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 Edexcel A Level Maths Differentiation with exam-style questions for A Level Maths. 327 questions covering 12.1 Gradients of Curves, 12.2 Differentiation from first principles, 12.3 Differentiating x^n, 12.4 Differentiating Quadratics, 12.5 Differentiating functions with two or more terms, 12.6 Gradients, Tangents and Normals, 12.7 Increasing and Decreasing Functions, 12.8 Second Order Derivatives, 12.9 Stationary Points, 12.10 Sketching Gradient Functions, and 12.11 Modelling with Differentiation, 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.
