← Calculus methods

Rates of change in a modelled context

Population, concentration, followers, spectators — the model changes every year, but the method never does: differentiate, substitute, then say what the number MEANS.

The three-step method for any rate-of-change question

  • 1. Differentiate the given model with respect to its variable (usually time) to get the rate-of-change function.
  • 2. Substitute the given value to get a number, with the correct unit (per hour, per minute, per day).
  • 3. INTERPRET the number in the context: a negative rate means the quantity is DECREASING at that instant, and the size tells you how fast.
Worked example Interpreting a negative rate of change The number of spectators at a match is modelled by P = 100t² − 2t⁴ + 750 for 0 ≤ t ≤ 7.5, where t is hours since the match started. Show the rate of change is −528 people per hour at t = 6, and explain what this means. Differentiate: dP/dt = 200t − 8t³. Substitute t = 6: dP/dt = 200(6) − 8(6)³ = 1200 − 1728 = −528. Interpret: a NEGATIVE rate means spectators are LEAVING, not arriving — specifically, 528 more people were leaving than arriving during that hour, six hours into the match. ✓ −528 people per hour, meaning the crowd was shrinking at that instant by 528 people per hour — the negative sign is not a mathematical inconvenience, it is the entire content of the answer.

Finding WHEN a rate is at its own maximum or minimum

  • The rate of change is itself a function (the first derivative) — to find when THAT rate is fastest, differentiate AGAIN to get the second derivative, and set it to zero.
  • This is the same second-derivative-test logic as any other optimisation, just applied one level up: you are now optimising the RATE, not the original quantity.
  • State the answer in context: "the population was growing fastest after t = 4 hours" is worth more than a bare value of t with no interpretation.
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A rate-of-change question always needs a sentence, not just a number

Every rate-of-change question in this paper is graded partly on INTERPRETATION, not just the calculus. Stopping at "dP/dt = −528" without saying what a negative value of −528 means for spectators leaving the venue leaves the context half of the mark unclaimed, however correct the arithmetic.

Practise just this