Answer to Question #265899 in Quantitative Methods for sagun poudel

Question

Find the values of the first and second derivatives of y = 4x2+2x-1 for

x=1.25 from the following table using Lagrange’s interpolation

formula.

x : 1 1.5 2 2.5

y : 5 11 19 29

Answer 1

We have 4 points, so the Lagrange polynomial will be of degree 3. The basic Lagrange polynomials are: 

$$L_0(x)=\dfrac{(x-1.5)(x-2)(x-2.5)}{(1-1.5)(1-2)(1-2.5)}$$

$$L_1(x)=\dfrac{(x-1)(x-2)(x-2.5)}{(1.5-1)(1.5-2)(1.5-2.5)}$$

$$L_2(x)=\dfrac{(x-1)(x-1.5)(x-2.5)}{(2-1)(2-1.5)(2-2.5)}$$

$$L_3(x)=\dfrac{(x-1)(x-1.5)(x-2)}{(2.5-1)(2.5-1.5)(2.5-2)}$$

$$P(x)=5L_0(x)+11L_1(x)+19L_2(x)+29L_3(x)$$

$$P(x)=-\dfrac{20}{3}(x-1.5)(x-2)(x-2.5)$$

$$+44(x-1)(x-2)(x-2.5)$$

$$-76(x-1)(x-1.5)(x-2.5)$$

$$+\dfrac{116}{3}(x-1)(x-1.5)(x-2)$$

$$=-\dfrac{20}{3}x^3+40x^2-\dfrac{235}{3}x+50$$

$$+44x^3-242x^2+418x-220$$

$$-76x^3+380x^2-589x+282$$

$$\dfrac{116}{3}x^3-40x^2+174x-116$$

The approximation for the $p^{th}$ derivative of some function can be found by taking the $p^{th}$ derivative of a polynomial approximation of the function

.

$$P'(x)=276x-\dfrac{226}{3}$$

$$P''(x)=276$$