Theorem: Taylor's Theorem (Lagrange Form of the Remainder)

Let $f$ have $n+1$ continuous derivatives on an interval containing $c$ and $x$ , and let $R_n(x) = f(x) - T_n(x)$ be the remainder after the -th degree Taylor polynomial . Then there exists a number strictly between and such that

11 viewsMathematics · Single Variable Calculus

Theorem: Taylor's Theorem (Lagrange Form of the Remainder)

Let $f$ have $n+1$ continuous derivatives on an interval containing $c$ and $x$ , and let $R_n(x) = f(x) - T_n(x)$ be the remainder after the $n$ -th degree Taylor polynomial $T_n$ . Then there exists a number $z$ strictly between $c$ and $x$ such that

R_n(x) = \frac{f^{(n+1)}(z)}{(n+1)!}(x-c)^{n+1}.

Consequently, if $|f^{(n+1)}(t)| \le M$ for all $t$ between $c$ and $x$ , then

|R_n(x)| \le \frac{M}{(n+1)!}|x-c|^{n+1}.

11 viewsMathematics · Single Variable Calculus

Theorem: Taylor's Theorem (Lagrange Form of the Remainder)

11 viewsMathematics · Single Variable Calculus