TEST 1 Prep

CONCEPTS THAT YOU MUST KNOW FOR TEST:

  • use Euler’s Method with a step size

  • evaluate a composition (for example, (f(g(x))(fog)) of a piecewise function

    • EX: f(x) = {2 x<0}

      •           {1 x=0}

        •     {3 x>0}

  • determine attrition and transmission coefficients for an SIR model (in terms of business, for example, marketing)

  • label a graph of the SIR model

  • draw a tangent line on a graph of a given curve

  • approximate using linear approximation

  • understand what is linear approximation AKA microscope equation

The test will be 5 questions with parts (a, b, c, d ,e ) each. 50 mins total time

all questions are FRQ, each 20 pts

You CANNOT use a graphing calculator

memorize the SIR formulas

similar questions to excercises in The Six Pillars of Calculus Business Edition Textbook by Lorenzo Sadun

remember that the microscope equation is…

  • f(x) is approximately f(a) + f’(a)(x-a)

NOTES ON TEXTBOOK PROVIDED

CH1

If something is real and it’s changing, then we can understand it better with calculus.

Calculus is the study of things that change.

If gas costs $4/gallon and we need to buy 500 gallons this year, what is the total cost?

If the price of gas keeps changing, total cost involves adding up the variable costs for the next 365 days. (Calculus)

You manufacture widgets and every widget costs $20 but sells for $23 so you make how much profit on each widget?

What if making more widgets causes the price to drop, thanks to the law of _____________?

How many widgets should we make to maximize profit? (Calculus)

Your company has 13% share of the market and market share increases at exactly 1%/year. How much will the market share be in 5 years?

What if market penetration slows down as you gain market share? (Calculus)

Six Pillars of Calculus

  1. Close is good enough. We are not looking for exact answers to a problem. For instance solve

    are not looking for exact answers. We are looking for approximations. We use this reasoning in calculus.

When we can’t solve problems right away we simplify things and look for an approximate answer. We improve it until we find something accurate.

Sometimes we need an exact answer. In that case, we use the limit of better and better approximations. This is calculus.

  1. Track the changes. We can learn things about something looking at the way it’s changing rather than looking at where it is right now.

Compare these two trends.

Rate of change of a quantity is related to the slope of the graph of that quantity. In math, this is called computing a derivative.

  1. What goes up has to stop before it comes down.

If we are making

widgets/month should we increase production? Why?

If we are making

widgets should we increase production? Why?

What Is the optimal production level? How can you describe the graph?

So where the rate of change is 0 is the optimal production level.

  1. The whole is the sum of the parts.

What is the national debt next year? It is the national debt this year plus this year’s budget deficit.

What about this year’s debt?

We can see that the national debt this year is the sum of the national budget deficits every year going back to 1776.

Look at the graph above. This is the budget deficit over time. If we add up all the values of the budget deficit then we get the same as the area under the curve. We can break the quantity we’re studying into little pieces, estimate each piece and add up the pieces. This is called integration.

It is about adding up the pieces.

  1. One step at a time.

Rather than asking what the national debt is, we can ask how much it changed in a short period of time. The change per year is what part of the debt curve?

If the steeper the curve then how is the debt changing?

How was the debt in the 1990s?

How about 2009-2012?

How about 2020?

And between 1950 and 1975?

Compare this figure and the one with the budget deficit. How are they related?

If we invest $1000 at 6% interest how can we know how much money we will have in 30 years? One year at a time.

One step at a time.

  1. One variable at a time. Many functions involve two or more input variables. What does boiling point of water depend on?

What does heat index depend on?

What do price of widgets depend on?

How about value of an oil field?

To understand functions of two or more variables, we hold everything except one variable fixed and study just that variable.

CH2.1

Chapter 2: Predicting the Future: The SIR Model

Section 2.1: A Problem of Market Penetration

You Work for a company that has just introduced a hot new phone app. People learn about it by word of mouth, and more and more people are using your product. Growth can’t go on forever. Eventually people get tired of your app and will move on to the Next Great Thing.

To make the most of your product’s popularity, you need to forecast usage for the next year or two.

We define our quantities:

t denotes time, measured in days. I(t) is the number of active users at time t.

t and I are called variables since they change.

Time t is the input variable and I is the output variable.

Describe the graph.

How can we be successful?

We need to understand what makes I change over time. So we need a mathematical model for what is going on.

Once we have our model, we have to analyze it.

Finally, we need to interpret our results.

