Direction fields

[NE2000]

a) what is the point of them (unless I haven't covered that yet, in which case just refer me to the area in which they are useful)

b) What's the best way to draw them, how many little sloping lines do you guys put in and how accurate do you get the gradients.

c) how do you draw them in the CAS calculator?

Thanks in advance

[kamil9876]

a.) it's just another visual way of representing how gradient changes just like a graph of dy/dx vs x.
b.)Usually they[VCAA or exam creators] provide you with an axis and so it's best to draw the slope field on that axis for integer co-ordinates(e.g: 1,-1). Also, if you can see an obvious turning point then maybe include some flat line there even though it doesn't meet my previous advice of integer co-ordinates. Most likely, you're field isn't going to be perfect, just like a graph. All it is is a 'sketch', 'rough intuitive depiction' of what's goin on. So if you notice some pattern where if x increases, dy/dx increases, then make sure that u draw a steeper thing next to it so the comparison is evident.

[shinny]

b. Best way to draw it to scale is simply to count boxes. If the gradient you're meant to have is two, from point you're at now, go two points right and then one up, and draw a line at that angle. If it's negative half, go two points down and one square right etc. This helps to keep it all basically exactly to scale with very little time wasted in technical matters.
c. Depends on which CAS you have. On the TI-89, go into mode and change it to differential equations and go to F1 and type it in. There were some tricks regarding how you type it in though. From what I remember, I think 'x' is typed as 't' and 'y' is typed as 'y(1)' or whatever y line you're working with.

[NE2000]

Thanks, that pretty much covered my concerns. I still find them slightly pointless though , anticipate their usefulness in future

[/0]

Thanks, that pretty much covered my concerns. I still find them slightly pointless though , anticipate their usefulness in future

Well, they can let you graph the antiderivative of any function. So if there is something which you can't integrate, you can draw a slope field to get the general shape of the antiderivative.