bloodhound Posted June 27, 2004 Share Posted June 27, 2004 Taken from my calculus exam The function f satisfies [math]f(2x)=xf(x)+1[/math] 1)Show that [math]f(0)=1[/math] 2)Show using induction that [math]2^{n}f^{n}(2x)=nf^{(n-1)}(x)+xf^{n}(x) \forall x \in R[/math] 3)Hence, derive the first three terms of the Maclaurin series of [math]f[/math] I managed to do it, but because of 1) and 2) if 3) was given by itself I would think i would be completely lost for some time. edit: [math]f^n[/math] denotes the nth derivative of f Link to comment Share on other sites More sharing options...
Dave Posted June 27, 2004 Share Posted June 27, 2004 Done (1) - pretty obvious. Working on 2, haven't quite got there yet, but I think I know how to do it Link to comment Share on other sites More sharing options...
jordan Posted June 27, 2004 Share Posted June 27, 2004 I have absolutly no idea what is going on there, but I was looking over it anyways. Just wondering, what do the last four symbols in number 2 mean (starting with what appears to be an up-side down A). Link to comment Share on other sites More sharing options...
Dave Posted June 27, 2004 Share Posted June 27, 2004 Read literally the upside-down A means "for all" - the entire thing means that that condition (the f(x) bit) is true for all x in the set of real numbers. btw, to generate a proper real numbers symbol, use \mathbb: [math]\forall x\in\mathbb{R}[/math] Link to comment Share on other sites More sharing options...
Dave Posted June 27, 2004 Share Posted June 27, 2004 Hmm. I did it, but I'm not sure it's entirely true. We know that [math]f(2x) = xf(x) + 1[/math]. Now say [math]g(x) = xf(x) + 1 - f(2x)[/math]. Then we have that [math]g'(x) = f(x) + xf'(x) - 2f(2x)[/math] by the chain rule (effectively this is implicit differentiation). By differentiating more and more times, it's pretty obvious that you get [math]g^{(n)}(x) = nf^{(n-1)}(x) + xf^{(n)}(x) - 2^n f^{(n)}(x)[/math] as required. Link to comment Share on other sites More sharing options...
admiral_ju00 Posted June 28, 2004 Share Posted June 28, 2004 Dave, I think you should start a Calc 101 section here, and kind of like instead of asking question / recieving answer thing, maybe you guys can start others (like myself) on our way to actually understand Calculus? (Kind of like a teaching/tutoring thing) Or maybe not.... Link to comment Share on other sites More sharing options...
Dave Posted June 28, 2004 Share Posted June 28, 2004 Sounds like a good idea; I'm sure I can draw up a "syllabus" of types Maybe if it kicks off properly I can get blike to create a sub-forum for it. Link to comment Share on other sites More sharing options...
jordan Posted June 28, 2004 Share Posted June 28, 2004 A daily calc lesson at 9:00 every morning taught by dave. Nice. Link to comment Share on other sites More sharing options...
bloodhound Posted June 28, 2004 Author Share Posted June 28, 2004 yes, and if it takes off, we can do others like Linear Mathematics 101, or Mathematical Analysis 101. etc Link to comment Share on other sites More sharing options...
Dave Posted June 28, 2004 Share Posted June 28, 2004 ack, linear algebra 101 Stick to calculus for now Link to comment Share on other sites More sharing options...
jordan Posted June 28, 2004 Share Posted June 28, 2004 Number theory? Link to comment Share on other sites More sharing options...
bloodhound Posted June 28, 2004 Author Share Posted June 28, 2004 I loved Linear algebra. all the vector spaces and subspaces. mind boggling. I havent done much number theory, except introductory group theory. Link to comment Share on other sites More sharing options...
Dave Posted June 28, 2004 Share Posted June 28, 2004 Nope, I haven't done any number theory. Maybe next year after I've taken the module Link to comment Share on other sites More sharing options...
Dave Posted June 28, 2004 Share Posted June 28, 2004 Btw, Calculus help is now UP, yippee (and this is my 2,000th post, woot) Link to comment Share on other sites More sharing options...
jordan Posted June 28, 2004 Share Posted June 28, 2004 Ok. No need for number theory then. I was interested because I read a book not to long ago that said it would be classified as number theory but never described what number theory was. I liked the book so I assumed I would have to like learning number theory as well. Link to comment Share on other sites More sharing options...
bloodhound Posted June 28, 2004 Author Share Posted June 28, 2004 I could put stuff up on group theory if u want. Link to comment Share on other sites More sharing options...
jordan Posted June 28, 2004 Share Posted June 28, 2004 How is group theory related to number thoery? Link to comment Share on other sites More sharing options...
bloodhound Posted June 28, 2004 Author Share Posted June 28, 2004 I think group theory is a part of number theory Link to comment Share on other sites More sharing options...
Dave Posted June 28, 2004 Share Posted June 28, 2004 Group theory is very interesting because you can use it to define basic addition and multiplication in a rigourous sense; indeed, if you start with set theory you can basically build up the natural numbers, the rationals, reals, etc and all the operations that can be done with them. Link to comment Share on other sites More sharing options...
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