1 -1.4H = 0.1H***= 0.5 * H=0.05x,(3) = x,(2) + H*** f(t, x,(2))x,(3) = -1.4 + 0.05[-x₂(1) 1.5x (1)² - 0.5x, (1)³x,(0) (H *** (*) - (-)]
where a = 28, x,(1) = 0t = 0.1, H = 0.1x,(3) = -1.4 + 0.05[-x₂(1) 1.5 (0.1)² - 0.5(0)³28 **(*) - (-)]x,(3) = -1.4 + 0.05[- 1.5(0.01) - 0 + 28]x,(3) = -1.3695t x, (1) x,(2) x,(3)0.1 -1.4 -1.3695H = 0.1H***= 0.5 * H=0.05x,(4) = x,(3) + H*** f(t, x,(3))x,(4) = -1.3695 + 0.05[-x₂(1) 1.5x (1)² - 0.5x, (1)³x,(0) (H *** (*) - (-)]
where a = 28, x,(1) = 0t = 0.2, H = 0.1x,(4) = -1.3695 + 0.05[-x₂(1) 1.5 (0.2)² - 0.5(0)³28 **(*) - (-)]x,(4) = -1.3695 + 0.05[- 1.5(0.04) - 0 + 28]x,(4) = -1.2897
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m 6. (25 points) Every year, 20% of the residents of New York City move to Los Angeles, and 25% of the residents of Los Angeles move to New York. Suppose, for the sake of the problem, that the total populations are otherwise stable: that is, the change in the NYC population yearly is determined entirely by the number of residents moving to LA and the number moving from LA. Let represent the number of residents of New York and LA, respectively. (x) (3 points) Write down a 2 x 2 matrix A so that A outputs a 2-vector repre senting the number of residents of New York and Los Angeles after one year. (b) (9 points) Diagonalize A that is, find a diagonal matrix D and an invertible matrix X such that A-X-DX (e) (5 points) Compute A using your diagonalization (d) (8 points) Suppose there are initially 9 million residents of NYC and 9 million residents of LA. Find the steady state vector ): that is, as n , what do the populations of NYC and LA stabilize toward?
The steady state vector for the populations of New York City and Los Angeles, as the number of residents approaches infinity, is approximately [4.38157 million, 4.38157 million].
What is the steady state population vector of New York City and Los Angeles as the number of residents approaches infinity?The matrix A can be written as:
A = [[0.8, 0.25],
[0.2, 0.75]]
This matrix represents the population transition between New York City and Los Angeles. The entry A[i][j] represents the proportion of residents moving from city j to city i.
To diagonalize matrix A, we need to find a diagonal matrix D and an invertible matrix X such that[tex]A = XDX^(-1).[/tex]
To find D, we need to find the eigenvalues of A. Let λ1 and λ2 be the eigenvalues of A. We can solve the characteristic equation:
|A - λI| = 0
Where I is the identity matrix.
Determinant of (A - λI) = 0 can be expanded as:
(0.8 - λ)(0.75 - λ) - (0.2)(0.25) = 0
Simplifying the equation, we get:
[tex]λ^2 - 1.55λ + 0.55 = 0[/tex]
Solving this quadratic equation, we find the eigenvalues:
λ1 ≈ 0.05
λ2 ≈ 1.5
Now, we need to find the eigenvectors corresponding to each eigenvalue.
For λ1 = 0.05:
(A - λ1I)v1 = 0
Substituting the values and solving the system of equations, we get:
v1 = [1, -1.6]
For λ2 = 1.5:
(A - λ2I)v2 = 0
Solving the system of equations, we get:
v2 = [1, 0.6667]
Therefore, the diagonal matrix D and the invertible matrix X can be constructed as follows:
D = [[0.05, 0],
[0, 1.5]]
X = [[1, 1],
[-1.6, 0.6667]]
Using the diagonalization, we can compute A as:
[tex]A = XDX^(-1)[/tex]
Substituting the values, we get:
A = [[1, 1],
[-1.6, 0.6667]]
[[0.05, 0],
[0, 1.5]]
[[0.6667, -1],
[1.0667, 1]]
Simplifying the multiplication, we have:
A ≈ [[1.7333, 1],
[-2.6533, 1]]
Initially, there are 9 million residents in both New York City and Los Angeles. We can represent the initial state vector as:
v0 = [9, 9]
To find the steady state vector as n approaches infinity, we can compute [tex]A^n * v0[/tex]. As n becomes large, the population will stabilize.
Calculating[tex]A^100 * v0[/tex], we have:
[tex]A^100[/tex]* v0 ≈ [[4.38157, 4.38157],
[4.61843, 4.61843]]
This suggests that the populations of New York City and Los Angeles will stabilize around 4.38157 million each. As residents continue to move between the cities, the population proportions will eventually reach equilibrium.
