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27 votes
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Natural occurrences of a to the (b to the c)?

If your students have learned some statistics, then you could point out that the normal distribution's probability density function uses this double exponential. $$f(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp\...
JRN's user avatar
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14 votes
Accepted

Geometric intuition for $D(e^x) = e^x$

Here is a method I use in all my calculus classes to show that it may be true that the derivative of $e^x$ is itself. Sketch a graph of $y=e^x$ high up on the board and sketch coordinate axes below ...
Rory Daulton's user avatar
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12 votes

Interesting settings for exponential growth or decay

(edited) a. For all of these, I would think to expand the category into subcategories. So not just "bank account" but time value of money and NPV, bond details, etc. You learn something when doing ...
guest's user avatar
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12 votes

Natural occurrences of a to the (b to the c)?

Count all possible functions mapping $i$ input bits to $o$ output bits: For each of the $2^i$ input combinations, each output has two possible outputs ($0$ and $1$), i.e. you have $2^{2^i}$, leading ...
Tobias Kienzler's user avatar
9 votes
Accepted

Why is a translated exponential function considered an exponential function?

To start with an opinion, I think that this classification exercise is kind of silly. The student is being asked to put functions into some categories without having a clear idea about what those ...
Xander Henderson's user avatar
  • 8,225
8 votes

Natural occurrences of a to the (b to the c)?

The number of undirected graphs of order $n$ is $2^{n \choose 2} = 2^{(n^2-n)/2}$ (e.g. consider the adjacency vector as a binary-encoded number). You can describe this in terms of the number of ...
Yonatan N's user avatar
  • 181
6 votes

Geometric intuition for $D(e^x) = e^x$

It may help to compare several exponential curves, e.g., $y=5^x$, $y=e^x$, and $y=2^x$,             and calculate slopes at a few points, as in ...
Joseph O'Rourke's user avatar
6 votes

Exponential & logarithm in a high school calculus class

I normally go other way around. Start with a logarithm, which by definition maps multiplication to addition: $\ln{ab} = \ln{a} + \ln{b}$. A detour into a history could be also useful: multiplication ...
user58697's user avatar
  • 171
5 votes

Geometric intuition for $D(e^x) = e^x$

As mentioned in other answers, the starting point is the definition of the exponential you are using. One possible definition is as the solution of $f'=f$ taking value $1$ at $0$ (or other equivalent ...
Benoît Kloeckner's user avatar
5 votes

Geometric intuition for $D(e^x) = e^x$

But, a geometric definition for the exponential function is the following: The function $f$ for which $f(0)=1$ and is such that its value is equal to its slope at each point is the exponential ...
James S. Cook's user avatar
4 votes

Interesting settings for exponential growth or decay

One basic model of atmospheric pressure has it decaying exponentially as a function of height above sea level. Two places to look for this are https://people.clas.ufl.edu/kees/files/...
KCd's user avatar
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4 votes

What do the Common Core Standards expect secondary students to learn about logarithms or the number $e$?

The standards that you identify actually do cover the things you assume are not covered. The formal properties of logarithms, for example, are proved using exponents, thus: F-BF.5: Understand the ...
James S.'s user avatar
  • 1,023
4 votes

Geometric intuition for $D(e^x) = e^x$

here is a way i teach about the natural exponential function. we look at the slope of the graphs $y = 2^x$ and $y = 3^x$ at $x = 0, y = 1$ by evaluating the average rate of change(slope of the secant ...
abel's user avatar
  • 251
4 votes

Clearest verb phrases for operations

If the domain was the real numbers, I would say that it raises two to the power of its input. If the domain was positive integers, I would break your model and say that it multiplied together $x$ ...
Matthew Daly's user avatar
  • 5,619
4 votes

Why is a translated exponential function considered an exponential function?

The working definition I have in my head doesn't fit the more rigorous definitions others have put in their answers. I think of exponential growth and decay as being constant percentage growth or ...
Sue VanHattum's user avatar
  • 20.8k
4 votes

Natural occurrences of a to the (b to the c)?

