Mathematics - Application of Derivative Question with Solution | TestHub

If the function $f(x)$ is continuous and has continuous derivatives through order $n-1$ on the interval $[a,b]$ and has a finite derivative of the $n$ᵗʰ order at every interior point of the interval, then at $x\in [a,b]$,

$$f(x)=f(a)+f^{\prime }(a)(x-a)+\frac{f^{\prime \prime }(a)}{2!}(x-a{)}^2+\dots$$

$$+\frac{f^{(n-1)}(a)}{(n-1)!}(x-a{)}^{n-1}+\frac{f^{(n)}(\theta x)}{n!}(x-a{)}^n$$

where $\theta \in (0,1)$ is called TAYLOR’S FORMULA of the function $f(x)$.

If we set $a=0$, we obtain MACLAURIN’S FORMULA. The last term in both Taylor and Maclaurin's formulae is called the REMAINDER.

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