# Different definitions of e: Index

I’m doing something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my series on the different definitions of $e$ that appear in Precalculus and Calculus.

Part 1: Justification for the formula for discrete compound interest

Part 2: Pedagogical thoughts on justifying the discrete compound interest formula for students.

Part 3: Application of the discrete compound interest formula as compounding becomes more frequent.

Part 4: Informal definition of $e$ based on a limit of the compound interest formula.

Part 5: Justification for the formula for continuous compound interest.

Part 6: A second derivation of the formula for continuous compound interest by solving a differential equation.

Part 7: A formal justification of the formula from Part 4 using the definition of a derivative.

Part 8: A formal justification of the formula from Part 4 using L’Hopital’s Rule.

Part 9: A formal justification of the continuous compound interest formula as a limit of the discrete compound interest formula.

Part 10: A second formal justification of the continuous compound interest formula as a limit of the discrete compound interest formula.

Part 11: Numerical computation of $e$ using Riemann sums and the Trapezoid Rule to approximate areas under $y = 1/x$.

Part 12: Numerical computation of $e$ using $\displaystyle \left(1 + \frac{1}{n} \right)^{1/n}$ and also Taylor series.

# Different definitions of logarithm: Index

I’m doing something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my series on how the different definitions of logarithm are in fact equivalent.

Part 1: Introduction to the two definitions: an antiderivative and an inverse function.

Part 2: The main theorem: four statements only satisfied by the logarithmic function.

Part 3: Case 1 of the proof: positive integers.

Part 4: Case 2 of the proof: positive rational numbers.

Part 5: Case 3 of the proof: negative rational numbers.

Part 6: Case 4 of the proof: irrational numbers.

Part 7: Showing that the function $f(x) = \displaystyle \int \frac_1^x \frac{dt}{t}$ satisfies the four statements.

Part 8: Computation of standard integrals and derivatives involving logarithmic and exponential functions.

# Improvisation in the Mathematics Classroom

Every so often, I’ll publicize through this blog an interesting article that I’ve found in the mathematics or mathematics education literature that can be freely distributed to the general public. Today, I’d like to highlight “Improvisation in the Mathematics Classroom” by Andrea Young. Here’s the abstract:

This article discusses ways in which improvisational comedy games and exercises can be used in college mathematics classrooms to obtain a democratic and supportive environment for students. Using improv can help students learn to think creatively, take risks, support classmates, and solve problems. Both theoretical and practical applications are presented.

Full reference: Andrea Young (2013) Improvisation in the Mathematics Classroom, PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 23:5, 467-476, DOI: 10.1080/10511970.2012.754809

# What to Expect When You’re Expecting to Win the Lottery

I can’t think of a better way to tease this video than its YouTube description:

