Interactive resources in Further Pure
Maclaurin series
Maclaurin series (Geogebra)
This resource is on the Integral website in the following sections: MEI FP2 Power series 1,
AQA FP3 Series 1, Edexcel FP2 Maclaurin series 1, OCR FP2 Maclaurin series 1, WJEC
FP3 Power Series 1.
It’s useful for students to connect the derivation of a Maclaurin series with the graphical
approximations: the linear approximation has the same first derivative, the quadratic has the
same first and second derivatives, and so on. The resource can also be used to look at
functions for which the Maclaurin series is valid only for a limited range of values.
To create a simple version of this resource yourself using Geogebra:
1. Enter the equation f(x)=e^x.
2. Create a slider called n, with minimum 1, maximum 10 and increment 1.
3. Enter the equation g(x) = TaylorPolynomial[f,0,n]. This creates n terms of the Taylor
polynomial for f at x = 0 (i.e. the Maclaurin polynomial).
4. Change the colours and thicknesses of the two graphs if you wish.
5. You may want to use a text box to put the approximation on the screen, e.g. create a
text box and type in "Approximation = " +g

Moving the slider shows successive approximations. Look at how the approximation
improves for a wider range of values of x as the value of n increases.

Double click on the function f(x) (either in the Algebra window or the graph itself) to
change the function, e.g. f(x)=sin(x), f(x)=ln(1+x). In particular, look at the range of
values of x for which the approximation is valid.

The version on the website also allows you to compare the values of the derivatives
of the function and the approximation. It can be helpful to use a different graph for
this, so that the values of the derivatives are more interesting!
Alternatively, you can create a similar resource using Autograph:
1. Enter the equation y = e^x.
2. Select the graph, right-click (or go to Object) and choose ‘Maclaurin series’
3. Uncheck ‘Show Progressive terms’ and click OK
4. Select the Maclaurin graph and click the ‘lightning bolt’ button
5. Use the arrows to increase or decrease the number of terms in the expansion.
Interactive resources in Further Pure
Matrices and simultaneous equations
Intersecting planes (Autograph)
This resource is on the Integral website in the following sections: MEI FP2 Matrices 3, AQA
FP4 Matrices 4, OCR FP1 Matrices 4, WJEC FP1 Matrices 2.
It can be difficult for students to visualise what is happening geometrically when they solve
three simultaneous linear equations. This resource allows them to explore the different
configurations of three planes, particularly cases in which there is no unique solution.
To create this resource yourself using Autograph:
1. Open a new 3D graphs page
2. Choose Equation|Enter equation. Enter the equation of a plane and click ‘Plot as 2D
equation’. Enter two other equations of planes in the same way.

Drag the axes round to look at where the graphs intersect, and whether there is a
unique solution, infinitely many solutions or no solutions.

Choose suitable sets of equations for students to explore. By including variables in
the equations you can use the Constant controller.
e.g. the default set of planes in the resource on the website are:
x + 3y – 2z = 7
2x – 2y + az = 2
3x + y – z = k
This gives a triangular prism initially (with a = k = 1). You can get a sheaf of planes by
changing k, and a single point by changing a.

Have a calculator that handles matrices available, and ask students to check the
determinant of the matrices corresponding to each set of equations.

Ask students to think about how they might recognise different configurations from
the equations of the planes. (Parallel or coinciding places are easily spotted, but a
triangular prism or a sheaf of planes requires some algebraic manipulation!)

You could ask students to find a set of equations corresponding to each possible
configuration.
Interactive resources in Further Pure
Matrices – eigenvalues and eigenvectors
Eigenvalues and eigenvectors (Autograph)
There is a Flash resource Eigenvalues and eigenvectors on the Integral website in the
following sections: MEI FP2 Matrices 4, AQA FP4 Matrices 5, Edexcel FP3 Matrices 2.
Students can usually learn the ‘recipe’ for finding eigenvalues and eigenvectors without too
much trouble. However, they often have very little idea of what an eigenvector is. This
resource introduces the idea geometrically.
To create a similar resource in Autograph:
1. In the “Object” menu, select “Enter shape”.
Enter the coordinates (0, 0) and (1, 1). This should give you a line segment from the
origin to the point (1, 1).
2. With the line segment selected, select “Matrix Transformation” in the “Object” menu.
Enter the matrix you want to. This should give you the image of the line segment
under this transformation.

Move the point (1, 1) around until you find a position for which the original line
segment and its image lie in the same straight line (for some matrix transformations,
they may be in opposite directions). When you have found an appropriate position,
look at the results box at the bottom of the page to see the coordinates of the point,
and hence write down a vector in the direction of the line segment (or you can rightclick on each point and add a text box to label each point with its coordinates). This is
an eigenvector for the matrix. Try to find two different positions for the eigenvector.

Note the position vectors of both the original point and its image, and hence find the
constant by which the eigenvector must be multiplied to obtain its image. This is the
eigenvalue corresponding to the eigenvector you have found.

Double-click on the image line to change the matrix. You can also check the ‘Show
eigenvectors’ box to show the direction of both eigenvectors.

