Describing the average rate of change using the concavity of the graph
About this video
The video looks at a graph of H that is concave down from x = -1 to x = 1. It explains that concave down means the average rate of change over smaller consecutive intervals is decreasing, shown by slopes of secant lines going from steeply positive to slightly negative.
You'll be able to
- identify an interval where a graph is concave down.
- explain that concave down means rates of change over consecutive intervals are decreasing.
- connect average rate of change to the slope of a secant line.
Key words
- Concave down
- A graph that bends downward like an upside-down bowl; its rate of change is decreasing.
- Average rate of change
- The slope of the straight line joining two points on a graph.
- Consecutive intervals
- Intervals that follow on one after another, like -1 to 0 and then 0 to 1.
Transcript
- [Instructor] We're told the figure below shows the graph of function H. Fair enough. What does the concavity of H over the interval from negative one to one mean about the average rate of change over smaller consecutive intervals within it? So pause this video and have a go at that before we do it together.
All right, so we're gonna be looking at the interval from negative one to one. And if we look at the graph over that interval, it looks like the part that I am tracing with my little digital marker right over here. That is the graph over that interval, and it's clearly downward opening, or we could say concave downwards.
Concave downwards. So what does concave downwards imply about the rate of change over smaller consecutive intervals within it? Well that means that our rate of change is going to be decreasing. And let's make sure that we intuitively understand why that is the case.
Because if we look at, let's just pick two different points on the graph. So if we look at the rate of change between this point and, let's say, well actually let's just do it this way. Let's look at the rate of change between negative one and zero, and then our average rate of change, it would be, we could think about it as the slope of this line right over here.
And then if we go between zero and one, I'll do that in a different color. Between zero and one, we've gone from a very high positive slope to a slightly negative slope. And that's because I'm doing this over intervals of one. But if you did it over even smaller intervals, you could take even more and you'll see that the rate of change is decreasing the entire time over this interval.
Now it starts to change once you get down here when we start getting concave upwards. But over that interval, our rate of change over small consecutive intervals is decreasing.
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