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Differentiation

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Question 144

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. 311 questions covering 9.1 Differentiating sin x and cos x, 9.2 Differentiating exponentials and logarithms, 9.3 The Chain Rule, 9.4 The Product Rule, 9.5 The Quotient Rule, 9.6 Differentiating Trigonometric Functions, 9.7 Parametric Differentiation, 9.8 Implicit Differentiation, 9.9 Using Second Derivatives, and 9.10 Rates of Change, 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.

Question bank