$π$, φ, $\sqrt2$, $e$, $i$

You can count the number of mathematical constants that you are likely to have come across in secondary school on one hand - introducing a new one is pretty rare and special opportunity. For me, Euler's number is the most challenging of these constants to get to grips with. I wanted to make sure that when I introduced e for the first time to my A2 group that I would be able to give them a feel for why this number is so special and not just give them them skills to deal with exam questions in which it features. Here are some ideas on the topic that I would like to share;

A FUNCTION THAT IS ITS OWN DERIVATIVE

As a warm up get pupils doing a bit of differentiating, if they don't yet know about $e$ we're probably just talking polynomials. Tricky question:

Well they were struggling so I gave them a hint:

If students are going to understand the significance of e to growth they first need a solid understanding of exponential growth, lets recap...

I'd start by posing a couple of simple problems -

A FUNCTION THAT IS ITS OWN DERIVATIVE

*Is there a function that is its own derivative?*

*Could a polynomial function ever differentiate back to itself?*

Most likely you will get a reason based on the power law for why polynomials can't but you may wish to use this sketch to illustrate the point graphically:

An interesting attempt from one of my students based on this sketch was; where $x>0$, $\lim\limits_{ a \to \infty } x^a=f'(x^a)$ ... a nice reminder of how misleading graphs can be if you neglect the scale of the axes and just look at one part of the graph!

An interesting attempt from one of my students based on this sketch was; where $x>0$, $\lim\limits_{ a \to \infty } x^a=f'(x^a)$ ... a nice reminder of how misleading graphs can be if you neglect the scale of the axes and just look at one part of the graph!

Another valiant attempt having not studied derivatives of trigonometric functions was $y=Sinx$.

*What's the gradient of the tangent where $Sinx$ is at a*

**maximum? What is the value of $Sinx$ at this point?... Nice try**
$y=a^x$ .... Desmos

You may need to explain how to plot a derivative in Desmos - after defining f(x) you can type in d/dx followed by f(x) or go here:

then type $f(x)$ so you have $\frac{d}{dx}f(x)$.

Rather than Desmos you might want to use this GeoGebra sketch at the board;

This sketch enables you recap differentiation; to take a step back and examine what we are really looking for here.

Set the slider to 4 to plot $y=4^x$ and move the red point (not the slider) along the curve.

*Whats happening to the gradient as x increases?*
Hit show tangent and move the red point again.

*Now if we plot these values we'd have the function $f'(x)$.*Hit trace tangent and again move the red-point.

*Now if I decrease the base number (a), what will happen to the the function $y=x^a$?... What will happen to the gradient function?...*Un-check 'Trace Gradient' move the slider to 2, then recheck 'Trace Gradient' and move the red dot to explore.

*So you as you probably found if we set a to a certain value then f(x)=f'(x)?***2.7 .... are you sure?**... Un-check 'Show Tangent' and 'Trace Tangent' and click 'Show Gradient Function'. Now set a to 2.7.

**This looks pretty**

**good, lets just look a bit closer.**Zoom in (one easy way to do this is by holding shift and using the cursor keys; left/right for the x-axis and up/down for the y)

See if the students can get any closer -

**Type in the letter e into the input box for the parameter 'a' (2.72 will appear) then zoom in to validate.(Note you can just type in e for Euler's number in GeoGebra as long as there is no other object defined as e - if you ever get a "redefinition error" press 'alt e' instead).***now that this value of a where f(x)=f'(x) is so important that we give it it's own letter just like*$π$; this time it's not a Greek letter (the first known algorithm for calculating*$π$ came from the ancient Greek mathematician Archimedes) it's simply the letter e coined by the Swizz mathematician Euler.**No matter how closely we zoom in these lines will always overlay precisely - there will never be any difference.*
Ah but is e actually 2.72, someone may have once told you pi was 3.14, but they were kind of selling you a bit short! Click options; rounding; 15 Decimal Places.

