Chemical Engineering Exam > Chemical Engineering Videos > Polynomial approximation of functions (part 2)
Polynomial approximation of functions (part 2) Video Lecture - Chemical Engineering
FAQs on Polynomial approximation of functions (part 2) Video Lecture - Chemical Engineering
| 1. What is polynomial approximation in chemical engineering? |  |
Ans. Polynomial approximation in chemical engineering refers to the use of polynomials to approximate complex functions or data sets. It is a mathematical technique used to simplify calculations and modeling in chemical engineering by representing the relationships between variables with polynomial equations.
| 2. How are polynomial approximations used in chemical engineering? | |
Ans. Polynomial approximations are used in chemical engineering to simplify complex functions or data sets. They can be used to approximate physical properties, such as heat capacity or viscosity, as well as to model chemical reactions or process dynamics. By using polynomial equations, engineers can make calculations and predictions more efficiently.
| 3. What are the advantages of using polynomial approximations in chemical engineering? | |
Ans. There are several advantages of using polynomial approximations in chemical engineering. Firstly, they provide a simple and accurate representation of complex functions or data sets. This simplifies calculations and modeling, making it easier to analyze and optimize chemical processes. Additionally, polynomial approximations can be easily integrated into computer simulations and mathematical models, allowing engineers to simulate and predict the behavior of chemical systems more effectively.
| 4. Are there any limitations to using polynomial approximations in chemical engineering? | |
Ans. Yes, there are limitations to using polynomial approximations in chemical engineering. One limitation is that polynomial approximations are only valid within a certain range of input values. If the input values exceed this range, the accuracy of the approximation may decrease significantly. Additionally, polynomial approximations may not capture non-linear relationships accurately, leading to errors in predictions. It is important for engineers to carefully validate and test polynomial approximations before relying on them for critical calculations or decisions.
| 5. How can one determine the degree of a polynomial approximation in chemical engineering? | |
Ans. The degree of a polynomial approximation in chemical engineering depends on the complexity of the function or data set being approximated. Generally, a higher degree polynomial can provide a more accurate representation of the data, but it also increases the complexity of calculations. Engineers often use techniques such as least squares regression or cross-validation to determine the optimal degree of a polynomial approximation. These techniques involve comparing the approximation's performance against a set of known data points or using statistical methods to find the best fit.
Text Transcript from Video
So where we left off in the
last video, we kept trying to
approximate this purple f
of x with a polynomial.
And we at first said we'll just
make the polynomial a constant
and set it -- it's just going
to intersect f of 0
at x is equal to 0.
So that's a first -- you can
kind of all think of it as a
0 of order approximation
of the function.
Then we said, oh, what if not
only do they intersect at x is
equal to 0, but let's say that
their slope is the same as x is
equal to 0, and that's
that approximation?
And that's about as good as
you're going to do with a
line, especially as
you get close to 0.
And we said, OK, that's good,
but what if their second
derivative is the same?
And that's where we ended
up with -- we added
this term here.
And I hinted that we'll just
keep doing this process.
And so you could imagine, if I
want the third derivative to be
the same, I could add another
term right here, plus, where I
know what the value of f of
x's third derivative is at 0.
So I'll write that as f to
the third derivative at
0 times x to the third.
Now what do you think is
going to be down here?
What's going to be
he denominator?
You might be tempted to say
that we'll put a 3 down here.
But it turns out you're going
to put a 3 times a 2, which
is a 6, or 3 factorial.
Now why is that?
Let me just take a little
departure here and I think
you'll start to understand
why you put a 6 down here.
Why this isn't a 3
and you put a 6.
Here you put a 2, but 2 is
also 2 factorial, right?
2 factorial is 2
times 1, right?
Hopefully you remember what
factorial -- actually, let
me tell you what factorial
is just in case.
10 factorial is equal to 10
times 9 times 8 times 7, dah,
dah, dah, dah, times 2 times 1.
So you're multiplying all of
the numbers up to that number.
4 factorial -- and the numbers
get big very, very fast -- is
4 times 3 times 2 times 1.
2 factorial is equal
to 2 times 1.
1 factorial is equal to 1.
Now this is kind of
a weird definition.
It comes out of combinatorics.
Actually it works for what
we're doing is, well, 0
factorial is also equal to 1.
I know that might be a
little un-intuitive.
This is just a definition.
It's like saying that i squared
is equal to negative 1.
It is a definition that it
makes formulas be more
general, I guess is a
simple way to put it.