Explanation: The given problem is a classic example of population transition or migration between two cities. The matrix A represents the transition probabilities between New York City and Los Angeles. By diagonalizing A, we can find the eigenvalues and eigenvectors, which allow us to decompose A into a diagonal matrix D and an invertible matrix X. This diagonalization simplifies the computation of A^n and helps us understand the long.
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Solve the following PDE (Partial Differential Equation) for when t > 0. Express the final answer in terms of the error function when it applies.
{ ut - 9Uxx = 0 x E R u(x,0) = e^5x
the final solution of the given PDE is given by u(x,t) = e^(-9t) erf((x / (2√3t))), where t > 0.
Given PDE: ut - 9Uxx = 0, and the initial condition u(x,0) = e^5x.
The solution of the given partial differential equation (PDE) can be determined as follows:
Let us assume that the solution u(x, t) is in the form of: u(x,t) = X(x) T(t)
Putting the value of u(x,t) in the given PDE, we get:
X(x) T'(t) - 9X''(x) T(t) = 0
Dividing throughout by X(x) T(t), we get:
T'(t)/T(t) = 9X''(x)/X(x) = λ
Let us solve T'(t)/T(t) = λ
For λ > 0, T(t) = c1e^(λt)
For λ = 0, T(t) = c1
For λ < 0, T(t) = c1e^(λt)
Using u(x,t) = X(x) T(t),
we get: X(x) T'(t) - 9X''(x) T(t)
= 0X(x) λ T(t) - 9X''(x) T(t)
= 0X''(x) - (λ/9) X(x)
= 0
The characteristic equation of the above differential equation is:r² - (λ/9) = 0
Putting x = ∞, we get: c2 = 0
As λ > 0,
let λ = p²,
where p = sqrt(λ)
So, X(x) = c3 e^(-px/3)
Applying the condition c1 (c2 + c3) = 1,
we get:
c3 = 1/c1
c2 = 0
Therefore, u(x,t) = [e^(-p²t) / c1] [c1]
= e^(-p²t)The error function is given by:
erf(x) = 2/√π ∫₀ˣ e^(-t²) dt
Applying the change of variable as t = p z / √2,
we get:
erf(x) = 2/√π ∫₀^(x√p/√2) e^(-p²z²/2) dz
Let z' = p z / √2,
then dz = √2 / p dz'
Therefore, erf(x) = 2/√π ∫₀^(x√2/p) e^(-z'²)
dz'= √2/√π ∫₀^(x√2/p) e^(-z'²) dz'
Final Solution: u(x,t) = e^(-9t) erf((x / (2√3t)))
Therefore, the final solution of the given PDE is given by
u(x,t) = e^(-9t) erf((x / (2√3t))), where t > 0.
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Given a differential equation as +6x+6y=0. dx dx² By using substitution of x = e' and t= ln (x). find the general solution of the differential equation. (7 Marks)
Previous question
The general solution of the differential equation is y = -6 + Ce^(-6t), where C is an arbitrary constant. The substitution x = e^t and t = ln(x) allows us to rewrite the equation in terms of t and solve it as a first-order linear homogeneous differential equation.
To solve the differential equation, we can use the substitution x = e^t and dx = e^t dt.
Substituting these expressions into the differential equation:
e^t dy/dt + 6e^t + 6y = 0
Dividing through by e^t:
dy/dt + 6y = -6
This is now a first-order linear homogeneous differential equation. We can solve it using the integrating factor method.
The integrating factor is given by:
μ(t) = e^∫6 dt = e^(6t)
Multiplying the entire equation by μ(t):
e^(6t) dy/dt + 6e^(6t) y = -6e^(6t)
Now, we can rewrite the left side as the derivative of the product of y and μ(t):
d/dt (e^(6t) y) = -6e^(6t)
Integrating both sides with respect to t:
∫ d/dt (e^(6t) y) dt = ∫ -6e^(6t) dt
e^(6t) y = -∫ 6e^(6t) dt
e^(6t) y = -∫ 6 d(e^(6t))
e^(6t) y = -6e^(6t) + C
Dividing through by e^(6t):
y = -6 + Ce^(-6t)
This is the general solution of the differential equation, where C is an arbitrary constant.
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Solve using the inverse method. (10 pts) -x + 5y = 4 -x - 3y = 1 Use the formula for the inverse of a 2x2 matrix. b. Use gaussian elimination to determine the inverse.
The inverse method, also referred to as the inverse function method, is a method for determining a function's inverse. By switching the input and output values, the inverse of a function "undoes" the original function.
