In a controlled chemical synthesis, the rate of production of a particular polymer, R kg h−1R\text{ kg h}^{-1}R kg h−1, is modelled by the equation:
R=20.4−20.0e−0.5t−0.4e0.25t R = 20.4 - 20.0e^{-0.5t} - 0.4e^{0.25t} R=20.4−20.0e−0.5t−0.4e0.25twhere ttt is the time in hours since the reaction vessel was activated.
Determine the maximum production rate predicted by the model, giving your answer to one decimal place. Fully justify your answer.
Derive an expression for the total mass of polymer produced, M kgM\text{ kg}M kg, in terms of ttt.
After 15 hours, the total mass of polymer produced was measured to be 200 kg. Comment on the accuracy of the model.
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.