Interpolation and Approximation

Question

A quadratic polynomial $p(x)$ is constructed by interpolating the data points $(0,1)$, $(1,e)$, $(2,e^2)$. If $\sqrt{e}$ is approximated by using $p(x)$ then its approximated value is.

Answer 1

Assume $p(x)=ax^2+bx+c$ this quadratic polynomial satisfies the given points

$$p(0)=1=c$$ $$p(1)=e=a+b+c$$ $$p(2)=e^2=4a+2b+c$$

you got $a=\frac{(e-1)^2}{2}$, $b=\frac{(e-1)(3-e)}{2}$, $c=1$

thus your polynomial is $$p(x)=\frac{(e-1)^2}{2}x^2+\frac{(e-1)(3-e)}{2}x+1$$

$$\frac{(e-1)^2}{2}x^2+\frac{(e-1)(3-e)}{2}x+1=0$$ $$(e-1)^2x^2+(e-1)(3-e)x+2=0$$