Tuesday, 26 April 2011
Test
\[ E(\mathbf{x}) = \sum_{i \in \mathcal{V}} \theta_i(x_i) + \sum_{ij \in \mathcal{E}} \theta_{ij}(x_i, x_j) \]
C3 Functions Round-Up
Reminder: this is really about AQA C3, as OCR and Edexcel are slightly different. They don't change the truths of mathematics though.
After quite a long break, I've started doing maths again. I've forgotten chunks of it, as you would expect. Here are some things I've forgotten (on the assumption that they're generally forgettable):
1. Inverse functions are simply functions reversed. The domain becomes the range and vice versa. However, not all functions can have an inverse. If a function is many-one (as quadratic functions generally are) then they can't have an inverse - it would be one-many, which is not a function at all.
2. To get an inverse function you can do two things. If x appears only once in the original function, then you can do a reverse flow chart, reversing the order and the operations. BUT BEWARE OF SELF INVERSE FUNCTIONS - if you meet a subtraction from or a divide into, those operations stay the same because they're self-inverse functions. IF x appears more than once you need to do some equation-fiddling. On a function, x is a/the domain value(s). y is the same as f(x) - ie it's the range of the function. So by re-arranging the equation to isolate x, you're turning the function INSIDE OUT and making x the f(x) instead of the variable at the beginning (the domain).
3. composite functions are two functions put together. They're usually written as fg(x). What this means is you do both operations. You write out g(x) first, say it's x2 + 4. Then you put it into f(x), which might be 2x - 3. The resulting composite function is 2(x2+4)-3.
4. On some functions, there are values that cannot be defined and that affects the range as well as the domain. Take f(x) = 1/x for example. X cannot be 0. But this means there are going to be impossible values of f(x) as well. A good way of estimating this is to draw the graph of f(x) on your calculator!! Always use the calculator wherever possible, even if you need to sketch the graph by more formal methods.
After quite a long break, I've started doing maths again. I've forgotten chunks of it, as you would expect. Here are some things I've forgotten (on the assumption that they're generally forgettable):
1. Inverse functions are simply functions reversed. The domain becomes the range and vice versa. However, not all functions can have an inverse. If a function is many-one (as quadratic functions generally are) then they can't have an inverse - it would be one-many, which is not a function at all.
2. To get an inverse function you can do two things. If x appears only once in the original function, then you can do a reverse flow chart, reversing the order and the operations. BUT BEWARE OF SELF INVERSE FUNCTIONS - if you meet a subtraction from or a divide into, those operations stay the same because they're self-inverse functions. IF x appears more than once you need to do some equation-fiddling. On a function, x is a/the domain value(s). y is the same as f(x) - ie it's the range of the function. So by re-arranging the equation to isolate x, you're turning the function INSIDE OUT and making x the f(x) instead of the variable at the beginning (the domain).
3. composite functions are two functions put together. They're usually written as fg(x). What this means is you do both operations. You write out g(x) first, say it's x2 + 4. Then you put it into f(x), which might be 2x - 3. The resulting composite function is 2(x2+4)-3.
4. On some functions, there are values that cannot be defined and that affects the range as well as the domain. Take f(x) = 1/x for example. X cannot be 0. But this means there are going to be impossible values of f(x) as well. A good way of estimating this is to draw the graph of f(x) on your calculator!! Always use the calculator wherever possible, even if you need to sketch the graph by more formal methods.
Wednesday, 6 April 2011
Polynomials
These are a very common sight in almost any study of mathematics. It’s worth taking a moment to define them properly and think about how they can be used.
A polynomial is a mathematical expression involving a whole, positive power of x. The type of polynomial depends on the highest power of x in the expression. The different bits of the expression are known as terms, and the polynomial as a whole is the sum of the terms it contains.
Take a quadratic expression: 4x² + 2x - 2. It has three powers of x: 2, 1 and 0. These are whole numbers and positive (except 0). This is therefore a polynomial. The same applies to cubic expressions: x³ + 4x² + 5x + 2 is a polynomial too.
