E IN PICTURES
TERRY A. LORING
0.1. Pictures of Money in a Bank. I am going to use matlab to make a lot of nice
pictures that should help you intuitively understand what e represents, and why f (x) = ex
has slope 1 at x = 0. The math that substantiates these pictures is actually not to hard, but
as it does not follow the book very closely, I am not going to include that here.
The departure I take from the book is I prefer to think of e as being defined by the limit
2n
1
=e
lim 1 + n
n→∞
2
and simultaneously by the limit
2n
1
lim 1 + n
= e.
n→∞
2 −1
These formulas arise naturally if you consider again $1 at 100% interest in the bank for a
year, with compounding done 2 times a year, then 4 times a year, then 8 times a year.
Twice a year:
time in years
x=0
x = 21
x=1
Four times a year:
time in years
x=0
x = 41
x = 24
x = 34
x=1
what is computed
1 1 1 + 21
1 1 + 12 1 + 12
your balance
1
what is computed
1 1 1 + 41
1 1 + 41 1 + 14
3
1 1 + 41
4
1 1 + 14
your balance
1
what .
is computed
2
1 1 + 12
1 1 + 21
1 1 1 + 21
1 1 + 12 1 + 12
your balance
3
2
9
4
5
4
25
16
125
64
625
256
But now I want you to imagine time running backwards a year as well:Twice a year:
Two times a year:
time in years
x = −1
x = − 21
x=0
x = 12
x=1
1
4
9
2
3
1
3
2
9
4
2
TERRY A. LORING
Four times a year:
time in years
x = −1
x = − 43
x = − 42
x = − 41
x=0
x = 14
x = 24
x = 34
x=1
what .
is computed
4
1 1 + 31
.
3
1 1 + 14
.
2
1 1 + 14
1 1 + 41
1 1 1 + 41
1 1 + 41 1 + 14
3
1 1 + 41
4
1 1 + 14
your balance
256
625
64
125
16
25
4
5
1
5
4
25
16
125
64
625
256
By connecting these data points by line segments, we get the blue functions shown here.
4
3.5
3
2.5
2
1.5
1
0.5
0
−1
−0.5
0
0.5
1
In black is the actual y = e . The green functions are above y = ex . This time, the banker is
compounding at 2, then 4, then 8 times a year, but the bonus comes from you getting 2n1−1
in interest when you really only deserve 21n .
x
E IN PICTURES
3
Two times a year, extra nice banker:
time in years
x = −1
x = − 21
x=0
x = 12
x=1
what .
is computed
2
1 1 + 11
1 1 + 11
1 1 1 + 11
1 1 + 11 1 + 11
your balance
what .
is computed
4
1 1 + 31
.
3
1 1 + 13
.
2
1 1 + 13
1 1 + 31
1 1 1 + 31
1 1 + 31 1 + 13
3
1 1 + 31
4
1 1 + 13
your balance
1
4
1
2
1
2
4
Four times a year, extra nice banker:
time in years
x = −1
x = − 43
x = − 42
x = − 41
x=0
x = 14
x = 24
x = 34
x=1
256
625
64
125
16
25
4
5
1
4
3
16
27
64
81
256
243
If we connect these data points with line segments, we get the green functions, in the graph
I am showing you again.
4
3.5
3
2.5
2
1.5
1
0.5
0
−1
−0.5
0
0.5
1
Here is the same with one more of each of the compounding functions, adding in the example
4
TERRY A. LORING
of 8 compounding intervals in a year.
4
3.5
3
2.5
2
1.5
1
0.5
0
−1
−0.5
0
0.5
1
0.2. The slope. Since the blue lines all have slope one just to the right of (0, 1), and the
green lines all have slope one just to the left of (0, 1), it seems most reasonable that the slope
of y = ex is one at x = 0.
This shows the wedges that trap y = ex between x = − 12 and x = 21 :
4
3.5
3
2.5
2
1.5
1
0.5
0
−1
−0.5
0
0.5
1
E IN PICTURES
This shows the wedges that trap y = ex between x = − 41 and x =
5
1
4
:
1
8
:
4
3.5
3
2.5
2
1.5
1
0.5
0
−1
−0.5
0
0.5
1
This shows the wedges that trap y = ex between x = − 18 and x =
4
3.5
3
2.5
2
1.5
1
0.5
0
−1
−0.5
0
0.5
1
You can zoom in on these images.
6
TERRY A. LORING
I hope these pictures help.
URL: http://www.math.unm.edu/~loring
E-mail address: [email protected]
Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM
87131, USA.