Understanding Engineering Mathematics

One rigorous method of defining the exponential functionexis by means of alimit.This
is quite an advanced concept that we only address in Chapter 14 and there is a temptation
to gloss over the idea here. However, the work is so important (and not without interest!)
that we will give it a try – don’t worry if it is too much for you at this stage, you should
be able to handle the exponential function well enough without studying the next couple of
pages. However, if youcanfind your way through this work it will be of great benefit to
you, as well as providing practice in some techniques of algebra. We are going to approach
the limit involved in the exponential function by considering a limiting process arising in
the study ofcompound interest. You can try to work out some of the details yourself.

Problem 2

Suppose you borrow£C at an interest ofI% compounded monthly, so

that the interest added at the end of each month is at

I

12

%, and you do

not pay off any of the loan until after a year

How much debt do you have at the end of the year?

At the end of the first month you owe

£

(

C+

I

12

C

100

)

=£C

(

1 +

I

1200

)

At the end of the second month you owe

£C

(

1 +

I

1200

)(

1 +

I

1200

)

=£C

(

1 +

I

1200

) 2

and so on.
So after 12 months you owe

£C

(

1 +

I

1200

) 12

Now suppose the terms were changed and interest was compounded daily. Then at the
end of a non-leap year you would owe

£C

(

1 +

I

36500

) 365

By the hour:

£C

(

1 +

I

876000

) 8760

and I will leave you to give the result if interest is calculated by the minute, or second
(RE 4.3.3A).
In general, if the interest is compounded at a constant rate over each ofnequal time
intervals in the year, then the amount owing at the end of the year will be

£C

(

1 +

I

100 n

)n