Give an alternative proof of Taylor's Theorem with a single...

Give an alternative proof of Taylor's Theorem with a single application of Rolle's Theorem by proceeding as follows. Let the notation and hypothesis be as in the statement of Taylor Theorem (Proposition $4.23$ ). Also, as in the proof of Taylor's Theorem, for $x \in[a, b]$, let $$ P(x)=f(a)+f^{\prime}(a)(x-a)+\frac{f^{\prime \prime}(a)}{2 !}(x-a)^{2}+\cdots+\frac{f^{(n)}(a)}{n !}(x-a)^{n} . $$ Define $g:[a, b] \rightarrow \mathbb{R}$ by $$ g(x)=f(x)+f^{\prime}(x)(b-x)+\frac{f^{\prime \prime}(x)}{2 !}(b-x)^{2}+\cdots+\frac{f^{(n)}(x)}{n !}(b-x)^{n}+s(b-x)^{n+1}, $$ where $s=[f(b)-P(b)] /(b-a)^{n+1} .$ Show that $g(a)=g(b)=f(b) .$ Apply Rolle's Theorem to $g$ to deduce Taylor's Theorem.

Give an alternative proof of Taylor's Theorem with a single application of Rolle's Theorem by proceeding as follows. Let the notation and hypothesis be as in the statement of Taylor Theorem (Proposition $4.23$ ). Also, as in the proof of Taylor's Theorem, for $x \in[a, b]$, let
$$
P(x)=f(a)+f^{\prime}(a)(x-a)+\frac{f^{\prime \prime}(a)}{2 !}(x-a)^{2}+\cdots+\frac{f^{(n)}(a)}{n !}(x-a)^{n} .
$$
Define $g:[a, b] \rightarrow \mathbb{R}$ by
$$
g(x)=f(x)+f^{\prime}(x)(b-x)+\frac{f^{\prime \prime}(x)}{2 !}(b-x)^{2}+\cdots+\frac{f^{(n)}(x)}{n !}(b-x)^{n}+s(b-x)^{n+1},
$$
where $s=[f(b)-P(b)] /(b-a)^{n+1} .$ Show that $g(a)=g(b)=f(b) .$ Apply Rolle's Theorem to $g$ to deduce Taylor's Theorem.

Key Concepts

Rolle's Theorem

Rolle's Theorem states that if a function is continuous on a closed interval and differentiable on the open interval, and if the function values at the endpoints are equal, then there exists at least one point within the interval where the derivative is zero. This theorem is instrumental in the alternative proof of Taylor's Theorem, where it is applied to a specially constructed function to demonstrate the existence of a point that leads directly to the formulation of the remainder term.

Remainder Term

The Remainder Term in Taylor's Theorem represents the error between the actual function and its Taylor polynomial approximation. It quantifies the deviation of the function from the polynomial and is crucial in establishing how accurately the polynomial approximates the function, often expressed in forms that include higher-order derivatives evaluated at some intermediate point.

Taylor's Theorem

Taylor's Theorem is a central result in real analysis that allows a function to be expressed as the sum of its Taylor polynomial and a remainder term. This theorem provides a way to approximate functions by polynomials and gives a quantifiable error term that helps in understanding how well the polynomial represents the function near a particular point.

Taylor Polynomial

A Taylor Polynomial is the polynomial part of the Taylor series expansion of a function, constructed using the function's values and its derivatives at a specific point. It serves as an approximation of the original function near that point and forms the main term in the expansion before the inclusion of the error or remainder term.

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