The altitude hhh (in meters) of a specialized research drone during a 10-second test flight is modeled by the function
h(t)=t3−kt2+15t−12 h(t) = t^3 - kt^2 + 15t - 12 h(t)=t3−kt2+15t−12where t t\,t is the time in seconds (0≤t≤100 \le t \le 100≤t≤10) and k k\,k is a constant.
Find, in simplest form, (i) h′(t)h'(t)h′(t) (ii) h′′(t)h''(t)h′′(t)
The curve with equation v=h′(t)v = h'(t)v=h′(t) (velocity) intersects the curve with equation a=h′′(t)a = h''(t)a=h′′(t) (acceleration) at the points P P\,P and QQQ.
Given that the ttt-coordinate of P P\,P is 5,
find the value of kkk.
Hence find the coordinates of QQQ.
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.