IB DP Maths Topic 9.7 The evaluation of limits of the form limx→af(x)/g(x) and limx→∞f(x)/g(x) . HL Paper 3

 

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Question

Find the value of \(\mathop {\lim }\limits_{x \to 1} \left( {\frac{{\ln x}}{{\sin 2\pi x}}} \right)\).

[3]
a.

By using the series expansions for \({{\text{e}}^{{x^2}}}\) and cos x evaluate \(\mathop {\lim }\limits_{x \to 0} \left( {\frac{{1 – {{\text{e}}^{{x^2}}}}}{{1 – \cos x}}} \right).\).

[7]
b.
Answer/Explanation

Markscheme

Using l’Hopital’s rule,

\(\mathop {\lim }\limits_{x \to 1} \left( {\frac{{\ln x}}{{\sin 2\pi x}}} \right) = \mathop {\lim }\limits_{x \to 1} \left( {\frac{{\frac{1}{x}}}{{2\pi \cos 2\pi x}}} \right)\)     M1A1

\( = \frac{1}{{2\pi }}\)     A1

[3 marks]

a.

\(\mathop {\lim }\limits_{x \to 0} \left( {\frac{{1 – {{\text{e}}^{{x^2}}}}}{{1 – \cos x}}} \right) = \mathop {\lim }\limits_{x \to 0} \left( {\frac{{1 – \left( {1 + {x^2} + \frac{{{x^4}}}{{2!}} + \frac{{{x^6}}}{{3!}} + …} \right)}}{{1 – \left( {1 – \frac{{{x^2}}}{{2!}} + \frac{{{x^4}}}{{4!}} – …} \right)}}} \right)\)     M1A1A1

Note: Award M1 for evidence of using the two series.

 

\( = \mathop {\lim }\limits_{x \to 0} \left( {\frac{{\left( { – {x^2} – \frac{{{x^4}}}{{2!}} – \frac{{{x^6}}}{{3!}} – …} \right)}}{{\left( {\frac{{{x^2}}}{{2!}} – \frac{{{x^4}}}{{4!}} + …} \right)}}} \right)\)     A1

EITHER

\( = \mathop {\lim }\limits_{x \to 0} \left( {\frac{{\left( { – 1 – \frac{{{x^2}}}{{2!}} – \frac{{{x^4}}}{{3!}} – …} \right)}}{{\left( {\frac{1}{{2!}} – \frac{{{x^2}}}{{4!}} + …} \right)}}} \right)\)     M1A1

\( = \frac{{ – 1}}{{\frac{1}{2}}} = – 2\)     A1

OR

\( = \mathop {\lim }\limits_{x \to 0} \left( {\frac{{\left( { – 2x – \frac{{4{x^3}}}{{2!}} – \frac{{6{x^5}}}{{3!}} – …} \right)}}{{\left( {\frac{{2x}}{{2!}} – \frac{{4{x^3}}}{{4!}} + …} \right)}}} \right)\)     M1A1

\( = \mathop {\lim }\limits_{x \to 0} \left( {\frac{{\left( { – 2 – \frac{{4{x^2}}}{{2!}} – \frac{{6{x^4}}}{{3!}} – …} \right)}}{{\left( {1 – \frac{{4{x^2}}}{{4!}} + …} \right)}}} \right)\)

\( = \frac{{ – 2}}{1} = – 2\)     A1

[7 marks]

b.

Examiners report

Part (a) was well done but too often the instruction to use series in part (b) was ignored. When this hint was observed correct solutions followed. 

a.

Part (a) was well done but too often the instruction to use series in part (b) was ignored. When this hint was observed correct solutions followed.

b.

Question

The function f is defined by \(f(x) = {{\text{e}}^{({{\text{e}}^x} – 1)}}\) .

(a)     Assuming the Maclaurin series for \({{\text{e}}^x}\) , show that the Maclaurin series for \(f(x)\)

is \(1 + x + {x^2} + \frac{5}{6}{x^3} + \ldots {\text{ .}}\)

(b)     Hence or otherwise find the value of \(\mathop {\lim }\limits_{x \to 0} \frac{{f(x) – 1}}{{f'(x) – 1}}\) .

Answer/Explanation

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.

Question

(a)     Using the Maclaurin series for \({(1 + x)^n}\), write down and simplify the Maclaurin series approximation for \({(1 – {x^2})^{ – \frac{1}{2}}}\) as far as the term in \({x^4}\)

(b)     Use your result to show that a series approximation for arccos x is

\[\arccos x \approx \frac{\pi }{2} – x – \frac{1}{6}{x^3} – \frac{3}{{40}}{x^5}.\]

(c)     Evaluate \(\mathop {\lim }\limits_{x \to 0} \frac{{\frac{\pi }{2} – \arccos ({x^2}) – {x^2}}}{{{x^6}}}\).

(d)     Use the series approximation for \(\arccos x\) to find an approximate value for

\[\int_0^{0.2} {\arccos \left( {\sqrt x } \right){\text{d}}x} ,\]

giving your answer to 5 decimal places. Does your answer give the actual value of the integral to 5 decimal places?

