Function approximation

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Newcastle University Mathematics and Statistics

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

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Question statement

Use the approximation $f(a+h) \approx f(a)+hf^{\prime}(a)$ to estimate

$\var{a+h}^{\frac{\var{m}}{\var{n}}}$ for a suitable function

$f(x)$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	16
c	decimal	dec("2")
est	decimal	dec("1.99931")
h	number	-0.022
s1	integer	-1
tr	number	1.99931
m	integer	1
n	integer	4
p	integer	-3

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
if(n=2,random(4,9,16,25,36),if(n=3,random(8,27,64),if(n=4,random(16,81),random(32,243))))

JME function reference

Description

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `integer`

← Depends on:

→ Used by:

Statement
Advice
"Unnamed part" → "Gap 1." - Minimum accepted value
"Unnamed part" → "Gap 1." - Maximum accepted value

Parts

Gap-fill
1. Gap Gap 0. Mathematical expression
  Add
2. Gap Gap 1. Number entry
  Add
3. Gap Gap 2. Number entry
  Add
4. Gap Gap 3. Number entry
  Add
5. Gap Gap 4. Number entry
  Add

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

In this case $f(x) =\;$ Gap 0..

$a= \;$ Gap 1. and $h=\;$ Gap 2.
Input your estimation to $5$ decimal places: Gap 3.

True value is: Gap 4. (input to 5 decimal places).

Advice

We have $f(x)=x^\frac{\var{m}}{\var{n}}$ and $a= \var{a}$ , $h=\var{h}$ .
Note that $\var{a}^\frac{1}{\var{n}}=\var{c}$ and so using the approximation :
$f(a+h)\approx f(a)+hf^{\prime}(a)$ and $f^{\prime}(x) = \frac{\var{m}}{\var{n}}\simplify[std]{x^({m-n}/{n})}$
we have:

$\simplify[std]{{a+h}^({m}/{n})}\approx \simplify[simplifyFractions]{{a}^({m}/{n})+{h}*({m}/{n})*{a}^({p}/{n})}=\simplify[std,!sqrtSquare]{{c}^{m}+ {h}*({m}/{n})*{c}^{m-n}}=\var{est}$
to 5 decimal places.
The true value to 5 decimal places is {tr}.

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Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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Checkpoint description

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

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Warning

Generated value: `integer`

Parts

Gap-fill

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Function approximation

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`