Derivation of Taylor Series in R^n

[mastrofoffi]

I was studying the derivation for taylor series in ℝ$^n$ on my book and I have some trouble understanding a passage; it's the very beginning actually:

$f : A$ ⊆ ℝ$^n$ → ℝ
$f$ ∈ $C^2(A)$
$x_0$ ∈ $A$

"be $g_{(t)} = f_{(x_0 + vt)}$ where v is a generic versor, then we have:
$g_{(t)} = g_{(0)} + g'_{(0)}t + \frac{1}{2}g''_{(\tau)}t^2$ where $\tau ∈ [0, t]$"

I don't understand why is it ok to do taylor expansion centered in t=0 and then use $g''_{(\tau)}$ instead of $g''_{(0)}$?
I'm actually fine with the rest of the demonstration which is quite easy but I'd like to understand what he's doing here; I was pretty sure it could be something coming from Lagrange's theorem(he uses it everywhere) but I can't really see it here.

 

[Ray Vickson]

I was studying the derivation for taylor series in ℝ$^n$ on my book and I have some trouble understanding a passage; it's the very beginning actually:

$f : A$ ⊆ ℝ$^n$ → ℝ
$f$ ∈ $C^2(A)$
$x_0$ ∈ $A$

"be $g_{(t)} = f_{(x_0 + vt)}$ where v is a generic versor, then we have:
$g_{(t)} = g_{(0)} + g'_{(0)}t + \frac{1}{2}g''_{(\tau)}t^2$ where $\tau ∈ [0, t]$"

I don't understand why is it ok to do taylor expansion centered in t=0 and then use $g''_{(\tau)}$ instead of $g''_{(0)}$?
I'm actually fine with the rest of the demonstration which is quite easy but I'd like to understand what he's doing here; I was pretty sure it could be something coming from Lagrange's theorem(he uses it everywhere) but I can't really see it here.

The Taylor expansion of first order with remainder for a univariate function is
$$g(t) = g(0) + t g'(0) + \frac{t^2}{2!} g''(\tau),$$
where $\tau$ is a value between $0$ and $t$.

In general, if $g \in C^{n+1}$ we have
$$g(t) = g(0) + t g'(0) + \frac{t^2}{2!} g''(0) + \cdots + \frac{t^n}{n!} g^{(n)}(0) + \frac{t^{n+1}}{(n+1)!} g^{(n+1)}(\tau).$$

See your textbook, or look on-line for "Taylor series with remainder".

 

[mastrofoffi]

Oh, so it is the remainder in lagrange form for the 1st order expansion.
Its so obvious now that i see it! I can't believe i got stuck on this ahah thank you ^^