Math 161

Sample Midterm 3

Problem 2

Dr. Wilson

2.

Let

Sketch the graph of F(x).

Let us put in a few points in the graph of f(x).

c and e are the places where f(x) = 0 so they are the places where F(x) has either a max or a min. Between a and c, f is positive so F is increasing until we get to c, at which point it stops increasing and starts decreasing which is to say that F has a max at c. h is where F crosses the axis. We will just have to eyeball it. h is the point where there is just as much area below the x axis and above the graph of f as there is area between a and c over the x axis under the graph of f. After c, f goes below the x axis so we are piling up negative area and the graph of F continues to descend until the graph of f returns to the x axis at point e where F has a local minimum. After e, we start piling up positive area under the graph of f, so F increases over the rest of the interval. By the time we get to x, however, F has not increased back up to 0. You should be able to see that between a and x, there is more area below the x axis under the graph of f than there is positive area. We will again have to eyeball it to determine the point on the x axis where the positive area under the graph of f balances out the negative area. In the picture, we have it at point j. In between at d where the graph of f has a min, the graph of F has a point of inflection, and at g where the graph of f has a max, the graph of F also has a point of inflection.