The Rate of Growth of Continuous Processes

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From the course by Duke University

Data Science Math Skills

1501 ratings

Duke University

Data Science Math Skills

1501 ratings

Data science courses contain math—no avoiding that! This course is designed to teach learners the basic math you will need in order to be successful in almost any data science math course and was created for learners who have basic math skills but may not have taken algebra or pre-calculus. Data Science Math Skills introduces the core math that data science is built upon, with no extra complexity, introducing unfamiliar ideas and math symbols one-at-a-time. Learners who complete this course will master the vocabulary, notation, concepts, and algebra rules that all data scientists must know before moving on to more advanced material. Topics include: ~Set theory, including Venn diagrams ~Properties of the real number line ~Interval notation and algebra with inequalities ~Uses for summation and Sigma notation ~Math on the Cartesian (x,y) plane, slope and distance formulas ~Graphing and describing functions and their inverses on the x-y plane, ~The concept of instantaneous rate of change and tangent lines to a curve ~Exponents, logarithms, and the natural log function. ~Probability theory, including Bayes’ theorem. While this course is intended as a general introduction to the math skills needed for data science, it can be considered a prerequisite for learners interested in the course, "Mastering Data Analysis in Excel," which is part of the Excel to MySQL Data Science Specialization. Learners who master Data Science Math Skills will be fully prepared for success with the more advanced math concepts introduced in "Mastering Data Analysis in Excel." Good luck and we hope you enjoy the course!

From the lesson

Measuring Rates of Change

This module begins a very gentle introduction to the calculus concept of the derivative. The first lesson, "This is About the Derivative Stuff," will give basic definitions, work a few examples, and show you how to apply these concepts to the real-world problem of optimization. We then turn to exponents and logarithms, and explain the rules and notation for these math tools. Finally we learn about the rate of change of continuous growth, and the special constant known as “e” that captures this concept in a single number—near 2.718.

Using Integer Exponents7:48

Simplification Rules for Algebra using Exponents11:08

How Logarithms and Exponents are Related12:36

The Change of Base Formula4:23

The Rate of Growth of Continuous Processes11:09

Meet the Instructors

Daniel Egger
Executive in Residence and Director, Center for Quantitative Modeling
Pratt School of Engineering, Duke University
Paul Bendich
Assistant research professor of Mathematics; Associate Director for Curricular Engagement at the Information Initiative at Duke
Mathematics

You will sometimes hear people refer to an exponential rate of growth, and

there are two different ways you can have an exponential rate of growth.

You can have a discrete exponential rate of growth, and

you can have a continuous exponential rate of growth.

A discreet rate is very straightforward.

Let's say that I have $1.00 and it's growing at a certain

rate, r, and I have a certain number of time intervals, t.

So the rate represents how much

my money would grow in discreet intervals of time, t.

So, if I had a rate of 100% per year,

that would be r and t = 1.

Then at the end of one year, I would have $2.00, and

that would be a 100% discreet exponential rate of growth.

At the end of two years, I would have $4.00.

At the end of three years, I would have $8.00 and so on.

But what we're especially interested in here is something

called continual exponential growth and

a very special constant known as Euler's constant, or e.

I'm going to show you right now how we can develop an intuition for

what this special number e is.

Let's suppose that we had our 100% interest rate.

And a clever man said, hey, if you're willing to pay me 100% interest for

a year, wouldn't you be willing to pay me 50% interest for six months?

And the bank would probably say, well, yeah, that seems fair.

And they say, okay, well great.

So I would like 50% interest for six months and

then I would like interest on my interest for another six months.

So, if I have a factor of 1 + 1 and

I repeat it one time, I will have 2.

But let's say I have a factor of (1.5) and

I repeat it 2 times, then I'm going to have a result of 2.25.

And the same clever man might say, well gee,

if you agreed to pay me interest twice a year, wouldn't you agree

to pay me 4 times a year including interest on my interest?

So what we're doing here is we're saying 1.25

raised to the 4th power or 1.5 raised to the 2nd power,

where this value is our rate per unit time and

these are our number of time intervals.

So you'll notice, this is decreasing and this is increasing, all right?

And if we were paid every 3 months,

we'd received $2.44 for every dollar.

So an obvious question to ask is does this number keep getting

bigger forever and ever?

As I make this time intervals smaller, does my potential wealth just increase and

increase in an unlimited way?

And the answer is surprisingly perhaps is no.

It does increase, but eventually it levels off.

So let's just take a look at what that would look like.

If I'm getting paid interest every month, my factor would be 1.08.

It would repeat 12 times, so I would have 1.08

to the 12th and the result would be 2.613.

If I'm getting paid interest every week,

that would be a factor of 1.019 times 52,

or 1.019 raised to the 52nd power, which would be,

2.693.

If I was paid interest everyday,

this would be 1.002739 times 365.

And that would be equal to 2.7146.

You notice, the numbers are not increasing as rapidly and

there are 8,760 hours in a year.

So, if I raised,

I would be receiving $2.71813.

And at the minutes of which

there are, let's see,

let's say 525,600,

I would receive 2.71828.

And then of the seconds of which there

are 31,536,000 in a year,

then I would receive 2.71828.

And although this number continues on, dot, dot, dot, to five

significant digits, it has stabilized as I get to a minute or a second.

This number is known as Euler's constant or

e, 2.71828.

So, let's consider an example of a problem.

Let's consider a baby elephant that grows continuously at a known rate.

So let's say I have a baby elephant.

It weighs 200 kg and I know that it grows

at a continuously compounded rate of 5%.

So it has a growth that is continuously compounded,

continuous at 5% per year.

I want to know what this elephant weighs in three years.

It will weigh (200 kg)

e to the (.05) times t, which is 3.

It would weigh 200 times e to the 0.15,

Which is =

232.4 kg.

In addition to this very, very valuable concept of e,

we also have the concept of log to the base e,

Which is actually written ln(x),

which stands for natural log.

Why is it the natural log?

Because it's the log that we use to calculate

naturally occurring continuous rates of growth.

So let's suppose that we have some rabbits.

And these rabbits with unlimited food increase in mass

with their babies at the rate of 200% per year.

So let's say that we start with a population of male and

female rabbits that weighs 10 kg.

And they have unlimited food and resources.

And what we want to know is if they're increasing at a continuously compounded

rate of 200% per year, How many years,

Until they weigh as much as the Earth,

which would be 5.972 x 10 to the 24 kg.

The way that we set up this problem is as follows.

We have 5.972 x 10 to the 24

kg = 10 kg times e to the 2t,

where 2 is equal to 200%,

that's the rate.

And t is equal to the number of years required to create

a 5.972 x 10 to the 24 kg worth of rabbits, okay?

So first, we can divide both sides by 10.

So now we have 5.972

x 10 to the 23 = e to the 2t, okay?

And now we're going to use a little trick which is that we're going to take

log to the base e of both sides of this equation.

So we have ln(5.972 x 10 to the 23rd) = 2t,

meaning that t is simply equal to the natural

log of this number divided by 2.

And you can use your pocket calculator, or Excel, or

your computer to calculate the natural log of this number and divide by 2.

And what you will find is it will take 27.37 years for

the rabbits to weigh as much as the Earth.

And that is all you need to know about continuously

compounded returns and continuous growth e and the natural log.

Thank you.

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