实物和抽象分析|Real and Abstract Analysis代写 MT4004

这是一份andrews圣安德鲁斯大学 MT4004作业代写的成功案例

实物和抽象分析|Real and Abstract Analysis代写 MT4004
问题 1.

$$
-\frac{f^{(N+1)}(\xi)}{N !}(x-\xi)^{N}=g^{\prime}(\xi)=\frac{g(x)-g(a)}{x-a} .
$$
Then, as $g(x)=0$,
$$
g(a)=\frac{f^{(N+1)}(\xi)}{N !}(x-\xi)^{N}(x-a)
$$


证明 .

The expression
$$
\sum_{n=0}^{N} \frac{f^{(n)}(a)}{n !}(x-a)^{n}
$$
is called the Taylor polynomial of degree $n$ at $a$, and
$$
f(x)-\sum_{n=0}^{N} \frac{f^{(n)}(a)}{n !}(x-a)^{n}
$$



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MT4004 COURSE NOTES :

Let $f$ be Riemann integrable over $I=[a, b]$, and define
$$
F(x)=\int_{a}^{x} f \quad(a \leq x \leq b) .
$$
Prove that $F$ is continuous on $I$. Prove also that if $x_{0} \in I$ and
$$
\lim {x \rightarrow x{0}, x \in I} f(x)=f\left(x_{0}\right),
$$
then
$$
\lim {x \rightarrow x{0}, x \in I} \frac{F(x)-F\left(x_{0}\right)}{x-x_{0}}=f\left(x_{0}\right)
$$





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