(Solutions based entirely on graphical or numerical methods are not acceptable.)
The power consumption of a high-performance computing cluster, P P\,P in kilowatts, is modeled by the function
P(t)=0.8t2.5−128t+2000,t>0 P(t) = 0.8t^{2.5} - 128t + 2000, \quad t > 0 P(t)=0.8t2.5−128t+2000,t>0where t t\,t is the time in hours since the start of a large-scale simulation.
Solve the equation P′(t)=0P'(t) = 0P′(t)=0.
Solve the equation P′′(t)=15P''(t) = 15P′′(t)=15.
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.