The velocity, vvv m s−1\text{m s}^{-1}m s−1, of a research drone at time ttt seconds (t>0t > 0t>0) is defined such that its acceleration aaa follows the equation:
a=dvdt=12t+kt3+10 a = \frac{dv}{dt} = \frac{12}{\sqrt{t}} + \frac{k}{t^3} + 10 a=dtdv=t12+t3k+10where kkk is a constant.
It is observed that the rate of change of the drone's acceleration is zero at the instant where t=4t = 4t=4.
Determine the value of kkk.
Given that the drone's velocity is 100100100 m s−1\text{m s}^{-1}m s−1 when t=1t = 1t=1, find an expression for vvv in terms of ttt, giving each term in simplest form.
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.