A biologist is monitoring the population of a specialized algae culture, PPP (in thousands), in a controlled environment. At time ttt hours (t>0t > 0t>0), the rate of change of the population is modeled by the derivative:
P′(t)=2t2+at+b3t P'(t) = \frac{2t^2 + at + b}{3\sqrt{t}} P′(t)=3t2t2+at+bwhere aaa and bbb are constants. It is observed that at time t=4t = 4t=4, the population is exactly 10,000 algae (P=10P = 10P=10) and the population is growing at a rate of 5,000 algae per hour (P′=5P' = 5P′=5).
Show that 4a+b=−24a + b = -24a+b=−2.
Given also that a+b=7a + b = 7a+b=7,
Determine, in simplest form, an expression for P(t)P(t)P(t).
A researcher adjusts the experiment such that the new population model is given by G(t)=P(t−3)G(t) = P(t - 3)G(t)=P(t−3). Given that the observation at t=4t = 4t=4 in the original model is recorded as point JJJ,
State the coordinates of the corresponding point KKK on the graph of G(t)G(t)G(t).
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.
