Help, please: A math teacher (me!) needs help with the concept of e?

Help, please: A math teacher (me! ) needs help with the concept of e I have never understood e yet I have to teach it. I can parrot all the good stuff re: the value of $1 compounded continuously for 1 year at 100%, but I don't "get it."

I understand pi as the circumference of a circle whose diameter is 1; I can understand how and why to apply pi to other circumstances. But why does the definition of e that I parroted before make e anything other than a curiosity? What is the big deal about e and natural logs?

Can you help me understand why e matters? And convince me in a way that I can convince my students when it comes up in the next few weeks? This is really, really frustrating me.

Thanks, I hope-- Asked by Pam_I_Am 35 months ago Similar questions: Help math teacher help concept Science > Math.

Similar questions: Help math teacher help concept.

I want to give you those scans I mentioned in the DB A really good book at an appropriate level for your students would be pearsonhighered.com/educator/academic/pr... This book is worth adding to your library (see if you can geta desk copy or used). (I am NOT the author). The techniques used are modeling ones rather than manipulation and are great for ALL students not just non math majors.(Many majors come through with great symbolic skills but little understanding.

) If you go through compound interest, e shows up rather naturally as the limiting value of an account with initial value $1 earning 100% interest compounded as often as you like. I am going to include the scans here. If they aren’t clear enough, please PM me and I’ll mail them to you some other way.

This introduction of e as a number which "naturally" occurs when studying growth is a nice one and ties in with the concept that our named constants (like pi) are waiting to be discovered rather than "made up". The common and natural logs follow from the idea of "reversing" our mathematical steps without having to go through the formal process of defining an inverse function. The modeling approach makes everything very "practical".

Why study exponential functions? Because we need to know how things grow (especially money!) or decay (radioactive waste). And here is a fun one: "The English language evolves naturally in such a way that 77% of all words disappear (or are replaced) every 1000 years.

Of a basic list of words used by Chaucer in 1400 A.D., what percentage should we expect to find still in use today?" (In 2000, the answer was 41.4%) Why look at logarithms? They help us solve those exponential problems and serve as models in their own right. The Richter scale is logarithmic which is why we care whether a quake was a 6 or a 7.

The difference sounds small there but is really a factor of 10 in size. I always had so much fun doing this stuff that you *almost* made me miss it. Sources: years of teaching this stuff unixcorn's Recommendations Elementary Mathematical Modeling (2nd Edition) Amazon List Price: $113.33 Used from: $69.50 This is the current edition.

I used the first and I believe they did a better job the second time when they really rewrote it with less nod to traditional algebra.

E matters because someone said it did. When I took Calc 2 in undergrad (I was a chemistry major), I never understood the concept, but I learned it as you did. I guess it's important in exponential growth factors which are not linear, such as cell URL1 matter how many questions I asked in math class, I never got a good answer.

A physics prof told me it was a button on my calculator (that was the most useful answer). So join the club. Your teachers and professors didn't understand it.

Your classmates didn't understand it. Your students won't understand it..

First words for yourself. Let me first tell you why e is important and where it comes from. The first definition that is stated in most books is the definition by limits.

Mostly students usually are told that e = lim (1+1/n)^n as n goes to infinity. But this definition is not really a good one for starters, IMO. The reason is that students will immediately ask: "where the heck did you get (1+1/n)^n from?!

" And it is a very legitimate question and not so easy to answer. I would prefer a different approach. Let me first try to convince you that it is an important number to look at.

1) Areas. One of the most important things for us is evaluating areas of regions in 2-D. We already know how to evaluate the area of a circle, a rectangle, a trapezoid and a triangle.

Now, if we look at a general region in 2-D how can we evaluate the area? This question is very complicated as stated. Lets try to look at a simpler region.

Lets try to look at the graph of a function and see if we can evaluate the area of the region under the graph of this function. Again this is not that easy and generally depends on the function. Lets look at simpler examples.

For example if we look at y = 3x which is a line the region under the graph of this function and above the x-axis and between lines x=0 to x=5, it is going to be a triangle and we already know how to evaluate its area. If you look at the region above the x-axis, below the line y = 4x and between vertical lines x=2 and x=5 this is going to be a trapezoid, and we know how to evaluate its area. Now lets look at a more complicated function.

For example lets look at y = 1/x. Fix a number a and try to find the area of the region above the x-axis below the graph of y=1/x and between vertical lines x=1 and x=a. This area turns out to be Ln(a) as shown in the following picture.

http://upload.wikimedia.org/wikipedia/commons/5/55/Log-pole-x.svg You can elaborate on that even more if they know how to differentiate, but even if they don’t this is a better approach than starting with an "unknown" sequence and telling them the limit of this sequence is e. You can make the connection between pi and e here. Tell them how pi shows up in calculating the area of a circle.

E shows up when we want to calculate the area of this region. 2) Anti-derivatives. You may or may not share this with students depending on whether they know what an anti-dvertent is or not.

But it certainly gives you a better understanding of natural logarithm. The idea is that in math we sometimes want to find an anti-derivative of a function.An anti-derivative of a function f(x) is a function g(x) such that when you take derivative of g(x) you get f(x). It is basically the reverse of what we do for differentiation.

For example the derivative of f(x)=x^2 is 2x.So, an anti-derivative of 2x would be x^2. The simplest functions are powers of x. That would be cool to see what an anti-derivative of x^n is for any n.

What is an anti-derivative of x^2. It is x^3 / 3. Because when you differentiate x^3/3 you would get 3x^2/3 and the 3’s would cancel and you would get x^2.

How about x^6? Its anti-derivative is x^7/7. So, basically if you want to find an anti-derivative of x^n you add 1 to the exponent and divide it by the new exponent.An anti-derivative of x^n is x^(n+1)/(n+1).

As you see we have (n+1) in the denominator, and we cannot divide by zero. So this only works when n is not -1.So we can find an anti-derivative of x^n for any n which is not -1. A natural question that one would immediately aks would be: What is an anti-derivative of x^(-1) = 1/x?

And the answer would be natural logarithm function: Ln(x).3) Differential equations. Don’t get scared by the above term. It is nothing complicated.

:) A natural question that may pass somebody’s mind is when can the derivative of a function be the same as the function itself? I.e. For what function f(x) do we have f(x) = f ’ (x)?

And the answer is exponential function e^x. This one is also a nice place that e shows up. 4) calculating e.

There are several ways of calculating e. One way that you can ask students to do with a calculator in the class is to use the limit definition of lim (1+1/n)^n. Tell them to do this with a calculator and see they get a number close to 2.7 but they can’t actually find the exact value.

There are also some other ways. For example the following sum gives us e. 1 + 1/1!

+ 1/2! + 1/3! + 1/4!

+ ... But I would prefer the first methos for student. ~~~~~~~~~~~~~~~~~~~~~ If I were to teach this to students I would start with #1 and the definition of Ln(x) in terms of areas. This may convice them and it actually is a helpful number.

Then I would jump to #4 and ask them to do the calculation. Each time I would remind them that those theorems are a bit deeper than I can explain why they are true, but they are. Good luck!

:) .

The best explanation that comes to my mind now is by anti-derivatives.

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