Malthus equation

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Question statement

Consider the Malthus equation

$x'(t)=r x(t), \qquad x(t_0)=x_0. \tag{M}$

Give any introductory information the student needs.

			Name	Type	Generated Value

population1	integer	118346
population2	integer	156739
year1	integer	2016
year2	integer	2017
rate	integer	-3
year3	integer	2005
fish	integer	102
fishes	integer	306

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Name

Data type

Value

xxxxxxxxxx
 
random(100000..120000)

JME function reference

Description

Describe what this variable represents, and list any assumptions made about its value.

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Generated value: `integer`

118346

→ Used by:

population2

Advice
"Part c)" - prompt

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Ask the student a question, and give any hints about how they should answer this part.

What should be instead of $A$ and $B$ in the expression for the function

$x(t)=A e^{rt-rB}$
to ensure that this function solves the Malthus equation (M)?

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$A=x_0$ , $B=t_0$

Marks: Distractor Message:
$A=t_0$ , $B=x_0$

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$A=x_0$ , $B=0$

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Advice

b) Note that the answer does not depend on $t_0$ , since 'in three years' means that $t-t_0=3$ that appears at the formula.

c) The solution to the given equation has the form

$u(t)=u(t_0)e^{\var{rate}t},$
where

$t_0$ is the initial moment of time, i.e.

$t_0=\var{year1}$ . Since

$\var{rate}<0$ ,

$u(t)$ is decreasing, i.e.

$u(t)<u(t_0)$ for all

$t>t_0$ . Since

$\var{population2}>\var{population1}$ , the population would not become larger than it was, and will not start to grow ever.

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Taxonomy: mathcentre

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Dmitri Finkelshtein 6 years, 6 months ago

Dmitri Finkelshtein 6 years, 6 months ago

Dmitri Finkelshtein 6 years, 6 months ago

Error in variable testing condition

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Generated value: `integer`

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Malthus equation

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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History

Comment

Checkpoint description

Dmitri Finkelshtein 6 years, 6 months ago

Dmitri Finkelshtein 6 years, 6 months ago

Dmitri Finkelshtein 6 years, 6 months ago

Error in variable testing condition

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Generated value: integer

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Advanced settings

Generated value: `integer`