Sketch the graph of y = f (x) for −3 ≤ x ≤ 6 and −10 ≤ y ≤ 10 showing clearly the axes intercepts and local maximum and minimum points. Use a scale of 2 cm to represent 1 unit on the x-axis, and a scale of 1 cm to represent 1 unit on the y-axis.

Find the value of f (−1).

Write down the coordinates of the y-intercept of the graph of f (x).

Find f '(x).

Show that \(f'( - 1) =  - \frac{{16}}{3}\).

Explain what f '(−1) represents.

Find the equation of the tangent to the graph of f (x) at the point where x is –1.

Sketch the tangent to the graph of f (x) at x = −1 on your diagram for (a).

P and Q are points on the curve such that the tangents to the curve at these points are horizontal. The x-coordinate of P is a, and the x-coordinate of Q is b, b > a.

Write down the value of

(i) a ;

(ii) b .

P and Q are points on the curve such that the tangents to the curve at these points are horizontal. The x-coordinate of P is a, and the x-coordinate of Q is b, b > a.

Describe the behaviour of f (x) for a < x < b.

(A1) for indication of window and labels. (A1) for smooth curve that does not enter the first quadrant, the curve must consist of one line only.

(A1) for x and y intercepts in approximately correct positions (allow ±0.5).

(A1) for local maximum and minimum in approximately correct position. (minimum should be 0 ≤ x ≤ 1 and –2 ≤ y ≤ –4 ), the y-coordinate of the maximum should be 0 ± 0.5.     (A4)

[4 marks]

\(-\frac{1}{3}(-1)^3 + \frac{5}{3}(-1)^2 - (-1) - 3 \)     (M1)

Note: Award (M1) for substitution of –1 into f (x)

= 0     (A1)(G2)

[2 marks]

(0, –3)     (A1)

OR

x = 0, y = –3     (A1)

Note: Award (A0) if brackets are omitted.

[1 mark]

\(f'(x) =  - {x^2} + \frac{{10}}{3}x - 1\)     (A1)(A1)(A1)

Note: Award (A1) for each correct term. Award (A1)(A1)(A0) at most if there are extra terms.

[3 marks]

\(f'( - 1) =  - {( - 1)^2} + \frac{{10}}{3}( - 1) - 1\)     (M1)

\(= -\frac{16}{3}\)     (AG)

Note: Award (M1) for substitution of x = –1 into correct derivative only. The final answer must be seen.

[1 mark]

f '(–1) gives the gradient of the tangent to the curve at the point with x = –1.     (A1)(A1)

Note: Award (A1) for “gradient (of curve)”, (A1) for “at the point with x = –1”. Accept “the instantaneous rate of change of y” or “the (first) derivative”.

[2 marks]

\(y = - \frac{16}{3} x + c\)     (M1)

Note: Award (M1) for \(-\frac{16}{3}\) substituted in equation.

\(0 = - \frac{16}{3} \times (-1) + c \)

\(c = - \frac{16}{3}\)

\(y =  - \frac{{16}}{3}x - \frac{{16}}{3}\)     (A1)(G2)

Note: Accept y = –5.33x – 5.33.

OR

\((y - 0) = \frac{{-16}}{3}(x + 1)\)     (M1)(A1)(G2)

Note: Award (M1) for \( - \frac{{16}}{3}\) substituted in equation, (A1) for correct equation. Follow through from their answer to part (b). Accept y = –5.33 (x +1). Accept equivalent equations.

[2 marks]

(A1)(ft) for a tangent to their curve drawn.

(A1)(ft) for their tangent drawn at the point x = –1.     (A1)(ft)(A1)(ft)

Note: Follow through from their graph. The tangent must be a straight line otherwise award at most (A0)(A1).

[2 marks]

(i) \(a = \frac{1}{3}\)     (G1)

(ii) \(b = 3\)     (G1)

Note: If a and b are reversed award (A0)(A1).

[2 marks]

f (x) is increasing     (A1)

[1 mark]

This question caused the most difficulty to candidates for two reasons; its content and perhaps lack of time.

Drawing/sketching graphs is perhaps the area of the course that results in the poorest responses. It is also the area of the course that results in the best. It is therefore the area of the course that good teaching can influence the most.

Candidates should:

• Use the correct scale and window. Label the axes.
• Enter the formula into the GDC and use the table function to determine the points to be plotted.
• Refer to the graph on the GDC when drawing the curve.
• Draw a curve rather than line segments; ensure that the curve is smooth.
• Use a pencil rather than a pen so that required changes once further information has been gathered (the turning points, for example) can be made.

This question caused the most difficulty to candidates for two reasons; its content and perhaps lack of time.

Drawing/sketching graphs is perhaps the area of the course that results in the poorest responses. It is also the area of the course that results in the best. It is therefore the area of the course that good teaching can influence the most.

Candidates should:

• Use the correct scale and window. Label the axes.
• Enter the formula into the GDC and use the table function to determine the points to be plotted.
• Refer to the graph on the GDC when drawing the curve.
• Draw a curve rather than line segments; ensure that the curve is smooth.
• Use a pencil rather than a pen so that required changes once further information has been gathered (the turning points, for example) can be made.

In part (b) the answer could have been checked using the table on the GDC.

