This is quite tricky to get your head round, I think.
Differentiation is what you do to a curve to obtain its gradient. It works by splitting the curve into infinitesimally tiny chunks and tending towards a limit (ie where you started from). I think I put the proof of it on an earlier post.
Basically it gives you the rate of change in the curve as a formula. Because curves are always changing, the gradient will be different at each point. You plug the x-coordinate into the differentiated gradient formula and there's your gradient at x=whatever.
But if you integrate that curve you get the area under the curve. It works in a similar way to differentiation: it takes the space under the curve and chops it up into tiny rectangles to approximate the area.
It seems to work out that in terms of a formula, integration is the opposite of differentiation (ie raise the power by one and divide by the value of the new power, instead of dropping the power by one and timesing by the new power).
The curve bounding an area with the x-axis is now the value of the area formula differentiated - ie it is the rate of change of the area itself.
Differentiation and integration are inverse processes: that's the theorem of calculus (which means stones used for counting, or something).
If you have a derivative, you can integrate it to get the indefinite integral - which you can then find the full equation for if you know a set of co-ordinates. So by knowing the derivative, you can find the equation of that curve.
If you have a derivative, often you can differentiate again to find a stationary point on the curve - the second derivative test.
I'm still not sure I get this, you know. I still find the connection between the two loose and hard to actually articulate. I can do it with the formulae no problem at all - but understanding the linkage and how rates of change relate to curves is much harder.
Monday, 16 February 2009
Sunday, 8 February 2009
More Basic Trig
Every angle has a sin or cosine or tan (well, not quite every angle, in the case of tan). You can use this value to find the angle, doing it easily on a calculator.
You just press sin-1 .456 or whatever it is, and that gives you the angle (if you have the calc. set to degrees that is).
You just press sin-1 .456 or whatever it is, and that gives you the angle (if you have the calc. set to degrees that is).
Saturday, 7 February 2009
Amen Trig Corner
Well I guess I don't get it, but then maths is like that. Someone tells you to do x, you do x, you get it right or you get it wrong: no one gives a toss. No one cares that you've just described the entire universe (less the supernovae or the black holes), no one cares that you've inscribed patterns across space - patterns which remain, whether you like them or not; whether you think the guy who made them matters or not: they remain.
They are not contingent like the semi colon, or culturally derived like the feminism or the theism: they are not subject to the whims of tenured professors: there they are and you can just something off if you don't like it, tbh.
But to just do x, if you have been told to: that defies and defeats the entire point of mathematics. They tell us this because they know we are thick and also because in a pyramidal structure like maths, you just cannot *understand* things that you need to know how to do, when you are little.
No-one tells an 8 year old about subjects, objects, predicates, the passive voice or the genitive case. Equally, you shouldn't need to tell an 8 year old about the mechanics of division in order to teach them how to do it. But so paranoid are we about maths, so in hoc to an educational vision based more in socialism than in intellectualism, that we cannot believe we can teach these skills without sending 8 year olds into A Level territory, and in the meantime, depriving them of the skills of actually dividing one number by another.
Division by repeated subtraction: for crying out loud are you REALLY going to stop bright 11 year olds dividing by decimals by using this infantile but ridiculously complex method? Are you REALLY going to prevent tough calculation by saying "well, just take away and take away and take away"?
More to the point: stop insisting on an "understanding" that defies most adults, let alone children. You do not need to know that a sentence consists of subject, object and predicate to write one. Why are we baffling kids with mathematical jargon? Why are we constructing entirely false notions of "understanding" which exclude the simple explanation "well, I am timesing x by y" or "well, I guess I am seeing how often z goes into a". - and hamstringing bright kids, and stopping them calculating?
It could'nt be because socialist academics who write the curriculum don't want kids to be able to do these things...could it?
They are not contingent like the semi colon, or culturally derived like the feminism or the theism: they are not subject to the whims of tenured professors: there they are and you can just something off if you don't like it, tbh.
But to just do x, if you have been told to: that defies and defeats the entire point of mathematics. They tell us this because they know we are thick and also because in a pyramidal structure like maths, you just cannot *understand* things that you need to know how to do, when you are little.
No-one tells an 8 year old about subjects, objects, predicates, the passive voice or the genitive case. Equally, you shouldn't need to tell an 8 year old about the mechanics of division in order to teach them how to do it. But so paranoid are we about maths, so in hoc to an educational vision based more in socialism than in intellectualism, that we cannot believe we can teach these skills without sending 8 year olds into A Level territory, and in the meantime, depriving them of the skills of actually dividing one number by another.