Three steps:

  • Model

  • solve

  • interpret

CH2.2

Chapter 2: Predicting the Future: The SIR Model

Section 2.2: Building the SIR Model

Let S(t) be the number of Potentials at time t,

Let I(t) be the number of Actives at time t, and

Let R(t) be the number of Rejecteds at time t.

  1. what rate do Potentitals adopt the product and become Actives?

  2. what rate do Actives stop using the product and become Rejecteds?

Losing customers: Attrition: Customers don’t use an app forever. They get tired of it or

switch to a competitor’s app. Some like navigation tools and browsers keep customers for months or years. Others, like games, can lose their customers after just a couple of weeks.

We are analyzing a new game that users keep using for an average of 30 days.

Among the Active users, roughly 1/30 of them will grow tired of the game and will be among

Rejecteds tomorrow.

I(t) and R(t) are numbers of people, and the number of new R’s on any given days is also measured in people. But the rate R’ at which R is changing is measured in people/day.

The ratio has units of people/day.

This is a rate equation. It describes the rate at which something is changing in terms of other data. What is it describing in this case?

  • T is the average length of time people use a product then we have

  • =?

  • we define b=1/T, then our rate equation is:

  • =

R’ has units people/day, I has units of people, and b has units of (1/day)s. We call b the attrition coefficient (boredom coefficient).

b is called a parameter. Is it a variable? Why or why not?

Gaining customers: Transmission Losing customers is grim. If our app has 2,100,000 users

and is only used for an average of 30 days. We are losing users at a rate of

  • need to get customers to replace the old ones.

  • our model, we assume we gain customers by word of mouth. We will see this from the perspective of a single Potential user called Joe. Joe hears about our app from a certain fraction p of the Active users each day, for a total of pI contacts. The fraction p is very small, but the Active population I can be very large.

The more Active users there are, the more times Joe will hear about our product. Every time Joe hears about our product, there is a probability q that he will be motivated to download it and start using it.

Therefore, there is a probability pqI per day of Joe becoming Active.

We define a=pq, and call a the transmission coefficient. This is a parameter.

Look at the entire population of Potentials. If there are S Potentials, each of whom has a probability aI (per day) of becoming Active, then each day there will be approximately aSI Potentials who become Actives.

S and I have units of people and S’ has units of people/day so a must have units of ?

Note the minus sign in our equation. The more Potentials become Actives, the fewer Potentials are left.

What about I? Completing the model.

S + I + R is the toal number of people out there and that doesn’t change. Since S’+I’+R’=0, we must have I’=-S’-R’.

t is the time. Depending on the setting, it can be measured in days, weeks, months, quarters, or years.

S(t) is the number of Potentials at time t, measured in people.

I(t) is the number of Actives at time t, also measured in people.

R(t) is the number of Rejecteds at time t, also measured in people.

b is the attrition coefficient. It is the reciprocal of the average time T that a user keeps using the product before moving on. The units of b are the reciprocal of whatever units we are using for time. The parameter can vary from product to product.

a is the transmission coefficient. It depends both on the product and the market. a will typically be bigger in smaller markets (where each Potential knows a greater fraction of the Actives) and smaller in the bigger markets. The units are 1/(time x people).

S’(t) = -aS(t)I(t)

I’(t) = aS(t)I(t) - bI(t)

R’(t)= bI(t)

CH2.3

Chapter 2: Predicting the Future: The SIR Model

Section 2.3: Analyzing the Model Numerically

How many Actives do we expect tomorrow?

How about day after tomorrow?

After a week?

After a month?

After a year?

So in general after t days we should have

We can use linear approximation to study the past as well as the future.

How many Actives do we have a week ago?

A month ago?

CH2.4

Page 1 of 2 Chapter 2: Predicting the Future: The SIR Model Section 2.4: Theoretical Analysis: What Goes Up Has to Stop Before it Comes Down What is happening when I(t) reaches its peak? What is happening when I’(t) >0? What is happening when I’(t) <0? What happens at the top of the curve? What is I’(t) equal to at the top of the curve? ICH increasing ICT decreasing Ict has stopped increasing and has not yet started I'LD 0 decreasing What happens if the threshold b/a is large and there aren’t many Potentials to begin with? What will happen if b/a is small and S(0) is large? our product will fizzle from the start we will have a long period of growth before we saturate the market

CH 2.5&2.6

Page 1 of 8 Chapter 2: Predicting the Future: The SIR Model Section 2.5: Epidemics Section 2.6: Covid-19 and the SIR Model