Reconstruct `f` from its First Derivative


Until now, we have addressed the question, "Given `f(x)`, what can we know about `f'(x)`?" But now we will reverse it.
Given the graph of `f'(x)` can we deduce what `f(x)` looks like?
You are given the graph of `f'(x)`, and your task is to reconstruct the graph of `f(x)`.
Once you can do this well, you are ready for the first derivative test.


  1. The graph of `f'(x)` is shown in red. Drag the blue points up and down so that together they follow the shape of the graph of `f(x)`. As a help, the three large green points are points on the graph of `f(x)`.
  2. Are the three green points necessary? Theoretically, could you reconstruct `f(x)` from only one green point? from no green points?
  3. How can you tell where the `f`-graph is increasing? decreasing?
  4. How can you tell where the `f`-graph has a max? a min?
  5. What information in the `f'`-graph would tell you the point where `f` increases the fastest?
  6. Keep practicing until you can get your accuracy consistently in the 90's!