4.2.2 Graphs and the Second Derivative

Now let's think about how the graph of a function $f\left(t\right)$ and its second derivative $f\prime \prime \left(t\right)$ are related. We know that the first derivative gives us information about slopes of small segments of the graph. The second derivative gives us information about the rate of change of these slopes. How is that information related to the shape of the graph?

Activity 1

1. Use your graphing tool to plot the graph of `f` and the graph of $f\prime \prime$ at the same time, where

   ${\displaystyle f\left(t\right)=t^3-t\mathrm{.}}$

2. Where is $f\prime \prime$ positive? What does the graph of `f` look like in this range?

3. Where is $f\prime \prime \left(t\right)$ negative? What does the graph of `f` look like in this range?

4. Where is $f\prime \prime \left(t\right)$ zero?

5. Print a graph of `f`, and then trace the graph from left to right with your finger. How does the curve change when you pass over the point where $f\prime \prime \left(t\right)$ is zero?

6. Repeat steps (a) - (e) for the function $g\left(t\right)=t^4-4t^2+2t\mathrm{.}$

Comment on Activity 1

Checkpoint 1

Finally, we note the interplay of first and second derivatives. For the function $f\left(t\right)=t^3-t$ (see Activity 1), there are two places where the first derivative changes sign, and they are important for locating the local maximum and the local minimum on the graph of `f`. But we found them both by solving $f\prime \left(t\right)=0.$ In this case it is pretty easy to tell which solution corresponds to the local maximum and which to the local minimum, because sketching the graph is easy. If the function were not so easily graphed — and we did not have a graphing tool at hand — how would we know which was which? Well, the maximum occurs where the graph is concave down (negative value of the second derivative) and the minimum where the graph is concave up (positive value of the second derivative). So the sign of the second derivative is sufficient to distinguish between peaks and valleys.

Checkpoint 2