SFY0004 Implicit 2

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Christian Lawson-Perfect

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Christian Lawson-Perfect 10 years, 1 month ago

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Bill Foster 12 years ago

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Name	Status	Author	Last Modified
SFY0004 Implicit 2	Ready to use	Bill Foster	01/06/2016 09:47
CATORCE DOS Derivacion implicita	draft	Marlon Arcila	08/09/2021 00:47
Implicit 2	draft	Leticija Dubickaite	07/06/2018 11:23
Implicit 2	draft	Leticija Dubickaite	05/07/2018 16:52
Nick's copy of Implicit 2	draft	Nick Walker	22/06/2018 00:48
Praneetha's copy of Nick's copy of Implicit 2	draft	Praneetha Singh	28/06/2018 05:50
Implicit Differentiation	Ready to use	Gareth Woods	25/01/2022 12:32
Aoife's copy of Implicit Differentiation	draft	Aoife O'Brien	21/09/2018 12:44
Simon's copy of SFY0004 Implicit 2	draft	Simon Thomas	11/03/2019 14:58
Implicit Differentiation	draft	Kevin Bohan	05/06/2019 14:44
Implicit Differentiation	draft	Kevin Bohan	05/06/2019 14:50

There are 14 other versions that do you not have access to.

Question statement

Given the following relation between $x$ and $y$

$\simplify[all,!collectNumbers]{x^2+y^2+{d}x y+{a}x+{b}y}=\var{c}$
answer the following questions.

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	-3
c	integer	7
b	integer	6
d	integer	-1

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
-random(1..9)

JME function reference

Description

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

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Generated value: `integer`

-3

This variable doesn't seem to be used anywhere.

Parts

Part a) Gap-fill
1. Gap Mathematical expression
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1. Step Information only
Part b) Gap-fill
1. Gap Gap 0. Number entry
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2. Gap Gap 1. Number entry
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Part c) Gap-fill

Gap-fill

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

Using implicit differentiation find $\displaystyle \frac{dy}{dx}$ in terms of $x$ and $y$ .

Input your answer here:

$\displaystyle \frac{dy}{dx}=$ Unnamed gap

Input all numbers as integers not as decimals.

If you want more help click on Show steps - you will not lose any marks.

Advice

a)

On differentiating both sides of the equation implicitly we get

$2x + \simplify[all,!collectNumbers]{2y*Diff(y,x,1) +{d}(y+x*Diff(y,x,1))+ {a} + {b} *Diff(y,x,1)} = 0$
Collecting terms in

$\displaystyle\frac{dy}{dx}$ and rearranging the equation we get

$( \simplify[all,!collectNumbers]{({b} + 2y+{d}x)} )\frac{dy}{dx} = \simplify[all,!collectNumbers]{{ -a} -2x-{d}y}$ and hence on further rearranging:

$\frac{dy}{dx} = \simplify[all,!collectNumbers]{({ - a} - 2 * x-{d}y) / ({b} + (2 * y)+{d}x)}$

b)

On putting $x=0$ in the relation we get:

$\simplify{y^2+{b}y={c}} \Rightarrow \simplify{y^2+{b}y-{c}=0 }\Rightarrow (y+\var{c})(y-1)=0$

Hence $a=-\var{c}$ and $b=1$ .

c)

First we find the tangent at the point $(0,-\var{c})$ .

We find using the formula we found for $\frac{dy}{dx}$ in part a) that the gradient at $(0,-\var{c})$ is:

$\frac{dy}{dx}=\frac{\simplify[all,!collectnumbers]{{-a}+{d*c}}}{\var{b}-\var{2*c}}=\simplify[all,fractionNumbers]{{a-d*c}/{c+1}}$

As the tangent goes through the point $(0,\var{-c})$ i.e. at $x=0,\;\;y=-\var{c}$ we see that the equation of the tangent is:

$y=\simplify[all,fractionNumbers]{{a-d*c}/{c+1}}x-\var{c}$

Next we find that the gradient at $(0,1)$ is:

$\frac{dy}{dx}=\frac{\simplify[all,!collectnumbers]{{-a}-{d}}}{\var{b}+2}=\simplify[all,fractionNumbers]{-{a+d}/{c+1}}$

As the tangent goes through the point $(0,1)$ i.e. at $x=0,\;\;y=1$ we see that the equation of the tangent is:

$y=\simplify[all,fractionNumbers]{-{a+d}/{c+1}}x+1$

Give a worked solution to the whole question.

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This question is used in the following exams:

Implicit Differentiation by Sarah Jowett in Sarah's workspace.
Maths Support: Implicit differentiation by Christian Lawson-Perfect in Maths Support Wiki.
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week2 by haifa abd in NEM1001.
week3 by haifa abd in NEM1001.
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week5 by haifa abd in NEM1001.
Maria's copy of mathcentre: Implicit differentiation by Maria Aneiros in Maria's workspace.
Implicit differentiation by Jinhua Mathias in Maths 3-2020.
mathcentre: Implicit differentiation by Michael Proudman in Mathcentre based Materials.
Maths 1B - Week 5 Homework (Applications of Derivatives 1) by Maria Aneiros in MY QUESTIONS.
Implicit differentiation by Adrian Jannetta in Foundation Maths.
For Matthew by Maria Aneiros in Maria's workspace.

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Taxonomy: mathcentre

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Checkpoint description

Christian Lawson-Perfect 10 years, 1 month ago

Bill Foster 12 years ago

Error in variable testing condition

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Warning

Generated value: `integer`

Parts

Gap-fill

a)

b)

c)

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SFY0004 Implicit 2

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Christian Lawson-Perfect 10 years, 1 month ago

Bill Foster 12 years ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

a)

b)

c)

Generated value: `integer`