(i)     Write down the value of \(y\) when \(x\) is \(2\).

(ii)    Write down the coordinates of the point where the curve intercepts the \(y\)-axis.

Sketch the curve for \( - 4 \leqslant x \leqslant 3\) and \( - 10 \leqslant y \leqslant 10\). Indicate clearly the information found in (a).

Find \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\) .

Let \({L_1}\) be the tangent to the curve at \(x = 2\).

Let \({L_2}\) be a tangent to the curve, parallel to \({L_1}\).

(i)     Show that the gradient of \({L_1}\) is \(12\).

(ii)    Find the \(x\)-coordinate of the point at which \({L_2}\) and the curve meet.

(iii)   Sketch and label \({L_1}\) and \({L_2}\) on the diagram drawn in (b).

It is known that \(\frac{{{\text{d}}y}}{{{\text{d}}x}} > 0\) for \(x < - 2\) and \(x > b\) where \(b\) is positive.

(i)     Using your graphic display calculator, or otherwise, find the value of \(b\).

(ii)    Describe the behaviour of the curve in the interval \( - 2 < x < b\) .

(iii)   Write down the equation of the tangent to the curve at \(x = - 2\).

(i)     \(y = 0\)     (A1)

(ii)    \((0{\text{, }}{- 2})\)     (A1)(A1)

Note: Award (A1)(A0) if brackets missing.

OR

\(x = 0{\text{, }}y = - 2\)     (A1)(A1)

Note: If coordinates reversed award (A0)(A1)(ft). Two coordinates must be given.

[3 marks]

     (A4)

Notes: (A1) for appropriate window. Some indication of scale on the \(x\)-axis must be present (for example ticks). Labels not required. (A1) for smooth curve and shape, (A1) for maximum and minimum in approximately correct position, (A1) for \(x\) and \(y\) intercepts found in (a) in approximately correct position.

[4 marks]

\(\frac{{{\text{d}}y}}{{{\text{d}}x}} = 3{x^2} + 3x - 6\)     (A1)(A1)(A1)

 Note: (A1) for each correct term. Award (A1)(A1)(A0) at most if any other term is present.

[3 marks]

(i)     \(3 \times 4 + 3 \times 2 - 6 = 12\)     (M1)(A1)(AG)

Note: (M1) for using the derivative and substituting \(x = 2\) . (A1) for correct (and clear) substitution. The \(12\) must be seen.

(ii)    Gradient of \({L_2}\) is \(12\) (can be implied)     (A1)

\(3{x^2} + 3x - 6 = 12\)     (M1)

\(x = - 3\)     (A1)(G2)

Note: (M1) for equating the derivative to \(12\) or showing a sketch of the derivative together with a line at \(y = 12\) or a table of values showing the \(12\) in the derivative column.

(iii)   (A1) for \({L_1}\) correctly drawn at approx the correct point     (A1)

(A1) for \({L_2}\) correctly drawn at approx the correct point     (A1)

(A1) for 2 parallel lines     (A1)

Note: If lines are not labelled award at most (A1)(A1)(A0). Do not accept 2 horizontal or 2 vertical parallel lines.

[8 marks]

(i)     \(b = 1\)     (G2)

(ii)    The curve is decreasing.     (A1)

Note: Accept any valid description.

(iii)   \(y = 8\)     (A1)(A1)(G2)

Note: (A1) for “\(y =\) a constant”, (A1) for \(8\).

[5 marks]

Many candidates managed to gain good marks in this question as they were able to answer the first three parts of the question. Good sketches were drawn with the required information shown on them. Very few candidates did not recognise the notation \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\) but they showed that they knew how to differentiate as in (d)(i) they found the derivative to show that the gradient of \({L_1}\) was \(12\). Candidates found it difficult to find the other \(x\) for which the derivative was \(12\). However, some could draw both tangents without having found this value of \(x\) . In general, tangents were not well drawn. The last part question did act as a discriminating question. However, those candidates that had the function drawn either in their GDC or on paper recognised that at \(x = - 2\) there was a maximum and so wrote down the correct equation of the tangent at that point.

Many candidates managed to gain good marks in this question as they were able to answer the first three parts of the question. Good sketches were drawn with the required information shown on them. Very few candidates did not recognise the notation \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\) but they showed that they knew how to differentiate as in (d)(i) they found the derivative to show that the gradient of \({L_1}\) was \(12\). Candidates found it difficult to find the other \(x\) for which the derivative was \(12\). However, some could draw both tangents without having found this value of \(x\) . In general, tangents were not well drawn. The last part question did act as a discriminating question. However, those candidates that had the function drawn either in their GDC or on paper recognised that at \(x = - 2\) there was a maximum and so wrote down the correct equation of the tangent at that point.

Many candidates managed to gain good marks in this question as they were able to answer the first three parts of the question. Good sketches were drawn with the required information shown on them. Very few candidates did not recognise the notation \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\) but they showed that they knew how to differentiate as in (d)(i) they found the derivative to show that the gradient of \({L_1}\) was \(12\). Candidates found it difficult to find the other \(x\) for which the derivative was \(12\). However, some could draw both tangents without having found this value of \(x\) . In general, tangents were not well drawn. The last part question did act as a discriminating question. However, those candidates that had the function drawn either in their GDC or on paper recognised that at \(x = - 2\) there was a maximum and so wrote down the correct equation of the tangent at that point.

Many candidates managed to gain good marks in this question as they were able to answer the first three parts of the question. Good sketches were drawn with the required information shown on them. Very few candidates did not recognise the notation \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\) but they showed that they knew how to differentiate as in (d)(i) they found the derivative to show that the gradient of \({L_1}\) was \(12\). Candidates found it difficult to find the other \(x\) for which the derivative was \(12\). However, some could draw both tangents without having found this value of \(x\) . In general, tangents were not well drawn. The last part question did act as a discriminating question. However, those candidates that had the function drawn either in their GDC or on paper recognised that at \(x = - 2\) there was a maximum and so wrote down the correct equation of the tangent at that point.

Many candidates managed to gain good marks in this question as they were able to answer the first three parts of the question. Good sketches were drawn with the required information shown on them. Very few candidates did not recognise the notation \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\) but they showed that they knew how to differentiate as in (d)(i) they found the derivative to show that the gradient of \({L_1}\) was \(12\). Candidates found it difficult to find the other \(x\) for which the derivative was \(12\). However, some could draw both tangents without having found this value of \(x\) . In general, tangents were not well drawn. The last part question did act as a discriminating question. However, those candidates that had the function drawn either in their GDC or on paper recognised that at \(x = - 2\) there was a maximum and so wrote down the correct equation of the tangent at that point.