| UBC Calculus Online Course Notes |

## What does a derivative tell us about a function?
You are probably wondering what information the derivative of a function gives us about a function we might be interested in. That's what we will discuss on this page. We have already learned that the derivative of a function tells us a lot about what happens when we inspect the graph of a function with a powerful microscope: specifically, it tells us how steep the tangent line to the graph would be at the point we are zooming in on. We also know that that steepness, or slope varies from point to point: some parts of the graph are flatter, and some parts slope up more steeply. On the graph below, we have plotted a graph of a familiar function,
x
appears with a minus sign. If you forgot this, have a look at
polynomials.)
We have decorated the graph of this polynomial with a few of its tangent lines. You can see by looking at these lines that their slope changes from point to point.
Now lets take a look at the slopes of the tangent lines we encountered on the graph. Here is what they might look like if we just string them up along the x axis and forget the exact point at which they were attached to the graph. Let's pretend for a minute that we have forgotten what the original function looked like exactly; but here is a record of all the "ups and downs" that it had.
In fact, we can even simplify our record further and forget about the exact values of the slopes of these tangent lines, and just record their signs. (We used the abbreviation "neg" below, because "-" looked too much like a tick mark on the y axis):
Getting back to our story, our function started out with positive
slope (somewhere
along the negative x axis). This tells us that
it must have grown for some time.
(Notice how the tangent lines are pointing towards increasing y values when their slope is positive. That's what we mean when we say that the function
must have been growing).
Eventually, it flattened out, and precisely at the point where the slope
of its tangent was zero, the function had a little "high".
(We will call that place a
- A
**positive**derivative means that the function is increasing - A
**negative**derivative means that the function is decreasing - A
**zero**derivative means that the function has some special behaviour at the given point. It may have a local maximum, a local minimum, (or in some cases, as we will see later, a "turning" point)
As a last remark we should remember that the derivative of a function is, itself, a function since it varies from point to point. If we want to, we could plot it on its own set of axes. You can compare the signs and slopes of the individual tangent lines of the original curve with the graph of the derivative. |