Concavity and Inflection Points on Graphs

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  • 0:06 Understanding Concavity
  • 2:40 Concave Up and Concave Down
  • 4:15 Examples of Concave Up…
  • 6:54 Lesson Summary
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Lesson Transcript
Instructor: Heather Higinbotham
You might not think of a cup when you think of an awesome skateboard ramp. But I'm sure a really bad ramp would give you a frown, right? Learn about cups and frowns in this lesson on concavity and inflection points.

Understanding Concavity

The ramp on the right has an increasing derivative
understanding concavity

The other day I was thinking about this: I really want a skateboarding ramp. So I asked a buddy of mine, 'Can you build me a skateboarding ramp? One that's, maybe, 10 feet tall and 10 feet long?' He came back to me with this. A nice, flat, 10-feet-tall, 10-feet-long ramp. It was a lot of fun, but that's not a skateboarding ramp. So I said, 'Hey, dude. Try again.' He came back with this ramp. It's also 10 feet tall and 10 feet wide, but it's not exactly what I wanted. When I tried it out, let's just say, I ended the day in the hospital. So what happened here? Why did he give me a ramp that fits all of what I asked, but isn't what I wanted. I didn't draw it out, but maybe there's a better way.

How can I get across that I want a ramp that slopes up like this? Let's compare the two ramps that he made and the ramp I wanted. If I graph them, their height as a function of distance, I have the boring, flat ramp; I have the bad, send-me-to-the-hospital ramp; and I have the ramp that I really wanted. Now, all of these are increasing. They're all 10 feet long, and they're all 10 feet tall. What's the difference between these ramps? Well, the derivatives are different. This boring, flat ramp has a constant derivative at every single value of x, so every single place along the width of the ramp has the same derivative. The send-me-to-the-hospital ramp, if I graph out the derivative, it's decreasing like this as I go up the ramp. The ramp that I really want has an increasing derivative as I go up the ramp, not a decreasing derivative. In math, we call this the concavity, how a derivative is changing. In the case of the ramp that I want, the derivative is getting bigger. The ramp is what we call concave up. It's a 'cup.' You can kind of see that here. If the derivative is getting smaller, as in the case of the ramp that sent me to the hospital here, then I have something that's concave down. I'm going to call that a 'frown.' It's sending me to the hospital, and it kind of looks like a frown. So we've got our frown and our cup, the concave down and the concave up.

When something is concave up, the second derivative is always positive
Concave Up Graph

Concave Up and Concave Down

Let's take a closer derivative at the case where we have a concave up line. This is the cup. Imagine that I'm sitting up here and about to go down my ramp. You can see that this looks a lot like a cup when you look at the cross section. Here I've got height as a function of distance. If I take a look at the derivative of the height as a function of distance, I see that sometimes it's negative, and sometimes it's positive. It's negative as I'm going down, and it's positive as I'm going up the other side of the ramp. If I take the second derivative - so if I take the derivative of the first derivative - that's always positive. This line - the derivative here - is always pointing up, so the second derivative is always going to be positive in a case where we have something that is concave up. For something that's concave down, well, this is our frown. See I'm sitting at the top of this ramp. You can see I'm about to fall off one side or the other, right? So that's a big frown. This is concave down. Here, as you move from left to right, the derivative is, at first, positive, and then it becomes negative. So the derivative is decreasing. If you look at the second derivative, the second derivative is always negative. So if you have a case where the second derivative is negative, you might be looking at something that is concave down, or a frown.

Examples of Concave Up and Concave Down

This graph has both concave up and concave down parts
Concave Up and Down Graph

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