If $S$ represents the set of $n$ students of a school, then $\mathscr P(S)$ is the set of rosters for all possible clubs that that school could host (if we assume that any two clubs with exactly the ...
Matthew Daly's user avatar
  • 5,619
4 votes

Natural occurrences of a to the (b to the c)?

Gompertz model was created in 1825 to study human mortality curves. From the 1920's, it was used in economic fields, and from there it was also used in Biology to study cells and microorganisms, such ...
FormerMath's user avatar
3 votes

Examples for "good" exponential growth versus linear growth

One example for "good" exponential growth is cryptographics. Linear key length growth vs. exponential growth for the effort to break the key. In this example we are kept safe by the exponential growth....
kutschkem's user avatar
  • 131
3 votes

Why is a translated exponential function considered an exponential function?

With exponential functions, the important thing to note is that the RATE of change is proportional. In a calculus class we might approach this in terms of taking the derivative and noticing that $f'(...
John Thompson's user avatar
3 votes

Interesting settings for exponential growth or decay

Catenary curves are the sum of two exponentials -- One concave upward but decreasing, and the other concave upward and increasing. They are a good model for the curve of suspension bridge cables. (...
Jasper's user avatar
  • 3,178
3 votes

Geometric intuition for $D(e^x) = e^x$

I would say to first sketch the curve $4^x$ and it's derivative $4^x \ln 4$, and the same for $3^x$ and $2^x$, then it will be easy to see that the case $e^x$ is a particular case. For the first two ...
blmayer's user avatar
  • 181
3 votes

Geometric intuition for $D(e^x) = e^x$

Here's an approach that can be illustrated with a simple picture, and also gives an actual bound on the derivative of $e^x$ as being, at the very least, fairly close to $e^x$ (which makes it much more ...
Daniel Hast's user avatar
  • 4,893
3 votes

Geometric intuition for $D(e^x) = e^x$

Bunny rabbit generations: look at the discrete case $P(n)=2^n$ with $P(0)=1$. It is easy to see that $\frac{\Delta P}{\Delta n}=P(n)$ where $\Delta P=P(n+1)-P(n)$ and $\Delta n=1$. Or more generally ...
user52817's user avatar
  • 11k
3 votes

Natural occurrences of a to the (b to the c)?

It is easy to square a number. So instead of computing $\exp(z)$ you can compute $\exp(z/2^k)^{2^k}$ for some suitably large positive integer $k$. This is a very simple way of accelerating the ...
user21820's user avatar
  • 2,649
3 votes

Natural occurrences of a to the (b to the c)?

What a nice collection of answers! I am inclined to use $\lfloor A^{3^n} \rfloor$ where $A$ is Mill's constant, $$A \approx 1.3063778838630806904686144926 \;, $$ just because it is astounding that ...
Joseph O'Rourke's user avatar
3 votes

Natural occurrences of a to the (b to the c)?

The maximum of a large number of independent, identically distributed random variables -- e.g., the highest flood observed over a long period -- has an extreme value distribution. One common case is ...
nanoman's user avatar
  • 271
3 votes

Exponential & logarithm in a high school calculus class

Here is the approach which seems most natural to me, and it wasn't a disaster the two times I used it in a classroom. The basic idea is to hold off on $e$ until it is motivated. Start with numerical ...
David E Speyer's user avatar
2 votes

Geometric intuition for $D(e^x) = e^x$

This is not so geometric, but if you have students who can actually do some difference quotients (numerically only, I mean) then just having them do a table where they estimate the derivative (say, $(...
kcrisman's user avatar
  • 5,976
2 votes

How can a layperson learn to intuit exponential growth?

In the 1970s in the US, there was a commercial for Faberge Organics shampoo that illustrated the power of doubling. The story was that a woman liked her new shampoo so much that she told two friends ...
guest's user avatar
  • 174

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