Understanding Theorem 6.1: Exploring C^3

[irishetalon00]

Homework Statement

(See attachment: if it doesn't work, see below for poorer formatting)[/B]
Theorem 6.1 (Centered Formula of Order O(h2)). Assume that f ∈ C^3[a, b] and that x − h, x, x + h ∈ [a, b]. Then (3) f (x) ≈ f (x + h) − f (x − h) 2h . Furthermore, there exists a number c = c(x) ∈ [a, b] such that (4) f (x) = f (x + h) − f (x − h) 2h + Etrunc( f, h), where Etrunc( f, h) = −h2 f (3) (c) 6 = O(h2).

Homework Equations

While reading Theroem 6.1, I have difficulty understanding (realms? "∈") in general, so I was wondering what is meant by C^3

The Attempt at a Solution

 

[haruspex]

what is meant by C^3

https://en.m.wikipedia.org/wiki/Smoothness

 

[Mark44]

realms? "∈"

This symbol, ∈, means "is an element of"

 

[Ray Vickson]

Homework Statement

(See attachment: if it doesn't work, see below for poorer formatting)[/B]
Theorem 6.1 (Centered Formula of Order O(h2)). Assume that f ∈ C^3[a, b] and that x − h, x, x + h ∈ [a, b]. Then (3) f (x) ≈ f (x + h) − f (x − h) 2h . Furthermore, there exists a number c = c(x) ∈ [a, b] such that (4) f (x) = f (x + h) − f (x − h) 2h + Etrunc( f, h), where Etrunc( f, h) = −h2 f (3) (c) 6 = O(h2).

Homework Equations

While reading Theroem 6.1, I have difficulty understanding (realms? "∈") in general, so I was wondering what is meant by C^3

The Attempt at a Solution

What you wrote makes no sense, and is almost always wrong. You wrote f (x) ≈ f (x + h) − f (x − h) 2h, which is generally false. What IS true is that f'(x) ≈ [f(x+h) - f(x-h)]/(2h) for small |h|. Parentheses are important!

 

[irishetalon00]

Thank you for pointing me to relevant pages and explaining symbols.