The current, xxx amps, at time ttt seconds after a switch is closed in an electric circuit is modelled by the differential equation
dxdt=k−2x \frac{dx}{dt} = k - 2x dtdx=k−2xwhere kkk is a constant. Initially, there is no current in the circuit.
Solve the differential equation to show that x=k2(1−e−2t)x = \frac{k}{2}(1 - e^{-2t})x=2k(1−e−2t).
In the long term, the current in the circuit approaches 12 amps.
Find the value of kkk.
Find the time, in seconds, for the current to reach 8 amps, giving your answer to 2 significant figures.
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.