In fact, the same applies, whatever the power of x, as long as it is not a fractional or negative number. A polynomial does not need to have several terms, like most cubics and quadratics do. So, 1 is also a polynomial, being a term in x raised to the power of zero. The same applies to any constant (ie any number). 4x + 1 is a polynomial too, as x is x raised to the power of one.
Polynomials are written in descending powers of x, though they do not have to be.
The numbers in front of the x variables are called coefficients, and can take any value, including fractional or negative values, as can constants.
The highest power of x in a polynomial defines the degree of that polynomial. So 1 is a polynomial of degree 0, because the highest power of x here is 0. 4x + 1 is a polynomial of degree 1, and quadratics are polynomials of degree 2.
Polynomials can be created by expanding brackets. (x+2)² gives x² + 4x + 4 when expanded. In fact, both expressions here are polynomials. If the brackets were to give us an expression with a fractional power, then of course this would not be the case.
When you make a polynomial like a quadratic equal to 0, then you have a polynomial equation. In the case of a quadratic, such as 4x² + 2x – 2, the resulting equation is usually written as: 4x² + 2x – 2 = 0. As an equation, it is now a description of a parabolic curve which intersects the x-axis at (-1,0) and (½, 0).
Polynomials can be added, subtracted, multiplied and divided. The Factor and Remainder Theorems can be used to find out additional information about them. For example, if a number assigned to the variable x causes the polynomial to equal zero, that number is a factor of the polynomial (a root, a value of x where the curve of the equation crosses the x-axis). So we can demonstrate that -1 is a factor of 4x² + 2x – 2 by putting it in place of x. 4(-1)² + (-1x2) – 2 = 0. Therefore -1 is a factor of this polynomial. This is the essence of the Factor Theorem. The Remainder Theorem is a little more involved but states that if we divide the polynomial by x-a then the remainder is f(a) – in other words, the remainder is the sum of the polynomial when a is put in place of the variable x.
These are the absolute basics. Of course it does get a lot more complicated! But it is worth familiarising yourself with these principles before moving on.
A polynomial is a mathematical expression involving a whole, positive power of x. The type of polynomial depends on the highest power of x in the expression. The different bits of the expression are known as terms, and the polynomial as a whole is the sum of the terms it contains.
Take a quadratic expression: 4x² + 2x - 2. It has three powers of x: 2, 1 and 0. These are whole numbers and positive (except 0). This is therefore a polynomial. The same applies to cubic expressions: x³ + 4x² + 5x + 2 is a polynomial too.
In fact, the same applies, whatever the power of x, as long as it is not a fractional or negative number. A polynomial does not need to have several terms, like most cubics and quadratics do. So, 1 is also a polynomial, being a term in x raised to the power of zero. The same applies to any constant (ie any number). 4x + 1 is a polynomial too, as x is x raised to the power of one.
Polynomials are written in descending powers of x, though they do not have to be.
The numbers in front of the x variables are called coefficients, and can take any value, including fractional or negative values, as can constants.
The highest power of x in a polynomial defines the degree of that polynomial. So 1 is a polynomial of degree 0, because the highest power of x here is 0. 4x + 1 is a polynomial of degree 1, and quadratics are polynomials of degree 2.
Polynomials can be created by expanding brackets. (x+2)² gives x² + 4x + 4 when expanded. In fact, both expressions here are polynomials. If the brackets were to give us an expression with a fractional power, then of course this would not be the case.
When you make a polynomial like a quadratic equal to 0, then you have a polynomial equation. In the case of a quadratic, such as 4x² + 2x – 2, the resulting equation is usually written as: 4x² + 2x – 2 = 0. As an equation, it is now a description of a parabolic curve which intersects the x-axis at (-1,0) and (½, 0).
