The Taylor series is one of a family of

approaches used for building approximations to functions.

So, before diving into the maths,

it's worth stopping for a minute to talk about when

it might be useful to have an approximation.

One example that seems to stick in peoples' minds is to do with cooking a chicken.

Imagine that you could write a function,

which describes the relationship between the mass of a chicken M,

and the amount of time T,

it should be put in the oven before it will be perfectly cooked.

Clearly, there are a lot of assumptions that

we'd need to have in place for such a function to even exist.

For example, this would only apply to a certain type of oven,

preheated to a certain temperature,

applied to a certain configuration of chicken i.e.

one that's in one piece rather than chopped up.

Furthermore, the heat transfer properties of a chicken will almost

certainly vary in a highly nonlinear fashion as a function of the mass.

Let's have a look at the sample function

that has some of the features that we've mentioned.

You can imagine that on the one hand,

the chicken cooking time T will be quite sensitive to

a lot of parameters that we might expect to vary case to case.

However, on the other hand if we'd like to sell a nice recipe book,

we're going to need to be pragmatic and simplify this monster as

people tend not to want to solve complicated equations while making dinner.

So, what do we do?

Firstly, we're going to make two assumptions,

one about ovens and the other about chickens.

So, let's assume that it's reasonable to suggest that everyone who buys

your cookbook will have

a sufficiently similar oven such that we can assume that they behave basically the same.

Secondly, that everyone will be cooking

a sufficiently similar chicken in

terms of the heat transfer properties as a function of mass,

that you can also assume that these will be the same in each case.

This allows us to remove

two potentially problematic terms that might

themselves have been a function of many other variables.

But we were still left with quite a messy function.

The next thing we're going to do is have a look at a plot of this function.

Now, already I'm telling you about what I view as relevant information in

the system as I'm not showing you timings for

chickens heavy than four kilograms or chickens of negative mass.

In fact, what I'm going to do now is go one step further and say that

considering a typical chicken from a supermarket is about 1.5 kilograms,

so let's focus here and build a nice straight line approximation of the form y = mx+c.

By using the Taylor series approach,

which we will be learning in the following videos,

it is possible for me to derive a function with

the same height and slope as one of the points in my graph.

This line is a fairly reasonable approximation to

the function in the region close to the point of interest,

but as you move further away,

the approximation becomes pretty poor.

However, this cookbook is not for people either roasting giant or miniature chickens.

So, we end up being able to write down

an expression in a much more palatable format where

our approximation T star is approximately

equal to 50 times the mass in kilograms plus 15.

So, if you'd like to roast a two kilogram bird,

leave it for about 115 minutes.

In the next few videos, we're going to go into a lot more detail about the Taylor series,

and also find out how to derive higher order terms.

I really don't know whether the famous chefs around the world are

knowingly using Taylor series approximations when they write their cookbooks,

but I like to think they are. See you next time.