Show that f(x) = x^3 and g(x) = 200x^3 grow at the same rate.

Question

solomonzelman · Answer

do they really?

anonymous · Answer

I'd honestly say g(x) grows faster but I'm not quite sure on how to justify it.

solomonzelman · Answer

You are not just shifting the functions to the side or up/down. You are multiplying times a scale factor, and that means that slope will differ.

solomonzelman · Answer

Well, your g(x)=f(x)•200

solomonzelman · Answer

What kind of prove.reasoning do you want?

maybe we can show that for every positive k, the function g(x) will have a bigger rate of change at every point?

solomonzelman · Answer

(or a bigger magnitude of the slope $\forall k\in \mathbb{Z} $)

solomonzelman · Answer

$\large\color{#000000 }{ \displaystyle f'(x)=3x^2 }$
$\large\color{#000000 }{ \displaystyle g'(x)=600x^2 }$

$\large\color{#000000 }{ \displaystyle 3x^2\le 600x^2 \quad \forall x\ne0}$

anonymous · Answer

So you just took the derivative? And the one with the larger one has a faster rate of growth?

anonymous · Answer

It asks me to show so I'm guessing what you did is what they're looking for.

solomonzelman · Answer

Yes, because the derivative itself is slope, by defnition.

anonymous · Answer

I don't really believe there is anything more to be said about this. We can't explain common sense

anonymous · Answer

Larger slope is equal to a faster rate of growth

solomonzelman · Answer

Yes, fabulous!

anonymous · Answer

Thanks for your help

solomonzelman · Answer

Anytime