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NationalTutoringObservatory
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MathEd-PII

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student
but on the notes he was confusing us saying that for example(watch what I'm doing on the board) is increasing
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student
how to know that a function is increasing?
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volunteer
He is not talking about the function increasing.
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volunteer
we will get to that.
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volunteer
at the blue point, is the curve concave up or concave down?
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student
Right now I'm not talking about the concavity yet, I'm just talking about that portion, but ok
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student
down
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volunteer
When a curve is concave down, is f" >0 or <0 ?
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student
that Idk sorry
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volunteer
OK, you haven't learned that yet.
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student
as I said, he did talk about it, but I'm js so confused with his explanations
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volunteer
I drew three tangent lines, A B and C
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student
ok
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volunteer
what might be an estimate for the slope of A, just roughly?
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student
-2, maybe?
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volunteer
OK
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student
0
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volunteer
How about at B?
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student
1.7
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volunteer
OK, so is the f', the first derivative, increasing of decreasing over that interval (we would call the curve there concave up, because it is facing up
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volunteer
Again, as we go from A to B to C, is f' increasing or decreasing
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student
it's increasing
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volunteer
right.
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volunteer
Now, f", the second derivative, is the derivative of f', right?
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student
yes
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volunteer
so if f' is increasiing as you go from A to C, would you expect f" to be >0 or <0?
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student
to find that out should I draw a second graph based on the estimated slopes we found or is there an easier way?
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volunteer
the point is that the derivative tells us if a function is increasing or decreasing. f' is itself a function. If you tell me that f' is increasing over an interval, then that means that the derivative of f' should be>0, and the derivative of f' is f".
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volunteer
so if f' is increasing over an interval, then f" on that interval would be >0
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student
so that make sense, if f' isincreasing on that interval then f'' is also increasing on that interval?
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volunteer
Exactly.
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volunteer
no, sorry
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volunteer
f" is positive over that interval.
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student
wym no sorry?
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student
so f'' is not increasing?
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volunteer
f">0, being >0 is not the same thing as increasing.
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volunteer
Using a similar argument, we can see that f" <0 where the blue point is, where the curve is facing down.
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student
okay how to know smth is positive vs increasing
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volunteer
In this case, they just want to know if f" is <0 or >0 at the blue point.
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student
I'm asking just in case in the future I'm asked that
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volunteer
Something happened to the site. Are you still there?
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volunteer
but since you can see that on the left f" >0 and on the right f"<0, then that that means that as you move from from A to the blue point, f" must be decreasing
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volunteer
Hello?
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student
sorry
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student
I got disconnected for a long time and js got back :(
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volunteer
Do you understand this question now?
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student
okay so let me recap and tell me if I'm correct
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student
so where the blue point is f' is decreasing because the slopes around that curve are going from estimate 1.7, 0 to -2, and this then means that f'' is <0
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volunteer
yes, that's right. It's the fact that f' is decreasing (which means f" <0) that gives the curve that concave/facing downward look.
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volunteer
concave down/facing downwards
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student
does being concave up or down say anything at all?
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volunteer
They describe the shape of the graph. That's something you couldn't do as well with just algebra
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student
what I'm asking is does the graph being concave up or down tell us that it's increasing, decreasing, positive or negative?
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student
at all?
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volunteer
It tells the shape, as I drew on the board. Do you see that?
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student
yes i see
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volunteer
It wouldn't tell if the function is positive or negative. the curve could be above or below the x axis. it could be anywhere
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student
ok
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student
another question i have is f' or f'' being <0 or >0, does that say if it's positive or negative?
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volunteer
Numbers that are <0 are negative and numbers that are >0 are positive
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volunteer
Are you all set now?
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student
okay that makes so much sense thx
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student
Do you have to go or you got a little bit of time?
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volunteer
I can do another question.
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student
okay thx one min
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volunteer
what do you think?
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volunteer
hello, are you there?
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student
sorry
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student
Do we have to create our own graph?
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volunteer
You can, but it's not necessary
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student
okay, so what do we do then?
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volunteer
First of all, is f'(2)) most likely postive or negative?
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student
maybe positive??
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volunteer
what does a derivative >0 mean for a graph?
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student
increasing
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volunteer
Yes. It is important to remember that. Does it look like the function is increasing?
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student
no it's decreasing
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volunteer
Yes. For a fn to be increasing, that means that as x increases (as you move left), f(x) would go up, But here, it looks like f(x) is going down, so f(x) is most likely decreasing.
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student
yes
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volunteer
We can estimate the slopes of the tangent lines at x =2, 2.1, 2.2
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volunteer
we can do that by figuring out the rate of change between the points that they give us.
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volunteer
What is the rate of change (the slope of the line) between (2,3) and (2.1,2.7) ?
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volunteer
That average rate of change gives us a rough estimate of the slope of the tangent line at x =2
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volunteer
Now let's move to the right.
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volunteer
let's get the rate of change between the next two points
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volunteer
One more
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volunteer
Good work. So, does it look like f' is increasing or decreasing over the interval from x =2 to x =2.3?
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student
it's increasing
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volunteer
Yes
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volunteer
so will this curve be concave up (f">0) or concave down (f"<0)?
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student
this will be concave up
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volunteer
that's right.
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volunteer
I also plotted them on Desmos, so you can see.
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volunteer
I think you can see that the curve looks concave up.
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student
yope
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student
to recap, first we found out that f(x) was decreasing, then we looked for the slopes and found out that f'(x) is increase which led us to the knowledge that f" is concave up or >0, correct?
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volunteer
yes, that's right
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volunteer
We used the average rate of change over small intervals to estimate the slope of the tangents lines
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student
okay now questions time, first question why did we need to find out if f(x) was increasing or decreasing?
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volunteer
because the question asked if f'(2)) most likely >0 or <0. If f(x) is increasing that means f'(2)) >0 and if f(x) is decreasing that means that f'(2) <0
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