4.1/

The derivative

1. Derivative:
   
2. The forward/backward difference formula:
   

   Derivative formulat [at \(x = x_0\)]

   This formula is known as the forward-difference formula if
   and the backward-difference formula if

   Error Bound:

3. The \((n + 1)\)-point formula to approximate \(f'(x_j)\):
   
   • Derivation:
4. Three-point Formula:
   

   for each \(j = 0, 1, 2\), where the notation \(\zeta_j\) indicates that this point depends on \(x_j\).

   • Derivation:

Three-Point Formulas

1. Equally Spaced nodes:
   

   The formulas from Eq. (4.3) become especially useful if

   \(x_1 =\)
   \(x_2 =\)

2. Three-Point Endpoint Formula:
   

   The approximation in Eq. (4.4) is useful at

   Errors: the errors in both Eq. (4.4) and Eq. (4.5) are

3. Three-Point Midpoint Formula:
   

   Errors:

   This is because Eq. (4.5) uses data on \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\) and Eq. (4.4) uses data \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\)
   Note also that f needs to be evaluated at \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\) in Eq. (4.5), whereas in Eq. (4.4) it \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\)

Five-Point Formulas

1. What?
   
2. Why?
   

3. Error:
   The error term for these formulas is

4. Five-Point Midpoint Formula:
   

   Used for approximation at

5. Five-Point Endpoint Formula:
   

   Used for approximation at

   Left-endpoint approximations are found using this formula with \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\) and right-endpoint approximations with \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\).

   The five-point endpoint formula is particularly useful for \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\)

Approximating Higher Derivatives

1. Approximations to Second Derivatives:
   
2. Second Derivative Midpoint Formula:
   
   • Derivation:

   Error Bound: If \(f^{(4)}\) is \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\) on \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\) it is \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\) , and the approximation is \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\) .

Round-Off Error Instability

1. Form of Error:
   
   • We assume that our computations actually use the values \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\) \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\)
   • which are related to the true values \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\) \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\) by:

     \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\) &
     \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\) \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\)

2. The total error:
   

   It is due to \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\)

3. Error Bound:
   

   • ASSUMPTION: \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\) \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\)

   • ERROR:

4. Reducing Truncation Error:
   
   • How? \(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\) \(\ \ \ \ \ \ \ \ \ \ \ \ \ \\)
   • Effect of reducing \(h\):
5. Conclusion: