#624 Use principle of inclusion-exclusion to solve

Use the principle of inclusion-exclusion to solve - Math

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Use the principle of inclusion-exclusion to solve the following question. Explain the steps of your solution briefly to show how did you use the principle of inclusion-exclusion.

How many arrangements of x, x, x, y, y, y, z, z, are there such that no three consecutive letters in these arrangements are the same?

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#605 Definition Solve one Interpolation problem for each

Definition Solve one Interpolation problem for each - Math

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Definition: Solve one “Interpolation” problem for each method given below:

1. Linear Interpolation

2. Lagrange Interpolation Polynomial

3. Newton's Divided Difference Interpolating Polynomials

4. Simple Polynomial Interpolation (Coefficients of an Interpolating Polynomials)

5. Cubic Spline Interpolation

The problem question must be given before the problem solution

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#600 Suppose that a drug is added to the body at a rate

Suppose that a drug is added to the body at a rate - Math

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Suppose that a drug is added to the body at a rate r(t), and let y(t) represent the concentration of the drug in the bloodstream at time t hours. In addition, suppose that the drug is used by the body at the rate ky, where k is a positive constant. Then, the net rate of change in y(t) is given by the equation

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#592 How would you solve for the unknown coefficients

How would you solve for the unknown coefficients - Math

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Given a set of N data pairs (x_i, y_i), how would you solve for the unknown coefficients a, b and c for the polynomial of the form y = a + bx³ + cx⁵ which best fits the data (in a least-squares sense).

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#591 Using the equations developed in class, solve for the

Using the equations developed in class, solve for the - Math

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1. Data from a permeability test on sand gave the following gradient-flow rate results (this is a test that you will conduct in the Geotechnical laboratory course in spring of Junior year):

gradient i 0.0 0.22 0.41 0.59 0.80 1.01 1.09

flow q (mm/s) 0.0 0.38 0.81 1.15 1.54 1.91 2.25

1. (a) Using the equations developed in class, solve for the "best fit" straight line through the data using least-squares regression, by hand, presuming i is the independent variable (ie use the equations based on the sums Σx, Σy, Σxy, Σx2, N).
2. (b) Solve for the best fit straight line again using the statistical form of the solution, with slope = sxy/sxx and y-intercept = (Mean y) - slope* (mean x). Compare with (a).
3. (c) It is known from Darcy's law in soil mechanics that the flow rate is directly proportional to the gradient and MUST be zero when the gradient is zero, ie q = k×i where k is a constant. How can we determine the best fit slope k and ensure that the line goes through the origin (0,0)?
4. (d) Compute an unbiased estimate of the variance of the error, 𝑠2= 𝑛𝑛−2 (1−𝑆𝑥𝑦2𝑆𝑥𝑥𝑆𝑦𝑦)𝑠𝑦𝑦
5. (e) Using a spreadsheet program, plot the data along with

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#590 Data from a permeability test on sand gave the

Data from a permeability test on sand gave the - Math 

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Data from a permeability test on sand gave the following gradient-flow rate results (this is a test that you will conduct in the Geotechnical laboratory course in spring of Junior year):

      gradient i ----------> 0.0       0.22     0.41     0.59     0.80     1.01     1.09

      flow q (mm/s)------> 0.0       0.38     0.81     1.15     1.54     1.91     2.25

a. Using the equations developed in class, solve for the "best fit" straight line through the data using least-squares regression, by hand, presuming i is the independent variable (ie use the equations based on the sums ∑x, ∑y, ∑xy, ∑x², N).

b. Solve for the best fit straight line again using the statistical form of the solution, with slope = s_xy/s_xx and y-intercept = (Mean y) - slope* (mean x). Compare with (a).

c. It is known from Darcy's law in soil mechanics that the flow rate is directly proportional to the gradient and MUST be zero when the gradient is zero, ie q = k×i where k is a constant. How can we determine the best fit slope k and ensure that the line goes through the origin (0,0)?

d. Compute an unbiased estimate of the variance of the error, 

 e. Using a spreadsheet program, plot the data along with the best fit straight line. Also plot the two parallel lines which are ±1 s away from the best fit line.

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#589 Given a set of N data pairs

Given a set of N data pairs - Math 

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Given a set of N data pairs (x_i, y_i), how would you solve for the unknown coefficients a, b, and c for the polynomial of the form which best fits the data (in a least-squares sense).

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#586 Data from a permeability test on sand gave

Data from a permeability test on sand gave - Math

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1. Data from a permeability test on sand gave the following gradient-flow rate results (this is a test that you will conduct in the Geotechnical laboratory course in spring of Junior year)

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#570 Use the following methods to solve the first-degree

Use the following methods to solve the first-degree - Math

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Definition: Use the following methods to solve the first-degree equation system with 3 unknowns below until the percent relative error falls below &=5%.

1. Gauss Elimination (25%)

2. LU Decomposition (25%)

3. Gauss-Seidel (25%)

General paper order and layout will make up 25% of your total homework grade.

Rules to be followed: •

The problem question must be given before the problem solution. •

If the problem is taken from a source (books, web addresses, online courses etc.), that source and the location of the question must be indicated. •

All calculations must be handmade and openly shown on your papers, only solution tables will not mean anything in terms of grading • Solutions must be clearly readable, sufficiently organized and include the necessary tables/iterations etc.

Using only one side of a paper for the solution of one question is highly recommended.

The uploading rules for the quizzes stated before are also valid for your homework.

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#568 Gauss-Seidel method until the percent relative

Gauss-Seidel method until the percent relative- Math

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A. LU decomposition method,

B. Repeat it for an alternative right-hand-side vector {B}^T = [12 18-6]

C. Gauss-Seidel method until the percent relative error falls below E₃= 5%

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#566 Determine the algebraic representation of the following

Determine the algebraic representation of the following - Math

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Determine the algebraic representation of the following piecewise function.

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