2022HSCShort answer

Q14 (b) (ii)

2 marks

Question

Let $J_n=\int_0^1x^ne^{-x}\,dx$ , where $n$ is a non-negative integer. Show that
$J_n\le\frac1{n+1}.$

Try the question before revealing the answer.

Solution

Criteria	Marks
Provides correct solution	2
States that $e^{-x}\le1$ for $0\le x\le1$ , or equivalent merit	1

Sample answer:

J_n=\int_0^1x^ne^{-x}\,dx\le\int_0^1x^n\,dx \quad\text{since }x^ne^{-x}\le x^n\text{ for }[0,1]

=\left[\frac{x^{n+1}}{n+1}\right]_0^1=\frac1{n+1}

$\therefore J_n\le\frac1{n+1}$

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Syllabus alignment

ProofCalculus

The Nature of ProofUse formal proof language, contradiction, counter-examples and rigorous arguments involving numbers and inequalities.
Further IntegrationApply substitution, partial fractions, integration by parts and recurrence relationships to theoretical and practical problems.

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