7 The diagram shows part of the graph of \(y = \mathrm { e } ^ { - x ^ { 2 } }\)
\includegraphics[max width=\textwidth, alt={}, center]{be9ad375-d3cd-488c-9e3b-a4cd035d0d1d-10_376_940_607_392}
The graph is formed from two convex sections, where the gradient is increasing, and one concave section, where the gradient is decreasing.
7
- Find the values of \(x\) for which the graph is concave.
7
- The finite region bounded by the \(x\)-axis and the lines \(x = 0.1\) and \(x = 0.5\) is shaded.
\includegraphics[max width=\textwidth, alt={}, center]{be9ad375-d3cd-488c-9e3b-a4cd035d0d1d-11_372_937_584_355}
Use the trapezium rule, with 4 strips, to find an estimate for \(\int _ { 0.1 } ^ { 0.5 } e ^ { - x ^ { 2 } } d x\) Give your estimate to four decimal places.
[0pt]
[3 marks]
7 - Explain with reference to your answer in part (a), why the answer you found in part (b) is an underestimate.
[0pt]
[2 marks]
7 - By considering the area of a rectangle, and using your answer to part (b), prove that the shaded area is 0.4 correct to 1 decimal place.
[0pt]
[3 marks]
\section*{END OF SECTION A
TURN OVER FOR SECTION B}