Differentiation: Cubic Stationary Points

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Clare Lundon 6 years, 5 months ago

Created this as a copy of Differentiation 16 - Applications.

Name	Status	Author	Last Modified
Leicester: Differentiation 3	draft	Bill Foster	06/08/2020 08:42
Differentiation 16 - Applications	Ready to use	Katie Lester	01/06/2016 09:47
Jinhua's copy of Differentiation 16 - Applications	Ready to use	Jinhua Mathias	27/08/2024 16:37
CF Maths Portfolio - Differentiation 16 - Applications	Ready to use	Jinhua Mathias	28/08/2024 12:15
Vicky's copy of Differentiation 16 - Applications	draft	Vicky Hall	17/10/2016 15:08
Stasjonaere punkter 2	draft	Ida Friestad Pedersen	19/07/2016 13:10
Ida Friestad's copy of Vicky's copy of Differentiation 16 - Applications	draft	Ida Friestad Pedersen	19/07/2016 13:23
Stasjonaere punkter 2 (kap 7 Sinus)	draft	Ida Friestad Pedersen	19/07/2016 14:12
Stasjonaere punkter 2 (kap 7 Sinus)	draft	Ida Friestad Pedersen	22/08/2016 08:52
Stasjonaere punkter 2 (kap 7 Sinus)	draft	Ida Friestad Pedersen	22/08/2016 08:51
Stasjonaere punkter 1 (kap 7 Sinus)	draft	Ida Friestad Pedersen	03/10/2016 11:44
Stasjonaere punkter 2 (kap 7 Sinus)	draft	Morten Brekke	14/10/2016 10:36
Stasjonaere punkter 2 (kap 7 Sinus)	draft	Morten Brekke	14/10/2016 10:37
Differentiation 16 - Applications	draft	Morten Brekke	14/10/2016 10:39
Morten's copy of Differentiation 16 - Applications	draft	Morten Brekke	14/10/2016 10:39
Morten's copy of Stasjonaere punkter 2 (kap 7 Sinus)	draft	Morten Brekke	19/10/2016 10:15
Morten's copy of Stasjonaere punkter 2 (kap 7 Sinus)	draft	Morten Brekke	24/10/2016 09:29
Stationary Points	draft	Robert Førland	05/11/2016 13:07
Stationary points	draft	Morten Brekke	02/11/2016 14:38
Differentiation 16 - Applications	draft	Marlon Arcila	08/11/2016 00:37
OCHO DOS Derivadas valor de f'	draft	Marlon Arcila	08/09/2021 00:45
Paul 's copy of Differentiation 16 - Applications	draft	Paul Howes	08/02/2017 13:40
Applications of differentiation 2	draft	cormac breen	23/02/2017 18:09
Harry's copy of Applications of differentiation 2	draft	Harry Flynn	08/05/2018 12:28
Differentiation: Cubic Stationary Points	draft	Clare Lundon	27/11/2018 15:43
Clare's copy of Applications of differentiation 2	draft	Clare Lundon	27/11/2018 15:42
Stationary points of cubics	Ready to use	Ian Davies	04/03/2019 11:44
Stationary points of rational functions	Ready to use	Ian Davies	04/03/2019 11:50
Maria's copy of Stationary points of rational functions	draft	Maria Aneiros	27/05/2019 05:47
Maria's copy of Clare's copy of Applications of differentiation 2	draft	Maria Aneiros	27/05/2019 05:47
Max & Mins	draft	Kevin Bohan	05/06/2019 11:18
Stationary points of rational functions	draft	Xiaodan Leng	10/07/2019 22:20
XiStationary points of cubics	draft	Xiaodan Leng	10/07/2019 22:20
Differentiation 16 - Applications	draft	Xiaodan Leng	11/07/2019 06:58

There are 34 other versions that do you not have access to.

Question statement

Find the coordinates of the stationary points of the function:

$f(x)=\simplify[all,!collectNumbers,!noleadingminus]{{a}x^3+{b}x^2+{c}x+{d}}$

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	-2
c	integer	104
b	decimal	dec("1")
r1	integer	-26
r2	integer	-4
mn	integer	0
d	integer	-4
lg2	string	$\lt$
lg1	string	$\gt$
type1	string	minimum
mx	integer	1
type2	string	maximum
f1	number	4.3333333333
f2	number	-4

Automatically regenerate variables?

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Value

xxxxxxxxxx
 
random(-2,-1)

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Description

Describe what this variable represents, and list any assumptions made about its value.

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Generated value: `integer`

-2

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Parts

Part a) Gap-fill
1. Gap Gap 0. Mathematical expression
  Add
2. Gap Gap 1. Mathematical expression
  Add
Part b) Gap-fill
1. Gap Gap 0. Mathematical expression
  Add
2. Gap Gap 1. Mathematical expression
  Add

Gap-fill

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

First differentiate:

$f'(x)=$ Gap 0.

$f''(x)=$ Gap 1.

Advice

On differentiating we get $\displaystyle \frac{df}{dx}=\simplify[std]{{3*a}x^2+{2*b}x+{c}}$ .

To find the stationary points we have to solve $\displaystyle \frac{df}{dx}=0$ for $x$ .

So we have to solve $\simplify[std]{{3*a}x^2+{2*b}x+{c}=0}$ .

Note that the quadratic factorises and the equation becomes $\simplify[std]{({3a}x-{r1})(x-{r2})=0}$ .

Hence we have two stationary points: $x=\simplify[std]{{r1}/{3a}}$ and $x=\var{r2}$ .

To find out the types of these stationary points we look at the sign of $\displaystyle \frac{d^2f}{dx^2} = \simplify{{6a}*x+{2*b}}$ at the stationary points.

If $\displaystyle \frac{d^2f}{dx^2} \lt 0$ at a stationary point then it is a MAXIMUM.

If $\displaystyle \frac{d^2f}{dx^2} \gt 0$ at a stationary point then it is a MINIMUM.

If $\displaystyle \frac{d^2f}{dx^2} = 0$ at a stationary point then we have to do more work!

At $x=\var{r2}$ we have $\displaystyle \frac{d^2f}{dx^2} = \simplify{{6*a*r2+2*b}}$ {lg1} $0$ hence is a {type1}.

At $\displaystyle x=\simplify[std]{{r1}/{3a}}$ we have $\displaystyle \frac{d^2f}{dx^2} = \simplify{{2*r1+2*b}}$ {lg2} $0$ hence is a {type2}.

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This question is used in the following exam:

Tutorial Work 4 EP2 Nov 2018 by Clare Lundon in Clare Year 2.

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Clare Lundon 6 years, 5 months ago

Error in variable testing condition

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Generated value: `integer`

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Gap-fill

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Differentiation: Cubic Stationary Points

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Clare Lundon 6 years, 5 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`