Question

Given the data below:

x 1 2 2.5 3 4 5
f(x) 0 5 6.5 7 3 1

Calculate f(3.4) using Lagrange polynomial of order 1 through 3. Choose the sequence of the points for your estimates to attain the best possible accuracy. That is, the points should be centered around and as close as possible to the unknown. (Round the final answer to four decimal places.)

Answer 1

5.16

Explanation:

Since after x=3, value of y starts decreasing for increasing values of x, let's choose. Also we need to find f(x) for x=3.4

x=3,4,5 and f(x)=7,3,1 to find lagrange polynomial

P(x)= ((x-x2)(x-x3)y1)/((x1-x2)(x1-x3) + ((x-x1)(x-x3)y2)/((x2-x1)(x2-x3) + ((x-x1)(x-x2)y3)/((x3-x1)(x3-x2)

P(x)= ((x-4)(x-5)7)/((3-4)(3-5)) + ((x-3)(x-5)3)/((4-3)(4-5)) + ((x-3)(x-4)1)/((5-3)(5-4))

P(x)= x² -11x+31

P(3.4)= 5.16