# Tag Info

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 ...

12

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 ...

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 ...

8

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 ...

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. 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 ...

2

There is only one way (that I know of) to solve this without knowledge of Calculus (and a calculator--that is without tables--although my solution will rely on the computational power of a calculator). First, I do not see the question as appropriate for the level of algebra or trigonometry (I certainly don't see the link to trigonometry). Frankly, I don't ...

2

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, $(f(3.001)-f(3))/(3.001-3)$) for various values, and then compare with the actual value of $e^x$, they might get surprised. No, this is not an original idea at ...

2

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 about it, and those two friends each told two friends, and so on... With each additional "and they told two friends", there was a doubling of the number of ...

2

The North Carolina School of Science and Mathematics has always done a great job of sharing projects. If you go here you'll find links to various courses and the modules that they use in them. I think the Algebra 2 and AFM courses are probably most relevant to you.

2

Some examples of exponential growth that don't seem to have been mentioned yet: Moore's law the expansion of the universe (when dominated by dark energy, as it essentially is right now) time required for monkeys banging on typewriters to produce $n$ words of Shakespeare, or for a brute-force search to crack a password of length $n$ motion of a pencil ...

2

Geometrically the relation $f^{\prime} = f$ means that the slope at $(x, f(x))$ of the graph of $f$ is equal to $f(x)$. We look for a function $f$ that has this property. Consider $g(x) = 1 + x$. Its graph is $(x, 1 + x)$ and the slope is $1$ at every point. Hence for $x$ very small this function almost has the desired property; it does have the desired ...

1

Here's the sequence I often use. The Smallpox Case Imagine terrorists bioengineer a new version of smallpox that infects 3 people. Each day, each infected person infects one more person. Make a table of values relating the number of days that have passed to the number of people infected for two weeks. Begin with the day 0 corresponding to 3 infections. ...

1

Geometricaly, ln(x) is the area from 1 to x under the graf of y=1/x. Geometricaly, derivation of the area under f(x) is f(x). Hence, d(ln(x))/dx = 1/x. exp(x) is the inverse of ln(x) i.e. if y=ln(x) then x=exp(y). Hence, if dy/dx=1/x then dx/dy=x i.e. the derivative of exp is exp.

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