The population NNN of a specific bacteria culture, in thousands, is modeled by the equation
N(t)=(2t+2)28+122t+23 N(t) = \frac{(2t+2)^2}{8} + 12\sqrt[3]{2t+2} N(t)=8(2t+2)2+1232t+2for t≥0t \ge 0t≥0, where ttt is the time in hours since the start of an experiment.
Find an expression for dNdt\frac{dN}{dt}dtdN.
The point PPP with coordinates (3,32)(3, 32)(3,32) lies on the graph of the population model. Find an equation of the tangent to the curve at the point PPP.
Show that the model predicts no stationary points for the population for t≥0t \ge 0t≥0.
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.