Shaheen's copy of Functions of two variables: Locate stationary point

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Shaheen Charlwood

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5 months ago

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Shaheen Charlwood 5 months ago

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Shaheen Charlwood 1 year, 5 months ago

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Shaheen Charlwood 1 year, 5 months ago

Created this as a copy of Functions of two variables: Stationary points 1.

Name	Status	Author	Last Modified
Functions of two variables: Stationary points 1	Ready to use	Bill Foster	01/06/2016 09:47
Katie's copy of Functions of two variables: Stationary points 1	Ready to use	Katie Dexter	01/06/2016 09:47
ma220_Tovariable	Ready to use	Tore Gaupseth	05/06/2016 10:09
Robert's copy of Functions of two variables: Stationary points 1	Ready to use	Robert Anderson	31/10/2024 10:29
George's copy of Functions of two variables: Stationary points 1	draft	George Loughlin	06/07/2017 15:09
Simon's copy of Functions of two variables: Stationary points 1	draft	Simon Thomas	14/03/2019 14:19
Shaheen's copy of Functions of two variables: Locate stationary point	Ready to use	Shaheen Charlwood	21/10/2024 11:28

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Question statement

In the following question find the $(x,y)$ coordinates of the single stationary point $(a,b)$ of the function

$f(x,y)=\simplify[std]{{a}*x^2+{b}*x*y+{c1}*y^2+{d}*x+{f}*y}$

Give any introductory information the student needs.

			Name	Type	Generated Value

rawstatval	number	-0.4
statval	number	-0.40
sol	string	by multiplying the second equa
lmin	string	Local minimum
q1	integer	2
dvalue	number	15
s3	integer	1
s2	integer	1
s1	integer	1
s5	integer	1
s4	integer	1
neither	string	Saddle point
b	integer	1
c1	integer	4
a	integer	1
c	integer	4
p1	integer	-1
d	integer	1
f	integer	2
lmax	string	Local maximum
y	number	-0.2
x	number	-0.4

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
a*x^2+b*x*y+c1*y^2+d*x+f*y

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: `number`

-0.4

← Depends on:

→ Used by:

statval

Advice

Parts

Gap-fill
1. Gap Gap 0. Mathematical expression
  Add
2. Gap Gap 1. Mathematical expression
  Add

Gap-fill

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

Input both cooordinates as fractions or integers and not decimals.

$x$ –coordinate, $a=$ Gap 0..

$y$ –coordinate, $b=$ Gap 1..

Advice

The $(x,y)$ coordinates of the stationary point of a function of 2 variables $f(x,y)$ are given by solving
the following 2 equations for $x$ and $y$ :

$\begin{eqnarray*} \partial f \over \partial x &=&0\\ \\ \partial f \over \partial y &=&0 \end{eqnarray*}$

In this case you get two linear equations to solve for $x$ and $y$ :

$\begin{eqnarray*} \simplify[std]{{2*a}x+{b}y+{d}}&=&0\\ \\ \simplify[std]{{b}x+{2*c1}y+{f}}&=&0 \end{eqnarray*}$
On solving these we get

$x = \simplify[std]{{2*c1*d-b*f}/{b^2-4*a*c1}},\;\;\;y=\simplify[std]{{2*a*f-b*d}/{b^2-4*a*c1}}$
On substituting these values into

$f(x,y)$ we get:

$f\left(\simplify[std]{{2*c1*d-b*f}/{b^2-4*a*c1}},\simplify[std]{{2*a*f-b*d}/{b^2-4*a*c1}}\right) = \var{rawstatval}\approx\var{statval}$
to 2 decimal places.

Give a worked solution to the whole question.

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Part	Test	Passed?
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Gap-fill		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet
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Mathematical expression		Hasn't run yet

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Shaheen's copy of Functions of two variables: Locate stationary point

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Taxonomy: mathcentre

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Taxonomy: Context

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Shaheen Charlwood 5 months ago

Shaheen Charlwood 1 year, 5 months ago

Shaheen Charlwood 1 year, 5 months ago

Error in variable testing condition

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Generated value: `number`

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Shaheen's copy of Functions of two variables: Locate stationary point

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Shaheen Charlwood 5 months ago

Shaheen Charlwood 1 year, 5 months ago

Shaheen Charlwood 1 year, 5 months ago

Error in variable testing condition

Error

Warning

Generated value: number

Parts

Gap-fill

Generated value: `number`