3.6. Worked Example

Try these first

1. 1.

   What does h represent in the context of finding a derivative using the central difference formula?

   Hint

   Think about how we calculate the difference between points.

2. 2.

   Given f(2) = 4 and f(2.2) = 5, what is the approximate value of f'(2)?

   Hint

   Use the central difference formula.

3. 3.

   What is the central difference formula used for?

   • Estimating integrals
   • Estimating derivatives
   • Solving equations

   Hint

   Recall its definition from our discussions.

4. 4.

   True or False: The step size (h) affects the accuracy of numerical differentiation.

   • True
   • False

   Hint

   Think about the implications of our previous example.

5. 5.

   Consider a function given by the points x = [0, 1, 2, 3] and f(x) = [0, 1, 4, 9]. Use numerical differentiation to calculate the approximate value of f'(2). What do the results tell you about the behavior of the function?

   Hint

   Consider the average slope between the points to determine the derivative at x = 2.

6. 6.

   Evaluate the error in your numerical differentiation for the given function values. If h = 0.5 and you find f'(2) using central difference, how would you verify the accuracy?

   Hint

   Remember to analyze the difference between the calculated derivative and the true derivative.