A high-performance motorboat starts from rest at a marker on a calm lake and accelerates in a straight line.
Using a simple model of constant acceleration a=2.4 m s−2a = 2.4 \text{ m s}^{-2}a=2.4 m s−2, a technician predicts that the velocity of the boat, exactly 5 seconds after starting from rest, is 12 m s−112 \text{ m s}^{-1}12 m s−1. Show how the technician obtained this prediction.
Using a refined model that accounts for water resistance, the boat's acceleration, a m s−2a \text{ m s}^{-2}a m s−2, at time ttt seconds after starting is given by the differential equation
dvdt=2.4−0.4v \frac{dv}{dt} = 2.4 - 0.4v dtdv=2.4−0.4vwhere v m s−1v \text{ m s}^{-1}v m s−1 is the velocity of the boat at time ttt. Find an expression for vvv in terms of ttt.
Compare the behavior of the velocity vvv as t→∞t \to \inftyt→∞ for both models.
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.