*Ah that's a bit better but still not the true number; sadly even if I was to write a number on every atom of the universe I still wouldn't be anywhere close to writing out e in full; what type of number do you think it might be?... yes just like**$π$ its irrational and also transcendental.*To change the degree of rounding, on the menu bar click - options; rounding. |

OK so that's nice, so you've convinced your students that the function $y=e^x$ has a cool property but where else does $e$ crop up and what is it useful for? Let's explore just one area; growth.

**RECAPPING THE CONCEPT OF EXPONENTIAL GROWTH**

I'd start by posing a couple of simple problems -

*e.g. You stick £1000 in a savings account offering 5% compound interest per annum. How much will you have in the account after a year?... 3 years?... 20 years?***You can use the following sketch to recap these ideas.**

*We're talking about discrete growth here, your bank has chosen to pay you interest once a year. What would happen if your bank was to pay you the interest in two, six month installments so that they put half of the annual interest into you account after six months, would this change anything?***Let's take a closer look....**

**THE CONCEPT OF A CONTINUOUS GROWTH FUNCTION**

The idea for this sketch came from reading Khalid Azad's fantastic Intuitive Guide to Exponential Functions and e - I would highly recommend this if you need to refresh your knowledge before teaching this topic. The idea is ask...

*If you have £1 in an account and are promised a return of 100% after a year (some return I know!) then how much would you have after a year?...£2... obviously. On this graph the initial investment is represented in blue and the return in green.*

*Now what about if half the interest half is paid after six months and left in the account until the end of the year, how much will you have at the end of the year?*

Move the slider to 2.

Ramp it up.

The other thing is you can't go for more than 99 increments - the model will sadly break.

COMPARING MODELLING FUNCTIONS

I considered a couple of ways to build rate into the equation but in the end took the textbook route of looking at equivalence between the $y=ab^t$ and $y=ae^{kt}$. To do this they need to be introduced to the natural logarithm and then it can be done in a few steps:

**What does the red bar represent?...It's the interest earn't on the green interest.**

*How would we calculate this?*
At this point you may need to recap the the basic idea of compound interest;

$$Total = Investment × (100\%+ Interest Rate(\%))^n$$

and discuss that interest rate here as a decimal will be 1 (as we are dealing with 100% return here) divided by n (the number of time periods we are splitting the year into). Hence you have the formula

$y=\left(1+\frac{1}{n}\right)^n$

**What would happen if we split the year into three chunks?***Move the slider to 3.*

*Can you see that little cyan bar?...What does that represent?***Just to make it clear how the colouring works; green is interest earn't on the blue, red is interest earn't on the green and cyan is interest earn't on the red.**

*What do you think will happen if we keep increasing the number of time chunks? ... Infinite money?...***Move the slider a bit more.**

*Are you sure?*Ramp it up.

*What is that number that the total interest is heading towards?***It was quite tricky to work out a a way to create this sketch in GeoGebra; in the end I ended up relying on using the spreadsheet function to generate a matrix to control the colored lines (GeoGebra nerds please let me know if yo have a more eloquent solution!). The downside to this method is that the sketch can get quite hungry so I've only shown the first six 'interests' as distinct colors and the rest are shown but are amalgamated into black. This only matters if you try to zoom in which you can do but just be aware of this limitation of the plot if students ask.**

The other thing is you can't go for more than 99 increments - the model will sadly break.

**Anyhow at this point I would plot up the formula:**

$y=\left(1+\frac{1}{x}\right)^x$

to show $e = \lim\limits_{ n \to \infty } \left(1+\frac{1}{n}\right)^n$.

*So e is the maximum amount you can get by compounding 100% growth on one unit (e.g. a pound), over a time period time period i.e by continually compounding the growth.*

**So what if you compounded over two time periods?**... $e^2$ ...

*Three time periods.... Half a time period?*

**Hang on a minute we seem to have something here...$y=e^x$ will give us the amount of growth (+the original amount) if we continuously****compound.**
You can link this back to a basic compound growth formula;

$Total = Investment × (100\%+ Interest Rate(\%))^n$

if students don't see why it's $e^x$ rather than $xe$.

You can now head back to the previous sketch and plot $y=e^x$ to see that this function perfectly models continuously compounded growth (just type $y=e^x$ into the input bar).

