Find the stationary point of a function on a disk

Error

There was an error loading the page.

Contributors

Christian Lawson-Perfect

Feedback

From users who are not members of Content created by Newcastle University :

Bill Foster

said

Ready to use

6 years, 6 months ago

History

○

Bill Foster 6 years, 6 months ago

Gave some feedback: Ready to use

○

Christian Lawson-Perfect 7 years, 9 months ago

Published this.

○

Christian Lawson-Perfect 7 years, 9 months ago

Created this.

Name	Status	Author	Last Modified
Find the stationary point of a function on a disk	Ready to use	Christian Lawson-Perfect	20/11/2019 14:51
Luis's copy of Find the stationary point of a function on a disk	draft	Luis Hernandez	30/11/2018 14:57

Question statement

In the following question find the $(x,y)$ coordinates of the single stationary point $(a,b) \in D$ of the continuous function $f: D \rightarrow \mathbb{R}$ :

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

where

$D = \{(x,y): \simplify[std]{(x-{c})^2+(y-{d})^2}\} \le \var{r}$

that is, $D$ is a disk of radius $\simplify[std]{sqrt({r})}$ and centre $(\var{c},\var{d})$ .

Input both cooordinates as fractions or integers and not decimals.

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	-6
c	integer	-3
b	integer	-6
d	integer	9
s3	integer	-1
s2	integer	-1
s1	integer	-1
s5	integer	-1
s4	integer	1
r	integer	3

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
s1*random(1..9)

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`

-6

← Depends on:

This variable doesn't seem to be used anywhere.

Parts

Gap-fill
1. Gap Gap 0. Number entry
  Add
2. Gap Gap 1. Number entry
  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.

$x$ – coordinate, $a=$ Gap 0.

$y$ – coordinate, $b=$ Gap 1.

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

$f(a,b)=$ Gap 2.

If you want some help, click on Show steps.

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 $y$

$\begin{align} \partial f \over \partial x &= 0 \\[1em] \partial f \over \partial y &= 0 \end{align}$

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

$\begin{align} \simplify[std]{{-2*b}*(x-{c})*e^(-(x-{c})^2-(y-{d})^2)} &= 0 \\[1em] \simplify[std]{{-2*b}*(y-{d})*e^(-(x-{c})^2-(y-{d})^2)} &= 0 \end{align}$

We can cancel off the term $\simplify[std]{e^(-(x-{c})^2-(y-{d})^2)}$ in both equations as $\simplify[std]{e^(-(x-{c})^2-(y-{d})^2)} \neq 0,\;\forall x,\;y$ .

On solving these, we get

$x = \var{c}, \quad \;y=\var{d}$

So the stationary point is $(\var{c},\var{d}) \in D$ .

On substituting these values into $f(x,y)$ we get:

$f(\var{c},\var{d})=\simplify[std,!zeropower,!othernumbers]{{a}+{b}*e^0={a+b}}$

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
Number entry
Number entry		Hasn't run yet
Number entry
Number entry		Hasn't run yet
Number entry
Number entry		Hasn't run yet
Information only

This question is used in the following exams:

Loading...

Error

Find the stationary point of a function on a disk

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Bill Foster 6 years, 6 months ago

Christian Lawson-Perfect 7 years, 9 months ago

Christian Lawson-Perfect 7 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

Loading...

Error

Find the stationary point of a function on a disk

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Bill Foster 6 years, 6 months ago

Christian Lawson-Perfect 7 years, 9 months ago

Christian Lawson-Perfect 7 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`