*Draft version 0.18*

Last quarter, I took a Marketing class at Stanford called Global Entrepreneurial Marketing (GEM – MS&E 271). The first exam was a strategic thinking paper (STP) that required students to analyze themselves as a startup using a marketing toolkit called DEDART which stands for:

**Diagnosis**Where is this market in its life cycle? How attractive is the market opportunity?**Experience**How will customers experience the benefits of the new product?**Decisions**What is your marketing (or business) plan?**Analysis**Why is your plan the best way to go?**Reality Test**What are the risks of your plan, and how can we manage those risks?

It was an awesome class and I learned a lot of valuable entrepreneurial lessons from great teachers and professionals while working with a talented team of students.

As part of my STP for the GEM class, I wrote about a mathematical model that I have been thinking about as a bonus exhibit. This post is about that model which I called “**Eventually Rising**“. Here goes:

**Bonus Exhibit I: Eventually Rising **

Given my engineering background, I often like to use mathematical concepts to illustrate my life view. I view constant change in life like a sinusoidal function that keeps falling and rising over time. A lot of people hate change and try to prevent it from happening, or they deny it or try to stop it once it happens. However, as John C. Maxwell once said “*Change is inevitable. Growth is optional.*” Success and failure are two examples of change, but unlike change, success and failure are not constant and they are not inevitable.

While change is inevitable, it’s not always forced upon us. Most of the time, we are the primary change inducers in our own lives. The sine and cosine functions are two functions that alternate between a peak positive value equals 1 (maxima – ultimate positive change) and a bottom negative value equaling -1 (minima – ultimate negative change). We all want more successes (maximas) and less failures (minimas) through our actions.

Our actions cause change to the world around us for better or for worse. In an ideal world, we could ensure that change is always positive and all outcomes of our actions are desirable. The world of mathematics offers a great function that represents this ideal world and that is the tangent function. The tangent function is a strictly monotonically increasing function over the period [-π/2, π/2] or [-180°, 180°]. Check Bonus Exhibit III for the definition of monotonic and non-monotonic functions.

** Source: http://goo.gl/erVwO*

I especially like the fact that the tangent function is the result of dividing two constantly changing functions that results in two exaggerated values -∞ and +∞ on the edges . We often hear, someone changed a 180 degrees and turned his or her life around. That’s exactly what the tangent function does, it goes from the absolute infinite bottom (-∞) to the absolute unattainable maximum (+∞) over a 180 degrees.

In Reid Hoffman and Ben Casanocha’s book, “The Startup of YOU“, they talk about introducing small risks to one’s life on regular basis to build up more resilience and achieve stability through low levels of volatility. That is the division of two changing functions to achieve growth 🙂 Its exactly like dividing the sine function by the cosine function. If we introduce changes to our lives at the right frequency and at the right magnitude, we will get a net positive constant growth. In reality, getting it right every single time is probably impossible. But if you if we introduce small changes to your life as a habit, you will see a lot of benefit.

We have to be careful though. Simply changing sin(x) into sin(-x) will result in a monotonically decreasing function (the exact opposite of what we want):

** sin(-x) / cos(x) using http://www.onlinefunctiongrapher.com/*

Or if you change the frequency of the changes, for example by dividing sin(-2x) by cos(0.2x), you will get a very bumpy ride:

** sin(-2x) / cos(0.2x) using http://www.onlinefunctiongrapher.com/*

We don’t live in an ideal world and we are never really at -∞ and we will never reach +∞. However, the lesson that I do take from that ideal model is that we should always strive to be growing. Its hard to be monotonically increasing, its probably impossible to be strictly monotonically increasing, but we can be eventually increasing. We are not always able to induce the right amount of change at the right time. Sometimes, change is induced upon us in ways that we have no control over. By investing in yourself, learning new things, gaining new skills, and making new connections as a habit, you maximize the chances of introducing the right change at the right time. When undesirable changes happen, you will take a fall. Those undesirable changes can be external (an earthquake for example) or internal (we all make mistakes).

That’s the math and philosophy behind the term “**Eventually Rising**“. The following image is a good visual representation of an eventually rising plot:

** Source: this site was tracking Google share value since inception*

By focusing on our own constant growth, we may be able to neutralize the undesirable change around us and turn it around to be rising just like the tangent function. By constantly taking any potential failure and flipping it a 180 degrees and making a success out of it.

Every company, large, medium or small, publicly traded or a private startup has its ups and downs. The same applies to people. No one is always winning or always losing. You want the net average over time to be heading up.

**Bonus Exhibit II: Functions of constant change (sine, cosine, and tangent)**

** Source: http://goo.gl/erVwO*

The Sine and Cosine functions represent the inevitable and constant changes in our lives.

** Source: http://goo.gl/erVwO*

The tangent function = Sin(x) / Cos(x) over the period [-π/2, π/2] or [-180°, 180°] is a strictly monotonically increasing function.

**Bonus Exhibit III: Monotonic and nonmonotonic functions **

“A function is called monotonically increasing if for all and such that one has , so preserves the order (see the left figure below). Likewise, a function is called monotonically decreasing if, whenever , then , so it reverses the order (see the right figure). If the order in the definition of monotonicity is replaced by the strict order , then one obtains a stronger requirement. A function with this property is called strictly increasing. Again, by inverting the order symbol, one finds a corresponding concept called strictly decreasing. Functions that are strictly increasing or decreasing are one-to-one (because for not equal to , either or and so, by monotonicity, either or , thus is not equal to .)” **Source: this Wikipedia page *

The following figure is a monotonically increasing function. It is strictly increasing on the left and right while just non-decreasing in the middle.

**Source: this Wikipedia page *

The following figure is a monotonically decreasing function. It is strictly decreasing on the left and right while just non-increasing in the middle.

** Source: this Wikipedia page *

The following plot compares a nonmonotonic function to a monotonic function:

** Source: http://goo.gl/Ol81E*