(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
Graph behaviors
(Khan Academy)
Seeing the graph of
| When | Its antiderivative |
|---|---|
| >0 | increasing |
| <0 | decreasing |
| increasing | concave up |
| decreasing | concave down |
| crosses x-axis (changes sign) | local extremum |
| goes from | local maximum |
| goes from | 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
