Four biologists are modeling the mass distribution m m\,m of a bacterial culture relative to the radial distance r r\,r from the center of a petri dish. They determine that the rate of change is given by dmdr=1r\displaystyle \frac{dm}{dr} = \frac{1}{r}drdm=r1 for r>0r > 0r>0. Each biologist proposes a different expression for the indefinite integral ∫1r dr\displaystyle \int \frac{1}{r} \, dr∫r1dr:
Dr. Aris: m=lnrm = \ln rm=lnr
Dr. Bond: m=Alnrm = A \ln rm=Alnr
Dr. Chen: m=ln(kr)m = \ln (kr)m=ln(kr)
Dr. Dyani: m=lnr+cm = \ln r + cm=lnr+c
Explain the error in Dr. Aris's proposal.
Explain the error in Dr. Bond's proposal.
Demonstrate why the expressions provided by Dr. Chen and Dr. Dyani are equivalent for certain values of the constants k k\,k and ccc.
Practise Edexcel A Level Maths Integration with exam-style questions for A Level Maths. 437 questions 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, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
