OCR MEI C2 (Core Mathematics 2)

2 Fig. 9 shows a sketch of the curve \(y = x ^ { 3 } - 3 x ^ { 2 } - 22 x + 24\) and the line \(y = 6 x + 24\). \begin{figure}[h] \end{figure}

1. Differentiate \(y = x ^ { 3 } - 3 x ^ { 2 } - 22 x + 24\) and hence find the \(x\)-coordinates of the turning points of the curve. Give your answers to 2 decimal places.
2. You are given that the line and the curve intersect when \(x = 0\) and when \(x = - 4\). Find algebraically the \(x\)-coordinate of the other point of intersection.
3. Use calculus to find the area of the region bounded by the curve and the line \(y = 6 x + 24\) for \(- 4 \leqslant x \leqslant 0\), shown shaded on Fig. 9.

3

1. The standard formulae for the volume \(V\) and total surface area \(A\) of a solid cylinder of radius \(r\) and height \(h\) are $$V = \pi r ^ { 2 } h \quad \text { and } \quad A = 2 \pi r ^ { 2 } + 2 \pi r h .$$ Use these to show that, for a cylinder with \(A = 200\), $$V = 100 r - \pi r ^ { 3 }$$
2. Find \(\frac { \mathrm { d } V } { \mathrm {~d} r }\) and \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} r ^ { 2 } }\).
3. Use calculus to find the value of \(r\) that gives a maximum value for \(V\) and hence find this maximum value, giving your answers correct to 3 significant figures.

4

1. Differentiate \(x ^ { 3 } - 6 x ^ { 2 } - 15 x + 50\).
2. Hence find the \(x\)-coordinates of the stationary points on the curve \(y = x ^ { 3 } - 6 x ^ { 2 } - 15 x + 50\).

5 Use calculus to find the \(x\)-coordinates of the turning points of the curve \(y = x ^ { 3 } - 6 x ^ { 2 } - 15 x\). Hence find the set of values of \(x\) for which \(x ^ { 3 } - 6 x ^ { 2 } - 15 x\) is an increasing function.