Interpreting the behavior of accumulation functions

In differential calculus we reasoned about the properties of a function $f$‍ based on information given about its derivative $f^{\prime }$‍. In integral calculus, instead of talking about functions and their derivatives, we will talk about functions and their antiderivatives.

This is the graph of function $f$‍.

Let $g(x)={\displaystyle {\int }_0^{x}f(t)dt}$‍. Defined this way, $g$‍ is an antiderivative of $f$‍. In differential calculus we would write this as $g^{\prime }=f$‍. Since $f$‍ is the derivative of $g$‍, we can reason about properties of $g$‍ in similar to what we did in differential calculus.

For example, $f$‍ is positive on the interval $[0,10]$‍, so $g$‍ must be increasing on this interval.

Furthermore, $f$‍ changes its sign at $x=10$‍, so $g$‍ must have an extremum there. Since $f$‍ goes from positive to negative, that point must be a maximum point.

The above examples showed how we can reason about the intervals where $g$‍ increases or decreases and about its relative extrema. We can also reason about the concavity of $g$‍. Since $f$‍ is increasing on the interval $[-2,5]$‍, we know $g$‍ is concave up on that interval. And since $f$‍ is decreasing on the interval $[5,13]$‍, we know $g$‍ is concave down on that interval. $g$‍ changes concavity at $x=5$‍, so it has an inflection point there.

It's important not to confuse which properties of the function are related to which properties of its antiderivative. Many students get confused and make all kinds of wrong inferences, like saying that an antiderivative is positive because the function is increasing (in fact, it's the other way around).

This table summarizes all the relationships between the properties of a function and its antiderivative.