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TRANSCRIPT
Surfaces of the form 𝑧 = 𝑓(𝑥, 𝑦)
Can you visualise these surfaces?
𝑧 = 𝑥2
𝑧 = 𝑦3
𝑧 = cos 𝑥 + sin 𝑦
Use GeoGebra to illustrate each of the surfaces.
HOW TO GUIDE
Open a new window / GeoGebra file
Click View → 3D graphics (close the Graphics window)
Type the function into the input bar in the form 𝑓(𝑥, 𝑦) = 𝑥2
For powers use ^
Click the toggle next to ‘3D graphics’ to get the toolbar of options underneath.
Click on the arrow next to the last small icon on the toolbar and select
‘projection for glasses’ to view the surface in 3D
Click on the ‘rotate 3D graphics’ button on the main toolbar then click in
the 3D graphics window and rotate the window to see the surface from
different perspectives.
Click on the down arrow on the same button and click ‘zoom out’. Then click
in the middle of the screen to zoom out – this allows you to see more of the
surface.
Note: In the latest version of GeoGebra instead of being along the top of the screen
the menus are accessible by clicking on the icon at the top right of the screen.
Select the 3D graphics windows and the input bar and deselect the 2D graphics
window.
To see the view in 3D click on at the top right of the screen and then to find
the ‘glasses’ tool.
Contours and sections
Create the section 𝑥 = 0 for the surface 𝑧 = cos 𝑥 + sin 𝑦. Vary this to sections of the
form 𝑥 = 𝑘 and 𝑦 = 𝑙.
HOW TO GUIDE
Type 𝑥 = 0 into the input bar
Right click on the plane and select ‘object properties’. Under colour drag the
‘opacity’ to 100 (you can also change the default colour if you wish). Close
the window.
Right click on the surface and select ‘object properties’. Under colour drag
the opacity to around 25 (again, change the colour if you wish).
Rotate the graphics window as before to see the image from different
perspectives. You should be able to clearly see the section formed where the
plane cuts the surface.
Click View and select Graphics. Resize this window so it is the same size as
the 3D graphics window.
In the graphics window select the slider tool and then click anywhere in
the Graphics window. In the box name the slider 𝑘 and ensure the option
‘number’ is selected. Change the interval to be between (say) -8 and 9 and
click OK. (You can move a slider by right clicking on it and dragging it to
where you want it to be)
In the algebra pane right click on 𝑥 = 0 and in the Object Properties, under
basic, change the equation to 𝑥 = 𝑘.
Drag the slider from left to right to see how the section changes for different
values of 𝑥.
Repeat the process for 𝑦 = 𝑙
What would the contours look like for this surface? Contours have the form 𝑧 = 𝑘
and indicate points that have the same height (𝑧-coordinate).
HOW TO GUIDE
Open a new window (this will be 2D graphics)
Input curves of the form cos 𝑥 + sin 𝑦 = 𝑘 (you will automatically be asked if
you want to create a slider for 𝑘).
Right click on the slider and select ‘animation on’.
Choose a sensible range of values for 𝑘
Finding stationary points and their nature
Question 1 (adapted from legacy FP3 June 2011)
A surface S has equation 𝑧 = 8𝑦3 − 6𝑥2𝑦 − 15𝑥2 + 36𝑥
Sketch the sections of S given by 𝑦 = −3 and 𝑥 = −6. From your sketches deduce
that (-6, -3, -324) is a stationary point on S and state the nature of the stationary
point.
Find the coordinates of the other three stationary points on S
Question 2 (adapted from legacy FP3 June 2015)
A surface has equation 𝑧 = 3𝑥2 − 12𝑥𝑦 + 2𝑦3 + 60
Show that the point A(8,4, −4) is a stationary point on the surface and find the
coordinates of the other stationary point B.
A point P with coordinates (8 + ℎ, 4 + 𝑘, 𝑝) lies on the surface. Show that
𝑝 = −4 + 3(ℎ − 2𝑘)2 + 2𝑘2(6 + 𝑘)
and deduce that the stationary point A is a local minimum.
By considering sections of the surface near to B in each of the planes 𝑥 = 0 and
𝑦 = 0 investigate the nature of the stationary point B.
Normal line and tangent plane
PREREQUISITE KNOWLEDGE
The equation of a plane can be written in the vector form 𝒓 = 𝒂 + 𝜇𝒃 + 𝜆𝒄 where 𝜆
and 𝜇 are constants, 𝒂 is the position vector of a point on the plane and 𝒃 and 𝒄 are
two non-parallel vectors in the plane. .
