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(a) Instead of writing the quadratic interpolation polynomial in terms of the monomials 1, a: and $2 . we shall write it in terms 01 three quadratic polynomials, which each have the property that they are equal to unity at one of the data points and zero at the others, respeotlyely. These quadratic polynomials are easily constructed. Let Lots) be a quadratic polynomal that satises Aquadratic polyornial that is equal to zero at a: = 4, 5 is (a; 4X93 5}. To make this equal to one at z = 1, just divide by (1 43(1 5). This yields L0{2:) = m = {34) (35). Enter a quadratic polynomial 131(3) that satisfies 1 =4 L1(ml : {0' Z: 1:5. 3 Use the "Plot" link to check your answer. L12J= _ lean (b) Enter a quadratic polynornial L2 (2:) that satisfies 1, 3:5, {9(3) 2 {0 321,4. ? Use the "Plot" link to check your answer. sm=' (c) The ultimate aim is still to write down a polynomial that passes precisely through the data points. This can easily be done by writing the quadratic interpolation polynomiat as 312(3) = foLntml + 1311(3) + 1315235}- ll pg (:12) is to pass precisely through {1, 2}, then 372(1) = f0L0(1) +f1L1{1]+ f2L2(1) = 2. Byoonstruction,L(1): l and 121(1) 2 132(1) : I], henoe f0 = 2. Similarly, "102(2) is to pass precisely through (4, 2) and (5,1), then f1 = I Number-I and f2 = I Number I . (d) Enter the interpolating polynomial 192 (to). Use the "Plot" link to check that your answer passes through the data points. mm: lash Find the polynomial pg (9:) : a0 + alas + (2932 that interpolates the points (0,3), (1, 5) and (3, *3). (a) The aim of interpolation is for the function to pass precisely through the data points. If pg (3:) is to pass precisely through (0, 3), then mmqu+mq+@@:s:s%:3 Write down the equations that must be satisfied for p2 (m) to pass precisely through the other two data points: p2(1):a,0+ 'Number 0.1+ Number 0,2: Number p2(3):a,0+ Number 0.1+ Number a2 : _Number (b) Solve the linear system in part~(a) for (10. a1 and (12. Enter the quadratic interpolating polynomial p2 (:8). Use the "Plot" link to check that your answer passes through the data points. aw: Ham