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Limits and Continuity · Paper 3 — Free Maths AI HL Practice Questions

1ChallengeLAQDefinition and calculation of limits10 marksPaper 3~15 min

A population of bacteria is growing in a laboratory dish. The number of bacteria,

N

(in thousands), after

t

hours is modelled by

N(t) = \frac{12t^2 + 5t + 1}{2t^2 + 1}, \quad t \geq 0.

(a)

Prove that

\displaystyle\lim_{t \to \infty} N(t)

exists and find its exact value. [3 marks]

(b)

Show that

N'(t) = \dfrac{-24t^2 + 10t + 5}{(2t^2+1)^2}

and hence evaluate

N'(0)

. [3 marks]

(c)

Interpret the value of

N'(0)

in the context of the model. [1 mark]

(d)

By considering

N'(t)

and the limit found in part (a), justify whether the model predicts that the bacterial population ever reaches or exceeds

6.5

thousand. [3 marks]

Solutions

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