The concentration C C\,C of a chemical catalyst in a specialized solvent depends on the depth ddd (in cm) according to the model
C=502(3d−k),d≠k3 C = \frac{50}{2(3d - k)}, \quad d \neq \frac{k}{3} C=2(3d−k)50,d=3kwhere k k\,k is a positive constant and k≠12k \neq 12k=12.
Find dCdd\displaystyle \frac{dC}{dd}dddC giving your answer in simplest form in terms of kkk.
The point P P\,P with ddd-coordinate 4 lies on the curve. Given that the rate of change of concentration with respect to depth at P P\,P is -3,
find the two possible values of kkk.
Given also that k<12k < 12k<12,
find the equation of the normal to the curve at PPP, writing your answer in the form ad+bC+c=0ad + bC + c = 0ad+bC+c=0, where aaa, bbb, and c c\,c are integers to be found.
Show, using algebraic integration, that
∫35502(3d−k) dd=λln2 \int_{3}^{5} \frac{50}{2(3d - k)} \, dd = \lambda \ln 2 ∫352(3d−k)50dd=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.