Exponential Function

1. Block 3
The Exponential Function

2. What is to be learned?
• The number that is vital in any sort of
natural growth (or decay)

3. Continuous Growth
Number can be modelled by
(1 + 1
/n)n
n = 1 (1 + 1
/1)1
= 2
= 2.25
n = 2 (1 + 1
/2)2
n = 10
n = 100
n = 1000
(1 + 1
/10)10
as n → ∞∞ (1 + 1
/n)n
→ 2.718281826……
known as exp or e
= 2.594
→ 2.705
→ 2.717
n = 10000 → 2.718
n = 100000 → 2.718

4. Harry’s hedge is 40 cm tall
His special fertilizer increases its height by 7% p.a.
1 year later
2 years later
3 years later
very simplistic
to model continual growth - use e

5. The Exponential Function
Natural growth or decay functions can be
expressed in terms of e
Usually something like
f(t) = f0ekt
f0
k
t
e
- initial value
- a number
- time
- e!!!!!!!!!!!!!!!!

6. A Proper Growth Calculation
The population P of some bacteria after t
days can be calculated using formula
P(t) = P0e0.6t where P0 is the original
population
There are 8 000 bacteria to start with.
How many will there be after a week?
P = 8 000(e(0.6X7)
)
= 8 000(e4.2
)
= 533 491 bacteria

7. The Exponential Function
The letter e (or exp) is a value that is used to
model growth ( or decay)
e = 2.718 (to 3d.p.)
usual type of formula
f0
k
t
e
f(t) = f0ekt
- initial value
- a number
- time
- e – which is a number

8. The mass of radioactive Goofyonium (M
grams)
after t years is given by
M = M0e-0.02t
If there is originally 80g of Goofyonium,
how much will there be after 100 years?
M = 80(e-0.02(100)
)
= 80(e-2
)
= 10.8 g
careful when using calculator

9. A proper growth function for the height (m)
of Henry’s hedge over t years is
H(t) = H0e0.4t
where H0 is the original height (m)
If the hedge was 40cm to start with, calculate the
height after 6 months.
H = 0.4(e0.2
)
= 0.49m
Key
Question