(Taylor's Theorem with Integral Remainder) Let n be a...

(Taylor's Theorem with Integral Remainder) Let $n$ be a nonnegative integer and let $f:[a, b] \rightarrow \mathbb{R}$ be such that $f^{\prime}, f^{\prime \prime}, \ldots, f^{(n+1)}$ exist and $f^{(n+1)}$ is continuous on $[a, b]$. Show that $$ f(b)=f(a)+f^{\prime}(a)(b-a)+\cdots+\frac{f^{(n)}(a)}{n !}(b-a)^{n}+\frac{1}{n !} \int_{a}^{b}(b-t)^{n} f^{(n+1)}(t) d t $$ Further, show that the remainder is equal to $$ \frac{(b-a)^{n+1}}{n !} \int_{0}^{1}(1-s)^{n} f^{(n+1)}(a+s(b-a)) d s $$ (Hint: Induction on $n$ and Integration by Parts.) [Note: The integral remainder does not involve an undetermined number $c \in(a, b) .]$

(Taylor's Theorem with Integral Remainder) Let $n$ be a nonnegative integer and let $f:[a, b] \rightarrow \mathbb{R}$ be such that $f^{\prime}, f^{\prime \prime}, \ldots, f^{(n+1)}$ exist and $f^{(n+1)}$ is continuous on $[a, b]$. Show that
$$
f(b)=f(a)+f^{\prime}(a)(b-a)+\cdots+\frac{f^{(n)}(a)}{n !}(b-a)^{n}+\frac{1}{n !} \int_{a}^{b}(b-t)^{n} f^{(n+1)}(t) d t
$$
Further, show that the remainder is equal to
$$
\frac{(b-a)^{n+1}}{n !} \int_{0}^{1}(1-s)^{n} f^{(n+1)}(a+s(b-a)) d s
$$
(Hint: Induction on $n$ and Integration by Parts.) [Note: The integral remainder does not involve an undetermined number $c \in(a, b) .]$

Key Concepts

Taylor's Theorem

Taylor's Theorem is a fundamental result in calculus that approximates a function by a polynomial whose coefficients are determined by the derivatives of the function at a specific point. The theorem shows that a function expressed as such can be written as the sum of its Taylor polynomial and a remainder term, thereby quantifying the error between the function and its polynomial approximation.

Integral Remainder (Lagrange form)

The integral form of the remainder in Taylor's Theorem expresses the error between the actual function and its Taylor polynomial as an integral involving higher order derivatives. This representation provides a precise measure of the approximation error and is directly linked to the smoothness and behavior of the function. It avoids dependence on an indeterminate point within the interval by integrating over the interval itself.

Integration by Parts

Integration by Parts is a technique derived from the product rule of differentiation that is used to evaluate the integral of the product of two functions. In the context of Taylor's Theorem with an integral remainder, it is instrumental in converting the remainder term into a form that clearly involves the higher order derivative, thereby facilitating an inductive argument in the proof.

Mathematical Induction

Mathematical Induction is a powerful proof technique used to establish the validity of a statement for all natural numbers. In the context of Taylor's Theorem, induction is used to prove the formula incrementally by showing that if the theorem holds for a polynomial of degree n, then it also holds for a polynomial of degree n+1.

Smoothness Conditions and Derivatives

The smoothness condition, which requires that the function's derivatives up to a certain order exist and are continuous, is critical in ensuring the validity of Taylor's Theorem. These conditions guarantee that the function behaves well enough for its Taylor polynomial to approximate it accurately, and they justify the interchange of differentiation and integration used in the derivation of the integral remainder.

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