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) = \dfrac{500}{1+4e^{-0.5 t}}

, find

P(0)

and the carrying capacity.

Example 4

medium

A model is

P(t)=\dfrac{100}{1+e^{-t}}

. What is its limit as

t\to\infty

Example 5

easy

Does a saturating curve ever exceed its limit?

Example 6

challenge

Two saturation curves both cap at

100

; curve A reaches

90

t=5

, curve B at

t=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

500

. What is its limiting value?

Example 9

easy

A lake can support at most

5000

fish. In a logistic model, what does

L = 5000

represent?

Example 10

hard

Which differential equation produces logistic saturation:

\dfrac{dP}{dt} = kP

\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)=\dfrac{100}{1+e^{-t}}

, find

P(0)

Example 14

challenge

A logistic model

P(t) = \dfrac{L}{1 + A e^{-kt}}

has

P(0) = 100

P(5) = 300

, and

L = 1000

. Find

k

Example 15

medium

A bacteria culture saturates at

10^6

cells. At time

t=0

there are

10^4

cells. Write a logistic model

P(t)

with growth rate

k=0.3

per hour and find

P(10)

Example 16

medium

Which of these models exhibit saturation: (i)

f(x)=e^{x}

, (ii)

g(x)=\dfrac{x}{1+x}

for

x\ge 0

, (iii)

h(x)=\ln x

Example 17

medium

The logistic function

P(t) = \dfrac{1000}{1+e^{-0.5(t-6)}}

models population growth. Find

P(0)

P(6)

, and

\lim_{t\to\infty}P(t)

Example 18

easy

Sketch-style: a saturation curve

f(t) = 100(1 - e^{-0.2 t})

has limit ___ as

t \to \infty

Example 19

medium

A logistic model has carrying capacity

1000

and is currently at

250

. What fraction of room-to-grow remains?

Example 20

medium

A learning curve is

A(t) = 100 - 80 e^{-0.3 t}

(accuracy in percent). Find

A(0)

A(10)

, and the saturation level.

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Asymptote Growth vs Decay