Question

7. State Taylor's theorem for a function f(x, y) of two variables and prove it by using Taylor's theorem for a single variable function.

Answer 1

Let k ≥ 1 be an integer , the function  f : R → R be k times differentiable at the point a ∈ R. Then there exists a function is h_k : R → R such that

$f(x)=f(a)+f'(a)(x-a)+{\frac {f''(a)}{2!}}(x-a)^{2}+\cdots +{\frac {f^{(k)}(a)}{k!}}(x-a)^{k}+h_{k}(x)(x-a)^{k},$

${\displaystyle {\mbox{and}}\quad \lim _{x\to a}h_{k}(x)=0}$ . This is called the Peano form of the remainder.

The polynomial appearing in Taylor's theorem is the k-th order Taylor polynomial

$P_{k}(x)=f(a)+f'(a)(x-a)+{\frac {f''(a)}{2!}}(x-a)^{2}+\cdots +{\frac {f^{(k)}(a)}{k!}}(x-a)^{k}$

of the function f at the point a. The Taylor polynomial is the unique "asymptotic best fit" polynomial in the sense that if there exists a function h_k : R → R and a k-th order polynomial p  such that $f(x)=p(x)+h_{k}(x)(x-a)^{k},\quad \lim _{x\to a}h_{k}(x)=0,$

then p = P_k. Taylor's theorem describes the asymptotic behavior of the remainder term

which is the approximation error when approximating f with its Taylor polynomial. Using the little-o notation, the statement in Taylor's theorem reads as

$R_{k}(x)=o(|x-a|^{k}),\quad x\to a.$