(Untitled)

[ethnic]

fuck, how dead is livejournal?

Comments

[anangrymoose]

you're dead

[ethnic]

microeconomics is dead

i wish

[anangrymoose]

Show that if a < b, then (b - a)/(1 + b^2) < tan^-1(b) - tan^-1(a) < )(b - a)/(1 + a^2)

Let f(x) = tan^-1(x)
so f'(x) = 1/(1 + x^2)

By the Mean Value Theorem, if a < b, then there is some point c (a < c < b) where f'(c) = (f(b) - f(a))/(b - a)

1/(1 + c^2) < (tan^-1(b) - tan^-1(a))/(b - a)

But a < c < b
so a^2 < c^2 < b^2
so 1 + a^2 < 1 +c^2 < 1 + b^2
so 1/(1 + b^2) < 1/(1 + c^2) < 1/(1 + a^2)

Therefore, 1/(1 + b^2) < (tan^-1(b) - tan^-1(a))/(b - a) < 1/(1 + a^2)
(b - a)/(1 + b^2) < tan^-1(b) - tan^-1(a) < (b - a)/(1 + a^2)

Q E FUCKING D CANAGASABEY

Re: < and =

[anangrymoose]

thanks cana

you fucking spastic