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8. (16 points) Suppose you use a quadratic curve y = ax? +b to fit the...
Question 4) Suppose that the (univariate) variable y is known to be a quadratic function of...
Question 4) Suppose that the (univariate) variable y is known to be a quadratic function of the variable x; that is, y = a x2 +bx+c, where the coefficients a, b, c are obtained by conducting an experiment in which values y1, .. , Yn of the variable y are measured for corresponding values 21,.. , Un of the variable x. Find the best least-squares fit of the quadratic polynomial using the data: {(-2,-5),(-1, -1),(0,4), (1,7), (2,6), (3,5), (4, -1)}....
Linear Algebra Question 25. Suppose you are looking at data which is supposed to fit an...
Linear Algebra Question
25. Suppose you are looking at data which is supposed to fit an exponential equation, i.e. a model of the form y= Cekz where C and k are constants. Suppose your data points are (2, 5), (3,8) and (4, 17). Use least-squares to find (decimal approximations to) the values of C and k which best fit this model How to proceed: First, take the natural log of both sides of the model to obtain what is called...
1. Use Cramer's rule to solve this system. X + 4z = 2 2x + y...
1. Use Cramer's rule to solve this system. X + 4z = 2 2x + y -z= 1 X +z=-1 on 910 2. Given the data points (0,), (1,3), (2,5) use the equation y=f(x) = mx +b to find the least square solution for best line fit. a. Evaluate the equation using the data points to obtain four linear equations. b. Write the system in 10a in a matrix form Ax=b. durants bhonebnih 516 10 boer word c. Write the...
6- Use a uniform quadratic B-spline curve with four control points to describe an ellipse whose m...
6- Use a uniform quadratic B-spline curve with four control points to describe an ellipse whose major axis a length of 6 units and minor axis 4 units
6- Use a uniform quadratic B-spline curve with four control points to describe an ellipse whose major axis a length of 6 units and minor axis 4 units
Exercise 5.10. Consider the set of n + 2 points: (1,1),(2, 1), (3,2), (3,2),...,(3,2) Suppose you...
Exercise 5.10. Consider the set of n + 2 points: (1,1),(2, 1), (3,2), (3,2),...,(3,2) Suppose you wish to best-fit these to a line y = mx + b using least-squares. (a) Write down the corresponding matrix equation. (b) Solve for using the method of least squares. Make sure you simplify: the answer should not be complicated. (c) Find limin (d) The line corresponding to your answer in (c) passes through (3,2). Why does this make sense?
(a) Suppose you are given the following (x, y) data pairs
(a) Suppose you are given the following (x, y) data pairs x136y217Find the least-squares equation for these data (rounded to three digits after the decimal) (b) Now suppose you are given these (x, y) data pairs. x217y136Find the least-squares equation for these data (rounded to three digits after the decimal). (c) In the data for parts (a) and (b), did we simply exchange the x and y values of each data pair? Yes No (d) Solve your answer from part (a) for x (rounded to...
9) Suppose you are given n points: (x,y)(, y). And we wish to fit a cirele to the data. A general...
9) Suppose you are given n points: (x,y)(, y). And we wish to fit a cirele to the data. A general circle, as we all know, is Cr-y+-k. So the question becomes: What are h, k, and r so that the circle becomes the best least squares fit? Show that this problem becomes Th .e. What is a, B and what is M? B, When fitting the cirele to the data points (0,2), (1,2),3,-),(0,-D,6,0) what are the normal equations? GIVE...
4. Consider the following data. 16 74 900 29810 (a) Suppose you would like to fit...
4. Consider the following data. 16 74 900 29810 (a) Suppose you would like to fit a model of the form yCes (C and k are constants) to this data. Transform the model in such a way that you can use linear least squares to determine C and k. (b) For this data set, use the following equations to find a best fit line. and (e) Use your calculation from part b) to find the model y - Cek for...
2 ax h-r 2. The following data represents a scientific model y 3b Use the Use...
