Before reading on, you should have a basic understanding of concavity. Review if you need to. Show/Hide Review.
- Concavity, as part of the second derivative test, gives us a way of classifying extreme values of a graph or function.
- Given a function f(x), we say that f(x) is concave up at a point 'c' if f''(c) > 0, and f(x) is concave down at a point 'c' is f''(c) < 0.
- To see where concavity changes sign values (that is, where f''(x) changes sign values), we look for where f''(x) = 0. These are known as inflection points.
- In addition to setting the second derivative equal to zero, we can also study concavity by looking at the graph of f(x). If, in an interval, f(x) appears to be shaped "up like a cup," we say it is concave up. If, in an interval, the graph of f(x) appears to be shaped "down like a frown," then it is concave down.
- Look at the example below for how concavity can vary in a given graph or function.
Connecting the Dots: Before we move on to experiments and fun, we should tie together the three major concepts that we’ve learned: position, velocity, and acceleration. From what we've gone over, we can easily deduce that a(t) = v’(t) = d’’(t). This should immediately indicate to you that the position and acceleration graphs are linked. Now, before getting formal, take a look at the graphic below to see how we connect position, velocity, and acceleration functions one-by-one. Notice that the position function is concave down - what do you notice about the associated acceleration function?
Concavity: As you can see in the example above, as time increases, the object is moving in the negative direction – this means that it has negative velocity. But what about acceleration? How can we "read" acceleration from the position function? Since acceleration is simply the rate at which velocity is increasing, let’s think about the graph like this: Is the slope decreasing at an increasing rate (more and more negative), or decreasing at a decreasing rate (approaching a constant slope)? The position graph has a negative slope, and it appears that the slope is getting more and more negative as time increases. This is where concavity helps us understand motion. Using the notions of "concave up" and "concave down," one can determine whether acceleration is positive or negative from the position function. Hence, because the position graph above is concave down, its acceleration is negative! From this, it’s clear that the graphs of position, velocity, and acceleration are interrelated!
Practice: Which position graph below represents an object that’s accelerating? Mouse over it to see if you’re correct!
Now, for a fun application in tying these notions together, click "Next" to head over to the experiment page!