Division by repeated subtraction: for crying out loud are you REALLY going to stop bright 11 year olds dividing by decimals by using this infantile but ridiculously complex method? Are you REALLY going to prevent tough calculation by saying "well, just take away and take away and take away"?
More to the point: stop insisting on an "understanding" that defies most adults, let alone children. You do not need to know that a sentence consists of subject, object and predicate to write one. Why are we baffling kids with mathematical jargon? Why are we constructing entirely false notions of "understanding" which exclude the simple explanation "well, I am timesing x by y" or "well, I guess I am seeing how often z goes into a". - and hamstringing bright kids, and stopping them calculating?
It could'nt be because socialist academics who write the curriculum don't want kids to be able to do these things...could it?
A Proper Go At Explaining Trig
Alright, well, clearly it is the study of angles and triangles and by use of the unit circle, of circles too.
We talk about sin, cosine, and tan a lot in the basic stuff.
Although we learned SohCahToa at school, it really only applies to right angled triangles. But by using the unit circle (radius 1, centre at the origin) and drawing a right angled triangle inside it, we can calculate basic values of sine, cosine and tan, which hold for any angle and can be used in calculations involving any triangle.
The sine of an angle is a measurement of its distance from the horizontal axis, which can be negative or positive and which can be between 1 and -1 inclusive. If you think of an angle sweeping up around the circle, it reaches its furthest point from the axis (ie sin of 1) at 90 degrees.
The cosine of an angle is a meaurement of the distance from the vertical y axis of a point on the end of the angle; again, between 1 and -1.
The tan of an angle is a measurement of whether a line of angle x would meet a tangent to the circle drawn at right angles to the radius (the x axis) near, far, or not at all. The tan can have any value. Calculators really don't like being asked tan 90 or tan 270 because the lines of these angles will never meet a tangent.
There are some cool diagrams on the wikipedia page that show these things in action.
We talk about sin, cosine, and tan a lot in the basic stuff.
Although we learned SohCahToa at school, it really only applies to right angled triangles. But by using the unit circle (radius 1, centre at the origin) and drawing a right angled triangle inside it, we can calculate basic values of sine, cosine and tan, which hold for any angle and can be used in calculations involving any triangle.
The sine of an angle is a measurement of its distance from the horizontal axis, which can be negative or positive and which can be between 1 and -1 inclusive. If you think of an angle sweeping up around the circle, it reaches its furthest point from the axis (ie sin of 1) at 90 degrees.
The cosine of an angle is a meaurement of the distance from the vertical y axis of a point on the end of the angle; again, between 1 and -1.
The tan of an angle is a measurement of whether a line of angle x would meet a tangent to the circle drawn at right angles to the radius (the x axis) near, far, or not at all. The tan can have any value. Calculators really don't like being asked tan 90 or tan 270 because the lines of these angles will never meet a tangent.
There are some cool diagrams on the wikipedia page that show these things in action.
Thursday, 5 February 2009
The Really Basic Basics of Trig
Trig is basically to do with triangles and circles. It works from the properties of angles relative to the sides of triangles and how these can be seen in circles (unit circles - with a radius of 1). It's used to tell you the size of angles and sides using other information you already know about the triangle.
The trig identities, I reckon, are probably best defined as ratios.
All fractions are ratios, as well as being processes.
For a right angled triangle (where we tend to begin with these)
SohCahToa
Sin =opposite/hypotenuse
Cos = adjacent/hypotenuse
Tan = opposite/adjacent
For each angle, there will be a different value for each one of these. So sin 42 will be the same whatever triangle you have. It regulates what the sizes of the opposite and hypotenuse will be only in terms of the ratio between them.
Sin and Cos (not sure about Tan) will always give a value between 1 and -1 inclusive.
The trig identities, I reckon, are probably best defined as ratios.
All fractions are ratios, as well as being processes.
For a right angled triangle (where we tend to begin with these)
SohCahToa
Sin =opposite/hypotenuse
Cos = adjacent/hypotenuse
Tan = opposite/adjacent
For each angle, there will be a different value for each one of these. So sin 42 will be the same whatever triangle you have. It regulates what the sizes of the opposite and hypotenuse will be only in terms of the ratio between them.
Sin and Cos (not sure about Tan) will always give a value between 1 and -1 inclusive.
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