Maclaurin series (three terms)

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Newcastle University Mathematics and Statistics

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Newcastle University Mathematics and Statistics 9 years, 3 months ago

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Name	Status	Author	Last Modified
Maclaurin series (three terms)	draft	Newcastle University Mathematics and Statistics	20/11/2019 14:50
Adrian's copy of Maclaurin series (three terms)	draft	Adrian Jannetta	16/08/2023 14:56
Maclaurin series (first three terms)	Needs to be tested	Gemma Crowe	18/08/2018 16:28
Maclaurin series (first three terms) - sine	Needs to be tested	Gemma Crowe	20/08/2018 10:06
Gemma's copy of Maclaurin series (first three terms) - cosine	Needs to be tested	Gemma Crowe	20/08/2018 10:07
Gemma's copy of Gemma's copy of Maclaurin series (first four terms) - exponential	Needs to be tested	Gemma Crowe	20/08/2018 10:34
Maria's copy of Maclaurin series (three terms)	Ready to use	Maria Aneiros	29/11/2021 01:12
Ugur's copy of Maclaurin series (three terms)	Ready to use	Ugur Efem	01/11/2023 13:41

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Question statement

You are asked to find the first 3 terms in the MacLaurin series for $f(x)=(\simplify[all]{{a}+{b}*x})^{1/\var{n}}$ i.e. up to terms in $x^2$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	49
tm0	decimal	dec("7")
tm2	number	-7
b	integer	-1
tm1	decimal	dec("-7")
s1	integer	-1
n	integer	2

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random(1,4,8,9,16,27,32,25,36,49)

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Generated value: `integer`

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The first three terms in the MacLaurin series are given by $a+bx+cx^2$ where $\displaystyle a=f(0),\;\;b=f'(0),\;\;c=\frac{f''(0)}{2}$
For this example,

$\begin{eqnarray*} f'(x)&=&\simplify[all,fractionNumbers]{{b}/{n}*({a}+{b}x)^(-{n-1}/{n})}\\ f''(x)&=&\simplify[all,fractionNumbers]{-{b^2*(n-1)}/{n^2}*({a}+{b}x)^(-{2*n-1}/{n})} \end{eqnarray*}$
and so we get:

$\begin{eqnarray*} a&=&f(0)=\simplify[all]{{a}^(1/{n})={tm0}}\\ b&=&f'(0)=\simplify[all,fractionNumbers]{{tm1}/{a*n}}\\ c&=&\frac{f''(0)}{2}=\simplify[all,fractionNumbers]{{tm2}/{2*a^2*n^2}} \end{eqnarray*}$
Hence the first three terms of the MacLaurin series are:

$\simplify[all,fractionNumbers,!collectNumbers]{{tm0}+{tm1}/{a*n}*x+{tm2}/{2*a^2*n^2}*x^2}$

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Maclaurin series (three terms)

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Newcastle University Mathematics and Statistics 9 years, 3 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Mathematical expression

Advanced settings

Checking accuracy

Variable value generators

Pattern restriction

Variables

String restrictions

Generated value: `integer`