Eritque Arcus Math Notes

differentiable

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Def 31.3

𝐹 is differentiable at π‘₯ if

(βˆƒπ‘Ž)(βˆ€πœ€>0)(βˆƒπ›Ώ)|π‘₯βˆ’π‘¦|<π›ΏβŸΉ|𝐹(𝑦)βˆ’πΉ(π‘₯)π‘¦βˆ’π‘₯βˆ’π‘Ž|<πœ€

where we can say π‘Ž=𝐹′(π‘₯)

Thm 31.4

differentiable 𝑓 on (π‘Ž,𝑏) and 𝑓′(π‘₯) bounded ⟹ 𝑓 is Lipschitz

Thm 32.1 Fundamental Theorem

  1. ∫π‘₯π‘Žπ‘”β€²(𝑑)𝑑𝑑=𝑔(π‘₯)βˆ’π‘
  2. if 𝐹′ is integrable, 𝐹(𝑏)βˆ’πΉ(π‘Ž)=βˆ«π‘π‘ŽπΉβ€²(𝑑)𝑑𝑑

Thm 32.2 Mean Value Theorem(MVT)

let 𝑓 be continue on [π‘Ž,𝑏] and 𝑓 is differentiable on (π‘Ž,𝑏)

(βˆƒπœ‰βˆˆ(π‘Ž,𝑏))𝑓(𝑏)βˆ’π‘“(π‘Ž)π‘βˆ’π‘Ž=𝑓′(πœ‰)

Rolle’s Lemma

Let 𝑓 be continue on [π‘Ž,𝑏] and differentiable on (π‘Ž,𝑏)

𝑓(π‘Ž)=𝑓(𝑏)⟹(βˆƒπœ‰)𝑓′(πœ‰)=0

Rules

  1. (𝑓+πœ†π‘”)β€²=𝑓′+πœ†π‘”β€²
  2. (π‘“βš¬π‘”)β€²=(π‘“β€²βš¬π‘”)𝑔′
  3. (𝑓𝑔)β€²=𝑓′𝑔+𝑓𝑔′
  4. π‘“βˆˆ[π‘Ž,𝑏] is strict monotone, π‘“βˆ’1 is continue

Thm 40.1 Chain rule

(π‘“βš¬π‘”)β€²=𝑓′(𝑔(π‘₯))𝑔′(π‘₯)

Prop 40.1

Let 𝑔:β„βŸΆβ„

π‘‘π‘‘π‘¦βˆ«π‘π‘Žπ‘”(𝑦,𝑑)𝑑𝑑=βˆ«π‘π‘Žπ‘‘π‘‘π‘¦π‘”(𝑦,𝑑)𝑑𝑑

Thm 40.2 differential

𝐻′(π‘Ž)=lim𝑛𝐻(π‘Ž+πœ‰π‘›)βˆ’π»(π‘Ž)πœ‰π‘›

Thm

Inverse of differentiable function is differentiable.

Thm

𝑓(π‘₯)βˆ’π‘“(π‘₯0)𝑔(π‘₯)βˆ’π‘”(π‘₯0)=𝑓′(πœ‰)𝑔′(πœ‰)

π‘₯0<πœ‰<π‘₯

Thm

β„Žβ€² is continue ⟺ β„Ž is continue and differentiable.

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