Feedback Examples: 44 Problems with Answers

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: 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

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.

Answer

x_1=8, x_2=7, x_3=6.5

; long-run value

x^*=6

First step

x_1 = 0.5(10)+3 = 8

;

x_2 = 0.5(8)+3 = 7

;

x_3 = 0.5(7)+3 = 6.5

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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.

Example 3

medium

Find the fixed point of

x_{n+1} = 0.6 x_n + 4

Example 4

medium

For the doubling map

x_{n+1} = 2 x_n

, what happens to any starting value

|x_0| > 0

Example 5

medium

A bank account grows by

1\%

per month and you withdraw $100 monthly. Write the iteration for the balance

B_{n+1}

and find its fixed point.

Example 6

hard

For the tent map

f(x) = 2x

[0, 1/2]

and

f(x) = 2 - 2x

[1/2, 1]

, find

f(f(0.3))

Example 7

hard

Use Newton's method

x_{n+1} = x_n - \dfrac{x_n^2 - 3}{2x_n}

to approximate

\sqrt{3}

starting at

x_0 = 2

. Compute

x_1

and

x_2

Example 8

hard

The logistic recursion

x_{n+1} = r x_n (1 - x_n)

has a nonzero fixed point. Find it as a function of

r

Example 9

challenge

For the logistic map

x_{n+1} = 3.2 x_n (1 - x_n)

, find a period-2 cycle by setting

f(f(x)) = x

and reducing to a quadratic in

x

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

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

P_{n+1} = P_n + 0.1 P_n

. Starting at

P_0 = 100

, find

P_2

and state whether the feedback is positive or negative.

Example 12

medium

A cooling object follows

T_{n+1} = T_n - 0.5(T_n - 20)

with

T_0 = 100

. Find

T_1

and

T_2

, and identify the equilibrium temperature.

Example 13

medium

In the loop

A

raises

B

B

raises

C

, and

C

lowers

A

, trace one full loop starting from an increase in

A

. Is the net feedback on

A

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 \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

S_{n+1} = 0.9 S_n + 50

. Find the equilibrium sales level.

Example 18

challenge

A feedback loop is

x_{n+1} = k\, x_n

. For which values of

k

does the loop's output stay bounded for every starting value, and what is the feedback behavior when

k = -1.5

Example 19

challenge

Find

c

so that the loop

x_{n+1} = x_n + c(10 - x_n)

reaches the equilibrium

x = 10

in exactly one step from any start, then state what happens if

c = 2

Example 20

challenge

In the loop

A \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:

v_{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

x_{n+1} = x_n + 2

from

x_0 = 1

. Find

x_3

Example 24

easy

Iterate

x_{n+1} = 3 x_n

from

x_0 = 2

. Find

x_2

Example 25

easy

Iterate

x_{n+1} = x_n - 5

from

x_0 = 20

. Find

x_4

Example 26

easy

Iterate

x_{n+1} = \tfrac{1}{2} x_n

from

x_0 = 16

. Find

x_3

Example 27

easy

Find the fixed point of

x_{n+1} = x_n + 0

Example 28

medium

For

x_{n+1} = 0.8 x_n + 6

with

x_0 = 0

, find

x_1, x_2, x_3

and the long-run value.

Example 29

medium

Find both fixed points of

x_{n+1} = x_n^2

on the real line.

Example 30

medium

Iterate

x_{n+1} = \cos(x_n)

from

x_0 = 1

. Compute

x_1, x_2, x_3

to four decimals.

Example 31

medium

For

x_{n+1} = 2 x_n - 6

, find the fixed point and decide if it is stable.

Example 32

medium

For

x_{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

x_{n+1} = x_n^2 - 2

Example 34

hard

Classify the stability of the fixed point of

x_{n+1} = 0.5 \sin(x_n) + 1

near

x^* \approx 1.398

Example 35

hard

A debt grows at

5\%

per year and you pay

\$2000

per year. Write the recursion and find the equilibrium debt.

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Related Concepts

Exponential Function Introduction to Differential Equations

Background Knowledge

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

exponential function