Date | November 2009 | Marks available | 10 | Reference code | 09N.3ca.hl.TZ0.2 |
Level | HL only | Paper | Paper 3 Calculus | Time zone | TZ0 |
Command term | Find, Show that, and Hence or otherwise | Question number | 2 | Adapted from | N/A |
Question
The function f is defined by f(x)=e(ex−1) .
(a) Assuming the Maclaurin series for ex , show that the Maclaurin series for f(x)
is 1+x+x2+56x3+… .
(b) Hence or otherwise find the value of lim .
Markscheme
(a) {{\text{e}}^x} - 1 = x + \frac{{{x^2}}}{2} + \frac{{{x^2}}}{6} + \ldots A1
{{\text{e}}^{{{\text{e}}^x} - 1}} = 1 + \left( {x + \frac{{{x^2}}}{2} + \frac{{{x^3}}}{6}} \right) + \frac{{{{\left( {x + \frac{{{x^2}}}{2} + \frac{{{x^3}}}{6}} \right)}^2}}}{2} + \frac{{{{\left( {x + \frac{{{x^2}}}{2} + \frac{{{x^3}}}{6}} \right)}^3}}}{6} + \ldots M1A1
= 1 + x + \frac{{{x^2}}}{2} + \frac{{{x^3}}}{6} + \frac{{{x^2}}}{2} + \frac{{{x^3}}}{2} + \frac{{{x^3}}}{6} + \ldots M1A1
= 1 + x + {x^2} + \frac{5}{6}{x^3} + \ldots AG
[5 marks]
(b) EITHER
f'(x) = 1 + 2x + \frac{{5{x^2}}}{2} + \ldots A1
\frac{{f(x) - 1}}{{f'(x) - 1}} = \frac{{x + {x^2} + 5{x^3}/6 + \ldots }}{{2x + 5{x^2}/2 + \ldots }} M1A1
= \frac{{1 + x + \ldots }}{{2 + 5x/2 + \ldots }} A1
\to \frac{1}{2}{\text{ as }}x \to 0 A1
[5 marks]
OR
using l’Hopital’s rule, M1
\mathop {\lim }\limits_{x \to 0} \frac{{{{\text{e}}^{({{\text{e}}^x} - 1)}} - 1}}{{{{\text{e}}^{({{\text{e}}^x} - 1)}} - 1' - 1}} = \mathop {\lim }\limits_{x \to 0} \frac{{{{\text{e}}^{({{\text{e}}^x} - 1)}} - 1}}{{{{\text{e}}^{({{\text{e}}^x} + x - 1)}} - 1}} M1A1
= \mathop {\lim }\limits_{x \to 0} \frac{{{{\text{e}}^{({{\text{e}}^x} + x - 1)}}}}{{{{\text{e}}^{({{\text{e}}^x} + x - 1)}} \times ({{\text{e}}^x} + 1)}} A1
= \frac{1}{2} A1
[5 marks]
Total [10 marks]
Examiners report
Many candidates obtained the required series by finding the values of successive derivatives at x = 0 , failing to realise that the intention was to start with the exponential series and replace x by the series for {{\text{e}}^x} - 1. Candidates who did this were given partial credit for using this method. Part (b) was reasonably well answered using a variety of methods.