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1.7 Differentiation

1.7 Differentiation

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Question 304

A biochemist is studying the production of a specific enzyme in a bioreactor. The mass of the enzyme, MMM mg, in the reactor, ttt hours after the reaction begins, is modelled by the equation

M=1200e0.5t5+e0.5tt≥0 M = \frac{1200e^{0.5t}}{5 + e^{0.5t}} \quad t \ge 0 M=5+e0.5t1200e0.5t​t≥0
a.

Determine the initial mass of the enzyme in the bioreactor.

[2]
b.

According to this model, find the limiting value of the enzyme's mass as ttt becomes very large.

[2]
c.

Calculate the time elapsed since the start of the reaction when the mass of the enzyme is exactly 900 mg. Give your answer in hours and minutes to the nearest minute.

[3]
d.

Show that

dMdt=Ke0.5t(5+e0.5t)2 \frac{dM}{dt} = \frac{Ke^{0.5t}}{(5 + e^{0.5t})^2} dtdM​=(5+e0.5t)2Ke0.5t​

where KKK is a constant to be determined.

[4]
e.

At time t=Tt = Tt=T, the rate of enzyme production is 40 mg/h. Find the value of TTT, giving your answer to one decimal place. (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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