Answer to Question #260856 in English for Zee

Question

A student went to test the motorbike one afternoon after class on the Parys Road. His velocity between two hills is listed in the table. The time is in seconds and the speed readings have been converted to m/s to ease calculations.

t(s) | 1 | 5 | 7 |

v(m/s)| 24 | 40 | 36 |

a) Determine the Lagrange functions L₀, L₁ and L₂ and the second degree Lagrange interpolation polynomial P₂(t) for the given data points. Simplify the polynomial.

b) formulate a theorem on the existence of a unique interpolation polynomial.

Answer 1

The given values are x0 = 0, y0 = -12 x1 = 1, y1 = 0 x2 = 3, y2 = 6 x3 = 4, y3 = 12 By Lagrange’s interpolaiton formula, ⇒ y = (x – 1) (x – 3) (x – 4) – x (x – 1) (x – 4) + x (x – 1) (x – 3) ⇒ y = (x – 1) (x – 4)[(x – 3) – x] + x (x – 1) (x – 3) ⇒ y = (x – 1) (x – 4) (-3) + x (x – 1) (x – 3) ⇒ y = (x – 1) [-3x + 12 + x2 – 3x] ⇒ y = (x – 1) (x2 – 6x + 12) ⇒ y = x3 – 6x2 + 12x – x2 + 6x – 12 y = x3 – 7x + 18x – 12 is the required polynomial formulae