The concentration of an enzyme, EEE micromoles per litre, in a bioreactor is monitored over a 6-hour cycle. The concentration at time TTT hours, where 0≤T≤60 \le T \le 60≤T≤6, is modeled by the equation:
E=T18(9+15T−2T2)+2 E = \frac{\sqrt{T}}{18}(9 + 15T - 2T^2) + 2 E=18T(9+15T−2T2)+2Given that EEE has a stationary value at T=αT = \alphaT=α:
Use calculus to show that α\alphaα satisfies the equation
10α2−45α−9=0 10\alpha^2 - 45\alpha - 9 = 0 10α2−45α−9=0Hence find the value of α\alphaα, giving your answer to 3 decimal places.
Use further calculus to prove that EEE is a maximum at this value of α\alphaα.
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.