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

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Simon Thomas 6 years 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	0.0416666667
statval	number	0.04
sol	string	by multiplying the second equa
lmin	string	Local minimum
q1	integer	0
dvalue	number	-24
s3	integer	-1
s2	integer	-1
s1	integer	1
s5	integer	-1
s4	integer	-1
neither	string	Saddle point
b	integer	-4
c1	integer	-2
a	integer	1
c	integer	-2
p1	integer	0
d	integer	-2
f	integer	-1
lmax	string	Local maximum
y	number	-0.4166666667
x	number	0.1666666667

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`

0.0416666667

← Depends on:

→ Used by:

statval

Advice

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 $y$ :

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

Remember that:

- to find $\partial f \over \partial x$ we differentiate with respect to $x$ , treating $y$ as a constant.

- to find $\partial f \over \partial y$ we differentiate with respect to $y$ , treating $x$ as a constant.

Hence we obtain:

$\partial f \over \partial x$ $=\simplify[std]{{2*a}x+{b}y+{d}}$

$\partial f \over \partial y$ $= \simplify[std]{{b}x+{2*c1}y+{f}}$

To find stationary points, we set each of these equations to be equal to 0, then solve simultaneously 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:

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

Give a worked solution to the whole question.

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This question is used in the following exam:

Partial derivatives and stationary points by Simon Thomas in Calculus.

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

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Taxonomy: mathcentre

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Simon Thomas 6 years ago

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Generated value: `number`

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

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Simon Thomas 6 years ago

Error in variable testing condition

Error

Warning

Generated value: number

Parts

Gap-fill

Generated value: `number`