27

If your students have learned some statistics, then you could point out that the normal distribution's probability density function uses this double exponential. $$f(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2\right)$$


21

From the perspective of mathematics education, I must ask: Why do you want to encourage students to solve such a problem without logarithms? There is a connection between square roots and logarithms (e.g., here) but I consider the mathematics that underlies it quite opaque without understanding $x\mapsto \log(x)$ beforehand. Personally, my inclination is ...


13

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 that on the board. Say that the lower graph will be the derivative of the upper graph. Start at a point on the far left on the $x$-axis. Ask the class what the ...


12

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 to a total of $2^{o\cdot 2^i}$ binary functions. And course if you're working with $n$ different values per input/output instead of binary you end up with $n^{o\...


11

(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 this finer description. b. Maybe it fits under your (2) but radioactive decay is a huge one you did not mention. Applications include carbon (and other) ...


10

First, I'll answer the question posed by Benjamin Dickman: Solving problems with limitations is good practice for working with algebraic structures that do not have analogous functions. For example, solving $5^x \equiv 326 \mod 331$ is a situation where the log button on your calculator isn't going to help at all. (Neither is the bisection method.) So you ...


9

One could solve this by "guess and check", together with the knowledge that $5^x$ is increasing. So start with your observation that $3 < x < 4$. Test $3.1, 3.2, 3.3, ...$ and observe that $3.5<x<3.6$. Repeat the process for the next two decimal places. I am not sure if this is what they intended. Another method which could be used if they ...


9

might i suggest a different approach that I have found very helpful when teaching about exponentiation rather than real world examples? I have found that until students understand why a rule works (i.e. the derivation or something similar), they won't be able to understand how to use it. Instead of teaching the rules to the students, have them expand all ...


8

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 different arrangements in which $n$ classmates can be friends. This also has the benefit of being easy to enumerate for small numbers.


7

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 categories mean or are used for. We introduce definitions and categorizations in order to help us understand abstract ideas. A definition without the underlying ...


6

Here's something I used to do in college algebra and precalculus classes, from the mid 1980s to the mid 2000s, which has the additional advantage of being an example in which estimation is used. I think you can adapt this to your case. (Use metric system units if appropriate.) Of course, you'll want to go a lot slower than I do below, which is written for ...


6

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 this nice drawing from wyzant.com: It becomes plausible that there is some curve between $2^x$ and $5^x$ such that the slope at each point is exactly the $y$-...


5

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 function. Or, in the language of differential equations, it is the unique solution of $\frac{dy}{dx}=y$ for which $y(0)=1$. As usual, the question really is... how ...


5

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 phrasings if you don't want to involve differential equation explicitly) and then there is nothing to explain. One more natural definition is that $\exp(x)=e^...


5

This method does not give 3 digits precision, so is not really an answer to the original question -- but if all one is looking for is a quick-and-dirty approximation, here is a strategy that is very elementary. First of all, let's change the problem to one that will be easier to solve: $5^x=325$. Since we are only looking for a reasonable approximation to ...


5

You could use combinatorics: How many words with 4 letters of 26 letter alphabet do exist? If you have the answer: How many passphrases of two word with 4 letters of the same alphabet do exist? How many passphrases of a 4 letter word and a 6 letter word do exist? The real-world problem arises from the security of passwords (for Facebook for example). ...


4

This problem is not hard with Briggs method. It is similar to the method described by Benjamin Dickman but converges with slightly less square roots. Take repeated square roots until the number is close enough to 1 to enable linear interpolation of the exponential. Do not take the same number of square roots for the other number. Instead take as many square ...


4

This is my method of solving. It uses the square function, and division. One then must accumulate the powers of 1/2, i.e 1/2,1/4,1/8, etc that are noted. Method - A) Divide by the base, in this case 5, and determine the exponent has a 3 prior to the decimal. B) Take the result and square it. C) If you can divide the result by 5, a binary 1 is noted. ...


4

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 inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. Exponential functions of ...


4

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/AtmosphericPressure.pdf and http://nova.stanford.edu/projects/mod-x/ad-expatm.html.


4

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 $x$ copies of $2$ because I think it is more intuitive to describe what exponentiation actually does that to imply that it is as natural as addition and multiplication.


4

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 same set of students merged their interests into a single club charter). In that same scenario, the yearbook that printed pictures of all of the clubs would be ...


4

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 as microbes, growth of tumours, and survival of cancer patients. The model is described by the differential equation $$\dfrac{dN}{dt} = rN \ln \left(\dfrac{K}{N}...


3

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. (The key assumptions are that the weight of the cables is small compared to the weight of the bridge deck, and the weight of the bridge deck is constant along the ...


3

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 functions the derivative graph is a little higher than the function, but with $2^x$ the derivative is lower, so there must be a number between 2 and 3 such that ...


3

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 line) of these functions on $[0, h]$ for small values of $h.$ conclude that there is a number $e$ with the property that the slope of the graph $y = e^x$ is one....


3

Take the graph of an exponential function such as $2^x$. If, for example, you shift it to the right by 3 units and then stretch it vertically by a factor of 8, you get the same graph back again. But the vertical stretch also increases the slope of the tangent line by a factor of 8. This holds for any shift, and therefore the derivative of this function is ...


3

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 plausible that they're in fact equal). Compare the slope of secant line through $(x - 1, e^{x-1})$ and $(x, e^x)$ with the slope of the secant line through $(...


3

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 for $P(n)=b^n$ where $b>1$ we have $\frac{\Delta P}{\Delta n}=cP(n)$ where $c=b-1$. This discrete case for exponential growth makes it reasonable to expect ...


3

I say the key descriptor of a exponential function is constant multiplicative rate of change, much as the descriptor of a linear function is constant additive rate of change. The function $f(x)=a(1.5)^x$ increases by 50% when $x$ increases by 1: $$\frac{f(x+1)}{f(x)} = \frac{a(1.5)^{x+1}}{a(1.5)^x} = 1.5$$ But adding a non-zero constant changes that: $$\frac{...


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