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:

1.2/

Binary Machine Numbers

1. Representing Real Numbers:

   A \(\ \ \ \ \ \ \ \ \ \ \ \ \\) (binary digit) representation is used for a real number.

   • The first bit is
   • Followed by:
   • and a \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\), called the
   • The base for the exponent is

2. Floating-Point Number Form:
3. Smallest Normalized positive Number:
   • When:
   • Equivalent to:
4. Largest Normalized positive Number:
   
   • When:
   • Equivalent to:
5. UnderFlow:
   • When numbers occurring in calculations have
6. OverFlow:
   • When numbers occurring in calculations have
7. Representing the Zero:
   
   • There are \(\ \ \ \ \\) Representations of the number zero:

Decimal Machine Numbers

1. What?
2. (k-digit) Decimal Machine Numbers:
3. Normalized Form:
4. Floating-Point Form of a Decimal Machine Number:
   
   • The floating-point form of y, denoted \(f_l(y)\), is obtained by:
5. Termination:
   There are two common ways of performing this termination:
   1. \(\ \ \ \ \ \ \ \ \ \\):
      

      This produces the floating-point form:

   2. \(\ \ \ \ \ \ \ \ \ \\): \(\ \\) which
      

      This produces the floating-point form:

      For rounding, when \(d_{k+1} \geq 5\), we
      When \(d_{k+1} < 5\), we
      If we round down, then \(\delta_i =\)
      However, if we round up,

6. Approximation Errors:
   

   • The Absolute Error: \(\ \ \ \ \ \ \\).

   • The Relative Error: \(\ \ \ \ \ \ \\).

7. Significant Digits:
   
8. Error in using Floating-Point Repr.:
   

   • Chopping:

     The Relative Error =

     The Machine Repr. [for k decimial digits] =

     \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\).

     \(\implies\)

     Bound \(\ \ \ \implies \ \ \ \ \ \ \ \ \ \ \ \\).

   • Rounding:

     In a similar manner, a bound for the relative error when using k-digit rounding arithmetic is

     Bound \(\ \ \ \implies \ \ \ \ \ \ \ \ \ \ \ \ \ \\).

9. Distribution of Numbers:
   The number of decimal machine numbers in \(\ \ \ \ \ \ \ \ \ \ \\) is \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\) for

Finite-Digit Arithmetic

1. Values:

   \[x = \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\]

   \[y = \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\]

2. Operations:
   
3. Error-producing Calculations:
   
   • First:
   • Second:
4. Avoiding Round-Off Error:
   
   • First:
   • Second:

     \(\implies \ \ \ \ \ \ \ \ \ \ \ \ \ \ x_1 = \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , \ \ \ \ \ \ \ \ \ \ \ \ \ \ x_2 = \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\)

Nested Arithmetic

1. What?

   Remember that chopping (or rounding) is performed:

   
   

   • \[\ \ \ \ \ \ \\]

   
   

   Polynomials should always be expressed \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\) , becasue, \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\)

2. Why?

1.3/

Main Idea

1. Algorithm:

Characterizing Algorithms

1. Stability:
   
   • Stable Algorithm:
     
     
   • Conditionally Stable Algorithm:
     
     
2. Error Growth:
   
3. Stability and Error-Growth:
   
   • Stable Algorithm:
   • UnStable Algorithm:

Rates of Convergence

1. Rate of Convergence:
   

   \(\beta_n \ = \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , \ \ \ \\) for

2. Big-Oh Notation: