Summary
Numerical methods are ways of calculating approximate solutions to equations. A good starting point is crucial to find an approximate solution to an equation.
- Graphical Approach — Using graphs to determine the number of real roots of an equation. Example: By sketching graphs of functions, determine the number of real roots by observing intersections.
- Change of Sign Approach — If a function is continuous in an interval and the function values at the endpoints have different signs, there is at least one root in that interval. Example: Evaluate function values at endpoints to show a root exists between them.
- Iterative Formula — A method to improve the approximation of a root by using a sequence of approximations. Example: Rearrange an equation to form x = F(x) and use an initial value to find a solution iteratively.
Exam Tips
Key Definitions to Remember
- Graphical Approach
- Change of Sign Approach
- Iterative Formula
Common Confusions
- Misinterpreting the graphical intersections as roots
- Forgetting to check for continuity in the change of sign approach
Typical Exam Questions
- How do you find the number of real roots using a graph? Sketch the graph and count intersections.
- How can you show a root exists between two points? Evaluate function values at the points and check for a sign change.
- What is the iterative process for finding a root? Use an initial value and apply the iterative formula repeatedly.
What Examiners Usually Test
- Ability to locate roots using graphical and sign change methods
- Understanding and application of iterative formulas to find roots