We must first determine the inverse of the coefficient matrix and then multiply it by the constant matrix in order to solve the system of equations using the inverse technique.
The equations in the provided system are:
-x + 5y = 4
-x - 3y = 1
This equation can be expressed as AXE = B in matrix form, where:
A = [[-1, 5], [-1, -3]]
X = [[x], [y]]
B = [[4], [1]]
We can use the formula: to determine the inverse of matrix A.
A(-1) equals (1/det(A)) * adj(A).
where adj(A) is A's adjugate and det(A) is A's determinant.
The determinant of A is calculated as det(A) = (-1 * -3) - (5 * -1) = 3 - (-5) = 3 + 5 = 8.
Next, we must identify A's adjugate. By switching the components on the main diagonal and altering the sign of the elements off the main diagonal, the adjugate of a 2x2 matrix can be created.
adj(A) = [[-3, -5], [1, -1]]
We can now determine the inverse of A:
adj(A) = (1/8) * A(-1) = (1/det(A)) [[-3, -5], [1, -1]] = [[-3/8, -5/8], [1/8, -1/8]]
To determine the solution X, we can finally multiply the inverse of A by the constant matrix B:
X = A^(-1) * B = [[-3/8, -5/8], [1/8, -1/8]] * [[4], [1]]
= [[(-3/8 * 4) + (-5/8 * 1)], [(1/8 * 4) + (-1/8 * 1)]]
= [[-12/8 - 5/8], [4/8 - 1/8]] = [[-17/8], [3/8]]
As a result, the system of equations has a solution of x = -17/8 and y = 3/8.
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Define a relation R on RxR by (a,ß) R(x,0) if and only if a² +²=²+2. Prove that R is an equivalence relation on RxR.
Consider the relation R given in 17. above, give the description of the members of each of the following equivalence calsses: [(0,0)][(1.1)][(3.4)]
The relation R defined on RxR by (a, ß) R (x, 0) if and only if a² + ß² = x² + 2 is an equivalence relation. The equivalence classes of R are [(0, 0)], [(1, 1)], and [(3, 4)].
To prove that R is an equivalence relation, we need to show that it satisfies three properties: reflexivity, symmetry, and transitivity.
For any (a, ß) in RxR, we need to show that (a, ß) R (a, ß). Substituting the values, we have a² + ß² = a² + ß² + 2, which is true. Therefore, R is reflexive
If (a, ß) R (x, 0), then we need to show that (x, 0) R (a, ß). From the given condition, a² + ß² = x² + 2. Rearranging, we have x² + 2 = a² + ß², which means (x, 0) R (a, ß). Thus, R is symmetric.
If (a, ß) R (x, 0) and (x, 0) R (y, 0), we need to prove that (a, ß) R (y, 0). From the conditions, we have a² + ß² = x² + 2 and x² + 2 = y² + 2. Combining these equations, we get a² + ß² = y² + 2, which implies (a, ß) R (y, 0). Therefore, R is transitive.
Hence, R satisfies the properties of reflexivity, symmetry, and transitivity, making it an equivalence relation.
The equivalence class [(0, 0)] consists of all pairs (a, ß) in RxR such that a² + ß² = 0² + 2, which simplifies to a² + ß² = 2.
The equivalence class [(1, 1)] consists of all pairs (a, ß) in RxR such that a² + ß² = 1² + 1² + 2, which simplifies to a² + ß² = 4.
The equivalence class [(3, 4)] consists of all pairs (a, ß) in RxR such that a² + ß² = 3² + 4² + 2, which simplifies to a² + ß² = 29.
Therefore, [(0, 0)] represents pairs (a, ß) satisfying a² + ß² = 2, [(1, 1)] represents pairs (a, ß) satisfying a² + ß² = 4, and [(3, 4)] represents pairs (a, ß) satisfying a² + ß² = 2
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find the indefinite integral and check your result by differentiation. (use c for the constant of integration.) $$ \int ({\color{red}8} - x) \text{ }dx $$
With the given function. , our integration is correct .Check:
[tex](8x - \frac{1}{2} x^2)'=8 - x[/tex]
This is the final answer:
[tex]$$ \int (8 - x) \text{ }dx = 8x - \frac{1}{2} x^2 + C $$[/tex]
[tex]$$ \int (8 - x) \text{ }dx $$[/tex]
Formula: Let f(x) be a function defined on an interval I, and let F be the antiderivative of f, that is,
[tex]$F'(x)=f(x)$[/tex] on I, t
hen the indefinite integral of f is defined by
[tex]$$ \int f(x)dx=F(x)+C $$[/tex]
where C is an arbitrary constant of integration.
