George's copy of Functions of two variables: Stationary points 1

Error

There was an error loading the page.

Contributors

George Loughlin

History

○

George Loughlin 7 years, 8 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

There are 33 other versions that do you not have access to.

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	1.0975609756
statval	number	1.10
sol	string	by multiplying the second equa
lmin	string	Local minimum
q1	integer	-1
dvalue	number	-41
s3	integer	-1
s2	integer	1
s1	integer	1
s5	integer	-1
s4	integer	1
neither	string	Saddle point
b	integer	5
c1	integer	-2
a	integer	2
c	integer	-2
p1	integer	0
d	integer	3
f	integer	-3
lmax	string	Local maximum
y	number	-0.6585365854
x	number	0.0731707317

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`

1.0975609756

← Depends on:

→ Used by:

statval

This variable doesn't seem to be used anywhere.

Parts

Gap-fill
1. Gap Gap 0. Mathematical expression
  Add
2. Gap Gap 1. Mathematical expression
  Add
3. Gap Gap 2. Number entry
  Add
1. Step Information only

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..

Input value of $f(x,y)$ at $(a,b)$ :

$f(a,b)=\;\;$ Gap 2. (Input to 2 decimal places).

If you want some help, click on Show steps. You will not lose any marks if you do so.

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.

Use this tab to check that this question works as expected.

There was an error which means the tests can't run:

Part	Test	Passed?
Gap-fill
Gap-fill		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet
Number entry
Number entry		Hasn't run yet
Information only

This question is used in the following exams:

Loading...

Error

George's copy of Functions of two variables: Stationary points 1

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

George Loughlin 7 years, 8 months ago

Error in variable testing condition

Error

Warning

Generated value: `number`

Parts

Gap-fill

Loading...

Error

George's copy of Functions of two variables: Stationary points 1

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

George Loughlin 7 years, 8 months ago

Error in variable testing condition

Error

Warning

Generated value: number

Parts

Gap-fill

Generated value: `number`