Date | May 2021 | Marks available | 3 | Reference code | 21M.2.AHL.TZ1.12 |
Level | Additional Higher Level | Paper | Paper 2 | Time zone | Time zone 1 |
Command term | Find | Question number | 12 | Adapted from | N/A |
Question
The function f has a derivative given by f'(x)=1x(k-x), x∈ℝ, x≠o, x≠k where k is a positive constant.
Consider P, the population of a colony of ants, which has an initial value of 1200.
The rate of change of the population can be modelled by the differential equation dPdt=P(k-P)5k, where t is the time measured in days, t≥0, and k is the upper bound for the population.
At t=10 the population of the colony has doubled in size from its initial value.
The expression for f′(x) can be written in the form ax+bk-x, where a, b∈ℝ. Find a and b in terms of k.
Hence, find an expression for f(x).
By solving the differential equation, show that P=1200k(k-1200)e-t5+1200.
Find the value of k, giving your answer correct to four significant figures.
Find the value of t when the rate of change of the population is at its maximum.
Markscheme
1x(k-x)≡ax+bk-x
a(k-x)+bx=1 (A1)
attempt to compare coefficients OR substitute x=k and x=0 and solve (M1)
a=1k and b=1k A1
f'(x)=1kx+1k(k-x)
[3 marks]
attempt to integrate their ax+bk-x (M1)
f(x)1k∫(1x+1k-x)dx
=1k(ln|x|-ln|k-x|)(+c)(=1kln|xk-x|(+c)) A1A1
Note: Award A1 for each correct term. Award A1A0 for a correct answer without modulus signs. Condone the absence of +c.
[3 marks]
attempt to separate variables and integrate both sides M1
5k∫1P(k-P)dP=∫1dt
5(ln A1
Note: There are variations on this which should be accepted, such as . Subsequent marks for these variations should be awarded as appropriate.
EITHER
attempt to substitute into an equation involving M1
A1
A1
A1
OR
A1
attempt to substitute M1
A1
A1
THEN
attempt to rearrange and isolate M1
OR OR
OR A1
AG
[8 marks]
attempt to substitute (M1)
(A1)
A1
Note: Award (M1)(A1)A0 for any other value of which rounds to
[3 marks]
attempt to find the maximum of the first derivative graph OR zero of the second derivative graph OR that (M1)
(days) A2
Note: Accept any value which rounds to .
[3 marks]