Practice Feedback in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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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...

Showing a random 20 of 50 problems.

Example 1

easy
Iterate xn+1=xn2 starting from x0=0.5. Compute x1,x2,x3 and determine the long-run behavior.

Example 2

medium
Find the fixed point of xn+1=0.6xn+4.

Example 3

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

Example 4

easy
Iterate xn+1=xn+2 from x0=1. Find x3.

Example 5

medium
Newton's method for finding 2 uses xn+1=12(xn+2xn). Starting from x0=1, compute x1,x2,x3 and compare to 2≈1.41421.

Example 6

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 7

easy
A signal that amplifies its own value each round is what kind of feedback?

Example 8

easy
A thermostat that turns off heat as the room warms uses what feedback?

Example 9

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

Example 10

challenge
In the loop A→B→C→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 11

easy
Iterate xn+1=12xn from x0=16. Find x3.

Example 12

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 13

medium
Iterate xn+1=cos⁡(xn) from x0=1. Compute x1,x2,x3 to four decimals.

Example 14

hard
Find all real fixed points of xn+1=xn2−2.

Example 15

hard
Use Newton's method xn+1=xn−xn2−32xn to approximate 3 starting at x0=2. Compute x1 and x2.

Example 16

medium
A predator-prey loop where more predators reduce prey, which in turn reduces predators, is what kind of feedback?

Example 17

hard
Classify the stability of the fixed point of xn+1=0.5sin⁡(xn)+1 near x∗≈1.398.

Example 18

challenge
A feedback loop is xn+1=k xn. 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

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

Example 20

easy
Iterate xn+1=3xn from x0=2. Find x2.
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