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=\frac12`,
`b=1`, and
`c=1`, so that the quadratic fit for
`f(x)=e^x` that satisfies the given points is


`g(x)=\frac12x^2+x+1`

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