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Mathematics 31 Online
OpenStudy (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 \[\lim_{x→0} \frac{f(x)}x=1\] (a) Find \(f(0)\) (b) Find \(f'(0)\) (c) Find \(f'(x)\)

OpenStudy (kainui):

First off, if it's true for all real numbers, we know: f(x+0)=f(x)+f(0)+x^3*0+x*0^3 f(x)=f(x)+f(0) f(0)=0 Now for the derivative I am still distracted by the other question, so I'll come back to this but I imagine it just involves the chain rule or something.

OpenStudy (turingtest):

cool, good on the first one :)

OpenStudy (turingtest):

I'll give a hint if someone begs, but only one

OpenStudy (ranga):

f'(0) = lim x->0 { f(x) - f(0) } / (x - 0) = lim x->0 f(x) / x = 1

OpenStudy (turingtest):

correct @ranga :)

OpenStudy (ranga):

Thanks.

OpenStudy (turingtest):

I should have made it three posts so I could give a medal for each, but I'll only give it for the last one I guess

OpenStudy (kainui):

Yeah exactly, good answer. L'H definitely won't work here.

OpenStudy (ranga):

c) f'(x) = x^2 + 1

OpenStudy (turingtest):

process please @ranga

OpenStudy (kainui):

I think I figured it out too late damn I'm writing furiously haha.

OpenStudy (ranga):

c) f'(x) = lim h-> 0 { f(x+h) - f(x) } / h = lim h-> 0 { f(x) + f(h) + x^2h + h^2x } / h = lim h-> 0 f(h) / h + x^2 + hx = 1 + x^2

OpenStudy (kc_kennylau):

\[f'(x)=\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h}=\lim_{h\rightarrow0}\frac{f(h)+x^2h+xh^2}{h}=\lim_{h\rightarrow0}\frac{f(h)}h+x^2+0=x^2+1\]

OpenStudy (kainui):

Yep.

OpenStudy (kc_kennylau):

lolz @ranga yay we have the same approach

OpenStudy (ranga):

Yay!!

OpenStudy (turingtest):

very nice @ranga and @Kainui , I like questions that remind us that single-variable calc can take you a bit off-guard

OpenStudy (ranga):

Thanks @TuringTest

OpenStudy (kainui):

Where do you find these questions?

OpenStudy (anonymous):

this is a very nice functional equation

OpenStudy (anonymous):

MIT OCW always throws the best equation,i had thought this questioin was a multivariable problem..not for single variable calculus

OpenStudy (turingtest):

they sure do :) I'll post another shortly. not from the same source, of course

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