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...

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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: In feedback, what a system puts out comes back as part of what it takes in next, either amplifying or damping the change.

Common stuck point: The procedure for feedback is the easy part; the trap is swapping positive and negative feedback. Asking "Does the system's output get fed back in as the input for the next step?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does the system's output get fed back in as the input for the next step?

Worked Examples

Example 1

medium
Iterate the map xn+1=0.5xn+3x_{n+1} = 0.5x_n + 3 starting from x0=10x_0 = 10. Compute x1,x2,x3x_1, x_2, x_3 and predict the long-run value.

Answer

x1=8,x2=7,x3=6.5x_1=8, x_2=7, x_3=6.5; long-run value x∗=6x^*=6

First step

1
x1=0.5(10)+3=8x_1 = 0.5(10)+3 = 8; x2=0.5(8)+3=7x_2 = 0.5(8)+3 = 7; x3=0.5(7)+3=6.5x_3 = 0.5(7)+3 = 6.5.

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Example 2

hard
Analyze the logistic map xn+1=3.5xn(1−xn)x_{n+1} = 3.5 x_n(1-x_n) by iterating from x0=0.5x_0=0.5 for five steps and commenting on the behavior.

Example 3

medium
Find the fixed point of xn+1=0.6xn+4x_{n+1} = 0.6 x_n + 4.

Example 4

medium
For the doubling map xn+1=2xnx_{n+1} = 2 x_n, what happens to any starting value ∣x0∣>0|x_0| > 0?

Example 5

medium
A bank account grows by 1%1\% per month and you withdraw $100 monthly. Write the iteration for the balance Bn+1B_{n+1} and find its fixed point.

Example 6

hard
For the tent map f(x)=2xf(x) = 2x on [0,1/2][0, 1/2] and f(x)=2−2xf(x) = 2 - 2x on [1/2,1][1/2, 1], find f(f(0.3))f(f(0.3)).

Example 7

hard
Use Newton's method xn+1=xn−xn2−32xnx_{n+1} = x_n - \dfrac{x_n^2 - 3}{2x_n} to approximate 3\sqrt{3} starting at x0=2x_0 = 2. Compute x1x_1 and x2x_2.

Example 8

hard
The logistic recursion xn+1=rxn(1−xn)x_{n+1} = r x_n (1 - x_n) has a nonzero fixed point. Find it as a function of rr.

Example 9

challenge
For the logistic map xn+1=3.2xn(1−xn)x_{n+1} = 3.2 x_n (1 - x_n), find a period-2 cycle by setting f(f(x))=xf(f(x)) = x and reducing to a quadratic in xx.

Practice Problems

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

Example 1

easy
Iterate xn+1=xn2x_{n+1} = x_n^2 starting from x0=0.5x_0 = 0.5. Compute x1,x2,x3x_1, x_2, x_3 and determine the long-run behavior.

Example 2

medium
Newton's method for finding 2\sqrt{2} uses xn+1=12(xn+2xn)x_{n+1} = \frac{1}{2}\left(x_n + \frac{2}{x_n}\right). Starting from x0=1x_0=1, compute x1,x2,x3x_1, x_2, x_3 and compare to 2≈1.41421\sqrt{2}\approx1.41421.

Example 3

easy
A microphone picks up sound from a speaker, which then plays that sound louder, which the microphone picks up again. Each loop the output is added back to the input. Is this positive or negative feedback?

Example 4

easy
A thermostat turns the heater OFF when the room gets too hot and ON when it gets too cold, pushing the temperature back toward the target. Is this positive or negative feedback?

Example 5

easy
Is the relationship 'I flip a switch and a light turns on' an example of feedback?

Example 6

easy
A savings account earns interest, and that interest is added to the balance, which then earns more interest. The growing balance increases future interest. Positive or negative feedback?

Example 7

easy
In a predator-prey system, more rabbits feed more foxes, but more foxes eat more rabbits, reducing the rabbit count. Does the fox population's growth push the rabbit population in the same or opposite direction?

Example 8

easy
True or false: positive feedback always causes a quantity to grow without bound.

Example 9

easy
A car's cruise control senses speed and increases throttle when the car slows on a hill, returning speed to the set point. Positive or negative feedback?

Example 10

easy
Which type of feedback tends to CREATE a stable equilibrium: positive or negative?