Answer/Explanation

Markscheme

(a)     using or obtaining \({(1 + x)^n} = 1 + nx + \frac{{n(n – 1)}}{2}{x^2} + \ldots \)     (M1)

\({(1 – {n^2})^{ – \frac{1}{2}}} = 1 + ( – {x^2}) \times \left( { – \frac{1}{2}} \right) + \frac{{{{( – {x^2})}^2}}}{2} \times \left( { – \frac{1}{2}} \right) \times \left( { – \frac{3}{2}} \right) + \ldots \)     (A1)

\( = 1 + \frac{1}{2}{x^2} + \frac{3}{8}{x^4} + \ldots \)     A1

[3 marks]

 

(b)     integrating, and changing sign

\(\arccos x = – x – \frac{1}{6}{x^3} – \frac{3}{{40}}{x^5} + C + \ldots \)     M1A1

put x = 0,

\(\frac{\pi }{2} = C\)     M1

\(\left( {\arccos x \approx \frac{\pi }{2} – x – \frac{1}{6}{x^3} – \frac{3}{{40}}{x^5}} \right)\)     AG

[3 marks]

 

(c)     EITHER

using \(\arccos {x^2} \approx \frac{\pi }{2} – {x^2} – \frac{1}{6}{x^6} – \frac{3}{{40}}{x^{10}}\)     M1A1

\(\mathop {\lim }\limits_{x \to 0} \frac{{\frac{\pi }{2} – \arccos {x^2} – {x^2}}}{{{x^6}}} = \mathop {\lim }\limits_{x \to 0} \frac{{\frac{{{x^6}}}{6} + {\text{higher powers}}}}{{{x^6}}}\)     M1A1

\( = \frac{1}{6}\)     A1

OR

using l’Hôpital’s Rule     M1

\({\text{limit}} = \mathop {\lim }\limits_{x \to 0} \frac{{\frac{1}{{\sqrt {1 – {x^4}} }} \times 2x – 2x}}{{6{x^5}}}\)     M1

\( = \mathop {\lim }\limits_{x \to 0} \frac{{\frac{1}{{\sqrt {1 – {x^4}} }} – 1}}{{3{x^4}}}\)     A1

\( = \mathop {\lim }\limits_{x \to 0} \frac{{ – \frac{1}{2} \times \frac{1}{{{{(1 – {x^4})}^{3/2}}}} \times – 4{x^3}}}{{12{x^3}}}\)     M1

\( = \frac{1}{6}\)     A1

[5 marks]

 

(d)     \(\int_0^{0.2} {\arccos \sqrt x {\text{d}}x \approx \int_0^{0.2} {\left( {\frac{\pi }{2} – {x^{\frac{1}{2}}} – \frac{1}{6}{x^{\frac{3}{2}}} – \frac{3}{{40}}{x^{\frac{5}{2}}}} \right){\text{d}}x} } \)     M1

\( = \left[ {\frac{\pi }{2}x – \frac{2}{3}{x^{\frac{3}{2}}} – \frac{1}{{15}}{x^{\frac{5}{2}}} – \frac{3}{{140}}{x^{\frac{7}{2}}}} \right]_0^{0.2}\)     (A1)

\( = \frac{\pi }{2} \times 0.2 – \frac{2}{3} \times {0.2^{\frac{3}{2}}} – \frac{1}{{15}} \times {0.2^{\frac{5}{2}}} – \frac{3}{{140}} \times {0.2^{\frac{7}{2}}}\)     (A1)

= 0.25326 (to 5 decimal places)     A1

Note: Accept integration of the series approximation using a GDC.

 

using a GDC, the actual value is 0.25325     A1

so the approximation is not correct to 5 decimal places     R1

[6 marks]

Total [17 marks]

Examiners report

Many candidates ignored the instruction in the question to use the series for \({(1 + x)^n}\) to deduce the series for \({(1 – {x^2})^{ – 1/2}}\) and attempted instead to obtain it by successive differentiation. It was decided at the standardisation meeting to award full credit for this method although in the event the algebra proved to be too difficult for many. Many candidates used l’Hopital’s Rule in (c) – this was much more difficult algebraically than using the series and it usually ended unsuccessfully. Candidates should realise that if a question on evaluating an indeterminate limit follows the determination of a Maclaurin series then it is likely that the series will be helpful in evaluating the limit. Part (d) caused problems for many candidates with algebraic errors being common. Many candidates failed to realise that the best way to find the exact value of the integral was to use the calculator.

Question

Find \(\mathop {\lim }\limits_{x \to 0} \left( {\frac{{1 – \cos {x^6}}}{{{x^{12}}}}} \right)\).

Answer/Explanation

Markscheme

METHOD 1

\(f(0) = \frac{0}{0}\), hence using l’Hôpital’s Rule,     (M1)

\(g(x) = 1 – \cos ({x^6}),{\text{ }}h(x) = {x^{12}};{\text{ }}\frac{{g'(x)}}{{h'(x)}} = \frac{{6{x^5}\sin ({x^6})}}{{12{x^{11}}}} = \frac{{\sin ({x^6})}}{{2{x^6}}}\)     A1A1

EITHER

\(\frac{{g'(0)}}{{h'(0)}} = \frac{0}{0}\), using l’Hôpital’s Rule again,     (M1)

\(\frac{{g”(x)}}{{h”(x)}} = \frac{{6{x^5}\cos ({x^6})}}{{12{x^5}}} = \frac{{\cos ({x^6})}}{2}\)     A1A1