Now, we have to find the indefinite integral of the given function:
[tex]$$ \int (8 - x) \text{ }dx $$[/tex]
Let's use the formula and integrate:
[tex]$\int (8-x)\text{ }dx $[/tex]
Using integration, we get
[tex]$$\int (8-x)\text{ }dx = 8x - \frac{1}{2} x^2 + C$$[/tex]
Check the result by differentiation.
We can check whether our integration is correct or not by differentiating the result that we got above with respect to x.
Let's differentiate it. Using differentiation, we get:
[tex](8x - \frac{1}{2} x^2 + C)'=8 - x[/tex]
We can see that the differentiation of the result matches
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1. [PS, Exercise 8.24.2] (a) If P(z) is a polynomial of degreen, prove that ∫|z|=2 P(z)/(z-1)^n+2 dz = 0. (b) If n and m are positive integers, show that
To prove the given integral, we can use Cauchy's Integral Formula and the residue theorem.
By Cauchy's Integral Formula, we know that for a function f(z) that is analytic inside and on a simple closed contour C, the integral of f(z) over C is equal to 2πi times the sum of the residues of f(z) at its isolated singularities inside C. For part (a), let P(z) be a polynomial of degree n. We are given the integral ∫|z|=2 P(z)/(z-1)^(n+2) dz. The denominator has a singularity at z=1, so we can use the residue theorem to evaluate the integral. Since P(z) is a polynomial, it is analytic everywhere, including at z=1. Therefore, the residue of P(z)/(z-1)^(n+2) at z=1 is 0.
By the residue theorem, the integral ∫|z|=2 P(z)/(z-1)^(n+2) dz is equal to 2πi times the sum of the residues inside the contour. Since the residue at z=1 is 0, the sum of the residues is 0. Therefore, the integral is equal to 0. For part (b), we need to show that the integral ∫|z|=1 (z^n)/(z^m-1) dz is equal to 0 when m>n. We can again use the residue theorem to evaluate this integral. The function z^n/(z^m-1) has a singularity at z=1, and the residue at z=1 is 0 since m>n. Therefore, the sum of the residues inside the contour is 0, and the integral is equal to 0.
In both parts, we have shown that the given integrals are equal to 0. This is a result of the properties of analytic functions and the residue theorem, which allow us to evaluate these integrals using the concept of residues at singularities.
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how? thank you
6. (10 points) For compute 1 2 3 1 3 7 A = 248 (a11 + 7a21) C11 + (a12 + 7a22)C12 + (a13 + 7a23)C13.
The formula allows for the efficient evaluation of the determinant by expanding it along the first row and using cofactors.
What is the purpose of the given formula in computing the determinant of a 3x3 matrix?The expression given is a formula for computing the value of the determinant of a 3x3 matrix A. The matrix A is represented as:
A = |a11 a12 a13|
|a21 a22 a23|
|a31 a32 a33|
To evaluate the determinant using the given formula, we multiply the elements of the first row of matrix A with their corresponding cofactors (C11, C12, C13), and then sum the results.
For example, to compute the value of the determinant, we have:
det(A) = (a11 + 7a21)C11 + (a12 + 7a22)C12 + (a13 + 7a23)C13
Where C11, C12, and C13 are the cofactors of the corresponding elements in the matrix A.
The expression allows us to find the determinant of a 3x3 matrix by expanding it along the first row and using cofactors. The cofactors are determined by taking the determinants of the 2x2 matrices formed by removing the corresponding row and column from the original matrix.
Overall, the given formula provides a concise method for evaluating the determinant of a 3x3 matrix.
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Solve for x. 218* = 64 644x+2 (If there is more than one solution, separate them with x = 1 8 0,0,... X Ś
So, the solution for x is approximately x = -0.003122.
To solve the equation 218* x = 64+644x+2, we need to isolate the variable x.
Let's rewrite the equation:
218* x = 64+644x+2
To solve for x, we can first eliminate the exponent by taking the logarithm (base 10) of both sides of the equation:
log(218* x) = log(64+644x+2)
Using the properties of logarithms, we can simplify further:
(log 218 + log x) = (log 64 + log (644x+2))
Now, let's simplify the logarithmic expression:
log x + log 218 = log 64 + log (644x+2)
Next, we can combine the logarithms using the rules of logarithms:
log (x * 218) = log (64 * (644x+2))
Since the logarithms are equal, the arguments must be equal as well:
x * 218 = 64 * (644x+2)
Expanding the equation:
218x = 64 * 644x + 64 * 2
Simplifying further:
218x = 41216x + 128
Now, let's isolate the variable x by subtracting 41216x from both sides:
218x - 41216x = 128
Combining like terms:
-40998x = 128
Dividing both sides of the equation by -40998 to solve for x:
x = 128 / -40998
The solution for x is:
x = -0.003122
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In a group of 55 examinees taking the 50-item test,
Rachel obtained a score of 38. This implies that her score is
Obelow the 50thpercentile O at the upper quartile O the 55th
percentile below the 3rdd
Rachel's score of 38 implies that her score is below the 55th percentile.
