Please note: I am not responsible for the content of external websites and cannot guarantee that these features will be working, available or accurate.

Shodor Graph Sketcherr

This enables you to plot most functions in Cartesian form. It requires you to enter the graph as a function of x and then clicking “graph” will plot it for you. Using the tools in the top right hand corner you can zoom in/out and more around the graph, you can also adjust the axis.

A screen shot of the graph y=x^2 using the Shodor Graph Sketcher

Polar Co-ordinates Graph Sketcher

This applet will allow you to sketch graphs using polar co-ordinates. You can enter a range and your graph as a function of t (the angle). It also allows you to zoom in and out of the graph and includes a list of the functions it supports and how to enter them.

Screenshot of the graph r=sin(t) as drawn by the polar co-ordinates graph sketcher

Examples

Some simple examples which contain only one variable are:

Solving and Manipulating Inequalities

Inequalities can be solved by rearranging them and isolating the variable you want to find in a similar way to normal equation (see the post quadratic inequalities to see how to solve quadratics). However, rather than getting an exact value such as x=3 we get a range (open or closed) of values such as x<2 or -3<-1.

Much of the manipulation is the same though there are slight variations when dividing or multiplying by negative numbers or taking the reciprocal. The important thing to remember is that like normal equations we must do the same to both sides.

Addition and Subtraction

Addition and subtraction are exactly the same to equalities. We can add or subtract whatever we like as long as we do the same to both the sides. This enables us to take expressions “to the other side” by reversing their sign. For example all the following manipulations are valid.

Multiplication and Division

Again we can perform multiplication and division in a similar way to the way we perform it with equalities by doing the same to both sides. However, if we are multiplying or dividing by a negative number we must reverse the direction of the inequality since

This means we must be careful when diving by an unknown since by definition we don’t know whether or not it is positive or negative. If this has to be done you should consider both the cases it is positive and negative separately and if it is only positive or negative then the other inequality should lead to a contradiction which can easily be spotted such as x<0 and x>3.

Examples of valid manipulation are below:


for and/or for

Reciprocals

When taking the reciprocal or “one over” of an expression you must reverse the inequality so

if then

or in the more simple case

if then

It is this idea that allows us to know, for example,

from the knowledge that

This makes much of integration easier as it is often much easier to work out the derivative a function than work out the integral of one so we can look for functions which when differentiated give us the function that we want to integrate and then know that the integral is that function plus a constant.

and the fundamental theorem of calculus we cay say that

where c is the integration constant

Simple Example

The most basic example of this is the integration of 1/x,

More complex example: Integration of tan(x)

A slightly more complicated example of this is the integration of tan(x). To do this we must remember that and notice that . This means that -tan(x) is of the form as required. Using this we can get

Trick for using this identity

Sometimes we get integrals that are almost in this form but not exactly, eg) , however to solve these we can often factorise a constant so that it is in the required form. In this example we can take out a 2 so we get

If a and b are both positive then the graph will be an upward curve which tends to infinity as x tends to infinity and tends to 0 as x tends to negative infinity and looks something like the below. Note that all exponential graphs cut the y axis at 1.

The graph of y = 2^x (y equals 2 to the power x)

If a is positive and b is negative the graph is simply a reflection of this about the y axis to give the following graph:

The graph of y=2^-x (y equals 2 to the power of minus x)

The most import exponential graph is because the gradient of this graph is always equal to the value of at that point.

…

The series is said to be taken about meaning we calculate the derivatives of the function at the point a and then from these we find the value of the function. Because the series is infinite we can never find the value of the function exactly but we can give it to any required degree of accuracy by taking the first i terms of the series. The Maclaurin Series is just the Taylor series about 0.

Example cos(x)

An example of the Taylor series is to find a power series for cos(x). We can choose to do this about any value of a so in this example we will use . In this example we will look at the first 2 no zero terms.

The first we need to find the value of the derivatives and the function at pi/2.
The first one is:

We now need to differentiate the function to get:

and then take its value at which is:

Similarly for the second term we differentiate again to find the second derivative is:

so at pi/2 this is

Since the last term was zero we need to find the next one which is:

so

Now we have found all the values of the function and its derivative we need for the level of accuracy required we can simply put these values into the series to get

…
…

To test this we can try
…

which is close to the 1/2 it really is.

the equation

has no integer (whole number) solutions for n>2

For example a solution for n=2 is x=3, y=4, z = 5 since

however the theorem states that for any n larger than 2 a set of integer solutions such as these cannot be found.

Despite the simplicity of this theorem it took 300 years until 1994 for it to be solved by Andrew Wiles using advanced maths.

A fascinating book on the problem is called Fermat’s Last Theorem (by Simon Singh) which goes through the history of the problem and many of the people who have attempted to solve it.

Using the identity tanx = sinx / cosx we can write

we can now substitue in
sin(A+B) = sinAcosB + sinBcosA
and
cos(A+B) = cosAcosB – sinAsinB
to get

We can now divide both the top and bottom by cosAcosB to get

or

We can now simplify this by cancelling any cosA and cosB to get

finally by substituting the identity we find our result

And it can be shown that this result can be extended to

The Maclaurin series for a function f(x) is as follows:

…
or

Example sin(x)

An expression for sin(x) can be calculated using this method. Firstly we must decide what level of accuracy to go to, for the example we will go to the 3rdterm. The next step is to differentiate sinx. Firstly let f(x) = sinx

then

We can now combine these into the series to get

…

…

which can be used to calculate the value of sin(x) — though only for the radian measure of angle.

eg)
… 0.8663

Which is approximately the value you would get if you type sin(pi/3) into a calculator.

This can be shown using De Moivres theorem. Firstly consider the complex number z = cosΘ + isinΘ and let

zⁿ = 1
(cosΘ+isinΘ)ⁿ = 1

which using De Moivres theorem gives

cos nΘ + isin nΘ = 1

We can now compare the real and imaginary parts to find the values of n

cos nΘ = 1 and sin nΘ = 0

Therefore nΘ = 2pπ for integral values of p
So

which gives

so the roots of unity are



etc

However what is interesting about these is when you plot them on an argand diagram they are evenly spaced around a circle of radius 1 and centred at the origin. An example for n=8 is shown below

The roots of unity are space evenly around a circle of radius 1 and centre at the origin

This happens because the increase in the angle for each successive root is equal since we divided 2pi by n.