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1.10 G: Differentiation

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Question 153

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]

1.10 G: Differentiation Questions

  1. A Level
  2. /Maths
  3. /1.10 G: Differentiation

Practise AQA A Level Maths 1.10 G: Differentiation with exam-style questions for A Level Maths. 321 questions covering 1.10.1 The derivative and second derivative, 1.10.2 Differentiating standard functions, 1.10.3 Applications of differentiation, 1.10.4 Product, quotient and chain rules (A-level only), 1.10.5 Implicit and parametric differentiation (A-level only), and 1.10.6 Constructing differential equations (A-level only), matched to the AQA A Level Maths (7357) 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