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Feb. 8 Lecture
2.5 and 4.4.
The exponential
function
e
x
and its
derivative.
Picture from Wikipedia
Compound interest
Consider a savings account that pays 100%
interest per year.
If the payment is made only once then $1
becomes $2 at the end of the year.
If the payment is made twice
1
→
1.5
→
(1.5)(1.5) = 2.25
once = 2,
twice = 2.25
If the payment is made 4 times
1
→
1.25
→
(1.25)
2
→
(1.25)
3
→
(1.25)
4
= 2.44
Monthly payments
(1
+
1/12 )
12
= 2.613
Daily payments
(1
+
1/365 )
365
= 2.7145
Daily payments
(1
+
1/365 )
365
= 2.7145
Hourly payments
(1
+
1/8,760 )
8760
= 2.718127
Every minute
(1
+
1/525,600 )
525,600
= 2.71827924
Every second
(1
+
1/31,536,000)
31,536,000
= 2.71828178
Limit
e = 2.7 1828 1828 45 90 45 …
The derivative of e
x
(
)
0
Rule 8
Algebra 4.
1
1
li
.
'
m
1
x
x
x
y
x
y
x
h
x
h
x
h
h
a
a
a
e
e
e
e
h
h
e
e
e
h
+
+
→
⋅
=
−
−
=
⋅
=
−
=
A differential equation
The chain rule says
(
)'
'(
)
'
If
(
)
then
'(
)
Rule 8B.
( )
solves
'( )
( )
(
)'
'
g
g
g
g
rt
f g
f
g
g
f g
e
f
g
e
e
h t
e
h
e g
t
r h t
=
⋅
=
=
=
⋅
=
=

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