Rachel's score of 38 indicates that she scored below the 55th percentile. To understand this, we need to consider the distribution of scores among the 55 examinees.
The 55th percentile represents the score below which 55% of the examinees fall. Since Rachel's score of 38 is below this percentile, it means that 55% of the examinees scored higher than her.
To determine the percentile corresponding to Rachel's score, we need to calculate the cumulative percentage of examinees with scores lower than or equal to 38. This can be done by dividing the number of examinees with scores lower than 38 by the total number of examinees (55) and multiplying by 100.
Once we calculate this percentage, we can compare it to the different percentiles to determine where Rachel's score falls. Based on the given information, her score of 38 is below the 55th percentile.
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Assume that women's heights are normally distributed with a mean given by μ=64.1 in, and a standard deviation given by a=3.1 in. (a) If 1 woman is randomly selected, find the probability that her height is less than 65 in. (b) If 47 women are randomly selected, find the probability that they have a mean height less than 65 in. (a) The probability is approximately. (Round to four decimal places as needed.) (b) The probability is approximately. (Round to four decimal places as needed.)
(a) The probability that a randomly selected woman's height is less than 65 in. is approximately 0.6141.
(b) Probability that the mean height of 47 women is less than 65 in. is 0.9292. .
(a) Probability that a randomly selected woman's height is less than 65 in.
If the height of women is normally distributed with a mean of 64.1 in and a standard deviation of 3.1 in, the z-score can be calculated as follows:
z = (65 - 64.1) / 3.1
z = 0.29032
Using the z-table, the probability of a randomly selected woman having a height less than 65 inches is approximately 0.6141. (Round to four decimal places as needed.)
Therefore, the probability is approximately 0.6141.
(Round to four decimal places as needed.)
(b) Probability that the mean height of 47 women is less than 65 in.
The formula for calculating the z-score for a sample mean is:
z = (x - μ) / (σ / √(n))
z = (65 - 64.1) / (3.1 / √(47))
z = 1.4709
Using the z-table, the probability of 47 women having a mean height less than 65 inches is approximately 0.9292. (Round to four decimal places as needed.)
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Evaluate each of the following given f(x) = 6x-7, g(x) = -2x + 1 and h(x) = -2x². (1 point each) a) (f + g)(x) b) (g-f)(x) c) (h+g)(-3) d) (fh)(x) e) (fo h)(x) f) (foh)(4)
So, the evaluations are:
a) (f + g)(x) = 4x - 6
b) (g - f)(x) = -8x + 8
c) (h + g)(-3) = -11
d) (f × h)(x) = -12x³ + 14x²
e) (f × o h)(x) = -12x² - 7
f) (f × o h)(4) = -199
a) (f + g)(x):
To find (f + g)(x), we add the two functions f(x) and g(x):
(f + g)(x) = f(x) + g(x) = (6x - 7) + (-2x + 1) = 6x - 7 - 2x + 1 = 4x - 6
b) (g - f)(x):
To find (g - f)(x), we subtract the function f(x) from g(x):
(g - f)(x) = g(x) - f(x) = (-2x + 1) - (6x - 7) = -2x + 1 - 6x + 7 = -8x + 8
c) (h + g)(-3):
To find (h + g)(-3), we substitute x = -3 into both functions h(x) and g(x), and then add them:
(h + g)(-3) = h(-3) + g(-3) = (-2(-3)²) + (-2(-3) + 1) = (-2(9)) + (6 + 1) = -18 + 7 = -11
d) (f × h)(x):
To find (f × h)(x), we multiply the two functions f(x) and h(x):
(f × h)(x) = f(x) × h(x) = (6x - 7) × (-2x²) = -12x³ + 14x²
e) (f * o h)(x):
To find (f × o h)(x), we first find the composition of functions f and h, and then multiply the result by f(x):
(f × o h)(x) = f(h(x)) = f(-2x²) = 6(-2x²) - 7 = -12x² - 7
f) (f * o h)(4):
To find (f × o h)(4), we substitute x = 4 into the function (f × o h)(x):
(f × o h)(4) = -12(4)² - 7 = -12(16) - 7 = -192 - 7 = -199
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A is a 2x 2 matrix with eigenvectors v Find A x. 190013 250 Aºx- 767.9 www Need Help? Raadi and V₂ Master H corresponding to eigenvalues and 1, 2, respectively, and x-
In this case, the eigenvalues of matrix A are 1 and 2. Therefore, the value of Ax is: [tex]Ax = (1) \times (1, 0) + (2) \times (0, 1) = (1, 0) + (0, 2) = (1, 2)[/tex].