A plane can also be written in the cartesian form 𝑎𝑥 + 𝑏𝑦 + 𝑐𝑥 + 𝑑 = 0.
The vector equation of a line can be written in the form 𝒓 = 𝒂 + 𝜆𝒅 where 𝜆 is a
constant, 𝒂 is the position vector of a point on the line and 𝒅 is the direction vector of
the line.
The vector product finds a vector which is perpendicular to two given vectors (
𝑎1
𝑎2
𝑎3
)
and (
𝑏1
𝑏2
𝑏3
) and is given by (
𝑎1
𝑎2
𝑎3
) × (
𝑏1
𝑏2
𝑏3
) = (
𝑎2𝑏3 − 𝑎3𝑏2
𝑎3𝑏1 − 𝑎1𝑏3
𝑎1𝑏2 − 𝑎2𝑏1
)
Use GeoGebra to construct a 3D graphic of the surface 𝑧 = 𝑥2 + 𝑦2 showing the
tangent plane and normal plane at the point A(1,2,5) and the direction vectors
parallel to the x-axis and the y-axis at that point.
HOW TO GUIDE
Click View – 3D graphics to open a 3D graphic window and input the function
𝑧 = 𝑥2 + 𝑦2 by typing 𝑓(𝑥, 𝑦) = 𝑥2 + 𝑦2
In the Graphics window click on the second button from the left (point) and
click anywhere on the 𝑥 axis to create the point A. Click anywhere on the 𝑦
axis to create a point B.
In the input bar enter 𝑎 = x(A) then enter 𝑏 = y(B). This creates two numerical
quantities 𝑎 and 𝑏 associated with the 𝑥 and 𝑦 coordinates of A and B
respectively.
Input the point (𝑎, 𝑏) – this creates point C in the 𝑥𝑦 plane
Input the point (a, b, 𝑓(𝑎, 𝑏)) – this creates the D point in 3D
You should now be able to drag the points A and B in the Graphics window
and see the effect in 3D in the 3D graphics window
Input 𝑓_𝑥(𝑥, 𝑦) = derivative [𝑓(𝑥, 𝑦), 𝑥] and 𝑓_𝑦(𝑥, 𝑦) = derivative [𝑓(𝑥, 𝑦), 𝑦] –
this will produce graphs which show the value of the partial derivatives 𝜕𝑓
𝜕𝑥 and
𝜕𝑓
𝜕𝑦. Hide these graphs by clicking on the circular button next to their equations
in the algebra pane. (continued)
Input vector[D, D+(1, 0, 𝑓_𝑥(𝑎, 𝑏))]. This will default to have the name 𝑢 and
represents a vector in the direction of the section parallel to the 𝑥 axis.
Input vector[D, D+(0, 1, 𝑓_𝑦(𝑎, 𝑏))]. This will default to have the name 𝑣 and
represents a vector in the direction of the section parallel to the 𝑦 axis.
If you wish, right click on the vectors 𝑢 and 𝑣 and go to object properties to
change the colour / thickness of each of the vectors. Move points A and B in
the graphics window to see how 𝑢 and 𝑣 behave.
Input 𝑛=vector[D, D+cross(𝑢, 𝑣)]. This creates a vector perpendicular to 𝑢
and 𝑣 which is the normal vector to the tangent plane at that point. You can
leave this visible or hide it, whichever you prefer.
Input line[D,𝑛] – this creates the normal line at the point D.
Input perpendicular plane[D,𝑛] to create the tangent plane at D.
Again move points A and B in the graphics window to see how the vectors
and plane behave. You can use object properties to change the colour or
opacity of the surface and the plane if this helps to visualise things more
clearly.
Mark Schemes for stationary points exam questions
Question 1 (adapted from legacy FP3 June 2011)
Question 2 (adapted from legacy FP3 June 2015)
A little trip ‘off spec’
The specification does not require students to know any rules for determining the
nature of stationary points, but they might be interested to explore a rule and learn
about second partial derivatives at the same time.
