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1.7 Differentiation

1.7 Differentiation

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Question 311

The profile of a suspended track for a high-speed transport system is modelled by the curve CCC. The path of the track starts at point P P\,P and ends at point QQQ, as defined by the parametric equations

x=12t2+2t+10 x = \frac{1}{2}t^2 + \frac{2}{t} + 10 x=21​t2+t2​+10 h=3t+12t h = 3t + \frac{12}{t} h=3t+t12​

where 1≤t≤61 \le t \le 61≤t≤6.

The horizontal distance from a sensor at the origin is x x\,x metres, and h h\,h is the height of the track above the ground in metres.

P P\,P is the point on the track where t=1t = 1t=1 and Q Q\,Q is the point where t=6t = 6t=6.

a.

Safety regulations require that the difference in height between the start point P P\,P and the end point Q Q\,Q must be less than 6 metres. Show that the track meets this requirement.

[2]
bi.

Find an expression for dhdx\displaystyle \frac{dh}{dx}dxdh​ in terms of ttt.

[3]
bii.

A vertical reinforcement pillar is placed between the ground and the lowest point R R\,R on the track. Find the height of this pillar.

[3]
biii.

Calculate the acute angle the track makes with the horizontal at the end point QQQ. Give your answer to the nearest degree.

[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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