The concentration CCC (in mg/L) of a chemical catalyst over time ttt (in minutes) is modeled by the function
C(t)=(2t−3)4e−2t,t≥1.5 C(t) = (2t - 3)^4 e^{-2t}, \quad t \ge 1.5 C(t)=(2t−3)4e−2t,t≥1.5Show that
C′(t)=A(2t−3)3(7−2t)e−2t C'(t) = A(2t - 3)^3 (7 - 2t) e^{-2t} C′(t)=A(2t−3)3(7−2t)e−2twhere A A\,A is a constant to be found.
Hence find the exact coordinates of the two stationary points on the curve with equation y=C(t)y = C(t)y=C(t).
A secondary reaction is modeled by the function HHH, defined by
H(t)=5C(t+0.5) H(t) = 5 C(t + 0.5) H(t)=5C(t+0.5)Find the coordinates of the maximum stationary point on the curve with equation y=H(t)y = H(t)y=H(t).
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.