Question 6 - A-Level Maths

1. Write down the derivative of \(\mathrm { e } ^ { \mathrm { kx } }\), where \(k\) is a constant.
2. A business has been running since 2009. They sell maths revision resources online. Give a reason why an exponential growth model might be suitable for the annual profits for the business. Fig. 6 shows the relationship between the annual profits of the business in thousands of pounds ( \(y\) ) and the time in years after \(2009 ( x )\). The graph of lny plotted against \(x\) is approximately a straight line. \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{a13f7a05-e2d3-4354-a0c7-ef7283eff514-07_1052_1157_751_242} \captionsetup{labelformat=empty} \caption{Fig. 6}
  \end{figure}
Show that the straight line is consistent with a model of the form \(\mathbf { y } = \mathrm { Ae } ^ { \mathrm { kx } }\), where \(A\) and \(k\) are constants.
Estimate the values of \(A\) and \(k\).
Use the model to predict the profit in the year 2020.
How reliable do you expect the prediction in part (d) to be? Justify your answer.