Question

Find quadratic polynomial which best fits the function f(x)=e^x and x=0, in the sense that g(0)= f(0), and g'(0) = f'(0), and g"(0) = f"(0).
g(x)=?

Answer 1

Let $g(x)=ax^2+bx+c$. Then

$\begin{cases}g(0)=c\\g'(0)=b\\g''(0)=2a\end{cases}$

Meanwhile, since $f(x)=e^x$, you have $f(0)=f'(0)=f''(0)=e^0=1$.

It follows that $a=\dfrac12$, $b=1$, and $c=1$, so that the quadratic fit for $f(x)=e^x$ that satisfies the given points is

$g(x)=\dfrac12x^2+x+1$

Note that this is just the second-order Taylor polynomial for $e^x$ about $x=0$.