The concentration of a specific contaminant in a groundwater sample, C C\,C mg/L, h h\,h hours after a purification agent is added, is modelled by the equation
C=250e−0.12h+15,h≥0 C = 250e^{-0.12h} + 15, \quad h \ge 0 C=250e−0.12h+15,h≥0Find the initial concentration of the contaminant in the sample.
Sketch the graph of C C\,C against hhh. On your sketch, state the equation of any asymptote to the curve.
Find the value of h h\,h for which C=50C = 50C=50, giving your answer to 2 decimal places.
Show by differentiation that
dCdh=A+BC \frac{\mathrm{d}C}{\mathrm{d}h} = A + BC dhdC=A+BCwhere A A\,A and B B\,B are constants to be determined.
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.