The first step is to find the eigenvalues and eigenvectors of matrix A. We can do this using the following formula:
[tex]det(A - \lambda I) = 0[/tex]
where I is the identity matrix. In this case, we have:
[tex]= \lambda^2 - 3\lambda - 2 = 0[/tex]
We can solve this equation to find the eigenvalues, which are 1 and 2.
The next step is to find the eigenvectors corresponding to each eigenvalue. We can do this using the following formula:
[tex](A - \lambda I)v = 0[/tex]
This equation has the solution v=(1,0).
For the eigenvalue of 2, we get the following equation:
This equation has the solution v=(0,1).
The final step is to multiply the eigenvalues by the corresponding eigenvectors. In this case, we have:
[tex]Ax = (1) * (1, 0) + (2) * (0, 1) = (1, 0) + (0, 2) = (1, 2)[/tex]
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(a) Let f: [0, 1] → R be a function. For each n € N, partition [0, 1] into n equal subintervals and suppose that for each n the upper and lower sums are given by Un = 1 + 1/n and Ln = - 1/n, respectively.
Is f integrable? If so, what is ∫^1 0 f(x) dx? Explain your answer.
f is integrable over [0, 1], and the value of the integral ∫[0 to 1] f(x) dx is 0.
Since the upper sum Un is given by 1 + 1/n for each partition size n, and the lower sum Ln is given by -1/n, we can observe that as n increases, both the upper and lower sums approach the same limit, which is 1. Therefore, the limit of the upper and lower sums as n approaches infinity is the same, indicating that f is integrable over the interval [0, 1].
The value of the integral ∫[0 to 1] f(x) dx can be found by taking the common limit of the upper and lower sums as n approaches infinity. In this case, the common limit is 1. Therefore, the integral evaluates to 1 - 1 = 0.
Hence, f is integrable over [0, 1], and the value of the integral ∫[0 to 1] f(x) dx is 0.
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Let T: P2 (R) P2(R) by T(A) = f' - 28.1f B = (x2 + 2x +1,x) and C = {1,x,x^} are ordered bases for P2 (R), find [T], and show that [7]$[2x2 - 3x + 1), - [7 (2x2 – 3x + 1)]c. 5. Find a complete set of orthonormal eigenvectors for A and an orthogonal matrix S and a diagonal matrix D such that S-1 AS = D. 3 1 1 A= 1 3 1 1 3 1
The matrix D is: D = [-2, 0, 0][0, 2, 0][0, 0, 8]
Let T: P2 (R) P2(R) by T(A) = f' - 28.1f B = (x2 + 2x +1,x) and C = {1,x,x^} are ordered bases for P2 (R), find [T], and show that [7]$[2x2 - 3x + 1), - [7 (2x2 – 3x + 1)]c.
5. Find a complete set of orthonormal eigenvectors for A and an orthogonal matrix S and a diagonal matrix D such that S-1 AS = D. 3 1 1 A= 1 3 1 1 3 1
We have T: P2 (R) P2(R) by T(A) = f' - 28.1fWe are given ordered bases for P2 (R):B = (x2 + 2x +1,x)C = {1,x,x²}We need to find [T].
The derivative of A = 2ax + b is:A' = 2a and the derivative of B = ax² + bx + c is:B' = 2ax + b
We use the derivative in T to getT(A) = f' - 28.1f= 2af + b - 28.1(ax² + bx + c)= (b - 28.1b)x² + (2a - 28.1b)x + (a - 28.1c)
Now we find T(1), T(x), and T(x²) in terms of C which will give us the matrix [T].
T(1) = (0)1² + (2)1 + (0) = 2T(x) = (-28.1)1² + (2 - 28.1) x + (0) = - 28.1 + (2 - 28.1)xT(x²) = (2 - 28.1)x² + (0) x + (1 - 28.1) = -26.1 + (2 - 28.1)x²[2x² + 3x - 1]C = [1, x, x²][2x² + 3x - 1]B= (2)(x² + 2x + 1) + (3)x - 1= 2x² + 7x + 1
Therefore, [7]$[2x² + 3x - 1]C - [7(2x² – 3x + 1)]B= 7[-2x² - 6x] + 7[21x + 35]= 7[-2x² + 21x] + 7[35]= 7[-2(x - 21/4)(x + 7/2)] + 7[35]= -14(x - 21/4)(x + 7/2) + 245
Complete set of orthonormal eigenvectors for A:
First, we need to find the eigenvalues of A:|A - λI|= 0= (3 - λ)[(3 - λ)² - 2] - [(3 - λ) - 2][(3 - λ) - 2]= λ³ - 9λ² + 24λ - 16= (λ - 1)(λ - 2)(λ - 8)λ₁ = 1λ₂ = 2λ₃ = 8
We know that the sum of squares of entries in an orthonormal matrix is equal to 1, so the square of the entries of the orthonormal eigenvectors will sum up to 1.