Recounting one of the stories included in his book How Not to Be Wrong: The Power of Mathematical Thinking, Jordan Ellenberg (University of Wisconsin-Madison) tells how a group of MIT students exploited a loophole in the Massachusetts State Lottery to win game after game, eventually pocketing more than $3 million. A personal note: though I haven’t talked with him in years, Dr. Ellenberg and I were actually in the same calculus class about 30 years ago. # The New York Knicks and perfect squares Funny but true: as the disaster of the 2014-15 season for the New York Knicks progressed, they had a record of: • 1-1 after 2 games • 2-4 after 6 games • 3-9 after 12 games • 4-16 after 20 games • 5-25 after 30 games, and • 6-36 after 42 games. http://www.sbnation.com/nba/2014/12/21/7431765/the-knicks-love-math-as-much-as-they-love-losing # Engaging students: Graphing the sine and cosine functions In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place. I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course). This student submission comes from my former student Jessica Trevizo. Her topic, from Precalculus: graphing the sine and cosine functions. How could you as a teacher create an activity or project that involves your topic? For this activity students can either work with a partner or work individually. I enjoyed this activity because students are able to derive the sine and cosine functions on their own using fun materials other than the original paper and pencil. The knowledge that students should gain from this activity is the relationship between the unit circle and the sine/cosine function. Along with this activity, students will be practicing previous concepts learned such as converting degrees to radians, finding the domain/rage, and finding the x-intercepts and y-intercepts. Also, amplitude, period, and wavelength are important vocabulary words that can be introduced and applied to the parent functions. To complete the activity assign the students to write a paragraph comparing and contrasting both functions. In their paragraph make sure students include a discussion of the intercepts, maxima, minimum, and period. It is essential for the students to know how to graph the parent functions of sine and cosine and where they come from before teaching the students about the transformations of the functions. http://illuminations.nctm.org/Lesson.aspx?id=2870 A.1 What interesting word problems using this topic can your students do now? Real life word problems that involve the sine and cosine function can be used to keep the students engaged in the topic. Both of the functions can used to model situations that occur in real life in a daily basis such as; recording the path of the electric currents, musical tones, radio waves, tides, and weather patterns. Here is an example of a word problem, “Throughout the day, the depth of the water at the end of a dock in Bar Harbor, Maine varies with tides. The table shows the depths (in feet) at various times during the morning.” With the data provided the students are able to do several things such as: be able to use a trigonometric function to model the data and find the depth of the water at any specific time. Also, if a boat needs at least 10 feet of water to moor at the dock, the students should be able to figure out safe dock times for the boat. How can technology be used to effectively engage students with this topic? Most of the students are familiar with sound waves. As an engage go to www.onlinemictest.com and have the students observe the sound waves that appear on the screen as you speak. Many students will recognize the various sine and cosine functions on the screen. With the online mic test students are also able to make relationships between the sound and the wave. Download several different tones and play them so the sound waves of the tones appear on the screen. Have the students sketch the graph of a soft high note, soft low note, loud high note, and a loud low note. The following graphs should look similar to the figure below. Once all of the students have recorded their own observations have the students work with a partner to compare their graphs. Also give the students a minute or two so they can compare and contrast the 4 different graphs by using the new vocabulary that they learned such as amplitude and period. Students are able to remember the new vocabulary when they have opportunities to have discussions that require them to use them. # Another poorly written word problem Textbooks have included the occasional awful problem ever since Pebbles Flintstone and Bamm-Bamm Rubble chiseled their homework on slate tablets while attending Bedrock Elementary. But even with the understanding that there have been children have been doing awful homework problems since the dawn of time (and long before the advent of the Common Core), this one is a doozy. There’s no sense having a debate about standards for elementary mathematics if textbook publishers can’t construct sentences that can be understood by elementary students (or their parents). # Engaging students: Compound interest In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place. I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course). This student submission comes from my former student Daniel Littleton. His topic, from Precalculus: compound interest. How has this topic appeared in the news? In a publication entitled Business Insider, Sam Ro published an article entitled “Every 25 Year Old In America Should See This Chart” on March 21, 2014. In this article Ro stated that in past times companies would offer pension plans to long term employees in order to support them in retirement. He goes on to state that in modern times employees need to contribute to retirement plans such as a 401K or an IRA in order to save for retirement. These plans function by the mathematical principle of compound interest. While the mechanics of compound interest are not presented in the article an illustration is shown how individuals who save their money through this formula accumulate a greater amount of money over time. He even presents a situation in which one individual can save money for a less amount of time than another and still accrue a greater total amount of savings because of compound interest. This illustration, presented below, can be a useful tool in engaging students in the possibilities that compound interest could have in their own futures. This information was collected from the following web page on Friday, April 04, 2014; http://www.businessinsider.com/compound-interest-retirement-funds-2014-3. How can this topic be used in your students’ future courses in mathematics or science? Compound interest is introduced at the Pre-Calculus level of secondary education. At the Post-Secondary Education level compound interest is a concept that is included in several areas of study. For example, students that wish to study business will need to have a mastery of compound interest. Additionally, those studying finance or economics will constantly use the principle of compound interest in their computations. Not only does this formula come into play in the mathematics of monetary systems, but also in the workings of political science as well. Those that wish to pursue political aspirations will need a firm understanding