Question

Estimating Derivatives from a Graph For the function g whose graph is given, arrange the following numbers in increasing order, and explain your reasoning.

$0g\text{'}(-2)g\text{'}\left(0\right)g\text{'}\left(2\right)g\text{'}\left(4\right)$

Short answer

The increasing order is $g\text{'}\left(0\right)$, 0, $g\text{'}\left(4\right)$, $g\text{'}\left(2\right)$, $g\text{'}(-2)$.

Step-by-step solution

Step 1.  Given

The given function is:

Step 2.  To determine

For the function g whose graph is given, arrange the following numbers in increasing order, and explain your reasoning.

$0g\text{'}(-2)g\text{'}\left(0\right)g\text{'}\left(2\right)g\text{'}\left(4\right)$

Step 3.  Calculation

At $x=-2$, the function is increasing, so $g\text{'}(-2)$ is positive.

At $x=0$, the function is decreasing, so $g\text{'}\left(0\right)$ is negative.

At $x=2$, the function is increasing, so $g\text{'}\left(2\right)$ is positive.

At $x=4$, the function is increasing, so $g\text{'}\left(4\right)$ is positive.

The rate of increment is in the increasing order $g\text{'}\left(4\right)$, $g\text{'}\left(2\right)$, $g\text{'}(-2)$ because of the steepness of the graph.

So, the increasing order is $g\text{'}\left(0\right)$, 0, $g\text{'}\left(4\right)$, $g\text{'}\left(2\right)$, $g\text{'}(-2)$.