A simple test involves finding 𝐷 =𝜕2𝑓
𝜕𝑥2×
𝜕2𝑓
𝜕𝑦2− (
𝜕2𝑓
𝜕𝑥𝜕𝑦)
2
Step 1: At each stationary point work out the values of the second partial derivatives
Step 2: Calculate D at each stationary point
Step 3: If D < 0 the stationary point is a saddle point
If D > 0 and 𝜕2𝑓
𝜕𝑥2 > 0 the stationary point is a (local) minimum
If D > 0 and 𝜕2𝑓
𝜕𝑥2< 0 the stationary point is a (local) maximum
If D = 0 an alternative method is needed
Explore the stationary points on the surfaces:
𝑧 = 8𝑥2 + 6𝑦2 − 2𝑦3 + 5
𝑧 = 2𝑥2 + 𝑦2 + 3𝑥𝑦 − 3𝑦 − 5𝑥 + 8
𝑧 = 𝑥3 − 3𝑥2𝑦
How is this guy involved with one of them?
MEI Extra Pure:
Multivariable
calculus
Claire Baldwin
FMSP Central Coordinator
MEI Further Mathematics A levelAssessment Overview
Mandatory unit:
Core Pure
144 raw marks
2 hrs 40 mins50% of A level
Major options:
Mechanics Major
Statistics Major
120 raw marks
2 hrs 15 mins33⅓% of A level
Minor options:
Mechanics Minor
Statistics Minor
Modelling with Algorithms
Numerical Methods
Extra Pure
Further Pure with Technology
60 raw marks
1 hr 15 mins
(1 hr 45 mins for FPT)
16⅔% of A level
Extra Pure is not one of the units that is designated as being co-teachable
with AS Further Mathematics or AS Mathematics
Modular MEI specification• Multivariable calculus is currently on the Further
Applications of Advanced Mathematics (FP3) module
• This is a 1½ hour examination where candidates choose
3 questions from 5, worth 24 marks each.
Option 1: Vectors
Option 2: Multi-variable calculus
Option 3: Differential Geometry
Option 4: Groups
Option 5: Markov Chains
• The content of the multivariable calculus section is
essentially the same in the new specification
Linear MEI specification• Multivariable calculus is in the Extra Pure minor option
along with Recurrence Relations, Matrices and Groups.
• There are no optional questions – candidates must
answer all the questions in the printed answer booklet.
• The four topics may not be evenly weighted in the
assessment e.g. on the sample assessment materials:
Q1 (10 marks) and Q2 (4 marks) – Groups
Q3 (12 marks) – Recurrence Relations
Q4 (16 marks) – Multivariable calculus
Q5 (18 marks) – Matrices
Total: 60 marks, 1 hour 15 mins
Key ideas
normal line
and tangent
plane
partial
derivatives
contours
and
sections
Any hikers in the room??
But first…… a quiz!!
http://www.rgs.org/webcasts/activities/contours/co
ntours.html
or
http://bit.ly/2tcN0yc
Applications of contour diagrams
Partial differentiation
Partial differentiation
Partial differentiation
Partial differentiation
Partial differentiation
Partial differentiation
Partial differentiation
𝜕
𝜕𝑦𝑦𝑒𝑥𝑦 = 𝑒𝑥𝑦 + 𝑥𝑦𝑒𝑥𝑦
Stationary points on surfacesIn two dimensions a stationary point can be:
a maximum a minimum a point of inflexion
What might a stationary point look like on a three
dimensional surface?
Stationary points on surfaces
Maximum
point
Minimum
point
Two types of
saddle point
How to find stationary points on surfaces
The method will be given if investigation of the nature of
the stationary points is required.
Explore the location and nature of the stationary points on
the surfaces 𝑧 = 𝑒−(𝑥2+𝑦2) and 𝑧 = 𝑥𝑦 𝑥 + 𝑦 − 1
Check your answers using GeoGebra
Explore the location and nature of the stationary points on the
surfaces𝑧 = 𝑒−(𝑥2+𝑦2) and 𝑧 = 𝑥𝑦 𝑥 + 𝑦 − 1
Check your answers using GeoGebra
Acknowledgements• Royal Geographical Society quiz
http://www.rgs.org/webcasts/activities/contours/contours.html
• Seabed image
http://www.infomar.ie/surveying/Bays/BayProfileImages/Dingle/BaysDingleBY1200.jpg
• Isobars image http://www.metoffice.gov.uk/public/weather/surface-
pressure/#?tab=surfacePressureColour&fcTime=1494849600
• Rainfall map
http://www.lordgrey.org.uk/~f014/usefulresources/aric/Resources/Teaching_Packs/Key_Sta
ge_4/Weather_Climate/11.html
• EEG image http://www.neurotherapytacoma.com/resources/brain-at-work
• Corneal topography image https://entokey.com/wp-content/uploads/2017/01/gr2-691.jpg
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