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Differentiation

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Question 159

A high-precision cam profile in a mechanical sensor follows a path defined by the equation

x=6sin⁡4θ0≤x≤6,0≤θ≤π8 x = 6 \sin 4\theta \quad 0 \le x \le 6, \quad 0 \le \theta \le \frac{\pi}{8} x=6sin4θ0≤x≤6,0≤θ≤8π​

where xxx is the horizontal displacement in millimetres and θ\thetaθ is the angular position of the cam in radians.

a.

Find dxdθ\frac{dx}{d\theta}dθdx​ in terms of θ\thetaθ.

[2]
b.

Hence show that

dθdx=k36−x2 \frac{d\theta}{dx} = \frac{k}{\sqrt{36-x^2}} dxdθ​=36−x2​k​

where kkk is a constant to be determined.

[3]
c.

A specific calibration point P(a,b)P(a, b)P(a,b) lies on the profile. At this point:

  • The rate of change of the angular position with respect to displacement, dθdx\frac{d\theta}{dx}dxdθ​, is exactly 1123\frac{1}{12\sqrt{3}}123​1​.
  • Both aaa and bbb are positive constants.

Determine the exact values of aaa and bbb.

[4]

Differentiation Questions

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

Practise Edexcel A Level Maths Differentiation with exam-style questions for A Level Maths. 327 questions covering 12.1 Gradients of Curves, 12.2 Differentiation from first principles, 12.3 Differentiating x^n, 12.4 Differentiating Quadratics, 12.5 Differentiating functions with two or more terms, 12.6 Gradients, Tangents and Normals, 12.7 Increasing and Decreasing Functions, 12.8 Second Order Derivatives, 12.9 Stationary Points, 12.10 Sketching Gradient Functions, and 12.11 Modelling with Differentiation, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.

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