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https:/.../2020-07-07_sds302_exponential-functions-part-one.mp4

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    Today we're going to start talking
    about exponential functions.
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    So, in order to introduce this,
    I have a short video.
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    ♪ (music) ♪
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    - This is an old story, but it reminds
    us of the surprises we can get
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    when even a small number like two
    is multiplied by itself many times.
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    King Shahram of India was so pleased when
    his Grand Vizier Sissa Ben Dahir
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    presented him with the game of chess that
    he asked Ben Dahir to name his own reward.
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    The request was so modest that
    the happy king immediately complied.
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    What the grand vizier had asked was this,
    that one grain of wheat be placed on
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    the first square of the chess board,
    two grains on the second square,
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    four on the third, eight on the fourth,
    16 on the fifth square, and so on.
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    Doubling the amount of wheat on each
    succeeding square until all 64 squares
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    were accounted for.
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    When the king's steward had gotten to
    the 17th square,
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    the table was well filled.
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    By the 26th square, the chamber held
    considerable wheat and a nervous king
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    ordered the steward to speed up the count.
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    When 42 squares were accounted for,
    the palace itself was swamped.
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    Now fit to be tied, King Shahram
    learns from the court mathematician
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    that had the process continued,
    the wheat required would have covered
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    all India to a depth of over 50 feet.
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    Incidentally, laying this many grains of
    wheat end to end also does something
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    rather spectacular.
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    They would stretch from the Earth,
    beyond the sun, past the orbits
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    of the planets, far out across the galaxy,
    to the star Alpha Centauri,
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    four light-years away.
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    They would then stretch back to Earth,
    back to Alpha Centauri,
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    and back to the Earth again.
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    ♪ (music) ♪
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    - So, what was going on in that video?
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    You can see that, rather than going up
    by the same amount for each square
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    that they advanced on the chess board,
    that they went up by a doubled amount.
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    So, that's a different type of function
    than we've been looking at.
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    Rather than going up by one and then
    another one and another one,
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    they went up by one and then
    they doubled and doubled again
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    and doubled again.
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    So, we cannot represent that with
    the type of linear function that
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    we've been looking at before.
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    In a linear function, such as this one,
    if I went up by one square,
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    I would increase here by 1.5.
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    So, if you think about how this plays out
    in a table, for each one unit increase
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    in t, I am increasing by the same amount.
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    Now, if I were to look at the percentage
    increase, therefore, I would actually
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    indicate that I'm changing the percent
    that I'm going up by each time.
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    So, going up by 1.5 from 2 is different
    than going up by 1.5 from 6.5.
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    So, if you can see that I've done these
    calculations already, going from 2 to 3.5
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    is an increase of the amount of 1.5,
    but it's actually a 75% increase.
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    But by the time I get up to these values,
    going from eight to 9.5,
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    that's only a 19% increase.
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    So, when I have a linear function, what
    you'll see is that you have a constant
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    amount of increase but you do not have
    a constant percentage of increase.
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    Now, what's going on in an exponential
    function and what was happening in
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    that video, is that for every one unit
    increase in t,
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    we will see a different constant.
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    So, in this case, if I use my exponential
    equation and I go up by one every time,
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    you can see that I go up
    by, first, just one.
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    Then I go up by 1.5 and then 2.25
    and then 3.36 and then 5.06.
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    So, I'm actually not increasing by
    the same amount each time.
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    That's because rather than adding
    a value each time,
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    I'm multiplying an additional time.
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    So, here I'm multiplying by 1.5
    an additional time each time I increase.
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    Well, because I'm multiplying by
    a constant amount, what that actually
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    does end up doing is increasing by
    a constant percentage.
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    So, going from 2 to 3 increases by 50%.
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    But also going from 10.13 to 15.19,
    I'm increasingly by 50%.
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    So, in an exponential function,
    I increase by a constant percentage,
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    but differing amounts.
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    Let's look at this with what
    we call a decay function.
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    So, this is also an exponential function,
    but one that's decreasing.
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    So, in this case, in my decay function,
    now you can see that I'm multiplying by
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    a decimal repeatedly.
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    Well, if I repeatedly multiply by
    a decimal, I should be getting
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    smaller and smaller values.
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    So, here, I decreased by one but then
    I decreased by half, by .25, by .125,
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    and so on.
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    What that means is is I'm basically
    slowing down my growth over time,
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    but I'm decreasing still by
    a constant percentage.
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    So, here, I decreased by 50% each time.
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    Now, if I just had one that I was
    multiplying by every time,
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    then I would actually have no change in
    percentage because I would just
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    have a constant amount because multiplying
    by one is the same every time.
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    So, what is an exponential function?
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    An exponential function can be one that is
    recognized by a constant
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    percentage rate of change.
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    Remember, this is as opposed to our linear
    functions which had a constant
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    amount of change.
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    So, rather than changing by one each time
    or by five each time,
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    maybe I'm changing by 5% each time.
    And those are different.
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    An exponential growth model is one that
    has a positive percent rate of change,
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    while a decay model is one that has
    a negative percent rate of change.
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    So, a exponential growth model will be
    increasing over time,
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    and a decay model will be
    decreasing over time.
