Examples from Taylor's theorem

Approximating $e^x$ by a quadratic

Take

for $f(x)=e^x$ about zero. The third derivative is $e^x$. On $[-1/2,1/2]$, an intermediate point satisfies $e^c\le e^{1/2}$, so

The bound holds for every $x$ in the interval.

Computing $e$ to within $10^{-6}$

At $x=1$, the Maclaurin polynomial gives

For $0<c<1$, the Lagrange remainder satisfies $e^c<3$, hence

With $n=9$,

Therefore

with the requested accuracy. The bound guarantees the accuracy; the actual error is about $3.03\times10^{-7}$.

Error for a sine approximation

Every derivative of $\sin x$ is one of $\pm\sin x$ or $\pm\cos x$, so its absolute value is at most $1$. For a degree-$n$ Taylor polynomial about $a$,

Near the expansion centre, the factorial makes the bound decrease rapidly.

A fifth-order stationary point

Let

At $x=2$,

The first nonzero derivative has odd order, so $x=2$ is a horizontal point of inflection. Since $f(2)=-10$, the point is $(2,-10)$.