Practice Feedback in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
Feedback occurs when the output of a system influences its future input — positive feedback amplifies changes; negative feedback stabilizes them.
Microphone feedback: sound → speaker → microphone → more sound → louder...
Example 1
mediumIterate the map x_{n+1} = 0.5x_n + 3 starting from x_0 = 10. Compute x_1, x_2, x_3 and predict the long-run value.
Example 2
hardAnalyze the logistic map x_{n+1} = 3.5 x_n(1-x_n) by iterating from x_0=0.5 for five steps and commenting on the behavior.
Example 3
easyIterate x_{n+1} = x_n^2 starting from x_0 = 0.5. Compute x_1, x_2, x_3 and determine the long-run behavior.
Example 4
mediumNewton's method for finding \sqrt{2} uses x_{n+1} = \frac{1}{2}\left(x_n + \frac{2}{x_n}\right). Starting from x_0=1, compute x_1, x_2, x_3 and compare to \sqrt{2}\approx1.41421.