But let me erase all of this
because that was just a
divergence just because I
realized I was going to use
factorial, so you should
know what a factorial is.
But I think that's a fairly
straightforward concept.
So going back to
what we were doing.
I was asking you why do I put
a 6 down here instead of a
3, like we put a 2 here?
Well, let's just take
this term alone and take
its third derivative.
So if I have the term and it's
f, the third derivative at 0, x
to the third over -- and
let me just write 6 as
3 times 2 or 2 times 3.
That'll make it a
little more clear.
What's the first
derivative here?
What happens when I take
the derivative once?
Well, I'm going to multiply the
whole thing by the 6 exponent
and decrement the
exponent, right?
So I'm going to multiply the
whole thing times 3 times f,
the third derivative, x
squared over 2 times 3.
So that first time I did
it, this 3 and this
3 cancel out, right?
That red's looking a
little bit too demonic.
Let me pick another color.
And then when I take the
second derivative what
am I going to get?
Well, the 3's gone, now I just
have a 2 in the denominator, so
I multiply the whole thing by 2
times f prime prime prime of 0
times, and I decrement the
exponent, x to the 1 over 2.
Well, now the 2's
cancel out, right?
So the reason why you're
putting a factorial there is
every time you take a
derivative you're decrementing
the exponent 1, and multiplying
the whole expression
by the exponent.
So if you're going to take n
derivative, you're essentially
going to be multiplying this
expression times n factorial.
So you don't want an n
factorial out here.
You put an n factorial
at the bottom.
Hopefully that makes sense.
Play around with it yourself
and it should start to make
a little bit more sense.
So in general, if we just
kept doing this process
forever, what would the
function look like?
The reason why I'm covering
this is because going this way
we're going to be able to prove
what I think is the most
mind-bending concept
in mathematics.
And it will make you love
mathematics, hopefully.
Some people actually --
well, I won't go into the
spiritual aspects of it.
So what would be this, if I
just kept saying that I'm
just going to keep taking
derivatives and adding them to
this term, this polynomial?
Well, the polynomial would
become p of x is equal to f
of 0 plus f prime of 0 x.
And let's just divide it by 1
factorial, just to make it
clear that that's a
1 factorial, right?
And that's an x
to the 1, right?
That's just this term,
but I just wrote it a
little differently.
This term right here, this
is f of 0 times x to the 0.
I know that's really
messy, but hopefully you
see what I'm saying.
And that's over 0
factorial, right?
0 factorial is 1, x to the 0
is 1, so it's just f of 0.
And then plus the second
derivative at 0 times x squared
over 2 factorial plus --
and we just keep adding.
The third derivative at x is
equal to 0 of x to the third
over 3 factorial, and
we just keep going on.
So we could do
this to infinity.
And actually we will do
it, and this is called
the Maclaurin Series.
So if we just wanted to
approximate this as hard as we
can, essentially take the
infinite derivatives of it, we
get the Maclaurin Series.
So we are going to define
this polynomial p of x.
It's going to be the infinite
series, the infinite sum.
Let's start with n is equal
to 0, and we're going
to go to infinity.
What is each term?
It's going to be f of -- well
it's going to be f, the nth
derivative of f evaluated at
0 times x to the n
over n factorial.
This is the Maclaurin Series.
We're later learn that the
Maclaurin Series is a specific
example of the Taylor Series,
which is a specific example
of a power series.
But anyway, this might seem
very complicated to you.
I have all the sigma notation.
Just remember, this is
essentially just that and I
just keep going to infinity.
And if you play around with
it it should make sense.
But I think this will become a
lot more concrete when I do
this with a specific f of x.
This is where it gets cool.
In case you don't think
it's already cool.
So let's pick f of x to be,
to me, the most amazing
function of them all.
If I ever built a shrine or
a church or something or
skyscraper, I would somehow
make this function show up all
over the place, and then years
from now people would be awed
by the mysticism of it all.
But anyway, let's try to
approximate e to the x
with a Maclaurin Series.
You know that sigma thing,
that's hard to memorize.
Just remember you want all the
derivatives to be the same.
So let's make the
approximation of this.
Actually, I won't prove it.
It's out of the scope of
what we're doing right now.
But the approximation, even
when it's centered at 0,
actually equals the function
when you take the infinite sum.
But let's just see
what it looks like.
Because this is pretty cool.
Before we start building the
polynomial, let's just figure
out a couple of things.
So what is f prime of x?
