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1.7 Differentiation

1.7 Differentiation

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Question 251

An ecologist is monitoring two distinct populations of insects.

The number of insects, NNN, in the first population is modelled by the equation

N=Aekt,t≥0 N = A e^{kt}, \quad t \ge 0 N=Aekt,t≥0

where A A\,A and k k\,k are positive constants and t t\,t is the time in days from the start of the observation.

Given that:

  • there were 4000 insects in this population at the start of the observation
  • there were 12000 insects 6 days later
a.

Find the exact value of A A\,A and the value of k k\,k to 4 significant figures.

[4]
b.

The number of insects, NNN, in the second population is modelled by the equation

N=80000e−0.4t,t≥0 N = 80000 e^{-0.4t}, \quad t \ge 0 N=80000e−0.4t,t≥0

where t t\,t is the time in days from the start of the observation.

Find the rate of decrease of insects in this second population exactly 4 days from the start. Give your answer to 3 significant figures.

[3]
c.

When t=Tt = Tt=T, the number of insects in the two populations was the same.

Find the value of TTT, giving your answer to 3 significant figures.

(Solutions relying entirely on calculator technology are not acceptable.)

[4]
Markscheme

1.7 Differentiation Questions

  1. A Level
  2. /Maths
  3. /1.7 Differentiation

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.

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