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. 311 questions covering 9.1 Differentiating sin x and cos x, 9.2 Differentiating exponentials and logarithms, 9.3 The Chain Rule, 9.4 The Product Rule, 9.5 The Quotient Rule, 9.6 Differentiating Trigonometric Functions, 9.7 Parametric Differentiation, 9.8 Implicit Differentiation, 9.9 Using Second Derivatives, and 9.10 Rates of Change, 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.
