Does the antiderivative of a function gives any information about it's graph?

This question may look a little obvious but here we go:

In every Calculus 1 course we learn that using a function's firth and second derivatives we can know a lot of things about it's graph.

But if we only have an antiderivative of a function($\displaystyle \int { f(x)dx=x({ x }^{ 3 } } +1)$ for example) can we know anything about it's graph before differentiating?

I guess that could it be something like $\displaystyle \Delta x$, since the classic Galileo equation for uniform acceleration $\displaystyle \Delta x={ x }_{ 0 }+{ v }_{ 0 }\Delta t+\frac { 1 }{ 2 } a{ \left( \Delta t \right) }^{ 2 }$.

Thanks for answering (Wink)

Re: Does the antiderivative of a function gives any information about it's graph?

I presume that you have learned that the anti-derivative (integral) gives the area under the curve.