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1.7 Differentiation

1.7 Differentiation

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Question 207

An industrial chemist is monitoring the degradation of a catalyst in a chemical reactor. The amount of catalyst, A A\,A units, remaining in the system t t\,t hours after it is introduced is modelled by the differential equation

dAdt=−λA \frac{dA}{dt} = -\lambda A dtdA​=−λA

where λ \lambda\,λ is a positive constant. Given that the solution to this differential equation is of the form A=A0e−λtA = A_0 e^{-\lambda t}A=A0​e−λt, where A0 A_0\,A0​ is the initial amount of catalyst:

a.

On average, the catalyst has a half-life of 8.4 hours. At 8:00 am, 500 units of the catalyst are added to the reactor. Use the model to calculate the amount of catalyst remaining at 6:30 pm on the same day.

[4]
b.

To maintain the reaction safely, the chemist must ensure that the total amount of catalyst in the reactor never exceeds 600 units. The chemist intends to add a second batch of 500 units of catalyst. Determine the earliest time at which this second batch can be added. Give your answer to the nearest minute.

[5]
c.

Suggest one limitation of using this model for the degradation of the catalyst in a real industrial environment.

[1]
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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