Example 1. Suppose that the function y = f(x) is given by the graph below, and
that the pieces of f are either portions of lines or portions of circles. In addition, let F
be an antiderivative of f and say that F(0) = -1. Finally, assume that f(x) = 0 for
x0 and x ≥ 7.
F>o y=f(x)
= F(x)
Fo
F(x)
2
f>o f -1-
FO FLO
F'LO
3
4
6
f>o fco
F">0 FLO
3
4
5
6
F' = f>0
F'=f<0
Determine the interval(s) on which F is increasing, decreasing.
F"=f' F" = f'=0
Determine the interval(s) on which F is concave up, concave down, or neither.
F"=f'>0
at x=5
at x=2
Determine the point(s) at which F has a relative minimum, a relative maximum.
Determine the values of F at x = 1, 2,..., 7. F(-1)=? F(8) =?
Sketch a complete and accurate graph of F.
If G is an antiderivative off with G(0) = 0. Sketch a graph of G.