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1.7 Differentiation

1.7 Differentiation

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Question 171

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]
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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