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    So, here's the general
    form of our equation.
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    You're going to see I use t in my general
    form and that's because, essentially,
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    we'll always be looking at models
    with time as our independent variable.
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    Here we have some new parameters
    we're looking at, as well.
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    Because we're talking about time,
    I'm going to call a,
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    which is my y-intercept, my initial value.
    And that's because with time,
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    when time is zero, I get my y-intercept,
    so I get a.
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    A is therefore my initial value.
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    The growth factor is, what am I
    multiplying by each time?
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    And when I know what I'm multiplying
    by each time,
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    I can get to what is more important
    to me,
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    which is my growth rate, or what is
    the percentage that I'm increasing
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    or decreasing by each time.
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    So, my growth rate is just related to that
    growth factor and can be calculated
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    from it.
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    So, if I have my growth factor,
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    I can just subtract one and I'll get
    my growth rate.
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    Now, this r is not the same as
    our correlation r,
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    so don't let that fool you.
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    So, here's our definitions.
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    Our growth factor, or b, is the amount
    that we multiply by each y-value
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    to get the new y-value.
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    Our growth rate is the percentage increase
    or decrease that we get
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    for each one unit increase in x.
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    And these are the two things that
    we can get from our b.
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    The one that we're going to be interested
    in interpreting primarily is
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    our growth rate when we do
    our parameter estimates.
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    So, here are three examples of
    exponential models.
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    I have a growth, a stagnant,
    and a decay model.
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    One thing to note is that none of
    these have negative values possible.
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    In an exponential function, we can
    actually never cross that axis.
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    So, we can never have a negative value.
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    Notice that these are all the same
    concavity, so they are all trending
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    that same direction because we cannot
    cross this axis.
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    And, in fact, in a decay model,
    we will continuously get closer
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    and closer and closer to zero,
    but we will never actually get to zero.
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    So, let's work with some general
    functions.
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    I want to know what the growth factor
    would be if I said that water usage is
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    increasing by 5% per year.
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    So, what am I giving you here?
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    Well, I'm giving you the growth rate,
    that percentage of change.
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    And if I want to convert this to b,
    remember that b equals 1 plus r.
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    And that is r written as a decimal.
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    So, if I write my r as a decimal,
    I would have .05,
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    which tells me that my b here,
    my growth factor, would be 1.05.
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    Now, you try.
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    So, in this case, I've given you
    a decay model.
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    If I have decay, where I know that it's
    shrinking by 78%,
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    that actually means that
    I have a negative r value.
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    So, here I would end up with a b of 0.22.
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    When I have a decay model,
    my r can be negative.
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    So, I will expect to see a negative rate,
    because I am changing
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    in a downward trajectory.
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    Now, remember I can never actually have
    a negative value in my chart,
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    so that means that b cannot be negative,
    even though r can.
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    Well, why can't b be negative?
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    Well, think about if I had a chart here
    and I had a negative b value.
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    Well, when I add b to the first,
    I'd have a negative result,
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    but then when I square b,
    I get a positive,
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    and when I cubed it, I'd have a negative,
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    and then when I had to the fourth,
    I'd be up here.
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    So, I'd end up with some kind of
    function like that.
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    Well, that is not an exponential model.
    So, I cannot have a negative b.
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    When I have a negative r,
    for a decay model, what that will do is
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    give me a b value that is between
    zero and one.
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    So, I will have a b value that is
    less than one but more than zero
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    when I have a decay model.
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    If I have a b value over one,
    then I have a growth model.
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    So, here is another example of
    an exponential function.
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    And I've gone ahead and drawn it out.
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    You can see here that from my equation,
    I have 34.3 as my initial value.
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    So, if I were to sketch this or put this
    into technology and get it,
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    you would see that that's
    my initial value, my starting point.
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    My growth factor for this problem
    would be 0.63,
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    indicating I have a decay model because
    that b is less than one.
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    If I calculate my growth rate, then,
    I can have my b, 0.63,
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    as equal to 1 plus r.
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    And when I subtract one,
    I get a decay of negative 37%.
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    So, now you try it.
    Identify the growth rate in this equation.
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    In this equation, you can see that
    our growth rate is very small, at 0.2%.
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    What else do you notice about this chart?
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    Well, you should notice that
    it kind of looks like a line,
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    and not like that exponential curve that
    we're so used to seeing.
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    When you have a very small growth rate,
    one that is very close to zero,
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    you will actually see a model that
    doesn't have a very striking curve to it.
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    It'll almost like linear.
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    Now, if I zoomed this out to
    a thousand points, you'd probably begin
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    to be able to determine the curve.
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    But for very small r values,
    we'll have a pretty flat curve.
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    The larger your r value is,
    the more significant your curve is,
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    either going up or down.
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    So, if you were comparing two models
    drawn on the same chart,
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    you could see that the more curving one
    should have a higher absolute value of r,
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    because it's changing at a faster rate.
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    Alright. As always,
    bring us your questions to class.
Pavadinimas:
https:/.../2020-07-07_sds302_exponential-functions-part-one.mp4
Video Language:
English
Duration:
13:47

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