Coming back to original question;

*What would happen if your bank was to pay you the interest in 6 months installments?*

**Well yes of course you get more interest and it would be even better if they paid out the amount accrued from continuously compounding interest. They certainly take all of this mathematics into consideration in their models****but they usually quote an Annual Equivalent Rate (AER) which tells you how much interest you will get calculated once per year even if they agree to deposit it more frequently.**

*So now we have got to the point where we have a function $y=e^t$ that can be used to model exponential growth but it's a bit limited in that so far it can only model continuously compounded growth with an initial amount of 1 and a growth rate of 100%. How could we make this more adaptable to model different scenarios?*I considered a couple of ways to build rate into the equation but in the end took the textbook route of looking at equivalence between the $y=ab^t$ and $y=ae^{kt}$. To do this they need to be introduced to the natural logarithm and then it can be done in a few steps:

*Let's modify our percentage multiplier ($b$) by taking its natural logarithm to give a new parameter ($k$).*

*$k=\ln{b}$*

*or in other words*

*$b=e^k$*

*Substituting this into $y=ab^t$ yields;*

*$y=ae^{k^{t}} = ae^{kt}$*

**Now we have something very similar to $y=e^{t}$ except it contains the extra parameters $a$ (our initial investment) and $k$ which is an alternative growth constant that can be calculated from $b$ using $k=\ln{b}$. $y=ab^t$ and $y=ae^{kt}$ are identical functions where $k=\ln{b}$ and can both be used to model exponential growth**- you can confirm they are the same by plotting them both and watching them overlay.**In fact you could use any base to model exponential growth but e is by far the most common and the one you need to be used to dealing with.**
In summary we have discovered

**A**$y=e^x$ is its own derivative

**B**The number e is reached as

$\lim\limits_{ n \to \infty } \left(1+\frac{1}{n}\right)^n$

**C**$e$ is the maximum possible result you get if you continuously compound 100% growth for one time period.

**D**$y=e^x$ is the function of continuously compounded growth

**E**$y=ae^{kt}$can be used to model any form of exponential growth.

and we've only scraped the surface!

I couldn't resist displaying Euler's Identity and the Gaussian Integral to round of this introduction.

$e^{i\pi} +1 = 0$

**$ \int^{\infty}_{-\infty} e^{-x^{2}}.dx = \sqrt{\pi} $**

Marvelous

One of the ways I like to approach recognizing function type is in a table. Not as quick as visual recognition from a graph, but it let's you have more detailed information - a peek under the hood.

ReplyDeleteLinear is easy, of course, constant difference in output if there's a constant difference in input. Quadratics students get because the differences form a linear pattern that they recognize. Eventually you can get to where if the nth difference is constant, it's an n-degree polynomial. When you do this with students who have calculus, they see connections to the derivative.

What happens when you apply this to other function families? Periodic functions have periodic differences. Exponential functions...

2,4,8,16,32, ...-> 2, 4, 8, 16, ... hey!

2, 1, .5, .25, .125... -> 1, .5, .25... so what's going to happen with their derivatives?

Anyhow, this was just a connection I was thinking about to your nice work here.

Thanks for the comment John, very interesting and thought provoking. I've never thought about looking at differences or 2nd differences in output in the context of calculus.

ReplyDeleteI guess what your saying here is that with exponential functions where y = a^x the first difference in output is given for integers x by;

a^(x+1) - a^x = (a-1).a^x

So a^x is proportional to the first difference in output or more precisely

a^x = (first difference)/(a-1) .... that's pretty cool and something I hadn't considered.

This makes y=2^x a special case where 1/(a-1) = 1 so 2^x = the first difference

Of course f'(x) = (a^x).lna and the coefficient ln(a) is comparable to the 1/(a-1) coefficient that we see appearing in the first difference of output table. Here in the world of calculus y=e^x is the special case.

I like these ideas but I'm not sure whether I would bring them into this lesson; I'm curious to know what others think!

Very clear, thank you! Awesome use of Geogebra for functions/ modelling!

ReplyDelete