1.1/

Limits and Continuity

1. Limit [of Function]:
2. Continuity:
3. Limit [of Sequence]:
4. Convergence and Continuity, Correspondance:

Differentiability

1. Differentiablity:
2. Differentiablity and Continuity, Correspondance:
3. Rolle’s Theorem:
4. Generalized Rolle’s Theorem:
5. Mean Value Theorem:
   • Proof.
6. Extreme Value Theorem:
7. Intermediate Value Theorem:

Integration

1. The Riemann Integral:
   • Or, for equally spaced intervals:
2. Integrability and Continuity, Correspondance:
3. Weighted Mean Value Theorem for Integrals:

   • Implications:
     

   • When \(g(x) ≡ 1\), Theorem 1.13 is:
     

   • It gives:
     

Taylor Polynomials and Series

1. Taylor’s Theorem:

   • \(P_n(x)\): is called the \(\ \ \ \ \ \ \ \ \ \ \ \ \\) for \(\ \ \ \ \\) about \(\ \ \ \ \ \ \\).

   • \(R_n(x)\): is called the \(\ \ \ \ \ \ \ \ \ \ \ \ \\) (or \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\)) associated with \(P_n(x)\).

   • Since the number \(ξ(x)\) in the truncation error \(R_n(x)\) \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\), it is a function of \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\)

   • Taylor’s Theorem, only, ensures that \(\ \ \ \ \ \ \ \ \ \ \ \ \\)\(\ \ \ \ \ \ \ \ \ \ \ \ \\), and that its value \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\), and not \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\)

2. Polynomials:
   • Taylor’s Polynomial: The polynomial definied by
   • Maclaurin Polynomial:
3. Series:

   • Taylor’s Series:
     

   • Maclaurin Series:
     

4. Truncation Error:
   • Refers to: