Mathematics AA HL
IB Questions
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Question 1 of 2
The function \(f\) has a derivative given by \(f^{\prime}(x)=\frac{2}{x(m+x)}\), where \(x \in \mathbb{R}, x \neq 0, x \neq -m\) and \(m\) is a positive constant.
The expression for \(f^{\prime}(x)\) can be written in the form \(\frac{p}{x}+\frac{q}{m+x}\), where \(p, q \in \mathbb{R}\).
Find \(p\) and \(q\) in terms of \(m\).
Find an expression for \(f(x)\).
Consider \(P\), the population of a colony of bacteria, which has an initial value of 800.
The rate of change of the population can be modelled by the differential equation \(\frac{dP}{dt}=\frac{P(m-P)}{4m}\), where \(t\) is the time measured in hours, \(t \geq 0\), and \(m\) is the upper bound for the population.
Show that \(P = \frac{800me^{\frac{t}{4}}}{(m-800+800e^{\frac{t}{4}})}\).
At \(t=8\) the population of the colony has doubled in size from its initial value.
Calculate the value of \(m\), giving your answer correct to four significant figures.
Question 2 of 2
Consider a function \(f(x)\) defined by \(f(x) = \frac{4x+8}{x-3}\) for \(x \in \mathbb{R}, x \neq 3\).
Find the zero of \(f(x)\).
For the graph of \(y=f(x)\), write down the equation of:
the vertical asymptote;
the horizontal asymptote.
Find \(f^{-1}(x)\), the inverse function of \(f(x)\).