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1.7 Differentiation

1.7 Differentiation

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Question 247

A crystalline deposit of mass MMM milligrams is formed on an electrode during a chemical process. At time ttt seconds after the process begins, the rate of increase of the mass is modeled by the differential equation

dMdt=M326t2,t>0 \frac{\text{d}M}{\text{dt}} = \frac{M^{\frac{3}{2}}}{6t^2}, \quad t > 0 dtdM​=6t2M23​​,t>0

Given that M=16M = 16M=16 when t=4t = 4t=4,

a.

show that

M=(ptqt+r)2 M = \left( \frac{pt}{qt + r} \right)^2 M=(qt+rpt​)2

where ppp, qqq, and rrr are integers to be found.

[6]
b.

According to the model, find the limiting value of the mass of the deposit as t→∞t \to \inftyt→∞.

[2]
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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