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The Chain Rule
November 21st, 2009
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The chain rule allows you to differentiate composite functions (functions of other functions) ie) f(g(x)) such as sin(3×2) or (5×3+2x+3)2. The rule is as follows
or to understand it more simply you differentiate the inner function and multiply it by the derivative of the outer function (leaving what’s inside alone).
Differentiating brackets raised to a power
The chain [...]
Find equation of tangent to a curve
November 19th, 2009
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The tangent to a curve is a line which touches the curve at a point without intersecting it at that point so the gradient of the curve at that point and the gradient of the tangent are the same. So we can work out the point the tangent passes though and the gradient of the [...]
Online Graph Sketchers
November 2nd, 2009
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Whilst you should always try to sketch any graphs you need to sketch yourself, since this is the only way you’ll get to understand why graphs look the way they do, it is often useful to use a computer program to check against. Here are two online graph sketching applets that you can use for [...]
Inequalities
November 1st, 2009
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An inequality(or inequation) is similar to an equation accept for instead of saying both side of the inequality are equal we say one side is greater than (or equal to depending upon the type of inequality) the other, this is done using the greater than (), greater than or equal to (), less [...]
Fundamental Theorem of Calculus
October 31st, 2009
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This theorem forms much of the basis of calculus and the uses of differentiation and integration. It basically states that differentiation and integration are opposites so if you differentiate and integral you’ll get the function you started with. This can be stated as follows:
if then
or in the more simple case
if [...]
Integrating Fractions – using the natrual logarithm – Example tan(x)
October 27th, 2009
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From result found be differentiating the natural logarithm,
for some function f(x),
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). [...]
Exponential Functions
October 24th, 2009
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Exponential functions are any function of the form
latex for some constants a and b.
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 [...]
Taylor Series with example cos(x)
October 13th, 2009
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The Taylor series is the general case of the Maclaurin Series for calculating the value of a function. It enables you calculate the value of a function at any point if you can find the value of the function and and all its derivatives at any point. This is done as a power series. The [...]
Fermat’s Last Theorem
October 1st, 2009
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Fermat was a 17th century mathematician who provided a number of theorems and some of their proofs. The most intriguing of hi theorems is Fermat’s Last Theorem which is as follows:
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 [...]
Compound tan – tan(A+B)
September 24th, 2009
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We can use expressions for cos(A+B) and sin(A+B) to help us find tan(A+B).
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 [...]
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