The concentration C C\,C of a catalyst in a chemical reaction vessel, measured in mg/L, is modelled by the equation
C=1005(3t−k),t≠k3 C = \frac{100}{5(3t - k)}, \quad t \neq \frac{k}{3} C=5(3t−k)100,t=3kwhere t t\,t is the time in seconds since the start of the reaction, k k\,k is a positive constant, and k≠3k \neq 3k=3.
Find dCdt\displaystyle \frac{dC}{dt}dtdC, giving your answer in simplest form in terms of kkk.
The rate of change of the concentration at time t=1t = 1t=1 is -15 mg/L/s.
Find the two possible values of kkk.
Given also that k<3k < 3k<3,
find the equation of the normal to the curve of C C\,C against t t\,t at the point where t=1t = 1t=1, writing your answer in the form at+bC+c=0at + bC + c = 0at+bC+c=0, where a,b a, b\,a,b and c c\,c are integers to be found.
Show, using algebraic integration, that
∫131005(3t−k) dt=λln2 \int_{1}^{3} \frac{100}{5(3t - k)} \, dt = \lambda \ln 2 ∫135(3t−k)100dt=λln2where λ \lambda\,λ is a constant to be found.
825 exam-style questions on OCR (MEI) A Level Maths 1.9 Calculus, covering 1.9.1 Gradient of a curve at a point, 1.9.2 Gradient as limit of chord gradient, 1.9.3 Derivative as gradient of tangent, 1.9.4 Sketch the gradient function, 1.9.5 Differentiate y = kx^n, 1.9.6 Second derivative as rate of change of gradient, 1.9.7 Stationary points: maxima and minima, 1.9.8 Increasing and decreasing functions, 1.9.9 Tangent and normal at a point, 1.9.10 Differentiate e^kx, a^kx and ln x (A-level only), 1.9.11 Differentiate trigonometric functions (A-level only), 1.9.12 Product rule (A-level only), 1.9.13 Quotient rule (A-level only), 1.9.14 Chain rule (A-level only), 1.9.15 Rates of change with the chain rule (A-level only), 1.9.16 Implicit differentiation (A-level only), 1.9.17 Concavity and the second derivative (A-level only), 1.9.18 Points of inflection (A-level only), 1.9.19 Integration as reverse of differentiation, 1.9.20 Integrate kx^n, 1.9.21 Constant of integration, 1.9.22 Indefinite and definite integrals, 1.9.23 Area between a graph and the x-axis, 1.9.24 Integrate e^kx, 1/x, sin kx, cos kx (A-level only), 1.9.25 Integration as the limit of a sum (A-level only), 1.9.26 Area between two curves (A-level only), 1.9.27 Integration by substitution (reverse chain rule) (A-level only), 1.9.28 Integration by substitution (other cases) (A-level only), 1.9.29 Integration by parts (A-level only), 1.9.30 Integration using partial fractions (A-level only), 1.9.31 Formulate first order differential equations (A-level only), 1.9.32 Solve first order differential equations (A-level only), and 1.9.33 Interpret solutions of differential equations (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.