The horizontal position x x\,x of a piston in a high-precision engine is modelled by the equation
x=18cos2(2θ)0<θ<π4 x = 18 \cos^2(2\theta) \qquad 0 < \theta < \frac{\pi}{4} x=18cos2(2θ)0<θ<4πwhere θ \theta\,θ is the crankshaft angle in radians.
Show that the rate of change of the angle with respect to the position, dθdx\displaystyle \frac{d\theta}{dx}dxdθ, can be expressed in the form
dθdx=−1ABx−x2 \frac{d\theta}{dx} = -\frac{1}{A\sqrt{Bx - x^2}} dxdθ=−ABx−x21where A A\,A and B B\,B are integers to be determined.
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.
