\(\def\Real{\mathbb{R}}\)

#### Course outline is here.

#### Taylor formula

**Spaces of differentiable functions** on an interval \(I=[a,b]\subset\Real\):

\[

C(I,\Real), C^1(I,\Real),\ldots, C^n(I,\Real),\ldots

\]

For a function in \(C^1(I,\Real)\),

\[

f(y)=f(x)+\int_x^yf'(s)ds.

\]

Iterating (for functions in \(C^n(I,\Real)\), we obtain

\[

f(y)=f(x)+f'(x)(y-x)+\frac{f”(x)}{2!}(y-x)^2+\ldots+\frac{f^{(n-1)}(x)}{(n-1)!}(y-x)^{n-1}+

\int_{x<s_1<s_2<\ldots<s_n<y}f^{(n)}(s_1)\frac{(y-s_1)^{n-1}}{(n-1)!}ds.

\]

**Implications**:

- Mean value theorem
- If \(f”\geq 0\) on \(I\) and \(f'(x)=0\), then \(x\) is a global minimum of \(f\).

#### Several variables

Domain: \(U\subset\Real^n\), an open set.

**Reminder**: open, closed, bounded, compact sets.

**Exercise**: is the set \(\{|x|\leq 1, |y| \lt 1\}\) open?…