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.
425 exam-style questions on OCR A Level Maths 1.7 Differentiation, covering 1.7.1 Derivative as gradient of the tangent, 1.7.2 Gradient of the tangent at a point, 1.7.3 Sketching the gradient function, 1.7.4 Second derivatives, 1.7.5 Second derivative as rate of change of gradient, 1.7.6 Convex, concave and points of inflection (A-level only), 1.7.7 Differentiation from first principles for powers of x, 1.7.8 Differentiation from first principles for sin x and cos x (A-level only), 1.7.9 Differentiating x^n, 1.7.10 Differentiating e^(kx) and a^(kx) (A-level only), 1.7.11 Differentiating trigonometric functions (A-level only), 1.7.12 Derivative of ln x (A-level only), 1.7.13 Tangents and normals, 1.7.14 Stationary points, 1.7.15 Increasing and decreasing functions, 1.7.16 Points of inflection (A-level only), 1.7.17 Product and quotient rules (A-level only), 1.7.18 Chain rule (A-level only), 1.7.19 Parametric and implicit differentiation (A-level only), 1.7.20 Constructing differential equations (A-level only), and 1.7 Differentiation. Each one has a worked solution and a mark scheme showing where the marks go.