1.1.1. Definition

Try these first

1. 1.

   What is a finite difference?

   Hint

   Think about how derivatives relate to changes in functions.

2. 2.

   How do you express a forward difference mathematically?

   Hint

   What happens when we increase x by a small step h?

3. 3.

   What does a finite difference represent?

   • The exact value of a function
   • The change in function value
   • The graphical representation of a function

   Hint

   Consider what happens when inputs vary slightly.

4. 4.

   Is the formula for a backward difference ∇f(x) = f(x) - f(x - h)?

   • True
   • False

   Hint

   Look at how we evaluate past function values.

5. 5.

   Given f(x) = sin(x), find the first and second forward differences at x = π/4 with a small increment h = 0.1.

   Hint

   Use calculator or table values to evaluate sin at the required points.

6. 6.

   Suppose you have the data values f(x) at x = 1, 2, 3, 4. Show how these can be used to create a difference table to find approximations for higher order derivatives.

   Hint

   Follow the systematic method for creating a difference table.