| Date | May 2011 | Marks available | 2 | Reference code | 11M.1.sl.TZ2.11 |
| Level | SL only | Paper | 1 | Time zone | TZ2 |
| Command term | Calculate | Question number | 11 | Adapted from | N/A |
Question
The figure shows the graphs of the functions \(f(x) = \frac{1}{4}{x^2} - 2\) and \(g(x) = x\) .
Differentiate \(f(x)\) with respect to \(x\) .
Differentiate \(g(x)\) with respect to \(x\) .
Calculate the value of \(x\) for which the gradients of the two graphs are the same.
Draw the tangent to the parabola at the point with the value of \(x\) found in part (c).
Markscheme
\(\frac{1}{2}x{\text{ }}\left( {\frac{2}{4}x} \right)\) (A1) (C1)
Note: Accept an equivalent, unsimplified expression (i.e. \(2 \times \frac{1}{4}x\)).
[1 mark]
\(1\) (A1) (C1)
[1 mark]
\(\frac{1}{2}x = 1\) (M1)
\(x = 2\) (A1)(ft) (C2)
Notes: Award (M1)(A0) for coordinate pair \((2{\text{, }} - 1)\) seen with or without working. Follow through from their answers to parts (a) and (b).
[2 marks]
tangent drawn to the parabola at the \(x\)-coordinate found in part (c) (A1)(ft)
candidate’s attempted tangent drawn parallel to the graph of \(g(x)\) (A1)(ft) (C2)
[2 marks]
Examiners report
Parts (a) and (b) were reasonably well attempted indicating that candidates are well drilled in the process of differentiation. Correct answers however in part (c) proved elusive to many as frequent attempts to equate the two given functions rather than the gradients of the given functions resulted in a popular, but incorrect, answer of \(x = - 1.46\) . Part (d) was poorly attempted with many candidates simply either not attempting to draw a tangent or drawing it in the wrong place.
Parts (a) and (b) were reasonably well attempted indicating that candidates are well drilled in the process of differentiation. Correct answers however in part (c) proved elusive to many as frequent attempts to equate the two given functions rather than the gradients of the given functions resulted in a popular, but incorrect, answer of \(x = - 1.46\) . Part (d) was poorly attempted with many candidates simply either not attempting to draw a tangent or drawing it in the wrong place.
Parts (a) and (b) were reasonably well attempted indicating that candidates are well drilled in the process of differentiation. Correct answers however in part (c) proved elusive to many as frequent attempts to equate the two given functions rather than the gradients of the given functions resulted in a popular, but incorrect, answer of \(x = - 1.46\) . Part (d) was poorly attempted with many candidates simply either not attempting to draw a tangent or drawing it in the wrong place.
Parts (a) and (b) were reasonably well attempted indicating that candidates are well drilled in the process of differentiation. Correct answers however in part (c) proved elusive to many as frequent attempts to equate the two given functions rather than the gradients of the given functions resulted in a popular, but incorrect, answer of \(x = - 1.46\) . Part (d) was poorly attempted with many candidates simply either not attempting to draw a tangent or drawing it in the wrong place.
