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.)
425 exam-style questions on OCR A Level Maths 1.7 Differentiation, covering 1.7.1 Derivative as gradient of the tangent, 1.7.2 Gradient of the tangent at a point, 1.7.3 Sketching the gradient function, 1.7.4 Second derivatives, 1.7.5 Second derivative as rate of change of gradient, 1.7.6 Convex, concave and points of inflection (A-level only), 1.7.7 Differentiation from first principles for powers of x, 1.7.8 Differentiation from first principles for sin x and cos x (A-level only), 1.7.9 Differentiating x^n, 1.7.10 Differentiating e^(kx) and a^(kx) (A-level only), 1.7.11 Differentiating trigonometric functions (A-level only), 1.7.12 Derivative of ln x (A-level only), 1.7.13 Tangents and normals, 1.7.14 Stationary points, 1.7.15 Increasing and decreasing functions, 1.7.16 Points of inflection (A-level only), 1.7.17 Product and quotient rules (A-level only), 1.7.18 Chain rule (A-level only), 1.7.19 Parametric and implicit differentiation (A-level only), 1.7.20 Constructing differential equations (A-level only), and 1.7 Differentiation. Each one has a worked solution and a mark scheme showing where the marks go.