Polynomials can be added, subtracted, multiplied and divided. The Factor and Remainder Theorems can be used to find out additional information about them. For example, if a number assigned to the variable x causes the polynomial to equal zero, that number is a factor of the polynomial (a root, a value of x where the curve of the equation crosses the x-axis). So we can demonstrate that -1 is a factor of 4x² + 2x – 2 by putting it in place of x. 4(-1)² + (-1x2) – 2 = 0. Therefore -1 is a factor of this polynomial. This is the essence of the Factor Theorem. The Remainder Theorem is a little more involved but states that if we divide the polynomial by x-a then the remainder is f(a) – in other words, the remainder is the sum of the polynomial when a is put in place of the variable x.
These are the absolute basics. Of course it does get a lot more complicated! But it is worth familiarising yourself with these principles before moving on.
Thursday, 28 January 2010
What is Multiplication?
Can be a tricky question, this.
Primary schoolchildren are taught that it is repeated addition, which makes a lot of sense.
4 x 4 =
4 + 4 + 4 + 4 =
16
Four lots of four/groups of/sets of.
But multiplication is also a scaling quantity. It ratchets things up massively quickly. If you type 2x2 into your calculator, then keep multiplying every answer by 2, the calculator will quickly run out of digits. To compare this with addition misses the point that multiplication is the centre of all geometric operations.
It does through the squaring and cubing and so on effect. If you take four centimetres, and for each of these four, you add another four, you get 4 x 4. It is known as squaring because the Greeks used to see it geometrically as the way to find the area of a square. For each centimetre across, there are four up (and vice versa).
x2 is a powerful, recurring idea, which has its role in pretty much the entire universe. (E = mc2)
It's also worth noting the effect of multiplication by a fractional quantity.
4 x 1/4 will obviously increase the fraction, but it will have a decreasing effect on the whole number. Unlike multiplication by two whole numbers, where the answer is greater than either, in this operation, the answer will always be smaller than one of the inputted values.
Primary schoolchildren are taught that it is repeated addition, which makes a lot of sense.
4 x 4 =
4 + 4 + 4 + 4 =
16
Four lots of four/groups of/sets of.
But multiplication is also a scaling quantity. It ratchets things up massively quickly. If you type 2x2 into your calculator, then keep multiplying every answer by 2, the calculator will quickly run out of digits. To compare this with addition misses the point that multiplication is the centre of all geometric operations.
It does through the squaring and cubing and so on effect. If you take four centimetres, and for each of these four, you add another four, you get 4 x 4. It is known as squaring because the Greeks used to see it geometrically as the way to find the area of a square. For each centimetre across, there are four up (and vice versa).
x2 is a powerful, recurring idea, which has its role in pretty much the entire universe. (E = mc2)
It's also worth noting the effect of multiplication by a fractional quantity.
4 x 1/4 will obviously increase the fraction, but it will have a decreasing effect on the whole number. Unlike multiplication by two whole numbers, where the answer is greater than either, in this operation, the answer will always be smaller than one of the inputted values.
Tuesday, 19 January 2010
Tips For Revising for a Module
It's exam season: in fact, it's just gone. So many of us will be turning our thoughts to retakes - or in my case, to reapplying for exams we had to drop out of.
So how do you study for an A Level maths exam?
1) Don't necessarily do all the questions. For some, like me, it's essential to do every question in the book and all the past papers. For many, that's just repetition. You need to do the first few and the last few. Past papers are good to study as well as to do, to work out which questions often come up.
2) Make sure your calculator has new batteries. Yes - graphical calculator batteries do NOT last for years!
3) MAke sure you know the formulae which are NOT in the formula book for the module you are doing. There are usually several really common ones - like trig identities for example. Check each chapter in the textbook to see which formulae you need. It is easy to forget these, especially if you have been *ahem* working on the same module a while.
4) DON'T do anything the night before. The formulae will start to jump around in your head.
5) Revise the previous modules a little. You will find some identities and formulae you studied before that are not explicitly repeated come in handy. For example, the cosine and sine rules from C2 can be very handy when doing the questions on reciprocal trig functions in C3. Also the rules and techniques of radians are very useful indeed, though quite a bit of practice on this is done through C3.
