A precision laser cutting tool follows a path defined by the relationship between its lateral displacement, xxx mm, and the angle of its guide-rail, yyy radians, given by
x=16sin2y+5,0⩽y⩽π2 x = 16\sin^2 y + 5, \quad 0 \leqslant y \leqslant \frac{\pi}{2} x=16sin2y+5,0⩽y⩽2πThe point P(k,π3)P\left(k, \frac{\pi}{3}\right)P(k,3π) lies on this path.
Verify that k=17k = 17k=17.
Find dxdy\frac{dx}{dy}dydx in terms of yyy.
Hence show that dydx=12x−521−x\frac{dy}{dx} = \frac{1}{2\sqrt{x-5}\sqrt{21-x}}dxdy=2x−521−x1.
The normal to the path at PPP cuts the xxx-axis at the point NNN.
Find the exact area of triangle OPNOPNOPN, where OOO is the origin. Give your answer in the form aπ+bπ2a\pi + b\pi^2aπ+bπ2 where aaa and bbb are constants.
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.
