(Taylor's Theorem) Use Corollary 6.13 to prove the following...

(Taylor's Theorem) Use Corollary 6.13 to prove the following version of Taylor's theorem: If $f \in C^{n+1}(I)$ and $t_0 \in I$, then
$$
f(t)=\sum_{k=0}^n f^{(k)}\left(t_0\right) \frac{\left(t-t_0\right)^k}{k !}+\int_{t_0}^t \frac{(t-s)^n}{n !} f^{(n+1)}(s) d s,
$$
for $t \in I$.

Key Concepts

Taylor's Theorem

A fundamental result in calculus that states a function which is sufficiently differentiable can be approximated near a point by a polynomial whose coefficients are determined by the function's derivatives at that point, plus a remainder term that accounts for the approximation error.

Taylor Polynomial

The polynomial formed by summing the products of the function's derivatives at a specific point with the corresponding powers of the difference from that point, each divided by the appropriate factorial. This polynomial provides an approximation of the function near that point and becomes more accurate as higher-order terms are included.

Integral Remainder

A specific form of the error term in Taylor's theorem where the difference between the function and its Taylor polynomial is expressed as an integral involving the (n+1)th derivative of the function. This form leverages the fundamental theorem of calculus to quantify the error in terms of a continuous accumulation of contributions from the higher derivative.

Smooth Functions (C^(n+1))

Functions that have continuous derivatives up to order n+1 on a given interval. This property guarantees the validity of Taylor's theorem and the associated remainder term, ensuring that the function can be closely approximated by its Taylor polynomial.

Fundamental Theorem of Calculus

A central theorem that connects differentiation and integration. In the context of Taylor's theorem, it is used to derive the integral form of the remainder by providing a means to relate the (n+1)th derivative of a function to an integral representation of the error between the function and its Taylor polynomial.

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