Given the points (1,0.7),(2,0.73),(3,0.8),(4,0.75),(5,0.6). Use Lagrange polynomials to find the polynomial that goes through all the points.
Complete the following table
x
1.0
1.3
1.6
1.9
2.2
y
0.7651977
0.6200860
0.4554022
0.2818186
0.1103623
Assuming x = 1.5, Using Neville’s Method.
x
0.6283185308
1.2566370616
1.8849555924
2.5132741232
y
0.587785252358846
0.951056516219097
0.587785252026982
3.
Complete the divided differences table
Find the approximation polynomial
Evaluate the polynomial in x = 1.5
√
Let f (x) = x4 + 2x3 + πx. Verify whether f [1, 2, 3, 4] = f [0, 1, π, e, −1]
Find the equations that are needed to find a quadratic spline SI(x) = aI + bI(x − xI) + cI(x − xI )2. In this case you will not need a condition on the second derivative. HINT: from the condition on the first derivative find cI and substitute in the other equation to find a system of equations for the bI’s.
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