3.4. Central Difference Formulas

Try these first

1. 1.

   What is the central difference formula for the first derivative?

   Hint

   Think about how we approximate using points around x.

2. 2.

   What data points are needed for the second derivative approximation?

   Hint

   Remember you need data from both sides.

3. 3.

   What does the central difference formula use for its calculation?

   • Only the point of interest
   • Points on both sides
   • Only previous points

   Hint

   Think about the formula used for estimation.

4. 4.

   True or False: The central difference formula is less accurate than the forward difference formula.

   • True
   • False

   Hint

   Consider how more data points influence accuracy.

5. 5.

   Given the following data points: x = [1, 1.2, 1.4, 1.6, 1.8] and corresponding f(x) values = [0, 0.128, 0.544, 1.296, 2.432], apply the central difference method to estimate f′(1.6) and f′(1.4).

   Hint

   Use the right data points and don’t forget the step-size in your calculations.

6. 6.

   Analyze the effect on the accuracy of the resulting derivative estimation if the spacing (h) between points is halved. Discuss how it relates to truncation and round-off errors.

   Hint

   Consider both types of errors when adjusting h.