of economics and the means by which funds can be grown over time. As is evident, compound interest is a mathematical formula, but like many realms of mathematics it affects multiple realms of interest and practice in a real world environment. What interesting word problems using this topic can your students do now? There are an innumerable amount of problems that can be presented to students involving compound interest. One could deal with the monetary worth of valuable or precious items. For instance, “A necklace is appraised at$7200. If the value of the necklace has increased at an annual rate of 7.2%, how much was it worth 15 years ago?” This question is asking the student to solve for the original principle of the necklace, rather than the accrued value which is given. Another problem could be “A sum of $7000 is invested at an interest rate of 7% per year. Find the time required for the money to double if the interest is compounded quarterly.” This problem requires the student to determine the amount of time necessary for the investment to yield the desired amount. These are only two problems that I have presented that will allow the students to practice the concept of compound interest. There are undoubtedly multiple others that could be written with the same effect. # Education is not Moneyball I initially embraced value-added methods of teacher evaluation, figuring that they could revolutionize education in the same way that sabermetricians revolutionized professional baseball. Over time, however, I realized that this analogy was somewhat flawed. There are lots of ways to analyze data, and the owners of baseball teams have a real motivation — they want to win ball games and sell tickets — to use data appropriately to ensure their best chance of success. I’m not so sure that the “owners” of public education — the politicians and ultimately the voters — share this motivation. An excellent editorial the contrasting use of statistics in baseball and in education appeared in Education Week: http://www.edweek.org/tm/articles/2014/08/27/fp_eger_valueadded.html?cmp=ENL-TU-NEWS1 I appreciate the tack that this editorial takes: the author is not philosophically opposed to sabermetric-like analysis of education but argues forcefully that, pragmatically, we’re not there yet. Both the Gates Foundation and the Education Department have been advocates of using value-added models to gauge teacher performance, but my sense is that they are increasingly nervous about accuracy and fairness of the new methodology, especially as schools transition to the Common Core State Standards. There are definitely grounds for apprehensiveness. Oddly enough, many of the reasons that the similarly structured WAR [Wins Above Replacement] works in baseball point to reasons why teachers should be skeptical of value-added models. WAR works because baseball is standardized. All major league baseball players play on the same field, against the same competition with the same rules, and with a sizable sample (162 games). Meanwhile, public schools aren’t playing a codified game. They’re playing Calvinball—the only permanent rule seems to be that you can’t play it the same way twice. Within the same school some teachers have SmartBoards while others use blackboards; some have spacious classrooms, while others are in overcrowded closets; some buy their own supplies while others are given all they need. The differences across schools and districts are even larger. The American Statistical Association released a brief report on value-added assessment that was devastating to its advocates. ASA set out some caveats on the usage on value-added measurement (VAM) which should give education reformers pause. Some quotes: VAMs are complicated statistical models, and they require high levels of statistical expertise. Sound statistical practices need to be used when developing and interpreting them, especially when they are part of a high-stakes accountability system. These practices include evaluating model assumptions, checking how well the model fits the data, investigating sensitivity of estimates to aspects of the model, reporting measures of estimated precision such as confidence intervals or standard errors, and assessing the usefulness of the models for answering the desired questions about teacher effectiveness and how to improve the educational system. VAMs typically measure correlation, not causation: Effects – positive or negative – attributed to a teacher may actually be caused by other factors that are not captured in the model. Under some conditions, VAM scores and rankings can change substantially when a different model or test is used, and a thorough analysis should be undertaken to evaluate the sensitivity of estimates to different models. VAMs should be viewed within the context of quality improvement, which distinguishes aspects of quality that can be attributed to the system from those that can be attributed to individual teachers, teacher preparation programs, or schools. Most VAM studies find that teachers account for about 1% to 14% of the variability in test scores, and that the majority of opportunities for quality improvement are found in the system-level conditions. Ranking teachers by their VAM scores can have unintended consequences that reduce quality. # The number of digits in n! (Part 4) When I was in school, I stared at this graph for weeks, if not months, trying to figure out an equation for the number of digits in $n!$. And I never could figure it out. However, even though I was not able to figure this out for myself, there is a very good approximation using Stirling’s approximation. The next integer after $\log_{10} n!$ gives the number of digits in $n!$, and $\log_{10} n! \approx \displaystyle \frac{\left(n + \displaystyle \frac{1}{2} \right) \ln n - n + \displaystyle \frac{1}{2} \ln (2\pi)}{\ln 10}$ The graph below shows just how accurate this approximation really is. The solid curve is the approximation; the dots are the values of $\log_{10} n!$. Not bad at all… the error in the curve is smaller than the size of the dots. The following output from a calculator shows just how close the approximation to $log_{10} 69!$ is to the real answer. There are also additional terms to Stirling’s series that would get even closer answers. As I mentioned earlier in this series, I’m still mildly annoyed with my adolescent self that I wasn’t able to figure this out for myself… especially given the months that I spent staring at this problem trying to figure out the answer. First, I’m annoyed that I didn’t think to investigate $\log_{10} n!$. I had ample experience using log tables (after all, this was the 1980s, before scientific calculators were in the mainstream) and I should have known this. Second, I’m annoyed that I didn’t have at the tips of my fingers the change of base formula $\log_{10} n! = \displaystyle \frac{\ln n!}{\ln 10}$ Third, I’m annoyed that, even though I knew calculus pretty well, I wasn’t able to get at least the first couple of terms of Stirling’s series on my own even though the derivation was entirely in my grasp. To begin, $\ln n! = \ln (1 \cdot 2 \cdot 3 \dots \cdot n) = \ln 1 + \ln 2 + \ln 3 + \dots + \ln n$ For example, if $n = 10$, then $\ln 10!$ would be the areas of the 9 rectangles shown below (since$\ln 1 = 0\$):

