The cross-section of a designer architectural arch is modelled by the curve shown in a coordinate plane, with the equation
x=16sin2y+3,0⩽y⩽π2 x = 16\sin^2 y + 3, \quad 0 \leqslant y \leqslant \frac{\pi}{2} x=16sin2y+3,0⩽y⩽2πwhere x x\,x and y y\,y are spatial coordinates measured in decimetres. The point P(k,π6)P\left(k, \frac{\pi}{6}\right)P(k,6π) lies on the curve.
Verify that k=7k = 7k=7.
(i) Find dxdy\frac{dx}{dy}dydx in terms of yyy.
(ii) Hence show that dydx=12(x−3)(19−x)\frac{dy}{dx} = \frac{1}{2\sqrt{(x-3)(19-x)}}dxdy=2(x−3)(19−x)1.
The normal to the curve at PPP intersects the xxx-axis at the point NNN.
Determine 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 to be found.
717 exam-style questions on Edexcel A Level Maths Differentiation, 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. Each one has a worked solution and a mark scheme showing where the marks go.