That's also e to the x, right?
What's f prime prime of x?
Well that also
equals e to the x.
We have learned and actually
recently did a proof that
the derivative of e to
the x is e to the x.
But that also needs a second
derivative and the third and
the fourth and the nth
derivative of e to the x
is equal to e to the x.
I could take an arbitrary
number of derivatives of e
to the x and it equals e to
the x, which is amazing.
The rate of change of the
function at any point is
equal to the function.
The rate of change of the rate
of change of the function at
any point is equal
to the function.
I mean that's -- I want to just
go some place and ponder it,
but I'm too busy making videos.
But anyway, back to
what we were doing.
So what is f of 0?
f of 0 is equal to e to the 0,
which is equal to 1, right?
Well that's also going
to be f prime of 0.
That's also e to the 0,
which is equal to 1.
So all of the derivatives, the
nth derivative at 0 is going to
equal 1 for this specific case
of f of x, for e to the x.
And this is why
this is so cool.
But actually, it actually
gets even more amazing.
So, you hopefully realize
that f of 0 and all of
its derivatives at
0 are equal to 1.
So now we can do the powers
of the Maclaurin Series
in the next video.
See you soon.
last video, we kept trying to
approximate this purple f
of x with a polynomial.
And we at first said we'll just
make the polynomial a constant
and set it -- it's just going
to intersect f of 0
at x is equal to 0.
So that's a first -- you can
kind of all think of it as a
0 of order approximation
of the function.
Then we said, oh, what if not
only do they intersect at x is
equal to 0, but let's say that
their slope is the same as x is
equal to 0, and that's
that approximation?
And that's about as good as
you're going to do with a
line, especially as
you get close to 0.
And we said, OK, that's good,
but what if their second
derivative is the same?
And that's where we ended
up with -- we added
this term here.
And I hinted that we'll just
keep doing this process.
And so you could imagine, if I
want the third derivative to be
the same, I could add another
term right here, plus, where I
know what the value of f of
x's third derivative is at 0.
So I'll write that as f to
the third derivative at
0 times x to the third.
Now what do you think is
going to be down here?
What's going to be
he denominator?
You might be tempted to say
that we'll put a 3 down here.
But it turns out you're going
to put a 3 times a 2, which
is a 6, or 3 factorial.
Now why is that?
Let me just take a little
departure here and I think
you'll start to understand
why you put a 6 down here.
Why this isn't a 3
and you put a 6.
Here you put a 2, but 2 is
also 2 factorial, right?
2 factorial is 2
times 1, right?
Hopefully you remember what
factorial -- actually, let
me tell you what factorial
is just in case.
10 factorial is equal to 10
times 9 times 8 times 7, dah,
dah, dah, dah, times 2 times 1.
So you're multiplying all of
the numbers up to that number.
4 factorial -- and the numbers
get big very, very fast -- is
4 times 3 times 2 times 1.
2 factorial is equal
to 2 times 1.
1 factorial is equal to 1.
Now this is kind of
a weird definition.
It comes out of combinatorics.
Actually it works for what
we're doing is, well, 0
factorial is also equal to 1.
I know that might be a
little un-intuitive.
This is just a definition.
It's like saying that i squared
is equal to negative 1.
It is a definition that it
makes formulas be more
general, I guess is a
simple way to put it.
But let me erase all of this
because that was just a
divergence just because I
realized I was going to use
factorial, so you should
know what a factorial is.
But I think that's a fairly
straightforward concept.
So going back to
what we were doing.
I was asking you why do I put
a 6 down here instead of a
3, like we put a 2 here?
Well, let's just take
this term alone and take
its third derivative.
So if I have the term and it's
f, the third derivative at 0, x
to the third over -- and
let me just write 6 as
3 times 2 or 2 times 3.
That'll make it a
little more clear.
What's the first
derivative here?
What happens when I take
the derivative once?
Well, I'm going to multiply the
whole thing by the 6 exponent
and decrement the
exponent, right?
So I'm going to multiply the
whole thing times 3 times f,
the third derivative, x
squared over 2 times 3.
So that first time I did
it, this 3 and this
3 cancel out, right?
That red's looking a
little bit too demonic.
Let me pick another color.
And then when I take the
second derivative what
am I going to get?
Well, the 3's gone, now I just
have a 2 in the denominator, so
I multiply the whole thing by 2
times f prime prime prime of 0
times, and I decrement the
exponent, x to the 1 over 2.
Well, now the 2's
cancel out, right?
