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Inflection points (graphical)

AP.CALC:
FUN‑4 (EU)
,
FUN‑4.A (LO)
,
FUN‑4.A.4 (EK)
,
FUN‑4.A.5 (EK)
Sal analyzes the graph of a function g to find all the inflection points of g.

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  • blobby green style avatar for user jmathew2424
    I'm confused about the slope increasing and decreasing. Graphically, it looks like the slope changes signs around -3,0, and 3. Instead, the video showed the points of inflection in the middle of the slopes. Could you explain why that is in more depth?
    (33 votes)
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  • leaf green style avatar for user Kartik Khandelwal
    how does it become apparent that a slope is increasing or decreasing?
    (14 votes)
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    • orange juice squid orange style avatar for user Vivek Anand
      An interesting trick that one can use for this is to draw the graph of the first derivative. Then identify all of the points in say f'(x) where the slope becomes zero. These points, where slope is zero are the inflection points. Instead of microanalyzing the graph for increasing or decreasing this is much more accurate and rigorous.
      (15 votes)
  • blobby green style avatar for user D. Ashley Nelson
    At , Sal states and writes there are 3 inflection points. I see 4 inflections points. Didn't he miss an inflection point at x = - 3.5? Over the interval (-4, -3.5) I estimated a slope of at least 18 and a slope of 10 over (-3.5, -3) ===> the slope is decreasing as the function enters a concave downward. Can anyone provide a counter argument?
    (8 votes)
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  • duskpin ultimate style avatar for user vtx
    I don't know how to find the exact point. even sal says "around here"
    (4 votes)
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    • male robot hal style avatar for user jhwalbright
      This is just finding inflection points graphically. To find the exact point you need the equation of the function and find the 2nd derivative. That will be explained in a later video. This seems to be just to get you comfortable with the concept of an inflection point.
      (5 votes)
  • leaf blue style avatar for user Tarun Akash
    i noticed something, as the slope increases the angle of the slope goes anti-clockwise and as the slope decreases angle of the slope goes clock-wise. And the point at which the 'clock' starts moving from anti-clockwise to clock-wise or vice versa, it is called point of inflection. Has anybody else noticed this?
    (4 votes)
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  • blobby green style avatar for user pk.mcdonald
    What if f'' changes sign at a point but the double derivative doesn't exist at that point, like a cusp in f'? The concavity changes at the point so would that still be considered a point of inflection?
    (3 votes)
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  • duskpin sapling style avatar for user Amulya M
    So between any 2 critical points there is always an inflection point?
    (3 votes)
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  • sneak peak blue style avatar for user Devin T. Nguyen
    An interesting trick that one can use for this is to draw the graph of the first derivative. Then identify all of the points in say f'(x) where the slope becomes zero. These points, where slope is zero are the inflection points. Instead of microanalyzing the graph for increasing or decreasing this is much more accurate and rigorous.
    (3 votes)
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  • blobby green style avatar for user Li Hans
    How exactly do you identify whether or not the slope is increasing or decreasing?The inflection point around the x value of -2 doesn't seem to have a change in slope from decreasing to increasing.
    (3 votes)
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    • female robot grace style avatar for user loumast17
      The first derivative is the slope. when the derivative is 0 (a crititical point) the slope is 0.

      The second derivative is the slope of the slope, or in other words if the slope is increasing or decreasing. when the second derivative is 0 (a critical point) the slope of the slope is 0, or in other words the slope goes from increasing to decreasing or the other way around. this does not mean the slope goes from positive to negative though (or the other way around.

      You can have an increasing slope that is still negative, if the slope goes from -4 ot -3 it is still negative but increasing.

      Let me know if that doesn't make sense.
      (1 vote)
  • blobby blue style avatar for user A lobster named Krill
    what does inflection mean?
    (2 votes)
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Video transcript

- [Voiceover] We're told let g be a differentiable function defined over the closed interval from negative four to four. The graph of g is given right over here, given below. How many inflection points does the graph of g have? So let's just remind ourselves, what are inflection points? So inflection points are where we change concavity. So we go from concave, concave upwards, upwards, actually let me just draw it graphically. We're going from concave upwards to concave downwards, or concave downwards to concave upwards. Or another way you could think about it, you could say we're going from our slope increasing, increasing, increasing, to our slope decreasing. To our slope decreasing, or the other way around. Any points where your slope goes from decreasing, our slope goes from decreasing, to increasing. To increasing. So let's think about that. So as we start off right over here, at the extreme left it's seems like we have a very high slope. It's a very steep curve, and then it stays increasing but it's getting less positive. So it's getting a little bit, it's getting a little bit flatter, so our slope is at a very high level but it's decreasing. It's decreasing, decreasing, decreasing, slope is decreasing, decreasing even more. It's even more and then it's actually going to zero, our slope is zero and then it becomes negative. So our slope is still decreasing. Then it's becoming more and more and more negative, and then right around, and then right around here, it looks like it starts becoming less negative, or starts increasing. So our slope is increasing increasing, it's really just becoming less and less negative, and then it's going close to zero, approaching zero, it looks like our slope is zero right over here, but then it looks like right over there our slope begins decreasing again. So it looks like our slope is decreasing again. So it looks like our slope is decreasing. It's becoming more and more and more and more negative, and so it looks like something interesting happened right over there, we got a transition point, and then right around here, it looks like it starts, the slope starts increasing again. So it looks like the slope starts increasing. It's negative but it's becoming less and less and less negative and then it becomes zero and then it becomes positive and then more and more and more and more positive. So inflection points are where we go from slope increasing to slope decreasing. So concave upwards to concave downwards, and so slope increasing was here to slope decreasing, so this was an inflection point, and also from slope decreasing to slope increasing. So that's slope decreasing to slope increasing, and this is also slope decreasing to slope increasing. So how many inflection points does the graph of g have that we can see on this graph? Well it has three over the interval that at least we can see.