Katie's copy of Max and Min 6

Error

There was an error loading the page.

Contributors

Katie Dexter

Feedback

From users who are not members of Katie's workspace :

Christian Lawson-Perfect

said

Ready to use

8 years, 9 months ago

History

○

Christian Lawson-Perfect 8 years, 9 months ago

Gave some feedback: Ready to use

○

Katie Dexter 9 years, 6 months ago

Created this as a copy of Max and Min 6.

Name	Status	Author	Last Modified
Max and Min 6	Ready to use	Bill Foster	01/06/2016 09:47
Katie's copy of Max and Min 6	Ready to use	Katie Dexter	01/06/2016 09:47
Paul 's copy of Max and Min 6	Ready to use	Paul Howes	11/06/2016 19:09
Paul 's copy of Max and Min 6	draft	Xiaodan Leng	11/07/2019 05:56

There are 6 other versions that do you not have access to.

Question statement

Let $I=[\var{l},\var{m}]$ be an interval and let $g: I \rightarrow \mathbb{R}$ be the function given by:

$g(x)=\simplify{x^2/(x-{c})^({a}/{b})}$

Answer the following questions. There are seven parts and you may need to scroll down to complete all parts.

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	9
c	decimal	dec("6")
b	integer	7
valmin	decimal	dec("13.241")
m	integer	19
l	integer	9
xma	integer	9
lmi	integer	-7
s	integer	1
u	integer	76
valend	decimal	dec("13.344")
tol	number	0.001
l1	integer	9
valbegin	decimal	dec("19.726")
lma	integer	7
statpoint	number	16.8
xmi	integer	-7
valsd	decimal	dec("0.029")

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
if(b=3,random(1,2,4,5),if(b=5,random(1,2,3,4,6,7,8,9),random(2,3,4,5,6,8,9)))

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`

← Depends on:

→ Used by:

c
lma
lmi
statpoint
valbegin
valend
valmin
valsd

This variable doesn't seem to be used anywhere.

Parts

Part a) Choose one from a list
Part b) Gap-fill
1. Gap Mathematical expression
  Add
Part c) Choose one from a list
Part d) Gap-fill
Part e) Gap-fill
1. Gap Gap 0. Mathematical expression
  Add
2. Gap Gap 1. Choose one from a list
  Add
Part f) Gap-fill
1. Gap Gap 0. Number entry
  Add
2. Gap Gap 1. Number entry
  Add
Part g) Gap-fill
1. Gap Gap 0. Number entry
  Add
2. Gap Gap 1. Mathematical expression
  Add

Choose one from a list

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

Is $g(x)$ continuous at all points of $I$ ?

Choose Yes or No.

Show choice feedback state?

Show correct answer on reveal?

Show score feedback icon?

Advanced settings

Selection type

Number of display columns

(not displayed in columns)

Custom marking matrix?

Yes

Marks: Distractor Message:
No

Marks: Distractor Message:

Advice

The function $g(x)$ is continuous and differentiable at all points in $\mathbb{R}$ as $\var{c} \notin I$ .

On differentiating we see that

$\begin{eqnarray*}g'(x)&=&\simplify{2*x/(x-{c})^({a}/{b}) - {a}*x^2/({b}(x-{c})^({a+b}/{b}))}\\ &=&\simplify{({2*b}x*(x-{c})-{a}*x^2)/({b}(x-{c})^({a+b}/{b}))}\\ &=&\simplify{(x*({2*b-a}x-{2*b*c}))/({b}(x-{c})^({a+b}/{b}))} \end{eqnarray*}$

Stationary Points.

The stationary points are given by solving $g'(x)=0$ .

$\displaystyle g'(x)=0 \Rightarrow \simplify{x*({2*b-a}x-{2*b*c})=0} \Rightarrow x=0 \mbox{ or } x=\simplify[std]{{2*b*c}/{2*b-a}}$

We see that $\displaystyle x=\simplify[std]{{2*b*c}/{2*b-a}}$ is the only stationary point in $I$ .

The second derivative can be found by applying the quotient rule to the derivative of $g(x)$ and we obtain:

Using the quotient rule for differentiation we see that

$g''(x)=\simplify[std]{({a^2-3*a*b+2*b^2}*x^2+{4*b*c*(a-b)}*x+{2*c^2*b^2})/({b^2}(x-{c})^({a+2*b}/{b}))}$

Hence

$r(x)=\simplify[std]{({a^2-3*a*b+2*b^2}*x^2+{4*b*c*(a-b)}*x+{2*c^2*b^2})}$

The nature of the stationary points are determined by evaluating $g''(x)$ at the stationary points.

There is only one stationary point $\displaystyle x=\simplify[std]{{2*b*c}/{2*b-a}}$ in $I$ and at that point we have:

$g''\left(\simplify[std]{{2*b*c}/{2*b-a}}\right)=\var{valsd} \gt 0$

Hence this point is a local minimum.

Evaluating at end points of the interval.

The values of $g$ at the endpoints are:

$g(\var{l})=\var{valbegin}$ and $g(\var{m})=\var{valend}$ , both to 3 decimal places.

Global Maximum and Minimum

Global Maximum: Since $g$ does not have a local maximum in the interval $I$ , it must take a global maximum value at one of the end points of the interval.

From the values found for $g(\var{l})$ and $g(\var{m})$ found above, we see that $x=\var{xma}$ is the global maximum for $g$ in the interval $I$ .

Global Minimum: The local minimum of $g$ given by $\displaystyle x=\simplify[std]{{2*b*c}/{2*b-a}} \in I$ is the only local minimum and must be a global minimum in $I$ .

Note that the global minimum value for $g$ on $I$ is:

$g\left(\simplify[std]{{2*b*c}/{2*b-a}}\right)=\var{valmin}$

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?
Choose one from a list
Choose one from a list		Hasn't run yet
Gap-fill
Gap-fill		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet
Choose one from a list
Choose one from a list		Hasn't run yet
Gap-fill
Gap-fill		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet
Choose one from a list
Choose one from a list		Hasn't run yet
Gap-fill
Gap-fill		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet
Choose one from a list
Choose one from a list		Hasn't run yet
Gap-fill
Gap-fill		Hasn't run yet
Number entry
Number entry		Hasn't run yet
Number entry
Number entry		Hasn't run yet
Gap-fill
Gap-fill		Hasn't run yet
Number entry
Number entry		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet

This question is not used in any exams.

Loading...

Error

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Christian Lawson-Perfect 8 years, 9 months ago

Katie Dexter 9 years, 6 months ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Choose one from a list

Advanced settings

Stationary Points.

Evaluating at end points of the interval.

Global Maximum and Minimum

Loading...

Error

Katie's copy of Max and Min 6

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Christian Lawson-Perfect 8 years, 9 months ago

Katie Dexter 9 years, 6 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Choose one from a list

Advanced settings

Stationary Points.

Evaluating at end points of the interval.

Global Maximum and Minimum

Generated value: `integer`