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.)
825 exam-style questions on OCR (MEI) A Level Maths 1.9 Calculus, covering 1.9.1 Gradient of a curve at a point, 1.9.2 Gradient as limit of chord gradient, 1.9.3 Derivative as gradient of tangent, 1.9.4 Sketch the gradient function, 1.9.5 Differentiate y = kx^n, 1.9.6 Second derivative as rate of change of gradient, 1.9.7 Stationary points: maxima and minima, 1.9.8 Increasing and decreasing functions, 1.9.9 Tangent and normal at a point, 1.9.10 Differentiate e^kx, a^kx and ln x (A-level only), 1.9.11 Differentiate trigonometric functions (A-level only), 1.9.12 Product rule (A-level only), 1.9.13 Quotient rule (A-level only), 1.9.14 Chain rule (A-level only), 1.9.15 Rates of change with the chain rule (A-level only), 1.9.16 Implicit differentiation (A-level only), 1.9.17 Concavity and the second derivative (A-level only), 1.9.18 Points of inflection (A-level only), 1.9.19 Integration as reverse of differentiation, 1.9.20 Integrate kx^n, 1.9.21 Constant of integration, 1.9.22 Indefinite and definite integrals, 1.9.23 Area between a graph and the x-axis, 1.9.24 Integrate e^kx, 1/x, sin kx, cos kx (A-level only), 1.9.25 Integration as the limit of a sum (A-level only), 1.9.26 Area between two curves (A-level only), 1.9.27 Integration by substitution (reverse chain rule) (A-level only), 1.9.28 Integration by substitution (other cases) (A-level only), 1.9.29 Integration by parts (A-level only), 1.9.30 Integration using partial fractions (A-level only), 1.9.31 Formulate first order differential equations (A-level only), 1.9.32 Solve first order differential equations (A-level only), and 1.9.33 Interpret solutions of differential equations (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.