The areas of these nine rectangles is closely approximated by the area under the curve $y = \ln x$ between $x =1\frac{1}{2}$ and $x = 10\frac{1}{2}$. (Indeed, I chose a Riemann sum with midpoints so that the approximation between the Riemann sum and the integral would be very close.)

In general, for $n!$ instead of $10!$, we have

$\ln n! \approx \displaystyle \int_{3/2}^{n+1/2} \ln x \, dx$

This is a standard integral that can be obtained via integration by parts:

$\ln n! \approx \bigg[ \displaystyle x \ln x - x \bigg]_{3/2}^{n+1/2}$

$\ln n! \approx \left[ \left(n + \displaystyle \frac{1}{2} \right) \ln \left(n + \displaystyle \frac{1}{2} \right) - \left(n + \displaystyle \frac{1}{2} \right) \right] - \left[ \displaystyle \frac{3}{2} \ln \frac{3}{2} - \frac{3}{2} \right]$

$\ln n! \approx \left(n + \displaystyle \frac{1}{2} \right) \ln \left(n + \displaystyle \frac{1}{2} \right) - n - \displaystyle \frac{3}{2} \ln \frac{3}{2} + 1$

We can see that this is already taking the form of Stirling’s approximation, given above. Indeed, this is surprisingly close. Let’s use the Taylor approximation $\ln(1+x) \approx x$ for $x \approx 0$:

$\ln n! \approx \left(n + \displaystyle \frac{1}{2} \right) \ln \left(n \left[1 + \displaystyle \frac{1}{2n}\right] \right) - n - \displaystyle \frac{3}{2} \ln \frac{3}{2} + 1$

$\ln n! \approx \left(n + \displaystyle \frac{1}{2} \right) \left[\ln n + \ln \left(1 + \displaystyle \frac{1}{2n} \right) \right] - n - \displaystyle \frac{3}{2} \ln \frac{3}{2} + 1$

$\ln n! \approx \left(n + \displaystyle \frac{1}{2} \right) \left[\ln n + \displaystyle \frac{1}{2n} \right] - n - \displaystyle \frac{3}{2} \ln \frac{3}{2} + 1$

$\ln n! \approx \left(n + \displaystyle \frac{1}{2} \right) \ln n + \left(n + \displaystyle \frac{1}{2} \right) \displaystyle \frac{1}{2n} - n - \displaystyle \frac{3}{2} \ln \frac{3}{2} + 1$

$\ln n! \approx \left(n + \displaystyle \frac{1}{2} \right) \ln n +\displaystyle \frac{1}{2} + \frac{1}{4n} - n - \displaystyle \frac{3}{2} \ln \frac{3}{2} + 1$

$\ln n! \approx \left(n + \displaystyle \frac{1}{2} \right) \ln n - n + \left(\displaystyle \frac{3}{2} - \displaystyle \frac{3}{2} \ln \frac{3}{2} \right) + \displaystyle \frac{1}{4n}$

By way of comparison, the first few terms of the Stirling series for $\ln n!$ are

$\ln n! \approx \left(n + \displaystyle \frac{1}{2} \right) \ln n - n + \displaystyle \frac{1}{2} \ln (2\pi) + \displaystyle \frac{1}{12n}$

We see that the above argument, starting with an elementary Riemann sum, provides the first two significant terms in this series. Also, while the third term is incorrect, it’s closer to the correct third term that we have any right to expect:

$\displaystyle \frac{3}{2} - \displaystyle \frac{3}{2} \ln \frac{3}{2} \approx 0.8918\dots$

$\displaystyle \frac{1}{2} \ln (2\pi) \approx 0.9189\dots$

The correct third term of $\displaystyle \frac{1}{2} \ln(2\pi)$ can also be found using elementary calculus, though the argument is much more sophisticated that the one above. See the MathWorld website for details.