| Date | May 2007 | Marks available | 3 | Reference code | 07M.1.sl.TZ0.11 |
| Level | SL only | Paper | 1 | Time zone | TZ0 |
| Command term | Differentiate | Question number | 11 | Adapted from | N/A |
Question
The figure below shows the graphs of functions \(f_1 (x) = x\) and \(f_2 (x) = 5 - x^2\).
(i) Differentiate \(f_1 (x) \) with respect to x.
(ii) Differentiate \(f_2 (x) \) with respect to x.
Calculate the value of x for which the gradient of the two graphs is the same.
Draw the tangent to the curved graph for this value of x on the figure, showing clearly the property in part (b).
Markscheme
(i) \(f_1 ' (x) = 1\) (A1)
(ii) \(f_2 ' (x) = - 2x\) (A1)(A1)
(A1) for correct differentiation of each term. (C3)
[3 marks]
\(1 = - 2x\) (M1)
\(x = - \frac{1}{2}\) (A1)(ft) (C2)
[2 marks]
(A1) is for the tangent drawn at \(x = \frac{1}{2}\) and reasonably parallel to the line \(f_1\) as shown.
(A1) (C1)
[1 mark]
Examiners report
Most candidates were able to differentiate correctly, but only a third were able to calculate the value of x for which the gradients of the graphs were the same and a similar number did not attempt to. Some found the x-coordinate of the point of intersection.
Most candidates were able to differentiate correctly, but only a third were able to calculate the value of x for which the gradients of the graphs were the same and a similar number did not attempt to. Some found the x-coordinate of the point of intersection.
c) Very few candidates were able to draw the tangent correctly. Some tangents were drawn horizontally and some at the point of intersection. The line could have been drawn without any knowledge of calculus so the indication here was that many of the candidates misunderstood the question.
