Applications of differentiation 2

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cormac breen

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cormac breen 8 years, 2 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.

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	-1
c	integer	1
b	decimal	dec("-1")
r1	integer	-1
r2	integer	-1
mn	integer	0
d	integer	1
lg2	string	$\lt$
lg1	string	$\gt$
type1	string	minimum
mx	integer	1
type2	string	maximum
f1	number	0.3333333333
f2	number	-1

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random(-2..2 except 0)

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Generated value: `integer`

-1

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

$f'(x)=$ Gap 2.

$f''(x)=$ Gap 3.

Find when $f'(x)=0$ , hence find:

$x$ -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 $\displaystyle \frac{d^2f}{dx^2} = \simplify{{6a}*x+{2*b}}$ 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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cormac breen 8 years, 2 months ago

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Applications of differentiation 2

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Taxonomy: mathcentre

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Taxonomy: Context

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cormac breen 8 years, 2 months ago

Error in variable testing condition

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Generated value: integer

Parts

Gap-fill

Generated value: `integer`