Loading [MathJax]/jax/output/CommonHTML/fonts/TeX/fontdata.js

User interface language: English | Español

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(ex1) .

(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.

Syllabus sections

Topic 9 - Option: Calculus » 9.7 » The evaluation of limits of the form \mathop {\lim }\limits_{x \to a} \frac{{f\left( x \right)}}{{g\left( x \right)}} and \mathop {\lim }\limits_{x \to \infty } \frac{{f\left( x \right)}}{{g\left( x \right)}} .

View options