- #1
mastrofoffi
- 51
- 12
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.
##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.