Matrices add in the natural way:
Example:
Multiplying Matrices
Multiplying is a little more complex. Remember that you multiply rows by columns.
Example:
Determinants of Matrices
To find the determinant of a matrix multiply diagonal corners together and subtract.
Example:
Inverses of Matrices
Each entry in the matrix can be divided byin the natural way. An example is shown below
]]>has periodifisorand periodifis
has maximum value of 1 and minimum value of -1 ifisorandhas no maximum or minimum value ifis
The transformation of graph rules are applied in the usual way. This is illustrated above for This means that the maximum and minimum values ofareandrespectively, and the period ofisthe frequency is
The period and frequency of each curve above are the same since the coefficient ofis the same. The normal transformation rules apply. Adding 1 to sin x moves the curve up 1 and subtractingfrom x moves the curve right
]]>We might multiply this out to obtainNow we equateto this and solve for the coefficients a, b, c.
Example: Complete the square for the expression
Equating coefficients of
Equating coefficients of
Equating constant terms:
Hencein completed square form is
Having completed the square we can solve the equation
Example: Complete the square for the expressionHence solve the equation
Equating coefficients of
Equating coefficients of
Equating constant terms:
Hencein completed square form is
Now we can solve
]]>Example:
This extends to multiples and sums of powers ofIf we remember thatand that when differentiating for these functions too we multiply by the power and subtract one from the power then,
The process is the same for all powers offractions and negative powers included:
Whatever form the question is given it we must expression the function to be differentiated in the formFor example,
We write this as then
These are some more examples,
Differentiation is used to find rates of change (when we differentiate with respect to time), gradients, slopes, tangents and normals.
]]>The Product Rule:
If a function h consists of two simpler functionsandmultiplied together, then
Example: Differentiate
It is a good habit to get into to write downand then you can just substitute them into the expression for
Example: Differentiate
The product rule can be used repeatedly with any number of products.
If a function h consists of three simpler functionsandmultiplied together, then
Example: Differentiate
]]>The Quotient Rule:
If a function h consists of two simpler functionsandwiththen
(1)
Proof:
At this point we perform the sort of trick common in maths. We takefrom the first term and add it to the second. We can then factorise and simplify.
andand if we we letthen in the denominator,We then obtain (1).
Example: Differentiate
It is a good habit to get into to write downand then you can just substitute them into the expression for
Example: Differentiate
]]>Ifis to be written in the formthenand Ifis to be written in the formthen use and
Calculus and Algebra
Ifhas a turning point (also called a stationary or maximum or minimum) atthenatIfatthenhas a minimum atIfat thenhas a maximum atIf then no conclusion may be drawn from this test. The nature of the stationary point may be deduced by sketching the curve in the vicinity of the point.
Ifthen the completed square form foris
The minimum value of y can then be read off. This minimum value isand this minimum value occurs at
Logarithms
]]>The task is to find the shaded area above between the two curvesandwhich intersect at the points I labelled the curves top and bottom. Using this sophisticated notation, the area is
Sometimes it is not quite so obvious what the equation of either the top or bottom curve is.
The graph isWe have to find the area of the region marked A. The top is obviously
W take as bottom the curvethenare the solutions tosoORso soWe have then
]]>
We subtract a term from the right hand side to give
and then integrate to givewhich is usually written as
It is important to chooseandthe right way round. If there is anterm, thenis usually chosen to be this power, so that the resulting integralterm is easier to integrate than the original integral. It is useful to write down all the terms of the equation first then it becomes very easy to substitute them all into the formula.
Example: Find
Example:
It is usually the case thatif this is possible but there are exceptions. The best example is when we have to integrate a
Example: Find
Example: Find
To integrate this we have to write
]]>There are three steps to inverting a function
1. Makethe subject so that you have another function
Interchange occurrencesofandsono you have
Replaceby:the answer is
Example:Find
Interchangeand:
If you draw the graphs ofandon the same axis you will notice something very striking. The lineis a line of symmetry: to obtain the graphjust reflect the graphin the– axis. To see why this is so, notice that steps 1 and 2 above interchangeand
This is illustrated above for the graphsandwhich are inverse to each other.
A problem may arise if you have a functionwhich gives the same value offor more than one value ofWhen you try to invert the function and you finda value of may return no value ofor more than one value ofIt is necessary in a case like this to restrict the domain of the inverse function to eliminate those “impossible”'s and “duplicate”'s. For example, if– we take the positive square root to ensure only one value offor each value ofand we must haveIfthen and the domain ofisWe take the range to beso that there is one value offor each