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Find partial derivatives of $f(x,y)$ and identify its stationary points

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Bill Foster

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Bill Foster 6 years, 5 months ago

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Christian Lawson-Perfect 7 years, 8 months ago

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Christian Lawson-Perfect 7 years, 8 months ago

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Name	Status	Author	Last Modified
Find partial derivatives of $f(x,y)$ and identify its stationary points	Has some problems	Christian Lawson-Perfect	20/11/2019 14:51
Patrick's copy of Find partial derivatives of $f(x,y)$ and identify its stationary points	draft	Patrick Joyce	07/08/2017 17:45
Harry's copy of Find partial derivatives of $f(x,y)$ and identify its stationary points	draft	Harry Flynn	09/03/2018 14:14
Luis's copy of Find partial derivatives of $f(x,y)$ and identify its stationary points	draft	Luis Hernandez	30/11/2018 01:09
Luis's copy of Find partial derivatives of $f(x,y)$ and identify its stationary points	draft	Luis Hernandez	30/11/2018 14:56

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Question statement

Answer the following questions about the function:

$f(x,y)=\simplify[std]{ ({a} / 3) * x ^ 3 + ({b} / 2) * x ^ 2 * y + {c} * y ^ 2 * x + {d} * y}$

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	2
c	integer	12
ch	integer	1
d	number	-12
m	integer	2
s5	integer	-1
a1	integer	1
b	integer	14
b1	integer	6
c2	integer	2
c1	integer	2
d1	integer	2

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
a1*c1

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:

Statement
Advice
"Part a)" → "Gap 0." - Correct answer

Parts

Part a) Gap-fill
1. Gap Gap 0. Mathematical expression
  Add
2. Gap Gap 1. Mathematical expression
  Add
Part b) Choose several from a list

Gap-fill

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

Find the partial derivatives of $f$ with respect to $x$ and $y$ .

Note that if you want to enter a product of two unknowns, such as $xy$ , then you input the expression in the form x*y.

$\displaystyle { \partial f \over \partial x} =$ Gap 0.

$\displaystyle {\partial f \over \partial y} =$ Gap 1.

Advice

a)

$\begin{align} \partial f \over \partial x &= \simplify[std]{(({a} * (x ^ 2)) + ({b} * x * y) + ({c} * (y ^ 2)))} \\[1em] \partial f \over \partial y &= \simplify[std]{((({b} / 2) * (x ^ 2)) + ({(2 * c)} * x * y) + {d})} \end{align}$

b)

$(a,b)$ is a stationary point for the function $f(x,y)$ if $f_x=0$ and $f_y=0$ , where the partial derivatives are evaluated at $x=a$ , $y=b$ .

So you have to make sure that both of these partial derivatives are $0$ at the stationary point.

For this example we have from the above equations that:

$\begin{align} \simplify[std]{(({a} * (x ^ 2)) + ({b} * x * y) + ({c} * (y ^ 2)))} &= 0, & \mathbf{(1)}\\ \simplify[std]{((({b} / 2) * (x ^ 2)) + ({(2 * c)} * x * y) + {d})} &= 0, & \mathbf{(2)} \end{align}$

The left hand side of equation (1) can be factorised as:

$\simplify[std]{({a1}x+{b1}y)*({c1}x+{d1}y)=0}$

and so we have:

$y=\simplify[std]{{-a1}/{b1}*x}, \text{ or } y= \simplify[std]{{-c1}/{d1}*x}$

First case: $y= \simplify[std]{{-a1}/{b1}*x}$

Substituting this into equation (2) gives:

$\simplify[std]{{b}/2*x^2-{2c*a1}/{b1}*x^2+{d}}=0 \implies \simplify[std]{{-b*b1+4*c*a1}/{2*b1}*x^2={d}}$

Hence $x=\var{m}$ or $x = \var{-m}$ . The corresponding stationary points are:

$\left(\var{m},\simplify[std]{-{a1*m}/{b1}}\right) \text{ and } \left(\var{-m},\simplify[std]{{a1*m}/{b1}}\right)$

Second case: $y= \simplify[std]{{-c1}/{d1}*x}$

Substituting this into equation (2) gives:

$\simplify[std]{{b}/2*x^2-{2c*c1}/{d1}*x^2+{d}}=0 \Rightarrow \simplify[std]{{-b*d1+4*c*c1}/{2*d1}*x^2={d}}$

There can be no more stationary points as this equation has no real solution.

Give a worked solution to the whole question.

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Part	Test	Passed?
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Mathematical expression
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Find partial derivatives of $f(x,y)$ and identify its stationary points

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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Comment

Checkpoint description

Bill Foster 6 years, 5 months ago

Christian Lawson-Perfect 7 years, 8 months ago

Christian Lawson-Perfect 7 years, 8 months ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

a)

b)

First case: $y= \simplify[std]{{-a1}/{b1}*x}$

Second case: $y= \simplify[std]{{-c1}/{d1}*x}$

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Find partial derivatives of f(x,y)f(x,y) and identify its stationary points

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Bill Foster 6 years, 5 months ago

Christian Lawson-Perfect 7 years, 8 months ago

Christian Lawson-Perfect 7 years, 8 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

a)

b)

First case: y={{−a1}{b1}x}y= \simplify[std]{{-a1}/{b1}*x}

Second case: y={{−c1}{d1}x}y= \simplify[std]{{-c1}/{d1}*x}

Find partial derivatives of $f(x,y)$ and identify its stationary points

Generated value: `integer`

First case: $y= \simplify[std]{{-a1}/{b1}*x}$

Second case: $y= \simplify[std]{{-c1}/{d1}*x}$