Leicester: Differentiation 3

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Bill Foster

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Bill Foster 55 years, 3 months ago

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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 and classify their types:

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

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	2
c	integer	-10
b	decimal	dec("-2")
r1	integer	10
r2	integer	-1
mn	integer	1
d	integer	-2
lg2	string	$\gt$
lg1	string	$\lt$
type1	string	maximum
mx	integer	0
type2	string	minimum

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1. Gap Gap 0. Mathematical expression
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2. Gap Gap 1. Mathematical expression
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$x$ -coordinate of the stationary point giving a minimum: Gap 0.

$x$ -coordinate of the stationary point giving a maximum: 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 $\displaystyle \frac{d^2y}{dx^2} = \simplify{{6a}*x+{2*b}}$ at the stationary points.

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

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

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

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

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

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Bill Foster 55 years, 3 months ago

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Leicester: Differentiation 3

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Taxonomy: mathcentre

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Bill Foster 55 years, 3 months ago

Error in variable testing condition

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Generated value: integer

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Generated value: `integer`