\(\frac{{g”(0)}}{{h”(0)}} = \frac{1}{2}\), hence the limit is \(\frac{1}{2}\)     A1

OR

So \(\mathop {\lim }\limits_{x \to 0} \frac{{1 – \cos {x^6}}}{{{x^{12}}}} = \mathop {\lim }\limits_{x \to 0} \frac{{\sin {x^6}}}{{2{x^6}}}\)     A1

\( = \frac{1}{2}\mathop {\lim }\limits_{x \to 0} \frac{{\sin {x^6}}}{{2{x^6}}}\)     A1

\( = \frac{1}{2}{\text{ since }}\mathop {\lim }\limits_{x \to 0} \frac{{\sin {x^6}}}{{2{x^6}}} = 1\)     A1 (R1)

METHOD 2

substituting \({{x^6}}\) for x in the expansion \(\cos x = 1 – \frac{{{x^2}}}{2} + \frac{{{x^4}}}{{24}} \ldots \)     (M1)

\(\frac{{1 – \cos {x^6}}}{{{x^{12}}}} = \frac{{1 – \left( {1 – \frac{{{x^{12}}}}{2} + \frac{{{x^{24}}}}{{24}}} \right) \ldots }}{{{x^{12}}}}\)     M1A1

\( = \frac{1}{2} – \frac{{{x^{12}}}}{{24}} + …\)     A1A1

\(\mathop {\lim }\limits_{x \to 0} \frac{{1 – \cos {x^6}}}{{{x^{12}}}} = \frac{1}{2}\)     M1A1

Note: Accept solutions using Maclaurin expansions.

 

[7 marks]

Examiners report

Surprisingly, some weaker candidates were more successful in answering this question than stronger candidates. If candidates failed to simplify the expression after the first application of L’Hôpital’s rule, they generally were not successful in correctly differentiating the expression a \({2^{{\text{nd}}}}\) time, hence could not achieve the final three A marks.

Question

Find the first three terms of the Maclaurin series for \(\ln (1 + {{\text{e}}^x})\) .

[6]
a.

Hence, or otherwise, determine the value of \(\mathop {\lim }\limits_{x \to 0} \frac{{2\ln (1 + {{\text{e}}^x}) – x – \ln 4}}{{{x^2}}}\) .

[4]
b.
Answer/Explanation

Markscheme

METHOD 1

\(f(x) = \ln (1 + {{\text{e}}^x});{\text{ }}f(0) = \ln 2\)     A1

\(f'(x) = \frac{{{{\text{e}}^x}}}{{1 + {{\text{e}}^x}}};{\text{ }}f'(0) = \frac{1}{2}\)     A1

Note: Award A0 for \(f'(x) = \frac{1}{{1 + {{\text{e}}^x}}};{\text{ }}f'(0) = \frac{1}{2}\)

 

\(f”(x) = \frac{{{{\text{e}}^x}(1 + {{\text{e}}^x}) – {{\text{e}}^{2x}}}}{{{{(1 + {{\text{e}}^x})}^2}}};{\text{ }}f”(0) = \frac{1}{4}\)     M1A1

Note: Award M0A0 for \(f”(x){\text{ if }}f'(x) = \frac{1}{{1 + {{\text{e}}^x}}}\) is used

 

\(\ln (1 + {{\text{e}}^x}) = \ln 2 + \frac{1}{2}x + \frac{1}{8}{x^2} + …\)     M1A1

[6 marks]

METHOD 2

\(\ln (1 + {{\text{e}}^x}) = \ln (1 + 1 + x + \frac{1}{2}{x^2} + …)\)     M1A1

\( = \ln 2 + \ln (1 + \frac{1}{2}x + \frac{1}{4}{x^2} + …)\)     A1

\( = \ln 2 + \left( {\frac{1}{2}x + \frac{1}{4}{x^2} + …} \right) – \frac{1}{2}{\left( {\frac{1}{2}x + \frac{1}{4}{x^2} + …} \right)^2} + …\)     A1

\( = \ln 2 + \frac{1}{2}x + \frac{1}{4}{x^2} – \frac{1}{8}{x^2} + …\)     A1

\( = \ln 2 + \frac{1}{2}x + \frac{1}{8}{x^2} + …\)     A1

[6 marks]

a.

METHOD 1

\(\mathop {\lim }\limits_{x \to 0} \frac{{2\ln (1 + {{\text{e}}^x}) – x – \ln 4}}{{{x^2}}} = \mathop {\lim }\limits_{x \to 0} \frac{{2\ln 2 + x + \frac{{{x^2}}}{4} + {x^3}{\text{ terms &  above}} – x – \ln 4}}{{{x^2}}}\)     M1A1

\( = \mathop {\lim }\limits_{x \to 0} \left( {\frac{1}{4} + {\text{powers of }}x} \right) = \frac{1}{4}\)     M1A1

Note: Accept + … as evidence of recognition of cubic and higher powers.

Note: Award M1AOM1A0 for a solution which omits the cubic and higher powers.

[4 marks]

 

METHOD 2

using l’Hôpital’s Rule

\(\mathop {\lim }\limits_{x \to 0} \frac{{2\ln (1 + {{\text{e}}^x}) – x – \ln 4}}{{{x^2}}} = \mathop {\lim }\limits_{x \to 0} \frac{{2{{\text{e}}^x} \div (1 + {{\text{e}}^x}) – 1}}{{2x}}\)     M1A1

\( = \mathop {\lim }\limits_{x \to 0} \frac{{2{{\text{e}}^x} \div {{(1 + {{\text{e}}^x})}^2}}}{2} = \frac{1}{4}\)     M1A1

[4 marks]

b.

