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).
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.