Functions of two variables: Stationary points 2

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Christian Lawson-Perfect

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9 years, 11 months ago

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Christian Lawson-Perfect 9 years, 11 months ago

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Bill Foster 55 years, 2 months ago

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Name	Status	Author	Last Modified
Functions of two variables: Stationary points 2	Ready to use	Bill Foster	01/06/2016 09:47
ma221_Tovariable	Ready to use	Tore Gaupseth	05/06/2016 10:09
Robert's copy of Functions of two variables: Stationary points 2	Ready to use	Robert Anderson	31/10/2024 10:29
Kjell's copy of ma221_Tovariable	draft	Kjell Hansen	31/08/2016 14:51
Functions of two variables: Stationary points 2	draft	Michal Kaukic	03/10/2016 21:18
Partial differentiation	draft	Julie Crowley	25/09/2018 11:30
Alison's copy of Partial differentiation	draft	Alison Loddick	16/01/2019 15:52
Simon's copy of Functions of two variables: Stationary points 2	draft	Simon Thomas	14/03/2019 14:12
Ann's copy of Partial differentiation	draft	Ann Smith	22/08/2019 09:04

There are 51 other versions that do you not have access to.

Question statement

Answer the following questions about the function:

$f(x,y)=\simplify[std]{ ({a} / 3) * x ^ 3 + ({b} / 2) * x ^ 2 * y + {c} * y ^ 2 * x + {d} * y}$

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	8
c	integer	8
ch	integer	1
d	number	-8
m	integer	2
s5	integer	-1
a1	integer	2
other	string	There can be no more stationar
b1	integer	4
c2	integer	4
c1	integer	4
b	integer	20
check	string
d1	integer	2

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
a1*c1

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:

This variable doesn't seem to be used anywhere.

Parts

Part a) Gap-fill
1. Gap Gap 0. Mathematical expression
  Add
2. Gap Gap 1. Mathematical expression
  Add
Part b) Choose several from a list

Gap-fill

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

Enter the partial derivatives here. Note if you want to enter a product of unknowns, such as $xy$ then you input the expression in the form x*y.

$\displaystyle { \partial f \over \partial x}=$ Gap 0.

$\displaystyle {\partial f \over \partial y}=$ Gap 1.

Advice

a)

$\begin{eqnarray*} {\partial f \over \partial x} &=&\simplify[std]{(({a} * (x ^ 2)) + ({b} * x * y) + ({c} * (y ^ 2)))}\\ \\ \partial f \over \partial y &=&\simplify[std]{((({b} / 2) * (x ^ 2)) + ({(2 * c)} * x * y) + {d})} \end{eqnarray*}$

b)

$(a,b)$ is a stationary point for the function $f(x,y)$ if $f_x=0,\;\;f_y=0$ ,where the partial derivatives are evaluated at $x=a,\;\;y=b$ .
So you have to make sure that both these partial derivatives are $0$ at the stationary point.

For this example we have from the above equations that:

$\begin{eqnarray*} \simplify[std]{(({a} * (x ^ 2)) + ({b} * x * y) + ({c} * (y ^ 2)))}&=&0,\qquad &\mathbf{(1)}&\\ \\ \simplify[std]{((({b} / 2) * (x ^ 2)) + ({(2 * c)} * x * y) + {d})}&=&0, \qquad &\mathbf{(2)}& \end{eqnarray*}$

The left hand side of equation (1) can be factorised as:

$\simplify[std]{({a1}x+{b1}y)*({c1}x+{d1}y)=0}$

and so we have:

$y=\simplify[std]{{-a1}/{b1}*x},\mbox{ or } y= \simplify[std]{{-c1}/{d1}*x}$

First case: $y= \simplify[std]{{-a1}/{b1}*x}$

Substituting this into equation (2) gives:

$\simplify[std]{{b}/2*x^2-{2c*a1}/{b1}*x^2+{d}}=0 \Rightarrow \simplify[std]{{-b*b1+4*c*a1}/{2*b1}*x^2={d}}$

Hence $x=\var{m}\mbox{ or } x = \var{-m}$ and the stationary points which are on the list and which you had to choose are:

$\left(\var{m},\simplify[std]{-{a1*m}/{b1}}\right)\mbox{ and }\left(\var{-m},\simplify[std]{{a1*m}/{b1}}\right)$

Second case: $y= \simplify[std]{{-c1}/{d1}*x}$

{check}

Substituting this into equation (2) gives:

$\simplify[std]{{b}/2*x^2-{2c*c1}/{d1}*x^2+{d}}=0 \Rightarrow \simplify[std]{{-b*d1+4*c*c1}/{2*d1}*x^2={d}}$

{other}

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Functions of two variables: Stationary points 2

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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Checkpoint description

Christian Lawson-Perfect 9 years, 11 months ago

Bill Foster 55 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

a)

b)

First case: $y= \simplify[std]{{-a1}/{b1}*x}$

Second case: $y= \simplify[std]{{-c1}/{d1}*x}$

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Error

Functions of two variables: Stationary points 2

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Christian Lawson-Perfect 9 years, 11 months ago

Bill Foster 55 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

a)

b)

First case: y={{−a1}{b1}x}y= \simplify[std]{{-a1}/{b1}*x}

Second case: y={{−c1}{d1}x}y= \simplify[std]{{-c1}/{d1}*x}

Generated value: `integer`

First case: $y= \simplify[std]{{-a1}/{b1}*x}$

Second case: $y= \simplify[std]{{-c1}/{d1}*x}$