We can use methods from linear algebra to find fits for various functions through data points,...

 

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We can use methods from linear algebra to find fits for various functions through data points, and learn about the nature of such fits. Use the digits of your stu- dent number to construct five numbers in the following way: if the digits are d₁d2d3d4d5d6d7, write down the two-digit numbers y₁ = d2d3, y2 = d3d4, y3 = d4d5, y4d5d6 and y5=d6d7. So, for example, if your student number was 5123406, your numbers would be y₁ = 12, y2 = 23, y3 = 34, y4 = 40 and y5 = 066. If your student number has fewer than seven digits, pre-pad it with Os (so 12345 becomes 0012345). Now consider the five points A = (-2, y₁), B = (-1, y2), C = (0, y3), D = (1, y4) and E=(2, y5). (a) In this first part, you must find the polynomial that passes through points A, B, C, D and E. (i) Set up a system of equations to find the unknown coefficients ao, a1, a2, a3 and a4 for the polynomial y = a₁x¹ + α3x³ + a₂x² + a₁x + ao that passes through A, B, C, D and E. (ii) Solve this system of equations to find the unknown coefficients. (iii) Confirm that the resulting polynomial does indeed pass through the required points. We can use methods from linear algebra to find fits for various functions through data points, and learn about the nature of such fits. Use the digits of your stu- dent number to construct five numbers in the following way: if the digits are d₁d2d3d4d5d6d7, write down the two-digit numbers y₁ = d2d3, y2 = d3d4, y3 = d4d5, y4d5d6 and y5=d6d7. So, for example, if your student number was 5123406, your numbers would be y₁ = 12, y2 = 23, y3 = 34, y4 = 40 and y5 = 066. If your student number has fewer than seven digits, pre-pad it with Os (so 12345 becomes 0012345). Now consider the five points A = (-2, y₁), B = (-1, y2), C = (0, y3), D = (1, y4) and E=(2, y5). (a) In this first part, you must find the polynomial that passes through points A, B, C, D and E. (i) Set up a system of equations to find the unknown coefficients ao, a1, a2, a3 and a4 for the polynomial y = a₁x¹ + α3x³ + a₂x² + a₁x + ao that passes through A, B, C, D and E. (ii) Solve this system of equations to find the unknown coefficients. (iii) Confirm that the resulting polynomial does indeed pass through the required points.

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To find the polynomial that passes through points A B C D and E we will set up a system of equations ... View the full answer