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.
864 exam-style questions on Edexcel A Level Maths Integration, covering 11.1 Integrating Standard Functions, 11.2 Integrating f(ax + b), 11.3 Using Trigonometric Identities, 11.4 Reverse Chain Rule, 11.5 Integration by Substitution, 11.6 Integration by Parts, 11.7 Partial Fractions, 11.8 Finding Areas, 11.9 The Trapezium Rule, 11.10 Solving Differential Equations, and 11.11 Modelling with Differential Equations. Each one has a worked solution and a mark scheme showing where the marks go.