The pressure, PPP kPa, inside a vacuum chamber ttt seconds after a suction pump is activated is modelled by the equation
P=15+Ae−kt P = 15 + Ae^{-kt} P=15+Ae−ktwhere AAA and kkk are positive constants.
Given that the pressure inside the chamber at the instant the pump was activated was 100100100 kPa,
find the value of AAA.
Given also that, exactly 15 seconds after the pump was activated, the pressure was 404040 kPa,
find the value of kkk to 3 significant figures.
Using the values for AAA and kkk,
find, according to the model, the rate of change of the pressure inside the chamber exactly 5 seconds after the pump was activated. Give your answer in kPa s−1\text{kPa}\,\text{s}^{-1}kPas−1 to 3 significant figures.
Explain why, according to the model, the pressure cannot fall to 101010 kPa.
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.