Answer the following questions based upon this graph of , the derivative of some function f(x):

(a) On what interval(s) is f increasing?

(b) Where is positive?

(c) On what interval(s) is f concave down?

(d) At what x-value(s) does f have a relative extrema point?

(e) If f(0) = 1, approximate the value of f(2).

Difficulty:

Solution:

(a) f is increasing whenever the graph above is positive. In other words, those are the following intervals: . The "u" symbol between the intervals is the union sign. It means the correct answer is comprised of both intervals. We're not going to draw any conclusions about when f ' = 0--there is some disagreement about that among AP readers and professors. You could avoid any controversy by saying: .

(b) is positive wherever f ' is increasing: .

(c) f is concave down wherever is negative. We already know (from part b) when is positive, so our answer will be the complete opposite of that: .

(d) f has an extrema point wherever f ' crosses the x-axis, either from positive to negative or negative to positive: .

(e) You need to find the area beneath f '(x) from x = 0 to x = 2. This isn't simple since you don't have the function which defines f. However, you can estimate area by drawing in grid lines and marking boxes which each represent one square unit:

We can only approximate, but to me, it looks as though the area is about 4 boxes, so the function f will increase 4 units from x = 0 to x = 2. Since f(0) = 1, then f(2) is approximately 1 + 4 = 5.