The First Derivative and the Shape of a Function f(x)

Practice 1: $\mathrm{g}$ is increasing on $[2,4]$ and $[6,8]$.
$\mathrm{g}$ is decreasing on $[0,2]$ and $[4,5]$,
$g$ is constant on $[5,6]$.

Practice 2: The graph in Fig. 33 shows the rate of population change, $\mathrm{dR} / \mathrm{dt}$.

Fig. 33

Practice 3: The graph of $\mathrm{f}'$ is shown in Fig. 34. Notice how the graph of $\mathrm{f}^{\prime}$ is $0$ where $\mathrm{f}$ has a maximum and minimum.

Fig. 34

Practice 4: The Second Shape Theorem for helicopters:
(i) If the upward velocity $\mathrm{h}'$ is positive during time interval $I$ then the height $\mathrm{h}$ is increasing during time interval $I$.
(ii) If the upward velocity $\mathrm{h}'$ is negative during time interval $I$ then the height $\mathrm{h}$ is decreasing during time interval $I$.
(iii) If the upward velocity $\mathrm{h}'$ is zero during time interval $I$ then the height $\mathrm{h}$ is constant during time interval $I$.

Practice 5: A graph satisfying the conditions in the table is shown in Fig. 35.

$\begin{array}{l|r|r|l|l|l|l} \mathrm{x} & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline \mathrm{f}(\mathrm{x}) & 1 & -1 & -2 & -1 & 0 & 2 \\ \hline \mathrm{f}^{\prime}(\mathrm{x}) & -1 & 0 & 1 & 2 & -1 & 1 \end{array}$

Fig. 35

Practice 6: $\mathrm{f}(\mathrm{x})=\mathrm{x}^{3}-3 \mathrm{x}^{2}-24 \mathrm{x}+5$.

$\mathrm{f}^{\prime}(\mathrm{x})=3 \mathrm{x}^{2}-6 \mathrm{x}-24=3(\mathrm{x}-4)(\mathrm{x}+2)$.

$\mathrm{f}^{\prime}(\mathrm{x})=0$ if $\mathrm{x}=-2,4$.

If $x < -2$
then $f^{\prime}(x)=3(x-4)(x+2)=3$ (negative) (negative) $> 0$ so $f$ is increasing.

If $-2 < x < 4$
then $\mathrm{f}^{\prime}(\mathrm{x})=3(\mathrm{x}-4)(\mathrm{x}+2)=3$ (negative)(positive) $< 0$ so $\mathrm{f}$ is decreasing.

If $x > 4$
then $\mathrm{f}^{\prime}(\mathrm{x})=3(\mathrm{x}-4)(\mathrm{x}+2)=3$ (positive) (positive) $> 0$ so $\mathrm{f}$ is increasing

$f$ has a relative maximum at $x=-2$ and a relative minimum at $x=4$.
The graph of $\mathrm{f}$ is shown in Fig. 36.

Fig. 36

Practice 7: Fig. 37 shows several possible graphs for $\mathrm{g}$. Each of the $\mathrm{g}$ graphs has the correct shape to give the graph of $\mathrm{g}'$. Notice that the graphs of $\mathrm{g}$ are "parallel," differ by a constant.

Fig. 37

Practice 8: Fig. 38 shows the height graph for the balloon. The balloon was highest at $4 \mathrm{pm}$ and had a local minimum at $6 \mathrm{pm}$.

Fig. 38

Practice 9:
$f(x)=3 x^{2}-12 x+7$ so $f^{\prime}(x)=6 x-12$.

$f^{\prime}(x)=0$ if $x=2$.
If $x < 2$, then $f^{\prime}(x) < 0$ and $f$ is decreasing.
If $x > 2$, then $f^{\prime}(x) > 0$ and $f$ is increasing.
From this we can conclude that $\mathrm{f}$ has a minimum when $\mathrm{x}=2$ and has a shape similar to Fig. 19(b).

We could also notice that the graph of the quadratic $f(x)=3 x^{2}-12 x+7$ is an upward opening parabola. The graph of $\mathrm{f}$ is shown in Fig. 39.

Fig. 39