Let the orthonormal eigenvectors be represented as[v₁v₂v₃]λ₁ = 1v₁ + 3v₂ + v₃ = 0(-1/√2)v₁ + (1/√2)v₂ = 0(-1/√2)v₁ - (1/√2)v₂ = 0v₁² + v₂² + v₃² = 1v₁ = - 3/√11, v₂ = 1/√22, v₃ = 5/√11
The matrix S, whose columns are the eigenvectors of A, is:S = [v₁v₂v₃]= [-3/√11, 1/√2, 5/√11][1, 0, 0][0, 1/√2, -1/√2]= [-3/√11, 0, 5/√11][1/√2, 1/√2, 0][-1/√2, 1/√2, 0]
Therefore, the matrix S is:S = [-3/√11, 1/√2, 5/√11][1/√2, 1/√2, 0][-1/√2, 1/√2, 0]
To find the diagonal matrix D, we need to first compute S^-1:D = S^-1AS= D= [0.49, -0.7, -0.49][1, 0, 0][0, 0.7, 0.7][0.49, 0.7, -0.49][-2, 0, 0][0, 2, 0][0, 0, 8]S^-1 = [0.49, -0.7, -0.49][0.7, 0.7, 0][-0.49, 0.49, -0.7]
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The curve y = 2/3 ^x³/² has starting point A whose x-coordinate is 3. Find the x-coordinate of 2 3 the end point B such that the curve from A to B has length 78.
Expert Answer
To find the x-coordinate of the end point B on the curve y = 2/3^x^(3/2) such that the curve from point A with x-coordinate 3 to point B has a length of 78, we need to determine the value of x at point B.
The given curve y = 2/3^x^(3/2) represents an exponential decay function. To find the x-coordinate of point B, we need to integrate the function from x = 3 to x = B and set the result equal to the given length of 78. However, integrating the function directly is quite complex. Alternatively, we can use numerical methods to approximate the value of x at point B. One such method is the midpoint rule, which involves dividing the interval into small subintervals and approximating the curve using rectangles.
By applying numerical integration techniques, we can approximate the x-coordinate of point B such that the length of the curve from point A to B is approximately 78. The specific value will depend on the chosen interval and the accuracy desired in the approximation.
Note that due to the complexity of the function, finding an exact algebraic solution for the x-coordinate of point B may be challenging. Therefore, numerical approximation methods provide a practical approach to solve this problem.
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Solve the system with the addition method.
7x-2y= 29
-3x+9y= -45
According to the statement we are given the system of equations with two variables. The solution of the system is (171/10, -9).
They are,7x - 2y = 29 -------(1)-3x + 9y = -45 ------(2)We need to solve the system with the addition method.So, we can see that we have -2y and 9y in the two equations, which can be eliminated by adding the two equations.Let's add equation (1) and equation (2) to eliminate y.7x - 2y = 29-3x + 9y = -45________________________4x + 7y = -16Now, let's eliminate y by multiplying equation (1) by 9 and equation (2) by 2, and then subtracting the second from the first.7x - 18y = 261(-6x + 18y = -90)________________________x = 171/10Now, we need to substitute the value of x in any one of the equations to find the value of y. Let's substitute in equation (1).7x - 2y = 297(171/10) - 2y = 2907/10 - 2y = 2902/10 - 2y = -16y = -18/2 = -9Therefore, the solution of the system is (171/10, -9).
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Suppose we have a sample of five values of hemoglobin A1c (HgbA1c) obtained from a single diabetic patient. HgbA1c is a serum measure often used to monitor compliance among diabetic patients. The values are 8.5%, 9.3%, 7.9%, 9.2%, and 10.3%.
(a) What is the standard deviation for this sample?
(b) What is the standard error for this sample?
a. Standard deviation = 0.8%
b. Standard error = 0.36%
How to determine the valuesFirst, calculate the mean of the data;
8.5%, 9.3%, 7.9%, 9.2%, and 10.3%.
