One last post before I move on from ‘continuous’ curves….
I was going to include this in the last post, but that was already too long.
We have looked at curves which are continuous everywhere, and some which are not – but are continuous for most of the way.
Is it possible for a curve to be discontinuous everywhere? In theory yes, though we need to consider rational and irrational numbers.
A rational number is any that can be written down, accurately, with numbers. This includes numbers there are ‘recurring’ like 0.3333333 because this can be written as 1/3, and be accurate
Pi is an example of an irrational number
So if we say that y = f(x) where f(x) = 1 when x is rational and f(x) = 1/x where x is irrational… then that would define a curve, but one that is so chopped up is would be continuous in only very small sections between rational numbers
In the last post I showed the difference between a Continuous curve and a Discontinuous curve, with a few examples from well known curves.
In this post I am going to show that this can be useful to know in answering a certain sort of question.
I’m not going to do this whole exam question.
The first part involves putting the value x = 3 into the given equation. You will get the answer y = -6
Use the value x = 4 and you get the value y = 20.
This is where the fact that this is a continuous line is so important. You know that the line must cross the line y = 0 somewhere between x=3 and x=4 , and that is what we mean by the root.
If you don’t believe that, think of the equation y=1/x.
If x = -2, the y = -1/2
If x = 1 then y = 1.
The same situation; y goes from negative to positive. Does that mean we have a value of x between -2 and 1 for which 1/x = 0?
No it doesn’t, because the curve for y=1/x is not continuous. we can only use that rule for continuous curves.
Note: Some curves, such as y=1/x in fact, are continuous for much of their length. There is just one place where it is not. So you can use the rule above if the curve is continuous in the RANGE of numbers you are working in.
I’ve been looking at exam questions a lot recently in this diary – well it is coming up to exam time!
In June, the focus is going to be on famous Mathematicians, Over the summer I’m looking to write some entries on mathematical puzzles – If you know some good ones please send them in!
Today though, I’m looking at curves – the lines we can draw to represent certain mathematical equations.
In particular, today and tomorrow I am writing about curves that are ‘continuous’ and curves which are not, which we call ‘discontinuous’. Also, what that means for how we can use these curves to answer questions – That bit is for tomorrow!
What do I mean by continuous? Basically this means that an ant can follow the curve and get to all parts of the curve, without flying off the page or leaving the line. There is a Mathematically precise way of defining ‘continuous’ but I think my ant gives you the idea.
Lets look at some examples. (For this I am going to borrow screenprints from my favourite websites, the link on the Links page)
y = x2
You can imagine the any being able to walk around this line. In fact any line of the form
Axa + Bxb + Cxc+ …. will be continuous so long as a, b, c etc are positive integers.
y = sin (x)
And again the ant can walk up and down these curves.
Since the line for y = cos x looks similar, we can say that is continuous too.
Lets have a look at some Non-Continuous curves now, The easiest to show is y = 1/x
This curve is in two parts. Our ant isn’t going to get from one part of the curve to the other without a jump.
Going back to our trigonometry, Tan (x) – unlike Sine and cosine – is discontinuous. There are many and regular breaks in the line.
In tomorrow’s post I will look at some more eccentric examples and show why its important to understand if a curve is continuous or not.
For today’s question I am looking at one in Linear algebra
I should start by saying what I mean by ‘Linear Algebra’, which is a term I sometimes use with GCSE students and find they haven’t heard of it… I guess that’s just me using language from ‘further along’ the Maths road, that will become second nature to you later on if you continue with the subject…… anyway it means what it sounds like.. algebra as its used to describe lines.
Anyway, on with the question – there is not a lot of work to do here, but to get to the answer you need to remember how we can get to two lines being parallel through their equations – and the secret lies in their gradients. Two lines will be parallel if their gradients are the same. on the graph below I show two lines, one with the equation y = 2/3 x + 2 and one with the equation y = 2/3 x – 1. As you can see on the graph, the lines are parallel. The equations are in the familiar form y = mx + c so we read off the gradients as the same, 2/3 in each case.
In the question we have from the exam, that is not so. Well, not for both. We can see the gradient from the first line – that is 3.
Some re-arrangement is needed, and that’s the small amount of work needed.
3y – 9x + 5 = 0 – Add 9x and take 5 from each side gives
3y = 9x – 5 – Divide throughout by 3
y – 3x – 5/3. We don’t really need the 5/3 part; we can see though from the 3x that the lines have the same gradient so are parallel.
Today’s question is one I did with a student last week about petrol consumption.. and it comes with a confession – I got a bit brain-tied when I first tried to do it. This can happen to anyone.
Its not a spectacularly difficult question – though all question can seem difficult if you can’t see how to do it straight off.
My student started with the right step, by highlighting the important information. That was good start, I said, commented on what the units for ‘consumption’ were, and then my brain froze.
I unfroze it later so lets have a look at what to do.
Deal with the first 9 minutes first. What is the speed? Well, whenever you see a situation where it is a ‘mile a minute’ – just think 60mph, since there are 60 minutes in an hour. Makes thinks quicker that way.