Examiners report

In (a), candidates who found the series by successive differentiation were generally successful, the most common error being to state that the derivative of \(\ln (1 + {{\text{e}}^x})\) is \({(1 + {{\text{e}}^x})^{ – 1}}\). Some candidates assumed the series for \(\ln (1 + x)\) and \({{\text{e}}^x}\) attempted to combine them. This was accepted as an alternative solution but candidates using this method were often unable to obtain the required series.

a.

In (b), candidates were equally split between using the series or using l’Hopital’s rule to find the limit. Both methods were fairly successful, but a number of candidates forgot that if a series was used, there had to be a recognition that it was not a finite series. 

b.

Question

Find \(\mathop {\lim }\limits_{x \to \frac{1}{2}} \left( {\frac{{\left( {\frac{1}{4} – {x^2}} \right)}}{{\cot \pi x}}} \right)\).

Answer/Explanation

Markscheme

using l’Hôpital’s Rule     (M1)

\(\mathop {\lim }\limits_{x \to \frac{1}{2}} \left( {\frac{{\left( {\frac{1}{4} – {x^2}} \right)}}{{\cot \pi x}}} \right) = \mathop {\lim }\limits_{x \to \frac{1}{2}} \left[ {\frac{{ – 2}}{{ – \pi {\text{cose}}{{\text{c}}^2}\pi x}}} \right]\)     A1A1

\( = \frac{{ – 1}}{{ – \pi {\text{cose}}{{\text{c}}^2}\frac{\pi }{2}}} = \frac{1}{\pi }\)     (M1)A1

[5 marks]

Examiners report

This question was accessible to the vast majority of candidates, who recognised that L’Hôpital’s rule was required. However, some candidates omitted the factor \(\pi \) in the differentiation of \({\cot \pi x}\). Some candidates replaced \({\cot \pi x}\) by \(\cos \pi x{\text{/}}\sin \pi x\), which is a valid method but the extra algebra involved often led to an incorrect answer. Many fully correct solutions were seen.

Question

Use L’Hôpital’s Rule to find \(\mathop {\lim }\limits_{x \to 0} \frac{{{{\text{e}}^x} – 1 – x\cos x}}{{{{\sin }^2}x}}\) .

Answer/Explanation

Markscheme

apply l’Hôpital’s Rule to a \(0/0\) type limit

\(\mathop {\lim }\limits_{x \to 0} \frac{{{{\text{e}}^x} – 1 – x\cos x}}{{{{\sin }^2}x}} = \mathop {\lim }\limits_{x \to 0} \frac{{{{\text{e}}^x} – \cos x + x\sin x}}{{2\sin x\cos x}}\)     M1A1

noting this is also a \(0/0\) type limit, apply l’Hôpital’s Rule again     (M1)

obtain \(\mathop {\lim }\limits_{x \to 0} \frac{{{{\text{e}}^x} + \sin x + x\cos x + \sin x}}{{2\cos 2x}}\)     A1

substitution of x = 0     (M1)

= 0.5     A1

[6 marks]

Examiners report

The vast majority of candidates were familiar with L’Hôpitals rule and were also able to apply the technique twice as required by the problem. The errors that occurred were mostly due to difficulty in applying the differentiation rules correctly or errors in algebra. A small minority of candidates tried to use the quotient rule but it seemed that most candidates had a good understanding of L’Hôpital’s rule and its application to finding a limit.

Question

The Taylor series of \(\sqrt x \) about x = 1 is given by

\[{a_0} + {a_1}(x – 1) + {a_2}{(x – 1)^2} + {a_3}{(x – 1)^3} +  \ldots \]

Find the values of \({a_0},{\text{ }}{a_1},{\text{ }}{a_2}\) and \({a_3}\).

[6]
a.

Hence, or otherwise, find the value of \(\mathop {\lim }\limits_{x \to 1} \frac{{\sqrt x  – 1}}{{x – 1}}\).

[3]
b.
Answer/Explanation

Markscheme

let \(f(x) = \sqrt x ,{\text{ }}f(1) = 1\)     (A1)

\(f'(x) = \frac{1}{2}{x^{ – \frac{1}{2}}},{\text{ }}f'(1) = \frac{1}{2}\)     (A1)

\(f”(x) = – \frac{1}{4}{x^{ – \frac{3}{2}}},{\text{ }}f”(1) = – \frac{1}{4}\)     (A1)

\(f”'(x) = \frac{3}{8}{x^{ – \frac{5}{2}}},{\text{ }}f”'(1) = \frac{3}{8}\)     (A1)

\({a_1} = \frac{1}{2} \cdot \frac{1}{{1!}},{\text{ }}{a_2} = – \frac{1}{4} \cdot \frac{1}{{2!}},{\text{ }}{a_3} = \frac{3}{8} \cdot \frac{1}{{3!}}\)     (M1)

\({a_0} = 1,{\text{ }}{a_1} = \frac{1}{2},{\text{ }}{a_2} = – \frac{1}{8},{\text{ }}{a_3} = \frac{1}{{16}}\)     A1

Note: Accept \(y = 1 + \frac{1}{2}(x – 1) – \frac{1}{8}{(x – 1)^2} + \frac{1}{{16}}{(x – 1)^3} +  \ldots \)

 

[6 marks]

a.

