After we looked at the last example today I realized I should have been more clear about the restrictions on the composition problems we looked at.

For the problems we looked at, like `sin(tan^-1(x))`, you need to make sure that `x` is in the domain of `tan^-1(x)`. But once you do that, then really the only question is – are any of those values not allowed in `sin(x)`? For this example since inverse tangent accepts all real numbers and sine can evaluate all real numbers, we have no restriction, and `x in RR`.

My statement that `x` has to be in the domain of the inside function and in the range of the outside function helps us to avoid mistakes like `cos^-1(cos(10))=10`. This is not true although very tempting. In order for this to be true, 10 would have to be in the range of `cos^-1(x)`, which it isn’t. The range of `cos^-1(x)` is `[0,pi]`.

I hope you all really enjoy your time off – you have certainly earned it. If you post or comment during break you may not see it show up right away. Don’t worry – it will. Don’t post or comment twice, just be patient. -Peace

I enjoyed reviewing the photographs you took for this assignment. If you can e-mail me the digital pics I’d like to put up a slide show on the blog so everyone can see your work. In fact Flickr has a neat tool which lets you annotate your photograph with click-able boxes that could, for example, reveal information about your sinusoid or its equation.

I thought Blair submitted a noteworthy entry. His friend has a snake that has been photographed by National Geographic. Cool!! You can see it here.

What mathematical theorem is represented by the following mosaic? Justify your answer.

Please take a look at the graph we made for #3.
We determined that the period was `pi` but when we graphed it I don’t think we actually had a period of `pi`. The final graph should look like this:

agraph
xmin=-3pi; xmax=3pi; xscl=pi/2; ymin=-5; ymax=20; yscl=5; axes();
plot(10+5sin(2x+pi));
endagraph

The graph uses 1.6 as an approximation of `pi/2`.