If you right-click on the image point, you can put a trace on the image (this can’t be
done in the Flash resource). Investigate!
Interactive resources in Further Pure
The conic as sections of a cone
Conic sections (Autograph)
This resource is on the Integral website in the following sections: MEI FP2 Curves 2, AQA
FP1 Graphs 3, Edexcel FP3 Coordinate systems 1, WJEC FP2 Loci 1.
Students are often not aware of the connections between the different members of the family
of conic sections. This resource helps them to visualise the conics as sections of a cone.
To create this resource yourself using Autograph:
1. Open a new 3D page in Autograph. Enter the equation r = z (this gives a double
cone, using cylindrical polar coordinates).
2. Enter the equation z = a + bx (this gives a general plane – initially the values of a and
b take their default values of 1).

Use the Constant controller to set b to be zero and try different values of a. Move the
view round so that you can see the cone from above – you should be able to see
circles of varying radii.

Set a = 2 and try different values of b, starting with values between 0 and 1. These
should give ellipses.

Next set b = 1. The plane is now parallel to the slope of the cone, and this gives a
parabola (you will need to change the view to see this properly).

Now try values of b greater than 1. These give hyperbolae. As b gets larger, the
plane becomes closer to vertical.

To look at a vertical plane, change the equation of the plane to x = c. and the
hyperbola approaches a rectangular hyperbola. For the default value of c = 1, you get
a rectangular hyperbola.

Decrease c to zero, and you have a vertical plane through the origin, which gives a
pair of straight lines – also part of the family of conic sections!
Interactive resources in Further Pure
Conic graphs
Conics (Geogebra)
This resource is on the Integral website in the following sections: MEI FP2 Curves 2, AQA
FP1 Graphs 3, Edexcel FP3 Coordinate systems 1, WJEC FP2 Loci 1.
This resource helps to show students that the ellipse, parabola and hyperbola are all part of
the same family, and explores the focus-directrix property. Not all specifications require this,
but it’s useful background knowledge which gives students a context for the work they need
to cover.
To create a simple version of this resource yourself using Geogebra:
1. Mark the point (1, 0) for the focus. The y-axis will be a directrix.
2. Create a slider for e, with e going from zero to 3.
3. Put in the equation (x - 1)² + y² = e²x².
4. Mark a point P on the curve, and draw a segment from P to the focus. Change the
label so that it shows the length.
5. Draw a perpendicular from P to the y-axis, and mark the point where this line crosses
the y-axis. Then hide the line, and draw a segment from P to the point on the y-axis.
Change the label so that it shows the length.

Use the slider to vary the value of e and produce hyperbolas, a parabola and
ellipses.

Divide the length of the first segment by the other (use their labels, e.g. a/b) to show
that the ratio is always the same and equal to the eccentricity.
Alternatively, you can create a similar resource using Autograph:
1. Create a point on the x-axis for the focus, and draw a vertical line such as x = -1 for
the directrix.
2. Select both the point and the line, right-click (or use the Object menu) and choose
Conic by Eccentricity.
3. Select the resulting conic and press the ‘lightning bolt’ button. You can then vary the
eccentricity of the conic.
Interactive resources in Further Pure
Newton-Raphson method
Newton-Raphson (Geogebra)
This resource is on the Integral website in the following sections: MEI C3 Numerical methods
1, AQA FP1 Numerical methods 1, Edexcel FP1 Numerical methods 1, OCR FP2 Numerical
methods 2, WJEC FP3 Numerical methods 1.
This resource illustrates the Newton-Raphson iterations graphically. Students can explore
how different starting positions result in convergence to different roots, and look at how
quickly the iterations converge.
To create this resource yourself using Geogebra:
1. Enter the graph in the input box.
2. Put a point A on the x-axis.
3. Go to Spreadsheet view and type x(A) in cell A1. This gives you the x-coordinate of
point A – the first approximation to the root.
4. In cell A2, type A1 – f(A1)/f’(A1) and copy this down the column as far as you like.
(You can change the number of decimal places shown on Options|Rounding.
5. In cell B1, type Segment[(A1, 0), (A1, f(A1))]. This creates a vertical line from point A
to the graph. Copy this down the column.
6. In cell C1, type Segment[(A1, f(A1)), (A2, 0)]. This creates the tangent line from the
curve to the second approximation. Copy down the column as far as one row before
the end of column B entries.

You can move the first approximation around to explore convergence to different
roots or divergence. Look at column A to see how quickly the iterations converge – in
some cases they diverge first and then converge.

You can change the graph by double-clicking on it.
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Useful links
These interactive resources, and many others, as well as a wide range of other resources
are available by subscribing to the Integral website http://integralmaths.org. Information
about subscriptions to the website can be found at
http://www.mei.org.uk/index.php?section=onlineresources.
Geogebra is freely available open-source software. It can be downloaded from
http://www.geogebra.org. The Geogebra website also contains introductory materials, help
documents and a user forum.
Autograph is available to purchase from http://www.autograph-math.com as a single user or
site licence. You can download a 30-day trial free of change.