METHOD 1

\(\mathop {\lim }\limits_{x \to 1} \frac{{\sqrt x  – 1}}{{x – 1}} = \mathop {\lim }\limits_{x \to 1} \frac{{\frac{1}{2}(x – 1) – \frac{1}{8}{{(x – 1)}^2} +  \ldots }}{{x – 1}}\)     M1

\( = \mathop {\lim }\limits_{x \to 1} \left( {\frac{1}{2} – \frac{1}{8}(x – 1) +  \ldots } \right)\)     A1

\( = \frac{1}{2}\)     A1

 

METHOD 2

using l’Hôpital’s rule,     M1

\(\mathop {\lim }\limits_{x \to 1} \frac{{\sqrt x  – 1}}{{x – 1}} = \mathop {\lim }\limits_{x \to 1} \frac{{\frac{1}{2}{x^{ – \frac{1}{2}}}}}{1}\)     A1

\( = \frac{1}{2}\)     A1

 

METHOD 3

\(\frac{{\sqrt x  – 1}}{{x + 1}} = \frac{1}{{\sqrt x  + 1}}\)     M1A1

\(\mathop {\lim }\limits_{x \to 1} \frac{1}{{\sqrt x  + 1}} = \frac{1}{2}\)     A1

 

[3 marks]

b.

Examiners report

Many candidates achieved full marks on this question but there were still a large minority of candidates who did not seem familiar with the application of Taylor series. Whilst all candidates who responded to this question were aware of the need to use derivatives many did not correctly use factorials to find the required coefficients. It should be noted that the formula for Taylor series appears in the Information Booklet.

a.

Many candidates achieved full marks on this question but there were still a large minority of candidates who did not seem familiar with the application of Taylor series. Whilst all candidates who responded to this question were aware of the need to use derivatives many did not correctly use factorials to find the required coefficients. It should be noted that the formula for Taylor series appears in the Information Booklet.

b.

Question

Let \(g(x) = \sin {x^2}\), where \(x \in \mathbb{R}\).

Using the result \(\mathop {{\text{lim}}}\limits_{t \to 0} \frac{{\sin t}}{t} = 1\), or otherwise, calculate \(\mathop {{\text{lim}}}\limits_{x \to 0} \frac{{g(2x) – g(3x)}}{{4{x^2}}}\).

[4]
a.

Use the Maclaurin series of \(\sin x\) to show that \(g(x) = \sum\limits_{n = 0}^\infty  {{{( – 1)}^n}\frac{{{x^{4n + 2}}}}{{(2n + 1)!}}} \)

[2]
b.

Hence determine the minimum number of terms of the expansion of \(g(x)\) required to approximate the value of \(\int_0^1 {g(x){\text{d}}x} \) to four decimal places.

[7]
c.
Answer/Explanation

Markscheme

METHOD 1

\(\mathop {{\text{lim}}}\limits_{x \to 0} \frac{{\sin 4{x^2} – \sin 9{x^2}}}{{4{x^2}}}\)     M1

\( = \mathop {{\text{lim}}}\limits_{x \to 0} \frac{{\sin 4{x^2}}}{{4{x^2}}} – \frac{9}{4}\mathop {{\text{lim}}}\limits_{x \to 0} \frac{{\sin 9{x^2}}}{{9{x^2}}}\)     A1A1

\( = 1 – \frac{9}{4} \times 1 =  – \frac{5}{4}\)     A1

METHOD 2

\(\mathop {{\text{lim}}}\limits_{x \to 0} \frac{{\sin 4{x^2} – \sin 9{x^2}}}{{4{x^2}}}\)     M1

\( = \mathop {{\text{lim}}}\limits_{x \to 0} \frac{{8x\cos 4{x^2} – 18x\cos 9{x^2}}}{{8x}}\)     M1A1

\( = \frac{{8 – 18}}{8} =  – \frac{{10}}{8} =  – \frac{5}{4}\)     A1

[4 marks]

a.

since \(\sin x = \sum\limits_{n = 0}^\infty  {{{( – 1)}^n}\frac{{{x^{(2n + 1)}}}}{{(2n + 1)!}}} \)   \(\left( {{\text{or }}\sin x = \frac{x}{{1!}} – \frac{{{x^3}}}{{3!}} + \frac{{{x^5}}}{{5!}} –  \ldots } \right)\)     (M1)

\(\sin {x^2} = \sum\limits_{n = 0}^\infty  {{{( – 1)}^n}\frac{{{x^{2(2n + 1)}}}}{{(2n + 1)!}}} \)   \(\left( {{\text{or }}\sin x = \frac{{{x^2}}}{{1!}} – \frac{{{x^6}}}{{3!}} + \frac{{{x^{10}}}}{{5!}} –  \ldots } \right)\)     A1

\(g(x) = \sin {x^2} = \sum\limits_{n = 0}^\infty  {{{( – 1)}^n}\frac{{{x^{4n + 2}}}}{{(2n + 1)!}}} \)     AG

[2 marks]

b.

let \(I = \int_0^1 {\sin {x^2}{\text{d}}x} \)