6)If you do past papers and you can't get a tutor to mark them for you, you can do it yourself via the online answer papers which are published. It's not an exact science because you are not amazing at maths, unlike the markers, which is why you're studying it now. So BE HARSH on yourself and be conservative. Markers will often be generous with follow-through marks so you can at least be prepared. That's the pessimistic side of me coming through.
So how do you study for an A Level maths exam?
1) Don't necessarily do all the questions. For some, like me, it's essential to do every question in the book and all the past papers. For many, that's just repetition. You need to do the first few and the last few. Past papers are good to study as well as to do, to work out which questions often come up.
2) Make sure your calculator has new batteries. Yes - graphical calculator batteries do NOT last for years!
3) MAke sure you know the formulae which are NOT in the formula book for the module you are doing. There are usually several really common ones - like trig identities for example. Check each chapter in the textbook to see which formulae you need. It is easy to forget these, especially if you have been *ahem* working on the same module a while.
4) DON'T do anything the night before. The formulae will start to jump around in your head.
5) Revise the previous modules a little. You will find some identities and formulae you studied before that are not explicitly repeated come in handy. For example, the cosine and sine rules from C2 can be very handy when doing the questions on reciprocal trig functions in C3. Also the rules and techniques of radians are very useful indeed, though quite a bit of practice on this is done through C3.
6)If you do past papers and you can't get a tutor to mark them for you, you can do it yourself via the online answer papers which are published. It's not an exact science because you are not amazing at maths, unlike the markers, which is why you're studying it now. So BE HARSH on yourself and be conservative. Markers will often be generous with follow-through marks so you can at least be prepared. That's the pessimistic side of me coming through.
Monday, 18 January 2010
e (2.71828)
No, this is not a post for luvved-up early 90s teenagers.
e is a bizarrely cool number, sometimes known as the natural logarithm.
Like π it is an irrational number - ie it cannot be expressed exactly as the ratio of two whole numbers (and thus as a fraction).
How do we get it then?
Imagine an exponential curve. A curve that is nx. Say, y= 3x.
As usual, the gradient will generally differ as x differs.
Is there a curve which has a gradient of 1, where x=0? There is, and, coolly, this curve has a gradient of 1 where x=0 and ALSO passes through y=1, ie the value of the function is 1.
This curve is y = e x. It's not just at y=1 where the value of the function equals the gradient, it's every point on the curve.
As you can imagine, that makes differentiating with e easy...
There is an excellent page here which shows you some interesting examples of e in action.
e is a bizarrely cool number, sometimes known as the natural logarithm.
Like π it is an irrational number - ie it cannot be expressed exactly as the ratio of two whole numbers (and thus as a fraction).
How do we get it then?
Imagine an exponential curve. A curve that is nx. Say, y= 3x.
As usual, the gradient will generally differ as x differs.
Is there a curve which has a gradient of 1, where x=0? There is, and, coolly, this curve has a gradient of 1 where x=0 and ALSO passes through y=1, ie the value of the function is 1.
This curve is y = e x. It's not just at y=1 where the value of the function equals the gradient, it's every point on the curve.
As you can imagine, that makes differentiating with e easy...
There is an excellent page here which shows you some interesting examples of e in action.
Reciprocal Trig Functions
Sine, cosine and tangent are functions. You input a value - an angle in radians, or degrees, and for each angle there is a value. They are repeating functions.
They also have reciprocal functions.
The reciprocal of sine x is cosec x and is (1/sin x)
The reciprocal of cosine x is secant x and is (1/cos x)
The reciprocal of tan x is cotangent x (cot x) and is (1/tan x).
They also have reciprocal functions.
The reciprocal of sine x is cosec x and is (1/sin x)
The reciprocal of cosine x is secant x and is (1/cos x)
The reciprocal of tan x is cotangent x (cot x) and is (1/tan x).
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