
Today we're going to start talking
about exponential functions.

So, in order to introduce this,
I have a short video.

♪ (music) ♪

 This is an old story, but it reminds
us of the surprises we can get

when even a small number like two
is multiplied by itself many times.

King Shahram of India was so pleased when
his Grand Vizier Sissa Ben Dahir

presented him with the game of chess that
he asked Ben Dahir to name his own reward.

The request was so modest that
the happy king immediately complied.

What the grand vizier had asked was this,
that one grain of wheat be placed on

the first square of the chess board,
two grains on the second square,

four on the third, eight on the fourth,
16 on the fifth square, and so on.

Doubling the amount of wheat on each
succeeding square until all 64 squares

were accounted for.

When the king's steward had gotten to
the 17th square,

the table was well filled.

By the 26th square, the chamber held
considerable wheat and a nervous king

ordered the steward to speed up the count.

When 42 squares were accounted for,
the palace itself was swamped.

Now fit to be tied, King Shahram
learns from the court mathematician

that had the process continued,
the wheat required would have covered

all India to a depth of over 50 feet.

Incidentally, laying this many grains of
wheat end to end also does something

rather spectacular.

They would stretch from the Earth,
beyond the sun, past the orbits

of the planets, far out across the galaxy,
to the star Alpha Centauri,

four lightyears away.

They would then stretch back to Earth,
back to Alpha Centauri,

and back to the Earth again.

♪ (music) ♪

 So, what was going on in that video?

You can see that, rather than going up
by the same amount for each square

that they advanced on the chess board,
that they went up by a doubled amount.

So, that's a different type of function
than we've been looking at.

Rather than going up by one and then
another one and another one,

they went up by one and then
they doubled and doubled again

and doubled again.

So, we cannot represent that with
the type of linear function that

we've been looking at before.

In a linear function, such as this one,
if I went up by one square,

I would increase here by 1.5.

So, if you think about how this plays out
in a table, for each one unit increase

in t, I am increasing by the same amount.

Now, if I were to look at the percentage
increase, therefore, I would actually

indicate that I'm changing the percent
that I'm going up by each time.

So, going up by 1.5 from 2 is different
than going up by 1.5 from 6.5.

So, if you can see that I've done these
calculations already, going from 2 to 3.5

is an increase of the amount of 1.5,
but it's actually a 75% increase.

But by the time I get up to these values,
going from eight to 9.5,

that's only a 19% increase.

So, when I have a linear function, what
you'll see is that you have a constant

amount of increase but you do not have
a constant percentage of increase.

Now, what's going on in an exponential
function and what was happening in

that video, is that for every one unit
increase in t,

we will see a different constant.

So, in this case, if I use my exponential
equation and I go up by one every time,

you can see that I go up
by, first, just one.

Then I go up by 1.5 and then 2.25
and then 3.36 and then 5.06.

So, I'm actually not increasing by
the same amount each time.

That's because rather than adding
a value each time,

I'm multiplying an additional time.

So, here I'm multiplying by 1.5
an additional time each time I increase.

Well, because I'm multiplying by
a constant amount, what that actually

does end up doing is increasing by
a constant percentage.

So, going from 2 to 3 increases by 50%.

But also going from 10.13 to 15.19,
I'm increasingly by 50%.

So, in an exponential function,
I increase by a constant percentage,

but differing amounts.

Let's look at this with what
we call a decay function.

So, this is also an exponential function,
but one that's decreasing.

So, in this case, in my decay function,
now you can see that I'm multiplying by

a decimal repeatedly.

Well, if I repeatedly multiply by
a decimal, I should be getting

smaller and smaller values.

So, here, I decreased by one but then
I decreased by half, by .25, by .125,

and so on.

What that means is is I'm basically
slowing down my growth over time,

but I'm decreasing still by
a constant percentage.

So, here, I decreased by 50% each time.

Now, if I just had one that I was
multiplying by every time,

then I would actually have no change in
percentage because I would just

have a constant amount because multiplying
by one is the same every time.

So, what is an exponential function?

An exponential function can be one that is
recognized by a constant

percentage rate of change.

Remember, this is as opposed to our linear
functions which had a constant

amount of change.

So, rather than changing by one each time
or by five each time,

maybe I'm changing by 5% each time.
And those are different.

An exponential growth model is one that
has a positive percent rate of change,

while a decay model is one that has
a negative percent rate of change.