\( = \sum\limits_{n = 0}^\infty  {{{( – 1)}^n}\frac{1}{{(2n + 1)!}}} \int_0^1 {{x^{4n + 2}}{\text{d}}x{\text{ }}\left( {\int_0^1 {\frac{{{x^2}}}{{1!}}{\text{d}}x – } \int_0^1 {\frac{{{x^6}}}{{3!}}{\text{d}}x + } \int_0^1 {\frac{{{x^{10}}}}{{5!}}{\text{d}}x –  \ldots } } \right)} \)     M1

\( = \sum\limits_{n = 0}^\infty  {{{( – 1)}^n}\frac{1}{{(2n + 1)!}}} \frac{{[{x^{4n + 3}}]_0^1}}{{(4n + 3)}}{\text{ }}\left( {\left[ {\frac{{{x^3}}}{{3 \times 1!}}} \right]_0^1 – \left[ {\frac{{{x^7}}}{{7 \times 3!}}} \right]_0^1 + \left[ {\frac{{{x^{11}}}}{{11 \times 5!}}} \right]_0^1 –  \ldots } \right)\)     M1

\( = \sum\limits_{n = 0}^\infty  {{{( – 1)}^n}\frac{1}{{(2n + 1)!(4n + 3)}}} {\text{ }}\left( {\frac{1}{{3 \times 1!}} – \frac{1}{{7 \times 3!}} + \frac{1}{{11 \times 5!}} –  \ldots } \right)\)     A1

\( = \sum\limits_{n = 0}^\infty  {{{( – 1)}^n}{a_n}} \) where \({a_n} = \frac{1}{{(4n + 3)(2n + 1)!}} > 0\) for all \(n \in \mathbb{N}\)

as \(\{ {a_n}\} \) is decreasing the sum of the alternating series \(\sum\limits_{n = 0}^\infty  {{{( – 1)}^n}{a_n}} \)

lies between \(\sum\limits_{n = 0}^N {{{( – 1)}^n}{a_n}} \) and \(\sum\limits_{n = 0}^N {{{( – 1)}^n}{a_n}}  \pm {a_{N + 1}}\)     R1

hence for four decimal place accuracy, we need \(\left| {{a_{N + 1}}} \right| < 0.00005\)     M1



 

since \({a_{2 + 1}} < 0.00005\)     R1

so \(N = 2\)   (or 3 terms)     A1

[7 marks]

c.

Examiners report

Part (a) of this question was accessible to the vast majority of candidates, who recognised that L’Hôpital’s rule could be used. Most candidates were successful in finding the limit, with some making calculation errors. Candidates that attempted to use \(\mathop {{\text{lim}}}\limits_{x \to 0} \frac{{\sin x}}{x} = 1\) or a combination of this result and L’Hôpital’s rule were less successful.

a.

In part (b) most candidates showed to be familiar with the substitution given and were successful in showing the result.

b.

Very few candidates were able to do part (c) successfully. Most used trial and error to arrive at the answer.

c.

Question

Consider the functions \(f\) and \(g\) given by \(f(x) = \frac{{{{\text{e}}^x} + {{\text{e}}^{ – x}}}}{2}{\text{ and }}g(x) = \frac{{{{\text{e}}^x} – {{\text{e}}^{ – x}}}}{2}\).

Show that \(f'(x) = g(x)\) and \(g'(x) = f(x)\).

[2]
a.

Find the first three non-zero terms in the Maclaurin expansion of \(f(x)\).

[5]
b.

Hence find the value of \(\mathop {{\text{lim}}}\limits_{x \to 0} \frac{{1 – f(x)}}{{{x^2}}}\).

[3]
c.

Find the value of the improper integral \(\int_0^\infty  {\frac{{g(x)}}{{{{\left[ {f(x)} \right]}^2}}}{\text{d}}x} \).

[6]
d.
Answer/Explanation

Markscheme

any correct step before the given answer     A1AG

eg, \(f'(x) = \frac{{{{\left( {{{\text{e}}^x}} \right)}^\prime } + {{\left( {{{\text{e}}^{ – x}}} \right)}^\prime }}}{2} = \frac{{{{\text{e}}^x} – {{\text{e}}^{ – x}}}}{2} = g(x)\)

any correct step before the given answer     A1AG

eg, \(g'(x) = \frac{{{{\left( {{{\text{e}}^x}} \right)}^\prime } – {{\left( {{{\text{e}}^{ – x}}} \right)}^\prime }}}{2} = \frac{{{{\text{e}}^x} + {{\text{e}}^{ – x}}}}{2} = f(x)\)

[2 marks]

a.

