Hi,
Students often find it difficult to sketch with GC curves that are defined parametrically.
The typical issues for students using TI-GC are in deciding Tmin, Tmax and Tstep.
We'll consider the following example for our discussion:
The curve C is defined parametrically by
x = (1 + t)^(2/3), y = ln (t^2), t <= -1.
Find the area of the region enclosed by C, the lines x = 0 and x = 1, and the x-axis.
First, we need to sketch the curve and identify the region visually before we can arrive at a definite integral.
The lines x = 0 and x = 1 give us some indication of what values to set for Tmin and Tmax.
When x = 0, t = -1 obviously.
When x = 1, (1 + t)^(2/3) = 1 from which t = -2 is obtained after rejecting the other possibility t = 0, as we are given that t <= -1 and we know that y = ln (t^2) is not well-defined when t = 0.
So, we set Tmin = -2 and Tmax = -1.
Never assume that Tmin and Tmax are going to take positive values, as many students tend to do so for the sake of simplicity. School exams often expect approaches contrary to students' thinking :P
What is Tstep, you may ask? It's the interval between two consecutive T values. Say, we set Tstep to be 0.1, then we'll have the GC to plot the curve for these values:
-2 (Tmin), -1.9, -1.8, -1.7, -1.6, -1.5, -1.4, -1.3, -1.2, -1.1, -1 (Tmax).
We have to be careful to select a reasonable Tstep, or we'll risk having an inaccurate graph. For example, if we were to let Tstep be 0.5, then the GC will plot the curve inaccurately for these values:
-2 (Tmin), -1.5, -1 (Tmax).
You may try and compare both graphs on your GC to see what I mean. The bottomline is more points, better accuracy.
If one were to use a Tstep value of 0.01, he/she risks precious time because the GC will run for some time to generate the curve over many values of T:
-2 (Tmin), -1.99, -1.98, ..., -1.03, -1.02, -1.01, -1 (Tmax).
Another bottomline is more points, but more to a resonable extent.
Thank you.
Cheers,
Wen Shih
this will be very very very helpful to students
I want to write some tips on how to draw 1/f(x) and f'(x) but haven't really found the time for it yet.... soon :D
Tried these tips on my students... worked well so far :D
Dear eagle,
Thanks for your encouragement :)
Share with me your tips and I'll help you write :)
Cheers,
Wen Shih
Originally posted by wee_ws:Dear eagle,
Thanks for your encouragement :)
Share with me your tips and I'll help you write :)
Cheers,
Wen Shih
my main tip for 1/f(x) is whatever is above the x-axis remains above
whatever is below remains below
By just using the above on top of knowing the new asymptotes, and whether the curve will tends to 0-, 0+, +∞, -∞, one can sketch out the 1/f(x) curve very simply.
Then need examples to guide on how to apply properly :D
Very easy one...
Something similar can be used for f'(x) as well, for the 0-, 0+, +∞, -∞ on how to see the gradient of the curve
Maybe should compile all these tiny tips, tricks, techniques and strategies into a book such that it complements all other H2 Maths books... More of like a book on applications of H2 Maths concepts to questions... Include ways to check for and minimize careless mistakes using calculators, using common sense checks, etc
Hi eagle,
I have written about the graphs of y = 1/f(x) and y = f'(x):
http://www.freewebs.com/weews/curvesketching2.htm
http://www.freewebs.com/weews/curvesketching4.htm
You may like to suggest improvements :)
Thanks in advance!
Cheers,
Wen Shih