So the reason why you're
putting a factorial there is
every time you take a
derivative you're decrementing
the exponent 1, and multiplying
the whole expression
by the exponent.
So if you're going to take n
derivative, you're essentially
going to be multiplying this
expression times n factorial.
So you don't want an n
factorial out here.
You put an n factorial
at the bottom.
Hopefully that makes sense.
Play around with it yourself
and it should start to make
a little bit more sense.
So in general, if we just
kept doing this process
forever, what would the
function look like?
The reason why I'm covering
this is because going this way
we're going to be able to prove
what I think is the most
mind-bending concept
in mathematics.
And it will make you love
mathematics, hopefully.
Some people actually --
well, I won't go into the
spiritual aspects of it.
So what would be this, if I
just kept saying that I'm
just going to keep taking
derivatives and adding them to
this term, this polynomial?
Well, the polynomial would
become p of x is equal to f
of 0 plus f prime of 0 x.
And let's just divide it by 1
factorial, just to make it
clear that that's a
1 factorial, right?
And that's an x
to the 1, right?
That's just this term,
but I just wrote it a
little differently.
This term right here, this
is f of 0 times x to the 0.
I know that's really
messy, but hopefully you
see what I'm saying.
And that's over 0
factorial, right?
0 factorial is 1, x to the 0
is 1, so it's just f of 0.
And then plus the second
derivative at 0 times x squared
over 2 factorial plus --
and we just keep adding.
The third derivative at x is
equal to 0 of x to the third
over 3 factorial, and
we just keep going on.
So we could do
this to infinity.
And actually we will do
it, and this is called
the Maclaurin Series.
So if we just wanted to
approximate this as hard as we
can, essentially take the
infinite derivatives of it, we
get the Maclaurin Series.
So we are going to define
this polynomial p of x.
It's going to be the infinite
series, the infinite sum.
Let's start with n is equal
to 0, and we're going
to go to infinity.
What is each term?
It's going to be f of -- well
it's going to be f, the nth
derivative of f evaluated at
0 times x to the n
over n factorial.
This is the Maclaurin Series.
We're later learn that the
Maclaurin Series is a specific
example of the Taylor Series,
which is a specific example
of a power series.
But anyway, this might seem
very complicated to you.
I have all the sigma notation.
Just remember, this is
essentially just that and I
just keep going to infinity.
And if you play around with
it it should make sense.
But I think this will become a
lot more concrete when I do
this with a specific f of x.
This is where it gets cool.
In case you don't think
it's already cool.
So let's pick f of x to be,
to me, the most amazing
function of them all.
If I ever built a shrine or
a church or something or
skyscraper, I would somehow
make this function show up all
over the place, and then years
from now people would be awed
by the mysticism of it all.
But anyway, let's try to
approximate e to the x
with a Maclaurin Series.
You know that sigma thing,
that's hard to memorize.
Just remember you want all the
derivatives to be the same.
So let's make the
approximation of this.
Actually, I won't prove it.
It's out of the scope of
what we're doing right now.
But the approximation, even
when it's centered at 0,
actually equals the function
when you take the infinite sum.
But let's just see
what it looks like.
Because this is pretty cool.
Before we start building the
polynomial, let's just figure
out a couple of things.
So what is f prime of x?
That's also e to the x, right?
What's f prime prime of x?
Well that also
equals e to the x.
We have learned and actually
recently did a proof that
the derivative of e to
the x is e to the x.
But that also needs a second
derivative and the third and
the fourth and the nth
derivative of e to the x
is equal to e to the x.
I could take an arbitrary
number of derivatives of e
to the x and it equals e to
the x, which is amazing.
The rate of change of the
function at any point is
equal to the function.
The rate of change of the rate
of change of the function at
any point is equal
to the function.
I mean that's -- I want to just
go some place and ponder it,
but I'm too busy making videos.
But anyway, back to
what we were doing.
So what is f of 0?
f of 0 is equal to e to the 0,
which is equal to 1, right?
Well that's also going
to be f prime of 0.
That's also e to the 0,
which is equal to 1.
So all of the derivatives, the
nth derivative at 0 is going to
equal 1 for this specific case
of f of x, for e to the x.
And this is why
this is so cool.
But actually, it actually
gets even more amazing.
So, you hopefully realize
that f of 0 and all of
its derivatives at
0 are equal to 1.
So now we can do the powers
of the Maclaurin Series
in the next video.
See you soon.
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Oct 19, 2025
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