1.1.4. Properties of Finite Differences

Try these first

1. 1.

   What is a finite difference?

   Hint

   Recall the definition.

2. 2.

   State the polynomial property of finite differences.

   Hint

   Think about the behavior of polynomials.

3. 3.

   What is the result of the (n+1)th finite difference of a polynomial of degree n?

   • Zero
   • The polynomial itself
   • A constant

   Hint

   Think about the definition and properties of polynomials.

4. 4.

   True or False: The linearity property states that Δ(af(x) + bg(x)) = aΔf(x) + bΔg(x).

   • True
   • False

   Hint

   Recall the properties discussed.

5. 5.

   Calculate the finite difference of the function f(x) = x^4 at x = 1 using a step size of h = 1.

   Hint

   Make sure to substitute in the values of the function.

6. 6.

   Using the property of finite differences, explain what happens when you increase the degree of a polynomial beyond n.

   Hint

   Consider how polynomial nature influences finite differences.