Wednesday, April 28, 2010

Graph from a graph of f ' (x)

First, pay attention: the graph provided on the assignment is the graph of the derivative f '(x) and not the graph of f. So you can't look at the picture and say that because the graph you are looking at is increasing that f '(x) is positive; if the graph is increasing, then that means f '(x) is increasing, and not f(x). (This is useful information, but you just need to think about what it does say.)

Second, the number line sign analysis summaries will help identify the shape of the graph. Imagine taking the unit circle and breaking it up according to quadrants. The signs of f '(x) and f ''(x) determine which of these four basic shapes the graph is most like.
  • f '(x) = + and f ''(x) = + means f(x) looks like Quadrant IV (incr, conc. up)
  • f '(x) = - and f ''(x) = + means f(x) looks like Quadrant III (decr, conc. up)
  • f '(x) = + and f ''(x) = - means f(x) looks like Quadrant II (incr, conc. down)
  • f '(x) = - and f ''(x) = - means f(x) looks like Quadrant I (decr, conc. down)
The graph is just formed by taking these shapes and putting them end-to-end. You wouldn't actually use the entire portion of the unit circle because we probably don't want vertical tangents like the unit circle has. The circle just helps us remember the basic shape. The points where we join the shapes together will probably be inflection points (concavity changes) or extreme values.

However, sign analysis does not tell us the heights of any points. The problem gives only one point: f(0) = 1. The rest of the points of interest (especially the local extreme values) can be found by thinking about the information relating to the areas of the graph of f '(x). (Again, think about the Fundamental Theorem of Calculus).

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