This question caused the most difficulty to candidates for two reasons; its content and perhaps lack of time.

Drawing/sketching graphs is perhaps the area of the course that results in the poorest responses. It is also the area of the course that results in the best. It is therefore the area of the course that good teaching can influence the most.

Candidates should:

• Use the correct scale and window. Label the axes.
• Enter the formula into the GDC and use the table function to determine the points to be plotted.
• Refer to the graph on the GDC when drawing the curve.
• Draw a curve rather than line segments; ensure that the curve is smooth.
• Use a pencil rather than a pen so that required changes once further information has been gathered (the turning points, for example) can be made.

In part (c) coordinates were required.

This question caused the most difficulty to candidates for two reasons; its content and perhaps lack of time.

Drawing/sketching graphs is perhaps the area of the course that results in the poorest responses. It is also the area of the course that results in the best. It is therefore the area of the course that good teaching can influence the most.

Candidates should:

• Use the correct scale and window. Label the axes.
• Enter the formula into the GDC and use the table function to determine the points to be plotted.
• Refer to the graph on the GDC when drawing the curve.
• Draw a curve rather than line segments; ensure that the curve is smooth.
• Use a pencil rather than a pen so that required changes once further information has been gathered (the turning points, for example) can be made.

The responses to part (d) were generally correct.

This question caused the most difficulty to candidates for two reasons; its content and perhaps lack of time.

Drawing/sketching graphs is perhaps the area of the course that results in the poorest responses. It is also the area of the course that results in the best. It is therefore the area of the course that good teaching can influence the most.

Candidates should:

• Use the correct scale and window. Label the axes.
• Enter the formula into the GDC and use the table function to determine the points to be plotted.
• Refer to the graph on the GDC when drawing the curve.
• Draw a curve rather than line segments; ensure that the curve is smooth.
• Use a pencil rather than a pen so that required changes once further information has been gathered (the turning points, for example) can be made.

The “show that” nature of part (e) meant that the final answer had to be stated.

This question caused the most difficulty to candidates for two reasons; its content and perhaps lack of time.

Drawing/sketching graphs is perhaps the area of the course that results in the poorest responses. It is also the area of the course that results in the best. It is therefore the area of the course that good teaching can influence the most.

Candidates should:

• Use the correct scale and window. Label the axes.
• Enter the formula into the GDC and use the table function to determine the points to be plotted.
• Refer to the graph on the GDC when drawing the curve.
• Draw a curve rather than line segments; ensure that the curve is smooth.
• Use a pencil rather than a pen so that required changes once further information has been gathered (the turning points, for example) can be made.

The interpretive nature of part (f) was not understood by the majority.

This question caused the most difficulty to candidates for two reasons; its content and perhaps lack of time.

Drawing/sketching graphs is perhaps the area of the course that results in the poorest responses. It is also the area of the course that results in the best. It is therefore the area of the course that good teaching can influence the most.

Candidates should:

• Use the correct scale and window. Label the axes.
• Enter the formula into the GDC and use the table function to determine the points to be plotted.
• Refer to the graph on the GDC when drawing the curve.
• Draw a curve rather than line segments; ensure that the curve is smooth.
• Use a pencil rather than a pen so that required changes once further information has been gathered (the turning points, for example) can be made.

This question caused the most difficulty to candidates for two reasons; its content and perhaps lack of time.

Drawing/sketching graphs is perhaps the area of the course that results in the poorest responses. It is also the area of the course that results in the best. It is therefore the area of the course that good teaching can influence the most.

Candidates should:

• Use the correct scale and window. Label the axes.
• Enter the formula into the GDC and use the table function to determine the points to be plotted.
• Refer to the graph on the GDC when drawing the curve.
• Draw a curve rather than line segments; ensure that the curve is smooth.
• Use a pencil rather than a pen so that required changes once further information has been gathered (the turning points, for example) can be made.

This question caused the most difficulty to candidates for two reasons; its content and perhaps lack of time.

Drawing/sketching graphs is perhaps the area of the course that results in the poorest responses. It is also the area of the course that results in the best. It is therefore the area of the course that good teaching can influence the most.

Candidates should:

• Use the correct scale and window. Label the axes.
• Enter the formula into the GDC and use the table function to determine the points to be plotted.
• Refer to the graph on the GDC when drawing the curve.
• Draw a curve rather than line segments; ensure that the curve is smooth.
• Use a pencil rather than a pen so that required changes once further information has been gathered (the turning points, for example) can be made.

Parts (i) and (j) had many candidates floundering; there were few good responses to these parts.

This question caused the most difficulty to candidates for two reasons; its content and perhaps lack of time.

Drawing/sketching graphs is perhaps the area of the course that results in the poorest responses. It is also the area of the course that results in the best. It is therefore the area of the course that good teaching can influence the most.

Candidates should:

• Use the correct scale and window. Label the axes.
• Enter the formula into the GDC and use the table function to determine the points to be plotted.
• Refer to the graph on the GDC when drawing the curve.
• Draw a curve rather than line segments; ensure that the curve is smooth.
• Use a pencil rather than a pen so that required changes once further information has been gathered (the turning points, for example) can be made.

Parts (i) and (j) had many candidates floundering; there were few good responses to these parts.