4.1: Derivatives and Graphs
As we’ve seen, one of the most important connections between a function and its derivative is that a positive derivative means the quantity is increasing, and a negative derivative means the quantity is decreasing.
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Outside temperature has a positive derivative from 3am to 3pm, and a negative derivative from 3pm to 3am. Draw a graph of this, and label each part of the graph as “increasing” or “decreasing”.
With the positive derivative from 3am to 3pm, this should go up and be labeled “increasing”. From 3pm to 3am, the graph is going down and labeled “decreasing”.
At the interface between increasing and decreasing, at 3pm, is when the temperature is the highest. This is the key to one of the most useful applications of calculus: optimization! Optimization is either finding out when a quantity is maximized, or as high as possible, or finding out when a quantity is minimized, or as low as possible. Often this is at the interface of increasing and decreasing, and thus at the where a function goes from positive derivative to negative derivative. Hence, one of the most important maxims of calculus: optimization happens when the derivative is zero! We will come back to this in a future section.
For now, using this idea of when the derivative is positive, negative, or zero, we can draw a rough sketch of the derivative based on the graph of a function. Let’s see an example
When sketching the derivative, keep this idea in mind: slopes become \(y\)-values . First, let’s mark where the derivative is zero:
These are the places that, in the derivative graph, have zero for the \(y\)-value. That means these are the \(x\)-intercepts!
Now let’s mark where the derivative is positive, and where it is negative.
Finally, we can use this as a rough guide for a sketch, again keeping in mind slope becomes \(y\)-values. Here, the derivative is in black, while the original function is in grey.
Let’s see another example.
In this case, the graph is always going down, so the derivative is always negative. That doesn’t really tell you a lot about what the derivative graph looks like. However, there is a special point called an inflection point right here:
It’s not where the slope is zero, but it is where the slope gets the closest to zero. Everywhere else the slope is more negative than at the inflection point. So in the derivative graph this becomes a maximum or highest point, like this:
As long as that inflection point is the highest point (but still a negative \(y\)-value due to the negative slope of the original graph), that’s about the best you can do on that one.