tom.h.wilson

tom.wilson@mail.wvu.edu

Department of Geology and Geography

West Virginia University

Morgantown, WV

Geology 351 - Geomath

Review of exponential growth problems and equation manipulation using isostacy

Return/Discuss problems 2.11, 2.13

Creating e

Exponentials!

Take the plunge and find out about Archimedes principle and isostacy. Why ice cubes float and what they share with mountain roots.

In class problem use and substitute h=r+e to solve for e as a function of h, i and w.

Hand that in along with answers to computer problems 2.11 and 2.12.

i

w

r h

Agenda/objectives

2.11

Tom Wilson, Department of Geology and Geography

Make sure you distinguish between rate and slope

D/t=1856cm/4175yr=0.

2.11

Tom Wilson, Department of Geology and Geography

Be quantitative in your responses

A few minutes on e

Tom Wilson, Department of Geology and Geography

Open up Excel and work through this as we go

For example, N=10.

You can give N a variable name if you wish or just refer to the cell carrying the value N

=(1+1/c4)^c4 or =(1+1/N)^N

Increase N from 10 to 100 to 1000

Tom Wilson, Department of Geology and Geography

What do you think this expression will converge to when N approaches infinity?

Note that you can get a reliable answer in Excel up to about 1e12

Let’s consider this as a compound interest problem

Tom Wilson, Department of Geology and Geography

You can also use Excel for this

Let’s say you have $1000 to invest and you purchase a stock that earns on average 6% per year.

What is your value in one year?

In 2 years?

In 3 years?

We just have $1000 (1+0.06)(1+0.06)(1+0.06) for the three years. In other words, each year the value of our investment increases 6%.

However, in this example, we are just compounding annually

Calculate earnings at different fractions of the year

Tom Wilson, Department of Geology and Geography

What if we compound or give out earnings every half year?

Our interest would be calculated at the half as $1000(1+0.03) =

And at the end of the year as $1000(1+0.03)(1+0.03) =

What would happen if we calculated the value every quarter, every month, or continuously.

The formula you are evaluating is

What is

interest1

N

N

xlim 1

N

x N

Evaluate ex

Tom Wilson, Department of Geology and Geography

Evaluate this expression for large N and then compute ex

0.06e

rtValue Pe

This gives rise to the relationship that

rtValue Pe

This relationship looks very much like the population growth relationship

Tom Wilson, Department of Geology and Geography

(at time t) rtValue PeWhere P is the initial investment and r is the fractional interest per anum and t is the number of years

toP P e

Where Po is the population at reference time 0 and r is the fractional growth per anum

Make a calculation and examine it

Tom Wilson, Department of Geology and Geography

toP P eUse Po =1000 and =0.06

What is P in one year?

and note that P increases by about 6.2% not 6%

We’ll come back to this in a minute.

Questions about population growth?

Tom Wilson, Department of Geology and Geography

Why does the approximation of as the fractional growth in one year give you an accurate result?

Using the series approximation for the natural log

Why does the fractional increase in population ?

Tom Wilson, Department of Geology and Geography

actually equals ln(1+the fractional increase) or2 3 4

or ln(1 ) ...2 3 4

where x (<1) is the fractional increase in population

x x xx x

Have a look at the Excel file ln_Expansion.xlsx

It’s a busy presentation

Tom Wilson, Department of Geology and Geography

We’re letting x= the fractional increase in the above

ln_Expansion.xlsx

It's nice to simplify, but be sure you understand why it works

Tom Wilson, Department of Geology and Geography

50(0.01188)

50(0.01181)

(50)

but - actually

ln(1.01188) or

0.01180999

so (50)

o

o

P P e

P P e

In this case, there is a very small error in making the one term approximation

Low order approximations are good for small x!

Tom Wilson, Department of Geology and Geography

Dissecting the exponential decay relationship

Tom Wilson, Department of Geology and Geography

toN N e

Let’s use our calculus for a minute to get another perspective on this relationship

The derivative of N is

or just

to

dNN e

dtdN

Ndt

This is actually the starting point for the radioactive decay problem

Integrate

Tom Wilson, Department of Geology and Geography

Working from that relationship as a starting point, divide both sides by N and multiply both sides by dt

to get 1

dN dtN

What do you get when you integrate?

1dN dt

N

Integration yields -

The decay equation in logarithmic form

Tom Wilson, Department of Geology and Geography

1 2ln( )N c t c

Take a few minutes and transform this relationship into its exponential equivalent, to get

toN N e

Half life

Tom Wilson, Department of Geology and Geography

The half life is just the time it takes for No to decay to No/2.

1/ 2

2to

o

NN e

We used the ln transform to solve for t1/2. Note that if the half-life were given, we could solve for the growth rate.

Make sure you can apply these ln transforms and their inverse.

Isostacy

Tom Wilson, Department of Geology and Geography

We’ll talk more about isostacy on Thursday. To introduce the problem consider the ice cube floating in

your favorite drink. Some basic terms …

Hand in the in-class problems before leaving

2.13 and 2.15 will be due Thursday

Read chapter 3 and look over problems 3.10 and 3.11 (also see today’s handout) for discussion this Thursday