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Proof of Taylors Theorem There are several proofs of
Chapter 7, Problem 84AE(choose chapter or problem)
Proof of Taylor's Theorem There are several proofs of Taylor's Theorem, which lead to various forms of the remainder. The following proof is instructive because it leads to two different forms of the remainder and it relies on the Fundamental Theorem of Calculus, integration by parts, and the Integral Mean Value Theorem. Assume that f has at least n + 1 continuous derivatives on an interval containing a.
a. Show that the Fundamental Theorem of Calculus can be written in the form
\(f(x)=f(a)+\int_{a}^{x} f^{\prime}(t) d t\).
b. Use integration by parts \(\left(u=f^{\prime}(t), d v=d t\right)\) to show that
\(f(x)=f(a)+(x-a) f^{\prime}(a)+\int_{a}^{x}(x-t) f^{\prime \prime}(t) d t\).
c. Show that n integrations by parts gives
\(\begin{aligned} f(x)=& f(a)+\frac{f^{\prime}(a)}{1 !}(x-a)+\frac{f^{\prime \prime}(a)}{2 !}(x-a)^{2}+\cdots \\ &+\frac{f^{(n)}(a)}{n !}(x-a)^{n}+\underbrace{\int_{a}^{x} \frac{f^{(n+1)}(t)}{n !}(x-t)^{n} d t .}_{R_{n}(x)} \end{aligned}\)
d. The result in part (c) looks like \(f(x)=p_{n}(x)+R_{n}(x)\), where \(p_{n}\), is the nth-order Taylor polynomial and \(R_{n}\), is a new form of the remainder term, known as the integral form of the remainder term. Use the Integral Mean Value Theorem to show that \(R_{n}\) can be expressed in the form
\(R_{n}(x)=\frac{f^{(n+1)}(c)}{(n+1) !}(x-a)^{n+1}\),
where c is between a and x.
Questions & Answers
QUESTION:
Proof of Taylor's Theorem There are several proofs of Taylor's Theorem, which lead to various forms of the remainder. The following proof is instructive because it leads to two different forms of the remainder and it relies on the Fundamental Theorem of Calculus, integration by parts, and the Integral Mean Value Theorem. Assume that f has at least n + 1 continuous derivatives on an interval containing a.
a. Show that the Fundamental Theorem of Calculus can be written in the form
\(f(x)=f(a)+\int_{a}^{x} f^{\prime}(t) d t\).
b. Use integration by parts \(\left(u=f^{\prime}(t), d v=d t\right)\) to show that
\(f(x)=f(a)+(x-a) f^{\prime}(a)+\int_{a}^{x}(x-t) f^{\prime \prime}(t) d t\).
c. Show that n integrations by parts gives
\(\begin{aligned} f(x)=& f(a)+\frac{f^{\prime}(a)}{1 !}(x-a)+\frac{f^{\prime \prime}(a)}{2 !}(x-a)^{2}+\cdots \\ &+\frac{f^{(n)}(a)}{n !}(x-a)^{n}+\underbrace{\int_{a}^{x} \frac{f^{(n+1)}(t)}{n !}(x-t)^{n} d t .}_{R_{n}(x)} \end{aligned}\)
d. The result in part (c) looks like \(f(x)=p_{n}(x)+R_{n}(x)\), where \(p_{n}\), is the nth-order Taylor polynomial and \(R_{n}\), is a new form of the remainder term, known as the integral form of the remainder term. Use the Integral Mean Value Theorem to show that \(R_{n}\) can be expressed in the form
\(R_{n}(x)=\frac{f^{(n+1)}(c)}{(n+1) !}(x-a)^{n+1}\),
where c is between a and x.
ANSWER:Solution 84AEStep 1:a. Show that the Fundamental Theorem of Calculus can be written in the form By fundamental theorem of Calculus, if is an antiderivative of then…(1)We know that is an antiderivative of Therefore by (1) we getHence proved.