TuringTest
Suppose that \(f\) satisfies the equation \[f(x+y)=f(x)+f(y)+x^2y+xy^2\]for all real numbers \(x\) and \(y\). Suppose further that\[~~~~~~~~~~~~~\lim_{x\to0}\frac{f(x)}x=1\]a) Find \(f(0)\)
b) Find \(f'(0)\)
c) Find \(f'(x)\)
Delete
Share
This Question is Closed
TuringTest
Best Response
You've already chosen the best response.
0
I found \(f(0)=0\) but now I am not seeing how to proceed. Just a hint would be nice.
TuringTest
Best Response
You've already chosen the best response.
0
I further think I proved that the function is odd, though I think that doesn't really help me :/
Zarkon
Best Response
You've already chosen the best response.
2
use the definition of a derivative
Zarkon
Best Response
You've already chosen the best response.
2
the fact that it is odd does seem to help with the last part
TuringTest
Best Response
You've already chosen the best response.
0
I was thinking of using the definition of the derivative somehow but couldn't figure out how to get it into the right form....
Zarkon
Best Response
You've already chosen the best response.
2
\[f'(0)=\lim_{x\to0}\frac{f(x)-f(0)}{x-0}\]
TuringTest
Best Response
You've already chosen the best response.
0
limit f(x)-f(0) something...
TuringTest
Best Response
You've already chosen the best response.
0
ah dang but... oh, okay I guess I was trying to get things algebraically manipulated there, but I see now that since x=f(x) at x=0 the limit \(is\) the definition
got it :)
TuringTest
Best Response
You've already chosen the best response.
0
okay so that is f'(0)=1, on to the last part
you say the oddness could help?
Zarkon
Best Response
You've already chosen the best response.
2
yes
TuringTest
Best Response
You've already chosen the best response.
0
can I use the derivative of an odd function is even?
Zarkon
Best Response
You've already chosen the best response.
2
don't need it
Zarkon
Best Response
You've already chosen the best response.
2
use the definition of a derivative again
TuringTest
Best Response
You've already chosen the best response.
0
that's what I'm doing, let me toy for a sec...
Zarkon
Best Response
You've already chosen the best response.
2
I found f(x) ;)
TuringTest
Best Response
You've already chosen the best response.
0
haha, dang
okay then go ahead and give me a little more help on finding f'(x)
they don't even ask for f(x), jeez
TuringTest
Best Response
You've already chosen the best response.
0
...if you would be so kind
Zarkon
Best Response
You've already chosen the best response.
2
if you know f'(x) and that f(0)=0 then finding f(x) is really easy :)
Zarkon
Best Response
You've already chosen the best response.
2
\[f'(x)=\lim_{y\to x}\frac{f(y)-f(x)}{y-x}\]
Zarkon
Best Response
You've already chosen the best response.
2
\[f(y)-f(x)=f(y)+f(-x)\]
Zarkon
Best Response
You've already chosen the best response.
2
does that help?
Zarkon
Best Response
You've already chosen the best response.
2
ok
KingGeorge
Best Response
You've already chosen the best response.
1
If it helps, it's easier for me to see it when I write the derivative as \[f'(x)=\lim_{y\to0} \frac{f(x+y)-f(x)}{y}\]
TuringTest
Best Response
You've already chosen the best response.
0
that would be \[\lim_{y\to0}{f(y)+x^2y+xy^2\over y}=1+\lim...\]ah... it's \(f'(x)=x^2+1\), right ?
:)
Zarkon
Best Response
You've already chosen the best response.
2
yes
TuringTest
Best Response
You've already chosen the best response.
0
sweet, thanks y'all
I still want to see how your way was gonna go @Zarkon , but I'll work on that myself for a while
Zarkon
Best Response
You've already chosen the best response.
2
gotta go...fun problem...I'll post it later if you like
TuringTest
Best Response
You've already chosen the best response.
0
always a pleasure :)