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

Example 3

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

Example 4

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

Example 5

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

Example 10

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 11

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

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

Example 14

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

Example 15

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 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)+1x_{n+1} = 0.5 \sin(x_n) + 1 near x∗≈1.398x^* \approx 1.398.

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

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 20

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