Priyanka's copy of Taylor series (three terms)

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Priyanka Pandya 5 years, 2 months ago

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Priyanka Pandya 5 years, 2 months ago

Created this as a copy of Taylor series (three terms).

Name	Status	Author	Last Modified
Taylor series (three terms)	Ready to use	Newcastle University Mathematics and Statistics	20/11/2019 14:50
Maria's copy of Taylor series (three terms)	Ready to use	Maria Aneiros	29/11/2021 01:12
Priyanka's copy of Taylor series (three terms)	draft	Priyanka Pandya	10/02/2020 16:03
Ugur's copy of Taylor 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 Taylor series at $x=\var{c}$ for $f(x)=(\simplify[all]{{a-b*c}+{b}*x})^{1/\var{n}}$ i.e. up to terms in $x^2$ .

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			Name	Type	Generated Value

a	integer	16
tm0	decimal	dec("2")
tm2	number	-294
b	integer	-7
tm1	decimal	dec("-14")
s1	integer	-1
c	integer	2
n	integer	4

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Value

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

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Generated value: `integer`

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Advice

The first three terms in the Taylor series are given by $\simplify[all]{a+b(x-{c})+c(x-{c})^2}$ where $\displaystyle a=f(\var{c}),\;\;b=f'(\var{c}),\;\;c=\frac{f''(\var{c})}{2}$
For this example,

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

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

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

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Priyanka's copy of Taylor series (three terms)

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Priyanka Pandya 5 years, 2 months ago

Priyanka Pandya 5 years, 2 months ago

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Priyanka's copy of Taylor series (three terms)

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Priyanka Pandya 5 years, 2 months ago

Priyanka Pandya 5 years, 2 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`