Two quick questions (2 Viewers)

[Nooblet94]

1- What's the easiest way to graph y=f(g(x)) if you're given the graph of f(x) (e.g. graph y=f(e^x), y=f(1/x)). I can do them most of the time, but it takes me forever compared to graphing y=g(f(x)).

2- Anyone got examples of harder polynomial questions? Stuff like the q8 of 2010. I'd never seen anything like that before, so I kind've freaked out when I looked at it (although it turned out to be easy after looking at solutions).

 

[RealiseNothing]

1- What's the easiest way to graph y=f(g(x)) if you're given the graph of f(x) (e.g. graph y=f(e^x), y=f(1/x)). I can do them most of the time, but it takes me forever compared to graphing y=g(f(x)).

2- Anyone got examples of harder polynomial questions? Stuff like the q8 of 2010. I'd never seen anything like that before, so I kind've freaked out when I looked at it (although it turned out to be easy after looking at solutions).

Do you mean q8 of 2011? The only other similar question that comes to mind is q7 of 2010. btw the solutions to that question aren't exactly the best, they don't take into account a very specific case.

 

[Nooblet94]

Do you mean q8 of 2011? The only other similar question that comes to mind is q7 of 2010. btw the solutions to that question aren't exactly the best, they don't take into account a very specific case.

Oh yeah sorry, 2011. 2010 was the basel problem, wasn't it?

 

[RealiseNothing]

Oh yeah sorry, 2011. 2010 was the basel problem, wasn't it?

Yes the question 8 was the basel problem. Have a look at the question 7 though, it's probably the closest thing to what you're after (more towards the end of question 7).

 

[Carrotsticks]

http://www.cs.iastate.edu/~cs577/handouts/polyroots.pdf

 

[deswa1]

^That's actually really neat thanks Did not know that Descartes sign change rule thingo

 

[Trebla]

1- What's the easiest way to graph y=f(g(x)) if you're given the graph of f(x) (e.g. graph y=f(e^x), y=f(1/x)). I can do them most of the time, but it takes me forever compared to graphing y=g(f(x)).

Should be similar approach to graphing y = g(f(x)) because you should know the graphs of y = f(x) and y = g(x) in either case. The way I normally do it is to list the relevant graphical features of the outer function y = f(x) then replace x with the inner function g(x) and reinterpret those features.

 

[Carrotsticks]

A tip for y=f(1/x), for the extremeties (or where the function is monotone), you can rotate the page 90 degrees clockwise, then pretend that is y=f(x), then draw 1/f(x) as 'usual'. Though of course I would prefer you actually understand what is going on.

Also, equation number (4) in the link I attached (Under "More bounds of roots"), that is the 2011 Q8 and is one of the most well-known theorems of Root Approximation.

 

[asianese]

Curse that

 

[Carrotsticks]

 

[RealiseNothing]

I had a look at that yesterday again.

And those last 5 marks are so easy

 

[RealiseNothing]

Curse that

 

[asianese]

l000l