At that speed – which is less than 65 mph, we use the first line of data.
1 gallon will take us 50 miles at that speed, but we are going nowhere near that far – just 9 miles. So the amount of petrol used will be 9/50 of a gallon – which is 0.18 gallons.
The next part we are given the speed, 70mph, so we know it is the second line of the data we need. First though we need to know the distance. 1 Hour 36 minutes = 1.6 Hours. 36/60 is 0.6 of an hour, and that makes its easier to do the calculation.
70 x 1.6 = 112 miles.
[You could do this on your calculator. I’m always looking for short cuts, and I notice that this is the same as 16 x 7 which I can do with my times tables]
At this speed, we would use more than 1 gallon, because 1 gallon will only get us to 40 miles; 2 gallons 80 miles…. or to put it another way, divide our number of miles by 40 which gives us 2.8 gallons.
Now look at the petrol used in bother parts of the journey. This gives us 2.98 gallons.
This is less tan 3 gallons, which is what the question asked us to do.
With many questions that say ‘show that’, its best to leave thinking about what you are ‘showing’ until the end.
Then split the journey in two. Don’t try doing a whole question in one when you can split it into parts.
Yesterday’s post built up some of the things we need to answer a question where the power is negative and a fraction.
One key point is that for all numbers n
n0 = 1 and n1 = n.
With negative powers we still need to maintain the rule of adding powers.
2x x 2-x = 2x-x (and since x – x x = 0) = 20 = 1
Re-arrange this and we get 2-x = 1/2x
And there the first new rule – a negative power is 1/the positive power
32 = 9 so 3-2 = 1/9
To get the second rule we need to consider how powers can be combined.
(n2)3 = n6 – When you raise a power to a power – multiply the powers
[Consider n x n x n x n x n x n]
Now look at
(n2)1/2 = n1 = n
So what does raise to power of half mean if this involves getting from n2 to n? Taking the square root! We could replace 2 and 1/2 with 3 and 1/3 – so see n1/3 means take the cube root – and so on.
n1/x means take the xth root of n.
Just before we get back to the question given, lets just complete the patterns by considering what x3/2 means. I have seen some exam questions that do pose questions like this.
I suggest you split the 3/2 into 1/2 x 3 or, n3/2 = (n1/2)3
so 43/2 = (sqrt(4)3) = 23 = 8. It is generally easier to take the ‘root part first. In a non calculator paper you’ll only be asked about roots you know.
Let’s get back, at last, to the original question.
64-1/2– Take the 1/2 part first, that means square root, and the square root of 64 is 8. The – part means take the reciprocal 1/8
What Carol did was take 1/2 of 64, not the square root of 64, so your answer should include a sentence. ‘Carol didn’t know that a power of 1/2 means square root, not multiply by 1/2’ – then give the correct answer of 1/8.
I’ve made this into 2 blogs posts with lots of background but if you can remember the rules given in these posts, this question need not take you long in an exam,
Not because I’ve been watching election results today, but because I was discussing ‘powers’ with a student this week – Today’s subject is ‘powers of numbers’… Its the first of a two-parter.
Raising a number by a fraction
Before starting on the matters of powers, let’s consider the form of this question.
It’s another of those ones that suggests someone has done the question, and you have to give comment on their answer. In this case the question does say that Carol has made an error, but this won’t always be the case. I have done a question recently where the ‘answer’ given was correct, and the marks were to be gained by saying so.
With these question types, I think you just have to do the question yourself. You may be able to spot an error in something you haven’t tried yourself, but I’ll be honest, as a Maths tutor I can’t always do so myself.
Let’s get back to the question in hand and consider what a power means if is a) negative and b) a fraction – since in this case the power is negative and a fraction. I could tell you both answers, but I like to show how these answers fit into the wider picture.
You will recall that if were want to multiply two ‘powers’ together you can do this by adding the powers – so long as the ‘base’ is the same. The base is the number ‘raised to’ the power. In this case the base is 2.
22 x 23 = 25.
This makes sense when you think of 22 as 2 x 2 and 23 as 2 x 2 x 2. Put the big multiply together as
2 x 2 x 2 x 2 x 2 and you can see this can be written as 25
That rule of powers works well when the powers are positive whole numbers. What we want is for it to work with other numbers.
Powers of 0 and 1
First we need to consider two special cases.
What does 21 mean? And what does 20 mean?
21 means 2 multiplied together one time – and that is just 2! Remember this holds for all base numbers and you can’t go wrong.
41 = 4 361 = 36 40001 = 4000
With 20 we want to keep to the ‘adding indexes’ rule so that
23 x 20 = 23+0 = 23
8 x 20 = 8 – We can soon see that for this to work, 20 must have the value 1
Again, that works for all numbers
40 = 1 360 = 1 40000 = 1
Now we know what powers of 0 and 1 are, can you see what consequence this has for negative and fraction powers? This is what I’ll be looking at in the next post.