Example 11

medium
A population model is Pn+1=Pn+0.1PnP_{n+1} = P_n + 0.1 P_n. Starting at P0=100P_0 = 100, find P2P_2 and state whether the feedback is positive or negative.

Example 12

medium
A cooling object follows Tn+1=Tn−0.5(Tn−20)T_{n+1} = T_n - 0.5(T_n - 20) with T0=100T_0 = 100. Find T1T_1 and T2T_2, and identify the equilibrium temperature.

Example 13

medium
In the loop AA raises BB, BB raises CC, and CC lowers AA, trace one full loop starting from an increase in AA. Is the net feedback on AA positive or negative?

Example 14

medium
A rumor spreads: each informed person tells others, increasing the informed count, which increases the rate of new tellings. Early on, is this positive or negative feedback, and what shape does the early growth curve have?

Example 15

medium
A scale auto-corrects an overload of 30 g, removing 80% of the remaining error each step. After step 1 the residual error is 0.2⋅30=60.2 \cdot 30 = 6 g. What is the error after the second correction?

Example 16

medium
Two mirrors face each other and an image reflects back and forth. Classify the feedback, and explain why a real version does not produce infinite brightness.

Example 17

medium
A model gives next-year sales as Sn+1=0.9Sn+50S_{n+1} = 0.9 S_n + 50. Find the equilibrium sales level.

Example 18

challenge
A feedback loop is xn+1=k xnx_{n+1} = k\, x_n. For which values of kk does the loop's output stay bounded for every starting value, and what is the feedback behavior when k=−1.5k = -1.5?

Example 19

challenge
Find cc so that the loop xn+1=xn+c(10−xn)x_{n+1} = x_n + c(10 - x_n) reaches the equilibrium x=10x = 10 in exactly one step from any start, then state what happens if c=2c = 2.

Example 20

challenge
In the loop A→B→C→AA \to B \to C \to A the link signs are ++, −-, and −-. Determine the net feedback sign, and explain why an even number of negative links makes a loop positive.

Example 21

medium
A bathtub fills from a tap and drains through a hole; the higher the water, the faster it drains. Does this drain mechanism act as positive or negative feedback on the water level?

Example 22

medium
A spreadsheet cell adds 10% of its own previous value each step plus a constant 20: vn+1=1.1vn+20v_{n+1} = 1.1 v_n + 20. Is the self-term feedback positive or negative, and does the value grow without bound?

Example 23

easy
Iterate xn+1=xn+2x_{n+1} = x_n + 2 from x0=1x_0 = 1. Find x3x_3.

Example 24

easy
Iterate xn+1=3xnx_{n+1} = 3 x_n from x0=2x_0 = 2. Find x2x_2.

Example 25

easy
Iterate xn+1=xn−5x_{n+1} = x_n - 5 from x0=20x_0 = 20. Find x4x_4.

Example 26

easy
Iterate xn+1=12xnx_{n+1} = \tfrac{1}{2} x_n from x0=16x_0 = 16. Find x3x_3.

Example 27

easy
Find the fixed point of xn+1=xn+0x_{n+1} = x_n + 0.

Example 28

medium
For xn+1=0.8xn+6x_{n+1} = 0.8 x_n + 6 with x0=0x_0 = 0, find x1,x2,x3x_1, x_2, x_3 and the long-run value.

Example 29

medium
Find both fixed points of xn+1=xn2x_{n+1} = x_n^2 on the real line.

Example 30

medium
Iterate xn+1=cos⁡(xn)x_{n+1} = \cos(x_n) from x0=1x_0 = 1. Compute x1,x2,x3x_1, x_2, x_3 to four decimals.

Example 31

medium
For xn+1=2xn−6x_{n+1} = 2 x_n - 6, find the fixed point and decide if it is stable.

Example 32

medium
For xn+1=−0.5xn+3x_{n+1} = -0.5 x_n + 3, find the fixed point and the long-run behavior.

Example 33

hard
Find all real fixed points of xn+1=xn2−2x_{n+1} = x_n^2 - 2.

Example 34

hard
Classify the stability of the fixed point of xn+1=0.5sin⁡(xn)+1x_{n+1} = 0.5 \sin(x_n) + 1 near x∗≈1.398x^* \approx 1.398.

Example 35

hard
A debt grows at 5%5\% per year and you pay $2000\$2000 per year. Write the recursion and find the equilibrium debt.
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Background Knowledge

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

exponential function