Exponential applications

growth/increase/appreciation or decay/decrease/depreciation

Discrete: n "steps" Continuous
A(n) = A₀(1±r)ⁿ A(t) = A₀e^kt
Amount A as a function of n steps.
the 1 means 100%, the same.
± growth/decay rate r per step.
multiply itself n times.
then of the initial amount A₀ Amount A as a function of continuous t (time, length, etc)
A₀ :initial amount
Growth: k>0. k = ln(2) / doubling time
Decay: k<0 k = ln(1/2) / halving time

Discrete: n "steps"	Continuous
A(n) = A₀(1±r)ⁿ	A(t) = A₀e^kt
Amount A as a function of n steps. the 1 means 100%, the same. ± growth/decay rate r per step. multiply itself n times. then of the initial amount A₀	Amount A as a function of continuous t (time, length, etc) A₀ :initial amount Growth: k>0. k = ln(2) / doubling time Decay: k<0 k = ln(1/2) / halving time

Setting A₀ = 1 means 100%, then the function results are multiples or fractions of the initial amount.

Exs.

Each film of tinted plastic blocks 10% of the light passing through it.
which means 90% of the light passes through.
Adding a film is a "step".
The fraction f of light that passes through n successive films is:
f(n) = (1-.10)ⁿ = (0.90)ⁿ
The A₀ is 1, meaning all of the light, 100% of it, without any films.

How much of the light passes through 2 films?    How much is blocked?
Is the amount blocked by two films twice the amount blocked by one film?
How much of the light passes through 3 films?    How much is blocked?
Is the amount blocked by three films thrice the amount blocked by one film?
How much of the light passes through 5 films?    How much is blocked?
How much of the light passes through 10 films?    How much is blocked?
Is the amount blocked by 10 films twice the amount blocked by 5 films?

An initial amount of stuff (money, viruses, microplastics) A₀ grows 10% each time unit (i.e. compound interest rate r is 10% per time unit).
After n time units, the stuff will accumulate to:
A(n) = A₀(1+.10)ⁿ
This means one compounding period per time unit.

Say the initial amount is 1000.
What is the amount after 1 time unit: How much did it grow by:
What is the amount after 2 time units: How much did it grow by this time unit:
Is the growth from the first to the second time unit the same as the growth from the beginning to the first time unit::

Atmospheric pressure p (in mm of Hg) as a function of height h (in km) above sea level.
p(h) = 760e^-0.145h

What is the pressure at sea level?
What is the pressure 1 km above sea level?
What is the pressure one mile high?
What is the pressure at the summit of Fuji-san (3776m)?
What is the pressure at the summit of Mt Everest (8849m)?

k = = ln(1/2) / halving_h
So: halving_h = ln(1/2) / k =

Our Atmosphere

An initial amount of stuff P grows continuously at a compound interest rate r of 10% for a certain time unit.
After t amount of time, the stuff will accumulate to:
A(n) = Pe^0.1t

Say the initial amount is 1000.
What is the amount after 1 time unit: How much did it grow by:
What is the amount after 2 time units: How much did it grow this time unit:

What is the amount after 0.5 time unit:

k = = ln(1/2) / doubling_t
So: doubling_t = ln(2) / k =