The function $f$ and $f\prime \,f″\,\dots \,f^{\left(n\right)}$ (its first $n$ derivatives) are continuous in $\left[\mathrm{\α}\,\mathrm{\β}\right]$ 

$f^{\left(n\right)}$ is differentiable in $\left(\mathrm{\α}\,\mathrm{\β}\right)$

Both $x$ and $a$ are in $\left[\mathrm{\α}\,\mathrm{\β}\right]$

There exists a value $c$ in $\left(\mathrm{\α}\,\mathrm{\β}\right)$ for which the following representation of $f$ holds.

For $f\left(x\right)\=e^x$ at $x\=0$, obtain Taylor polynomials of degree 1, 2, and 3.

Graphically compare these polynomials on $\left[-4\,4\right]$.

Use $R_3\left(x\right)$ to estimate the largest difference between $f$ and $P_3$ on $\left[-1\,1\right]$.

Find the actual value of the largest difference between $f$ and $P_3$ on $\left[-1\,1\right]$.

With $f$ and $P_3$ as in Example 3.3.1, obtain $R_3\left(x\right)$ and use it to estimate the largest difference between $f$ and $P_3$ on $\left[-1\,1\right]$. Hence, obtain the maximum value of $\left|R_4\right|$ = $\left|\frac{f^{\left(4\right)}\left(c\right)}{4\!}{x}^4\right|$, which in turn requires finding the maximal value of $\left|f^{\left(4\right)}\left(c\right)\right|$ for $-1\le c\le 1$.

Find the actual value of the largest difference between $f$ and $P_3$ on $\left[-1\,1\right]$.