An ecologist is monitoring the population of a certain species of bird in a woodland. The number of birds, BBB, in the population, ttt years after the start of the study, is modelled by the equation
B=800e0.2t3+e0.2tt≥0 B = \frac{800e^{0.2t}}{3 + e^{0.2t}} \quad t \ge 0 B=3+e0.2t800e0.2tt≥0Find the number of birds in the woodland at the start of the study.
Find the upper limit for the number of birds according to this model.
Find the time, after the start of the study, when there are predicted to be 600 birds in the woodland. Give your answer in years and months to the nearest month.
Show that
dBdt=Ke0.2t(3+e0.2t)2 \frac{dB}{dt} = \frac{Ke^{0.2t}}{(3 + e^{0.2t})^2} dtdB=(3+e0.2t)2Ke0.2twhere KKK is a constant to be found.
Given that when t=Tt = Tt=T, dBdt=10\frac{dB}{dt} = 10dtdB=10, find the value of TTT to one decimal place. (Solutions relying entirely on calculator technology are not acceptable.)
Practise AQA A Level Maths 1.10 G: Differentiation with exam-style questions for A Level Maths. 321 questions covering 1.10.1 The derivative and second derivative, 1.10.2 Differentiating standard functions, 1.10.3 Applications of differentiation, 1.10.4 Product, quotient and chain rules (A-level only), 1.10.5 Implicit and parametric differentiation (A-level only), and 1.10.6 Constructing differential equations (A-level only), matched to the AQA A Level Maths (7357) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