METHOD 1

statement and attempted use of the general Maclaurin expansion formula     (M1)

\(f(0) = 1;{\text{ }}g(0) = 0\) (or equivalent in terms of derivative values)   A1A1

\(f(x) = 1 + \frac{{{x^2}}}{2} + \frac{{{x^4}}}{{24}}\) or \(f(x) = 1 + \frac{{{x^2}}}{{2!}} + \frac{{{x^4}}}{{4!}}\)     A1A1

METHOD 2

\({{\text{e}}^x} = 1 + x + \frac{{{x^2}}}{{2!}} + \frac{{{x^3}}}{{3!}} + \frac{{{x^4}}}{{4!}} +  \ldots \)     A1

\({{\text{e}}^{ – x}} = 1 – x + \frac{{{x^2}}}{{2!}} – \frac{{{x^3}}}{{3!}} + \frac{{{x^4}}}{{4!}} +  \ldots \)     A1

adding and dividing by 2     M1

\(f(x) = 1 + \frac{{{x^2}}}{2} + \frac{{{x^4}}}{{24}}\) or \(f(x) = 1 + \frac{{{x^2}}}{{2!}} + \frac{{{x^4}}}{{4!}}\)     A1A1

 

Notes: Accept 1, \(\frac{{{x^2}}}{2}\) and \(\frac{{{x^4}}}{{24}}\) or 1, \(\frac{{{x^2}}}{{2!}}\) and \(\frac{{{x^4}}}{{4!}}\).

     Award A1 if two correct terms are seen.

 

[5 marks]

b.

METHOD 1

attempted use of the Maclaurin expansion from (b)     M1

\(\mathop {{\text{lim}}}\limits_{x \to 0} \frac{{1 – f(x)}}{{{x^2}}} = \mathop {{\text{lim}}}\limits_{x \to 0} \frac{{1 – \left( {1 + \frac{{{x^2}}}{2} + \frac{{{x^4}}}{{24}} +  \ldots } \right)}}{{{x^2}}}\)

\(\mathop {{\text{lim}}}\limits_{x \to 0} \left( { – \frac{1}{2} – \frac{{{x^2}}}{{24}} –  \ldots } \right)\)     A1

\( =  – \frac{1}{2}\)     A1

METHOD 2

attempted use of L’Hôpital and result from (a)     M1

\(\mathop {{\text{lim}}}\limits_{x \to 0} \frac{{1 – f(x)}}{{{x^2}}} = \mathop {{\text{lim}}}\limits_{x \to 0} \frac{{ – g(x)}}{{2x}}\)

\(\mathop {{\text{lim}}}\limits_{x \to 0} \frac{{ – f(x)}}{2}\)     A1

\( =  – \frac{1}{2}\)     A1

[3 marks]

c.

METHOD 1

use of the substitution \(u = f(x)\) and \(\left( {{\text{d}}u = g(x){\text{d}}x} \right)\)     (M1)(A1)

attempt to integrate \(\int_1^\infty  {\frac{{{\text{d}}u}}{{{u^2}}}} \)     (M1)

obtain \(\left[ { – \frac{1}{u}} \right]_1^\infty \) or \(\left[ { – \frac{1}{{f(x)}}} \right]_0^\infty \)     A1

recognition of an improper integral by use of a limit or statement saying the integral converges     R1

obtain 1     A1     N0

METHOD 2

\(\int_0^\infty  {\frac{{\frac{{{{\text{e}}^x} – {{\text{e}}^{ – x}}}}{2}}}{{{{\left( {\frac{{{{\text{e}}^x} + {{\text{e}}^{ – x}}}}{2}} \right)}^2}}}{\text{d}}x = \int_0^\infty  {\frac{{2\left( {{{\text{e}}^x} – {{\text{e}}^{ – x}}} \right)}}{{{{\left( {{{\text{e}}^x} + {{\text{e}}^{ – x}}} \right)}^2}}}{\text{d}}x} } \)     (M1)

use of the substitution \(u = {{\text{e}}^x} + {{\text{e}}^{ – x}}\) and \(\left( {{\text{d}}u = {{\text{e}}^x} – {{\text{e}}^{ – x}}{\text{d}}x} \right)\)     (M1)

attempt to integrate \(\int_2^\infty  {\frac{{2{\text{d}}u}}{{{u^2}}}} \)     (M1)

obtain \(\left[ { – \frac{2}{u}} \right]_2^\infty \)     A1

recognition of an improper integral by use of a limit or statement saying the integral converges     R1

obtain 1     A1     N0

[6 marks]

d.

Examiners report

[N/A]

a.

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b.

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c.

[N/A]

d.

Question

Consider the infinite spiral of right angle triangles as shown in the following diagram.

The \(n{\text{th}}\) triangle in the spiral has central angle \({\theta _n}\), hypotenuse of length \({a_n}\) and opposite side of length 1, as shown in the diagram. The first right angle triangle is isosceles with the two equal sides being of length 1.

Consider the series \(\sum\limits_{n = 1}^\infty  {{\theta _n}} \).

Using l’Hôpital’s rule, find \(\mathop {\lim }\limits_{x \to \infty } \left( {\frac{{\arcsin \left( {\frac{1}{{\sqrt {(x + 1)} }}} \right)}}{{\frac{1}{{\sqrt x }}}}} \right)\).

[6]
a.

(i) Find \({a_1}\) and \({a_2}\) and hence write down an expression for \({a_n}\).

(ii) Show that \({\theta _n} = \arcsin \frac{1}{{\sqrt {(n + 1)} }}\).

[3]
b.

Using a suitable test, determine whether this series converges or diverges.

