antiderivative

is the antiderivative of if is the derivative of for all in the domain of

antiderivative and indefinite integral

(Cliff Notes) There are an infinite number of antiderivatives could exist for . For example, and are all antiderivatives of . They only differ by a constant value , so the family of all antiderivatives of is the indefinite integral

If is the antiderivative of :

Graph behaviors

(Khan Academy) Seeing the graph of to derive about its antiderivative/indefinite integral

When	Its antiderivative
>0	increasing
<0	decreasing
increasing	concave up
decreasing	concave down
crosses x-axis (changes sign)	local extremum
goes from to	local maximum
goes from to	local minimum
extremum	inflection point

Example

• on the interval , so must be increasing here
• changes its sign at and , so should have two extrema
  • goes from negative to positive at , so is a local minimum
  • goes from negative to positive at , so is a local minimum