Year 11 Methods (Unit 1 & 2) Rates Of Change

Instantaneous Rate Of Change

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Questions

Question 1

28806

The population \(P\) million of bacteria after \(t\) minutes is give by \(P = 4 \times {2^t}\) as illustrated in the graph. Estimate the instantaneous rate of change of \(P\) when \(x = 1\)

Answer

\(5.5~\text{million/min} \)

Worked Solution

\begin{align}
&\text{Let the tangent line be }\varphi(t)\\
&\varphi(1)=8 \text{ and } \varphi(0)= 2.5\\
&\begin{aligned}
\text{Instantaneous rate }&= \frac{\varphi(1)-\varphi(0)}{1-0}\\
&= \frac{8-2.5}{1} \\
&= 5.5 \mathrm{~millon} / \mathrm{min}
\end{aligned}
\end{align}

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