[6]
c.
Answer/Explanation

Markscheme

\(\mathop {\lim }\limits_{x \to \infty } \left( {\frac{{\arcsin \left( {\frac{1}{{\sqrt {(x + 1)} }}} \right)}}{{\frac{1}{{\sqrt x }}}}} \right)\) is of the form \(\frac{0}{0}\)

and so will equal the limit of \(\frac{{\frac{{\frac{{ – 1}}{2}{{(x + 1)}^{ – \frac{3}{2}}}}}{{\sqrt {1 – \left( {\frac{1}{{x + 1}}} \right)} }}}}{{\frac{{ – 1}}{2}{x^{ – \frac{3}{2}}}}}\)     M1M1A1A1

Note: M1 for attempting differentiation of the top and bottom, M1A1 for derivative of top (only award M1 if chain rule is used), A1 for derivative of bottom.

\( = \mathop {\lim }\limits_{x \to \infty } \frac{{{{\left( {\frac{x}{{(x + 1)}}} \right)}^{\frac{3}{2}}}}}{{\sqrt {\frac{x}{{x + 1}}} }} = \mathop {\lim }\limits_{x \to \infty } \left( {\frac{x}{{x + 1}}} \right)\)    M1

Note: Accept any intermediate tidying up of correct derivative for the method mark.

\( = 1\)    A1

[6 marks]

a.

(i)     \({a_1} = \sqrt 2 ,{\text{ }}{a_2} = \sqrt 3 \)     A1

\({a_n} = \sqrt {n + 1} \)    A1

(ii)     \(\sin {\theta _n} = \frac{1}{{{a_n}}} = \frac{1}{{\sqrt {n + 1} }}\)     A1

Note: Allow \({\theta _n} = \arcsin \left( {\frac{1}{{{a_n}}}} \right)\) if \({a_n} = \sqrt {n + 1} \) in b(i).

so \({\theta _n} = \arcsin \frac{1}{{\sqrt {(n + 1)} }}\)     AG

[3 marks]

b.

for \(\sum\limits_{n = 1}^\infty  {\arcsin \frac{1}{{\sqrt {(n + 1)} }}} \) apply the limit comparison test (since both series of positive terms)     M1

with \(\sum\limits_{n = 1}^\infty  {\frac{1}{{\sqrt n }}} \)     A1

from (a) \(\mathop {\lim }\limits_{n \to \infty } \frac{{\arcsin \frac{1}{{\sqrt {(n + 1)} }}}}{{\frac{1}{{\sqrt n }}}} = 1\), so the two series either both converge or both diverge     M1R1

\(\sum\limits_{n = 1}^\infty  {\frac{1}{{\sqrt 2 }}} \) diverges (as is a \(p\)-series with \(p = \frac{1}{2}\))     A1

hence \(\sum\limits_{n = 1}^\infty  {{\theta _n}} \) diverges     A1

[6 marks]

c.

Examiners report

[N/A]

a.

[N/A]

b.

[N/A]

c.

Question

The function f is defined on the domain \(\left] { – \frac{\pi }{2},\frac{\pi }{2}} \right[{\text{ by }}f(x) = \ln (1 + \sin x)\) .

Show that \(f”(x) = – \frac{1}{{(1 + \sin x)}}\) .

[4]
a.

(i)     Find the Maclaurin series for \(f(x)\) up to and including the term in \({x^4}\) .

(ii)     Explain briefly why your result shows that f is neither an even function nor an odd function.

[7]
b.

Determine the value of \(\mathop {\lim }\limits_{x \to 0} \frac{{\ln (1 + \sin x) – x}}{{{x^2}}}\).

[3]
c.
Answer/Explanation

Markscheme

\(f'(x) = \frac{{\cos x}}{{1 + \sin x}}\)     A1

\(f”(x) = \frac{{ – \sin x(1 + \sin x) – \cos x\cos x}}{{{{(1 + \sin x)}^2}}}\)     M1A1

\( = \frac{{ – \sin x – ({{\sin }^2}x + {{\cos }^2}x)}}{{{{(1 + \sin x)}^2}}}\)     A1

\( = – \frac{1}{{1 + \sin x}}\)     AG

[4 marks]

a.

(i)     \(f”'(x) = \frac{{\cos x}}{{{{(1 + \sin x)}^2}}}\)     A1

\({f^{(4)}}(x) = \frac{{ – \sin x{{(1 + \sin x)}^2} – 2(1 + \sin x){{\cos }^2}x}}{{{{(1 + \sin x)}^4}}}\)     M1A1

\(f(0) = 0,{\text{ }}f'(0) = 1,{\text{ }}f”(0) = – 1\)     M1

\(f”'(0) = 1,{\text{ }}{f^{(4)}}(0) = – 2\)     A1

\(f(x) = x – \frac{{{x^2}}}{2} + \frac{{{x^3}}}{6} – \frac{{{x^4}}}{{12}} +  \ldots \)     A1

 

(ii)     the series contains even and odd powers of x     R1

[7 marks]

b.

\(\mathop {\lim }\limits_{x \to 0} \frac{{\ln (1 + \sin x) – x}}{{{x^2}}} = \mathop {\lim }\limits_{x \to 0} \frac{{x – \frac{{{x^2}}}{2} + \frac{{{x^3}}}{6} +  \ldots  – x}}{{{x^2}}}\)     M1

\( = \mathop {\lim }\limits_{x \to 0} \frac{{\frac{{ – 1}}{2} + \frac{x}{6} +  \ldots }}{1}\)     (A1)

\( = – \frac{1}{2}\)     A1

Note: Use of l’Hopital’s Rule is also acceptable.

 

[3 marks]

c.

Examiners report

[N/A]

a.

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b.

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c.

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