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 AQA A Level Maths 1.11 H: Integration with exam-style questions for A Level Maths. 436 questions covering 1.11.1 Fundamental Theorem of Calculus, 1.11.2 Integrating standard functions, 1.11.3 Definite integrals and areas, 1.11.4 Integration as the limit of a sum (A-level only), 1.11.5 Integration by substitution and by parts (A-level only), 1.11.6 Integration using partial fractions (A-level only), 1.11.7 Differential equations with separable variables (A-level only), and 1.11.8 Interpreting solutions of differential equations (A-level only), matched to the AQA A Level Maths (7357) 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.
