2 5 Concavity

Added: 2026-05-12

[00:00:06]okay today we're going to talk about concavity you should have heard of concave up and concave down before so let's just uh review that because a function f of x is called increasing graph blank from left to right a function is increasing if a graph rises so a positive slope positive slope a function is increasing a negative slope a function is decreasing so think of what a negative slope looks like what does the graph look like what is the what is the graph doing from left to right the graph is falling or the graph falls from left to right but let's talk about concave up concave up a function is concave up if its average rate of change increases from left to right the average rate of change well rate of change is just saying the slope between two points but since we don't always have lines we can't always call it slope so we call it rate of change okay if the function is concave down its average rate of change is decreasing from left to right so let's go and look at let's describe the shape of a graph of a function f of x that is concave up describe the shape okay well let's let's draw a shape first this is concave up so how do you describe that so let's say all four parts of a let's say uh right side up right side up uh bowl right so concave down something like this so how do you describe that how about an upside down bowl okay well let's look at the average rate of change let's let's just imagine the slope between these two points okay that would be negative let's say it's negative 10.

[00:02:54]what kind of slope would that be would that be a negative 20 or would that be like a negative four i would say negative four how about here wouldn't that be zero okay how about that wouldn't that be like uh let's say positive four and then positive ten so we went from an average rate of change or a slope of negative 10 negative 4 0 4 10.

[00:03:23]so what were those rates of change doing they were getting larger right they were increasing and that is the true definition of concave up so let's take a look at this what kind of slope is this wouldn't you say that's like an a positive slope and then as we go along we get to zero and then if we look at the slope between two points on here okay negative so our average rates of change are decreasing that's the true definition of concave down of a function okay and as i always say things you will be using in calculus okay so example number one go ahead and pause the video and read the video i'm sorry read the example and see if you can draw a picture based on the explanation what would the function look like just a sketch of what it's saying so pause the video now and give it a try hey i said pause the video try it on your own okay if you did great if not you should do that now okay so it talks about a new product rate of increase is faster eventually the rate of increase slows down okay but they're still using the product so over time over time you have a product and it's constantly increasing it's constantly increasing but it increases and then at some point it's still increasing but it slows down so like concave up to concave down okay this point here is where it changes so we can say the function is increasing as time goes on because more and more people use it however the rate at which people are using it slows down so the rate of increase still increasing but the rate of increase slows down and in this case that happens after point a concave up concave up means the rate of change is increasing okay concave down the rate of change is decreasing but you still have an increase in product as time goes on people are using the product okay let's take a look at at this okay part a is this concave up or concave down and is the function increasing or decreasing over time so those are the two questions i want you to answer so i want you to pause the video and answer is it concave up or down and the second question is the function in this case p of t if we use the correct variables is p of t increasing or decreasing okay so go ahead and stop the video and answer that for these okay so again i just wanted to answer these questions this is concave up and functions increasing this one's concave down function is increasing this one it's concave down and as x gets bigger y gets smaller so the function is decreasing this is concave up and the function is decreasing okay so let's just look at the descriptions p is doing blank and the rate of change of p is doing what so what is p doing well we already answered that question i already had you answer what p of t was doing okay so let's go back and look at our answers p of t is increasing on a what's it doing on b we said it was increasing c was decreasing and d was decreasing so the function p was increasing on the first two decreasing on the second two but what about the rate of change of p again its vocabulary the rate of change the slopes so concave up was that the slopes were increasing or was that the slopes were decreasing well you go back to here if it's concave up the average rate of change is increasing and if you forget on a test just do this okay so that's one this would be two this would be ten okay so these slopes it between any two points as i go from left to right is increasing that's the definition of concave up so this is concave down my slopes or my average rate of change is decreasing what about this one what are the slopes doing concave down so the slopes are decreasing and this one concave up your slopes are increasing you're afraid of change okay let's take a look at the last example decide whether each of the following functions are concave up concave down or neither so they're giving us some points in a table so we're looking at our rate of change rate of change is the again the slope between two points but again we call it rate of change because slope is only for a line so if we look from here to here if we look at our change in y over our change in x's because that is your average rate of change okay let's look from here to here our average rate of change from 1 to 3 is 2 and then our x's the next one would be three the next rate of change would be four over one the next rate of change is ten over one so our rates of change are increasing therefore our function f of x is concave up okay so look at this one that's negative one over one negative two over one negative three over one okay our average rate of change is decreasing so f of x is concave down now you could plot the points and try connecting them but i wouldn't do that i would just know the definition for concavity okay how about this what is the average rate of change of this the average rate of change is another word for the slope between two points in this case can we call it slope yeah absolutely this is a line what's the concavity of this none the function is increasing but there's no concavity why is that because the rate of change is constant it's always equal to three it's not increasing it's not decreasing so it's not concave up or concave down okay and then h of x concave up concrete down or neither pretty simple we're looking at the picture right that's concave down okay and that's the definition the true definition of concavity