tadaa.. Core 2 notes in the making. and ive found out about LaTex.. so formatting is better/will be once i get the chance. There are a few formulas for core 2 in the formula booklet, youd be glad to hear. This includes the sequences ones, cosine rule, binomial expansion etc. do have a look to see.
An index is the number that is the power to something else (the small superscript). The plural of index is indices.
When multiplying powers, indices are added together.
When dividing powers, indices are taken from each other.
When finding powers of powers, the indices are multiplied.
Anything to the index of 0 is 1.
A negative index means that its the reciprocal of the answer to the index.
a^n/m = m√a'^n = m√a^n'
[It spells the word man..]
Translation of [a over b] = x replaced by -x, y replaced by -y.
Reflection in x-axis = y replaced by -y
Reflection in y-axis = x replaced by -x
x-direction stretch by k = x replaced by 1/k x
y-direction stretch by k = y replaced by 1/k y
Sequences and Series
The numbers in a list of numbers are called the terms of the sequence.
Sequences that end are finite.
Sequences that never end are infinite.
Sequences where all the terms are the same is called a constant sequence.
Sequence notation is shown by a letter followed but a subscript number. The subscript shows the position of the number in the sequence. eg. x˅1 represents the first number in the sequence.
Sequences are defined by a rule, eg. x˅n = 2n+5. x is used to represent a number, the nth term of the sequence. So in the sequence a˅1=7 a˅2=9 a˅3=11
When a term can be found from a previous term in a sequence, we say it has a recurrence relation. A recurrence relation is expressed: u˅n+1 = u˅n+3 where u represents the previous term.
An inductive definition for a sequence includes the first term and the recurrence relation.
Sometimes in a sequence the terms get closer and closer to a value. It is converging to that limit.
When a sequence has a limit, using the limit as the starting value gives a constant sequence.
Arithmetic sequences are sequences where there is a common difference between each term.
Arithmetic sequence with first term a and common difference d can be written a, a+d, a+2d, a+3d... etc.. the nth term would be a+(n-1)d
The word series means the sum of the terms in a sequence.
The sum to n terms for an arithmetic sequence is 1/2 n(a+l) where a=first term and l=last term required.
The formula a+(n-1)d (above) works out the last term required, so it can be substituted into the arithmetic series formula, giving 1/2 n(2a (n-1)d)
The sigma sign looks like: ∑ it is used as shorthand to write the sum of the first number of terms.
The number above represents the final value of the variable (n), the number on the bottom the initial variable value, and the expression on the right gives each term in the series (in terms of the variable). The expression on the right can be used to work out a and l, so the formula 1/2 n(a+l) can be used. (Top works out l, bottom works out a)
[ a=1^2=1 l=4^2 ]
Sequences and series 2
Sequences that have a common ratio between each term are called geometric sequences.
Geometric sequences can be expressed ar^n-1 where a= first term, r= ratio, n= number of terms.
A geometric series, just like the arithmetic series is the sum of the terms.
The formula for adding the terms of a geometric sequences is a(r^n -1) /r-1 expressed in sigma notation its:
The sum of all the terms in an infinite sequence can be written S͚͚˅∞
For geometric sequences, it only has a sum to infinity/converges to a limit when -1<r<1 this can also be written as │r│<1. This is because r^n has to converge to 0
When it does converge to a limit, the formula is a/1-r
When arranging n variables there are n! combinations/orders/arrangements they can be in [n factorial n!= n(n-1)(n-2)(n-3)(n-4)...x3x2x1]
When coefficients of an expansion is taken out, an interesting pattern called Pascals triangle is formed. However, drawing out a large Pascals triangle is not always practical. The binomial theorem can be used to find expansions.
This can also be written:
For non-right angled triangles, with edges a, b, c, and angles opposite the corresponding edge A, B, C there are a few rules which can be applied to help work out angles and edges:
- The cosine rule (derived from Pythagoras) a^2=b^2+c^2-2bc cosA
- The sine rule a/sinA = b/sinB or sinA/a = sinB/b
The area of the triangle would be 1/2 ac sinB (derived from 1/2 base height)
Radians are another measure of angles that simplifies some maths. It is the angle at the centre of a segment where radius and arc length are the same. Therefore, when angle of ϴ in the centre or the circle created by the arc, the length of the arc is rϴ Usually when pi is in the number, it means it is in radians.
- 360=2pi radians
- 180=pi radians
Using Pythagoras on one unit triangles means that a few things can be worked out (very useful to remember):
- sin pi/4 = sin 45 = 1/√2
- sin pi/6 = sin 30 = 1/2
- sin pi/3 = sin 60 = √3/2
- cos pi/4 = cos 45 = 1/√2
- cos pi/6 = cos 30 = √3/2
- cos pi/3 = cos 60 = 1/2
- tan pi/4 = tan 45 = 1
- tan pi/6 = tan 30 = 1/√3
- tan pi/3 = tan 60 = √3
The area of a sector is: ϴ/2pi pi r^2 = 1/2 r^2 ϴ
The amplitude of a graph is half the total height of the graph.
Sine, cosine and tangent are periodic functions (the graph repeats itself after a period); there can be multiple answers although the calculator will only show one. Sine and cosine have a period of 360 degrees or radians; the period of tangent is 180 degrees or radians. A calculator will only give one solution to a trigonometric equation, though there will be numerous solutions between given interval. These can be found using a graph, or by using the fact that:
If we construct a right angle triangle from the center, with the vertices on the circumference, then the hypotenuse has length of 1 unit. The angle between the hypotenuse and the other sides is 45 degrees. If we take one of these angles, and call it , and we can calculate the length of the other sides, using .
In our triangle;
Exponentials and Logarithms
Exponential is another word for 'power' eg. a^x x is the index.
For all values of y=a^x the graph passes through (0,1)
The logx of a number is the power you have to raise x by in order to get that number, eg. y=a^b <=> b=log˅a y therefore the index b is log˅a y
[I remember it as yab = blog-ay.. or.. float and swap... float the b, and swap it with the y...]
log˅base number = index is equivalent to base^index = number
log˅a (1/a) = -1
log˅a a^x = x
a^(log˅a x) = x
The laws of logarithms are quite easy, they follow the rules from indices:
log˅a m + log˅a n = log˅a mn
log˅a m - log˅a n = log˅a(m/n)
k log˅a m = log˅a m^k
[float and sink.... k floats and sinks to the front..
Logs are useful. it means equations like a^x = b can be solved by taking togs of both sides.
Differentiation and Integration
You might find recapping integration and differentiation from core 1 first useful.
[Differentiation = Decreasing power multiplying by old power, and INtegrating = INcreasing the power, then dividing by new power]
In core two, you'll have to differentiate and integrate numbers too. You simply turn them into a single power of x.
Remember that if it is a minus power, then the number will go up, eg. x^-3 => 1/ -2x^-2 as it
Integrating also works the same when it is negative accept when n=-1.
However, sometimes you will come across graphs that cannot be integrated with the methods that have been covered so far. You will need the trapezium rule for this.
The trapezium rule is a method of estimating the area under a curve by dividing it into a series of strips, each of which is approximately a trapezium.
If you substitute all the numbers in finding the areas of each trapezium you can simplify the formula; you end up with 1/2 x h(first and last ordinates + double the rest of the ordinates in the middle)