The Lagrange error bound

Three quick questions from earlier lessons.

$P_2$ matches $f$ , $f'$ and $f''$ at $a$ . Which derivative is responsible for the error?

The next term, with the worst derivative, bounds the error.

Lagrange error bound: if

|f^{(n+1)}(t)| \le M

between

a

and

x

, then

|f(x) - P_n(x)| \le \dfrac{M\,|x - a|^{n+1}}{(n + 1)!}

Bound the error of $P_3$ for $e^x$ at $x = 0.5$ (center 0).

M = 2

f⁽⁴⁾ = eˣ ≤ e^0.5 < 2 on [0, 0.5].

|R_3| \le \dfrac{2 \cdot 0.5^4}{24}

|x|⁴/4!.

\approx 0.0052

Compute.

Your turn: $|f^{(3)}| \le 6$ ; bound the error of $P_2$ at $|x - a| = 1$ .

|R_2| \le 1

6 · 1³/3!.

$|f^{(3)}(x)| \le 9$ on $[0, 1]$ . Bound the error of the degree-2 Maclaurin polynomial at $x = 1$ .
$|f^{(3)}(x)| \le 8$ on $[0, 1]$ . Bound the error of the degree-2 Maclaurin polynomial at $x = 1$ .

The Lagrange error bound for the degree-4 Taylor polynomial uses the maximum of which derivative?
1. $f^{(5)}$ , the next derivative
2. $f^{(4)}$
3. $f$ itself

On the interval, $|f^{(2)}(x)| \le 2$ . Bound the error of the degree-1 Taylor polynomial centered at $a$ , at a point with $|x - a| = \tfrac{1}{2}$ .
On the interval, $|f^{(3)}(x)| \le 1$ . Bound the error of the degree-2 Taylor polynomial centered at $a$ , at a point with $|x - a| = \tfrac{1}{2}$ .

Three questions to check you have got it.

$|f^{(3)}(x)| \le 9$ on $[0, 1]$ . Bound the error of the degree-2 Maclaurin polynomial at $x = 1$ .
$|f^{(3)}(x)| \le 8$ on $[0, 1]$ . Bound the error of the degree-2 Maclaurin polynomial at $x = 1$ .
$|f^{(3)}(x)| \le 2$ on $[0, 1]$ . Bound the error of the degree-2 Maclaurin polynomial at $x = 1$ .

Lagrange error bound

$|R_n| \le \dfrac{M|x - a|^{n+1}}{(n + 1)!}$ , $M \ge |f^{(n+1)}|$ .

Watch out: $n + 1$ everywhere.

End of the lesson