Practice Saturation in Math

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

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Quick Recap

Saturation is the phenomenon where a growing quantity approaches a limiting value asymptotically, with the rate of growth decreasing as the limit is approached.

Room fills until no more people fit. Growth can't continue forever.

Showing a random 20 of 50 problems.

Example 1

easy
Filling a room until no more people fit illustrates what behavior?

Example 2

easy
In logistic growth, what name is given to the limiting population?

Example 3

medium
For P(t)=5001+4e−0.5tP(t) = \dfrac{500}{1+4e^{-0.5 t}}, find P(0)P(0) and the carrying capacity.

Example 4

medium
A model is P(t)=1001+e−tP(t)=\dfrac{100}{1+e^{-t}}. What is its limit as t→∞t\to\infty?

Example 5

easy
Does a saturating curve ever exceed its limit?

Example 6

challenge
Two saturation curves both cap at 100100; curve A reaches 9090 at t=5t=5, curve B at t=20t=20. Which saturates faster, and what does that say about their early rates?

Example 7

easy
A drug concentration in the bloodstream rises toward a steady level and then changes very slowly. This behavior is called ___.

Example 8

easy
A logistic model levels off near 500500. What is its limiting value?

Example 9

easy
A lake can support at most 50005000 fish. In a logistic model, what does L=5000L = 5000 represent?

Example 10

hard
Which differential equation produces logistic saturation: dPdt=kP\dfrac{dP}{dt} = kP or dPdt=kP(1−P/L)\dfrac{dP}{dt} = kP(1 - P/L)?

Example 11

easy
True or false: in a saturating curve, the function's value can exceed its asymptotic limit briefly.

Example 12

medium
What kind of horizontal asymptote does a saturating function have on the right side: rising-toward, falling-toward, or oscillating-around?

Example 13

medium
For P(t)=1001+e−tP(t)=\dfrac{100}{1+e^{-t}}, find P(0)P(0).

Example 14

challenge
A logistic model P(t)=L1+Ae−ktP(t) = \dfrac{L}{1 + A e^{-kt}} has P(0)=100P(0) = 100, P(5)=300P(5) = 300, and L=1000L = 1000. Find kk.

Example 15

medium
A bacteria culture saturates at 10610^6 cells. At time t=0t=0 there are 10410^4 cells. Write a logistic model P(t)P(t) with growth rate k=0.3k=0.3 per hour and find P(10)P(10).

Example 16

medium
Which of these models exhibit saturation: (i) f(x)=exf(x)=e^{x}, (ii) g(x)=x1+xg(x)=\dfrac{x}{1+x} for x≥0x\ge 0, (iii) h(x)=ln⁡xh(x)=\ln x.

Example 17

medium
The logistic function P(t)=10001+e−0.5(t−6)P(t) = \dfrac{1000}{1+e^{-0.5(t-6)}} models population growth. Find P(0)P(0), P(6)P(6), and lim⁡t→∞P(t)\lim_{t\to\infty}P(t).

Example 18

easy
Sketch-style: a saturation curve f(t)=100(1−e−0.2t)f(t) = 100(1 - e^{-0.2 t}) has limit ___ as t→∞t \to \infty.

Example 19

medium
A logistic model has carrying capacity 10001000 and is currently at 250250. What fraction of room-to-grow remains?

Example 20

medium
A learning curve is A(t)=100−80e−0.3tA(t) = 100 - 80 e^{-0.3 t} (accuracy in percent). Find A(0)A(0), A(10)A(10), and the saturation level.
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