Numerical Analysis Define a spline function Bx consisting po

Numerical Analysis:

Define a spline function B(x) consisting polonimials of deree 3 as follows:

Assume B(x) = B(-x) for all x, that is B(x) is symmetric about the y-axis

Show that B is a cubic spline of degree 3.

Professor\'s hint : this function is bell-shaped, and symmetric about y-axis. So just check on the right side of y-axis. Check the continuity of B(x), B\'(x), B\'\'(x) at interior points on the two consecutive splines B₀ and B_1.

Numerical Analysis: Define a spline function B(x) consisting polonimials of deree 3 as follows: B(x) = (2- x)^3 for 1 geq 2, Assume B(x) = B(-x) for all x, that is B(x) is symmetric about the y-axis Show that B is a cubic spline of degree 3. Professor\'s hint : this function is bell-shaped, and symmetric about y-axis. So just check on the right side of y-axis. Check the continuity of B(x), B\'(x), B\'\'(x) at interior points on the two consecutive splines B0 and B1. leq 1, B(x) = 0 for x leq x leq 2, B(x)= 1 + 3(1- x) + 3(1- x)2 -3(1- x)3 for 0 leq x

Solution

<p>Numerical Analysis:</p> <p>Define a spline function B(x) consisting polonimials of deree 3 as follows:</p> <p>B(x) =&nbsp;&nbsp; (2- x)³ for 1<img alt=\"\\leq\" src= \"https://latex.codecogs.com/gif.latex?%5Cleq\" /> x <img alt=\"\\leq\" src=\"https://latex.codecogs.com/gif.latex?%5Cleq\" /> 2,</p> <p>B(x)= 1 + 3(1- x) + 3(1- x)² -3(1- x)³&nbsp;&nbsp;&nbsp; for 0 <img alt=\"\\leq\" src= \"https://latex.codecogs.com/gif.latex?%5Cleq\" /> x <img alt=\"\\leq\" src=\"https://latex.codecogs.com/gif.latex?%5Cleq\" /> 1,</p> <p>B(x) = 0 &nbsp;&nbsp; for x <img alt=\"\\geq\" src= \"https://latex.codecogs.com/gif.latex?%5Cgeq\" /> 2,</p> <p>Assume B(x) = B(-x) for all x, that is B(x) is symmetric about the y-axis</p> <p>Show that B is a cubic spline of degree 3.</p> <p>Professor\'s hint : this function is bell-shaped, and symmetric about y-axis. So just check on the right side of y-axis. Check the continuity of B(x), B\'(x), B\'\'(x) at interior points on the two consecutive splines B₀ and B_1.</p>