Feedback Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Feedback.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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...

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Positive feedback creates exponential growth or runaway behavior; negative feedback creates equilibrium and oscillation. Most stable systems rely on negative feedback.

Common stuck point: "Positive feedback" does not mean "good feedback" — it means the feedback reinforces change, which can be destabilizing.

Sense of Study hint: Trace the loop: write down what the output is, then ask how that output changes the next input. Repeat for 2-3 cycles to see the pattern.

Worked Examples

Example 1

medium
Iterate 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.

Solution

  1. 1
    x_1 = 0.5(10)+3 = 8; x_2 = 0.5(8)+3 = 7; x_3 = 0.5(7)+3 = 6.5.
  2. 2
    The sequence appears to converge. Fixed point: solve x^* = 0.5x^*+3 \Rightarrow 0.5x^*=3 \Rightarrow x^*=6.
  3. 3
    Since |f'(x)|=|0.5|<1 at the fixed point, x^*=6 is stable. The sequence converges to 6.

Answer

x_1=8, x_2=7, x_3=6.5; long-run value x^*=6
A feedback map repeatedly applies a function to its previous output. If the map has a stable fixed point, iteration converges there regardless of starting value (within the basin of attraction).

Example 2

hard
Analyze 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.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Iterate 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 2

medium
Newton'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.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

exponential function