Practice Stability in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
A system is stable at an equilibrium if small perturbations cause it to return toward that equilibrium; unstable if small perturbations cause it to move away.
A ball in a bowl returns to center; a ball on a hill rolls away.
Example 1
mediumFind all fixed points of f(x) = x^2 - x + 1 and determine their stability using the derivative criterion |f'(x^*)| < 1.
Example 2
hardFor the map g(x) = \cos(x), find the fixed point (Dottie number) approximately and determine its stability.
Example 3
easyClassify the fixed points of f(x) = 2x(1-x) as stable or unstable using the derivative criterion.
Example 4
mediumFor the map h(x) = \frac{1}{2}x + 4, find the fixed point and verify stability by iterating from x_0 = 0 and x_0 = 20 for three steps each.