The First Derivative and the Shape of a Function f(x)
Read this section to learn how the first derivative is used to determine the shape of functions. Work through practice problems 1-9.
Practice Answers
Practice 1: is increasing on and .
is decreasing on and ,
is constant on .
Practice 2: The graph in Fig. 33 shows the rate of population change, .
Fig. 33
Practice 3: The graph of is shown in Fig. 34. Notice how the graph of is where has a maximum and minimum.
Fig. 34
Practice 4: The Second Shape Theorem for helicopters:
(i) If the upward velocity is positive during time interval then the height is increasing during time interval .
(ii) If the upward velocity
is negative during time interval then the height is decreasing during time interval .
(iii) If the upward velocity is zero during time interval then the height is constant during
time interval .
Practice 5: A graph satisfying the conditions in the table is shown in Fig. 35.
Fig. 35
If
then (negative) (negative) so is increasing.
If
then (negative)(positive) so is decreasing.
If
then (positive) (positive) so is increasing
has a relative maximum at and a relative minimum at .
The graph of is shown in Fig. 36.
Fig. 36
Practice 7: Fig. 37 shows several possible graphs for . Each of the graphs has the correct shape to give the graph of . Notice that the graphs of are "parallel," differ by a constant.
Fig. 37
Practice 8: Fig. 38 shows the height graph for the balloon. The balloon was highest at and had a local minimum at .
Fig. 38
if .
If , then and is decreasing.
If , then and
is increasing.
From this we can conclude that has a minimum when and has a shape similar to Fig. 19(b).
We could also notice that the graph of the quadratic is an upward opening parabola. The graph of is shown in Fig. 39.
Fig. 39