2 ax h-r 2. The following data represents a scientific model y 3b Use the Use the method of least squares to 3b- x find coefficients a and b of the equation (15 points) 10.8 13.2 15.616.818 21.627.6 8.1 9.5 9.8 10 10.3 10.5
Task 3 - Simpson's Rule (Maximum Mark 15) Find the equation the quadratic y = ax?...
Task 3 - Simpson's Rule (Maximum Mark 15) Find the equation the quadratic y = ax? + bx + c which passes through the points AC-h, Y.), B(0,y), and C(h.yc). Use your quadratic to find an expression for the area under the quadratic between the points A and C. Explaining carefully how your result can be used to prove the general formula for Simpson's Rule. Notes: . You will need to research independently to find the formula for Simpson's Rule....
Question 4) Suppose that the (univariate) variable y is known to be a quadratic function of the variable x; that is, y = a x2 +bx+c, where the coefficients a, b, c are obtained by conducting an experiment in which values y1, .. , Yn of the variable y are measured for corresponding values 21,.. , Un of the variable x. Find the best least-squares fit of the quadratic polynomial using the data: {(-2,-5),(-1, -1),(0,4), (1,7), (2,6), (3,5), (4, -1)}....
Linear Algebra Question
25. Suppose you are looking at data which is supposed to fit an exponential equation, i.e. a model of the form y= Cekz where C and k are constants. Suppose your data points are (2, 5), (3,8) and (4, 17). Use least-squares to find (decimal approximations to) the values of C and k which best fit this model How to proceed: First, take the natural log of both sides of the model to obtain what is called...
1. Use Cramer's rule to solve this system. X + 4z = 2 2x + y -z= 1 X +z=-1 on 910 2. Given the data points (0,), (1,3), (2,5) use the equation y=f(x) = mx +b to find the least square solution for best line fit. a. Evaluate the equation using the data points to obtain four linear equations. b. Write the system in 10a in a matrix form Ax=b. durants bhonebnih 516 10 boer word c. Write the...
6- Use a uniform quadratic B-spline curve with four control points to describe an ellipse whose major axis a length of 6 units and minor axis 4 units
6- Use a uniform quadratic B-spline curve with four control points to describe an ellipse whose major axis a length of 6 units and minor axis 4 units
Exercise 5.10. Consider the set of n + 2 points: (1,1),(2, 1), (3,2), (3,2),...,(3,2) Suppose you wish to best-fit these to a line y = mx + b using least-squares. (a) Write down the corresponding matrix equation. (b) Solve for using the method of least squares. Make sure you simplify: the answer should not be complicated. (c) Find limin (d) The line corresponding to your answer in (c) passes through (3,2). Why does this make sense?
9) Suppose you are given n points: (x,y)(, y). And we wish to fit a cirele to the data. A general circle, as we all know, is Cr-y+-k. So the question becomes: What are h, k, and r so that the circle becomes the best least squares fit? Show that this problem becomes Th .e. What is a, B and what is M? B, When fitting the cirele to the data points (0,2), (1,2),3,-),(0,-D,6,0) what are the normal equations? GIVE...
4. Consider the following data. 16 74 900 29810 (a) Suppose you would like to fit a model of the form yCes (C and k are constants) to this data. Transform the model in such a way that you can use linear least squares to determine C and k. (b) For this data set, use the following equations to find a best fit line. and (e) Use your calculation from part b) to find the model y - Cek for...
2 ax h-r 2. The following data represents a scientific model y 3b Use the Use the method of least squares to 3b- x find coefficients a and b of the equation (15 points) 10.8 13.2 15.616.818 21.627.6 8.1 9.5 9.8 10 10.3 10.5
Task 3 - Simpson's Rule (Maximum Mark 15) Find the equation the quadratic y = ax? + bx + c which passes through the points AC-h, Y.), B(0,y), and C(h.yc). Use your quadratic to find an expression for the area under the quadratic between the points A and C. Explaining carefully how your result can be used to prove the general formula for Simpson's Rule. Notes: . You will need to research independently to find the formula for Simpson's Rule....
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