Pass Check: Iteration & Trial and Improvement
Use a recurrence relation repeatedly to find increasingly accurate solutions to equations.
Iteration
Repeating a calculation using the output of one step as the input for the next.
Iterative formula
A rearranged equation of the form xₙ₊₁ = ... used to generate a sequence of approximations.
Converge
When repeated iterations settle towards a fixed value, which is the approximate root.
Root
A value of x where f(x) = 0, which iteration is used to estimate.
Quick check
Mixed quiz
1. Which formula describes 'add 3 to the previous term'?
2. Which condition suggests an iterative sequence is converging?
3. To find a root to 2 decimal places, why must you test the midpoint (e.g. 1.65) between two 1dp values?
4. The equation x³ + x - 5 = 0 can be rearranged to x = (5-x)^(1/3)... which formula below correctly rearranges x³ = 5 - x for iteration?
5. Which pair of values would you test next to narrow down a root known to be between 2 and 3?
6. An iteration formula gives x₃ = 3.162, x₄ = 3.162 (to 3 dp). What can you conclude?
7. For x³-x-3=0, if f(1.65)=0.106 (positive), what does this tell us about the root to 2dp, given f(1.5)<0?
8. For f(x) = x², why must the domain be restricted (e.g. x ≥ 0) to have an inverse?