So, a exponential growth model will be
increasing over time,

and a decay model will be
decreasing over time.

So, here's the general
form of our equation.

You're going to see I use t in my general
form and that's because, essentially,

we'll always be looking at models
with time as our independent variable.

Here we have some new parameters
we're looking at, as well.

Because we're talking about time,
I'm going to call a,

which is my yintercept, my initial value.
And that's because with time,

when time is zero, I get my yintercept,
so I get a.

A is therefore my initial value.

The growth factor is, what am I
multiplying by each time?

And when I know what I'm multiplying
by each time,

I can get to what is more important
to me,

which is my growth rate, or what is
the percentage that I'm increasing

or decreasing by each time.

So, my growth rate is just related to that
growth factor and can be calculated

from it.

So, if I have my growth factor,

I can just subtract one and I'll get
my growth rate.

Now, this r is not the same as
our correlation r,

so don't let that fool you.

So, here's our definitions.

Our growth factor, or b, is the amount
that we multiply by each yvalue

to get the new yvalue.

Our growth rate is the percentage increase
or decrease that we get

for each one unit increase in x.

And these are the two things that
we can get from our b.

The one that we're going to be interested
in interpreting primarily is

our growth rate when we do
our parameter estimates.

So, here are three examples of
exponential models.

I have a growth, a stagnant,
and a decay model.

One thing to note is that none of
these have negative values possible.

In an exponential function, we can
actually never cross that axis.

So, we can never have a negative value.

Notice that these are all the same
concavity, so they are all trending

that same direction because we cannot
cross this axis.

And, in fact, in a decay model,
we will continuously get closer

and closer and closer to zero,
but we will never actually get to zero.

So, let's work with some general
functions.

I want to know what the growth factor
would be if I said that water usage is

increasing by 5% per year.

So, what am I giving you here?

Well, I'm giving you the growth rate,
that percentage of change.

And if I want to convert this to b,
remember that b equals 1 plus r.

And that is r written as a decimal.

So, if I write my r as a decimal,
I would have .05,

which tells me that my b here,
my growth factor, would be 1.05.

Now, you try.

So, in this case, I've given you
a decay model.

If I have decay, where I know that it's
shrinking by 78%,

that actually means that
I have a negative r value.

So, here I would end up with a b of 0.22.

When I have a decay model,
my r can be negative.

So, I will expect to see a negative rate,
because I am changing

in a downward trajectory.

Now, remember I can never actually have
a negative value in my chart,

so that means that b cannot be negative,
even though r can.

Well, why can't b be negative?

Well, think about if I had a chart here
and I had a negative b value.

Well, when I add b to the first,
I'd have a negative result,

but then when I square b,
I get a positive,

and when I cubed it, I'd have a negative,

and then when I had to the fourth,
I'd be up here.

So, I'd end up with some kind of
function like that.

Well, that is not an exponential model.
So, I cannot have a negative b.

When I have a negative r,
for a decay model, what that will do is

give me a b value that is between
zero and one.

So, I will have a b value that is
less than one but more than zero

when I have a decay model.

If I have a b value over one,
then I have a growth model.

So, here is another example of
an exponential function.

And I've gone ahead and drawn it out.

You can see here that from my equation,
I have 34.3 as my initial value.

So, if I were to sketch this or put this
into technology and get it,

you would see that that's
my initial value, my starting point.

My growth factor for this problem
would be 0.63,

indicating I have a decay model because
that b is less than one.

If I calculate my growth rate, then,
I can have my b, 0.63,

as equal to 1 plus r.

And when I subtract one,
I get a decay of negative 37%.

So, now you try it.
Identify the growth rate in this equation.

In this equation, you can see that
our growth rate is very small, at 0.2%.

What else do you notice about this chart?

Well, you should notice that
it kind of looks like a line,

and not like that exponential curve that
we're so used to seeing.

When you have a very small growth rate,
one that is very close to zero,

you will actually see a model that
doesn't have a very striking curve to it.

It'll almost like linear.

Now, if I zoomed this out to
a thousand points, you'd probably begin

to be able to determine the curve.

But for very small r values,
we'll have a pretty flat curve.

The larger your r value is,
the more significant your curve is,

either going up or down.

So, if you were comparing two models
drawn on the same chart,

you could see that the more curving one
should have a higher absolute value of r,

because it's changing at a faster rate.

Alright. As always,
bring us your questions to class.