Mean = 8.9%
The formula for standard deviation is expressed as;
SD = [tex]\sqrt{\frac{(x - mean)^2}{n} }[/tex]
Such that;
SD is the standard deviationn is the number of values in the sampleSubstitute the values, we have;
SD = √(8.5 - 8.9)² + (9.3 - 8.9)² + (7.9 - 8.9)² + (9.2 - 8.9)² + (10.3 - 8.9)²) / 5)
Subtract the value and square, we have
SD = √(0.16 + 0.16 + 1 + 0.09 + 1.96)/n
SD = √0.674
SD = 0.8%
For standard error, we have;
SE = SD / √n
SE = 0.8% / √5
SE = 0.36%
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Fix a non-singular matrix B E Mmxn. Then we can define a function : Mnxn+R by det(AB) (A) = det(B) Show that f satisfies the four conditions used to define the determinant in Def. 2.1 on pp. 324. Use this to prove that for any non-singular matrix B, det(AB) = det(A) det(B). (b) (1 pt) Using the result from (a), for a non-singular matrix C, what is det(C-1) in terms of det (C)? (c) (6 pts) Does the result from (a) still hold if B is singular? Give a counterexample, or prove that it's still true. 2 a 2.1 Definition Anxn determinant is a function det: Mnxn → R such that (1) det(21,..., k.ſi + Pj,...,n) = det(1, ... ,,..., Pn) for i ti (2) det(1, ..., , ..., Pi..., Pn) = -det(1, ..., P,..., , ..., Pn) for i #j (3) det(1, ..., kp,..., Pn) = k det(1, ...,,...,n) for any scalar k (4) det(I) = 1 where I is an identity matrix (the p's are the rows of the matrix). We often write |T| for det(T).
A = [12]. Then det(AB) = det([10] [12]) = 0, while det(A) det(B) = -2. Hence, det(AB) = det(A) det(B) is not true in general if B is singular. Given a non-singular matrix B E Mmxn, the function Mnxn+R by det(AB) (A) = det(B) satisfies the four conditions used to define the determinant in Definition 2.1 on pp. 324.
Using the results from part (a), we can prove that for any non-singular matrix B, det(AB) = det(A) det(B).a
Let A = [aij] be an n x n matrix. Given B, a non-singular matrix, define f by f(A) = det(BA). We know that f satisfies the four properties of the determinant from definition 2.1, namely:Linearity in the columns of A: If B is fixed, then f is linear in the columns of A, since det(BA) is linear in the columns of A.
Multiplicativity in a column of A: If we have two matrices A1 and A2 that differ in only one column, say the j-th column, then det(BA1) = det(BA2), since the j-th column contributes to the determinant in the same way in both cases. Hence, f satisfies property (2) of Definition 2.1. Normalization: det(BI) = det(B), where I is the n x n identity matrix. Hence f satisfies property (4) of Definition 2.1.
Invariance under transposition: If we interchange two columns of A, then the determinant changes sign, and hence f satisfies property (3) of Definition 2.1.Now, for any non-singular matrix B, det(AB) = det(A) det(B).b) Let C be a non-singular matrix. We want to express det(C-1) in terms of det(C). Using the result from part (a), we have det(C C-1) = det(I) = 1, i.e., det(C) det(C-1) = 1.
Hence, det(C-1) = 1/det(C).c) If B is singular, the result from part (a) need not hold. Consider the matrix B = [10]. This is a singular matrix, and has determinant 0.
Let A = [12].
Then det(AB)
= det([10] [12]) = 0,
while det(A) det(B) = -2.
Hence, det(AB) = det(A) det(B) is not true in general if B is singular.
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ONLY ANS B(ii)
ONLY ans b(ii)
In this question, I is the surface integral 1 = Swods where w=(y + 5x sin z)i + (x+5 y sin =) j+10 coszk, and S is that part of the paraboloid z =4 - *° - y?with :20.
In this question, the surface integral I is given by the expression 1 = ∬S w · ds, where w = (y + 5x sin z)i + (x + 5y sin z)j + 10cos(z)k, and S represents the part of the paraboloid z = 4 - x² - y² that lies above the xy-plane, i.e., z ≥ 0 and x² + y² ≤ 4.
The surface S is defined as the part of the paraboloid z = 4 - x² - y² that lies above the xy-plane. This means that the values of z are non-negative (z ≥ 0) and the x and y coordinates lie within a circle of radius 2 centered at the origin (x² + y² ≤ 4).
To evaluate the surface integral, we need to compute the dot product of the vector field w with the differential surface element ds and integrate over the surface S. The differential surface element ds represents a small piece of the surface S and is defined as ds = n · dS, where n is the unit normal vector to the surface and dS is the differential area on the surface.
By calculating the dot product w · ds and integrating over the surface S, we can determine the value of the surface integral I, which represents a measure of the flux of the vector field w across the surface S.
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