An aerodynamic profile for a glider wing is designed to be symmetric about the chord line (the xxx-axis). The upper boundary of the wing's cross-section is modelled by the parametric equations:
x=−1.5t2 x = -1.5t^2 x=−1.5t2 y=15t−0.75t2 y = 15t - 0.75t^2 y=15t−0.75t2for 0≤t≤200 \le t \le 200≤t≤20, where xxx and yyy are measured in millimetres.
Determine the total length of the glider wing along the xxx-axis.
Obtain an expression for dydx\frac{dy}{dx}dxdy in terms of ttt.
Hence, prove that the maximum thickness (total width) of the wing is exactly one-quarter of its total length.
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.