Continuing Education | Math Camp

About this course

you I’m sorry Jade bows I’m a lecturer in discipline at Columbia University’s School of Continuing Education and the Earth Institute what you’re about to see

is a set of lectures on the essentials of mathematics necessary for students who take our master of science in sustainability management program this is the first of three sessions being

everyone I’m sorry for my state I’m a bit of a klutz and but a week ago I stepped into a pothole so so this is why nothing is broken and so on and mostly it’s fine but it does get me a lot of

sympathy which I otherwise would not have okay great my name is Satya Geoff Bowes I am a lecturer in the mssm program which you’re all starting tomorrow which you’re sort of starting

today previous to this I used to teach in the MPA for environmental science and policy program run by the School of International Affairs the mssm program in some sense is an offspring of that

mpa in ESP program and i’ll give you a fuller introduction to myself into my courses tomorrow at orientation today we’re going to talk about mathematics I’d like this to be fairly interactive

this is a large group but please feel free to shout out questions when you don’t understand this is a group that’s probably going to be together for a long time so I’d like you to get

start to feel at home with each other and and with me okay let’s see this is going to work Wow okay so i put up there a little quote from an essay by gary snyder anybody knows who gary snyder is

yes he was a beat poet yes and also a sort of a poet of the deep ecology movement and this is an essay that he wrote about by the way you should have received slides for tonight but I know

that some of you certainly did not so when I get back to tomorrow I’ll make sure that you all received these slides so there’s no need to take notes and brush and so on the the references in

there or actually hyperlinked into the slide so you’ll be able to get to the essay if for some reason you want to actually read the whole essay but so this essay is about a time when Gary

Snyder was taking a trip in the Australian outback and he’s writing about going around on a tour on the back of a pickup truck and he was with an aboriginal historian who was telling him

various stories about the hills over there and the river that they just passed he realized that the stories were going very quickly he could barely make out what was going on in one story

before the storyteller started narrating something else and after about 20 minutes he realized that the stories were meant to be told while walking not while driving around in a pickup truck

and they were really the stories by which the recorded memory of that tribe was kept and they were attached to symbols so mountains certain rivers and so on and I

tell you the story because this has a link to mathematics believe it or not mathematics is not something to be done while driving around in the back of a pickup truck which is effectively what

we’re going to be doing today because we’ve got nine hours off of mathematics and statistics and probability which we really ought to take a couple of years to do so this is obviously not the way

to learn mathematics but that’s the nature of our our culture and civilization so you have to learn it in nine hours so that’s one thing the other thing is that mathematics is a language

for us for for our civilization in the same way that in the same way that writing that remembering stories about particular places might have been for this aboriginal narrator to Gary Snyder

and so many of you may have wondered why it was so important for you to learn mathematics since you’re presumably interested in sustainability management and it’s not immediately obvious that

mathematics is so relevant and today we will be today and the next couple of days we will be doing things that may not be quite so related but I want you to remember that it is useful and there

will be times certainly in the next nine hours figuratively speaking and also if you’re taking my economics of sustainability management class in the next 14 weeks where some of the things

you learn might seem technical or arcane but at some point you will find them useful okay the other thing is you you’ll need to self-select so some of you for some of you mathematics will not

be a problem statistics will not be a and you can safely ignore most of what I say for the next nine hours but those of you who know that it is a problem and need a revision you should be working

through the problems that are in the primer that was sent out the primer will cover a lot of what I will cover but not the calculus pot and I’ll tell you where to go to do problems there in a second

of course as I’ve said you you’re going to need mathematics in order to do sustainability management well certainly it’s a very important part of economics but also an important part of

quantitative analysis in general and the science courses that you’ll need to take as part of the program in having worked in interdisciplinary fields for a long time I found that the intersection

between disciplines is always used to intimidate so for example when I was studying for my PhD an important field was law in economics you know so looking at laws looking at the economic

foundations of laws and since economics was a relatively quantitative subject and lawyers tended to ask you mathematics and economics I found that economists would use mathematics to

obfuscate their analysis for lawyers in the same way in economics there were those who were very mathematical and those who were not so mathematical and so the mathematicians who did economics

use mathematics obfuscate to the economist so there’s this chain of obfuscation which often takes place in interdisciplinary work and it’s to break through that chain

that it is important for you to learn the disciplines and not be intimidated by them so the key point of this math camp but also of the sustainability management program is to prevent the

sort of approach where you read an article and you gloss over the equations or you read a policy piece and you ignore the tables or you engage with them only slightly okay okay so here’s

an example of mystification you may have seen probably a few days ago there was an op het about called math lessons for Locke of wars anybody see that a couple of people you don’t have time to read

New York Times our peds but you can again you can link to the dr pad and the letters from the slides when once you get them and you’ll see an example of how this is used ok here are the

resources that you want to investigate and use if you need them obviously the primer that was sent out there is also a mat primer at the School of International Affairs where I previously

taught and the link is over there if you go to that link there are actually problem sets which you can work through and then you can show the solutions immediately so I encourage you to

actually try to do some of the problems before you show the solution and also there are primers in in most economics textbooks those will tend to be more concise and are more appropriate

if you already know the material and want a a quick review okay let me give you the outline here so today we’re going to go very slowly we’ll simply talk about functions and equations this

should be high school mathematics so it really should be review for everybody and just a way to get those mental processes working again on Thursday we are going to attack calculus and that

will be a very dense session I expect that there may be some students on Thursday who have not met calculus before or are quite rusty on calculus so it’ll be quite a lot to cover in one day

but we will we’ll go through it and I will certainly at least figuratively hold your hand through through any pot that may seem obscure day 3 which is which is after schools which which is

after classes start is also going to be quite dense and it’ll be the second part of calculus that is integration and also probability and statistics in the primer that went out you will have covered

everything in functions in equations and a lot of what’s in probability and statistics there’s not much in the primer on calculus and and there’s a little bit on the statistics which I’ll

show you on the third day we is not actually in the primer also the aim in the primer is not intimidate too much in the summer when you’re sitting on the beach okay any questions so far

okay today what we’re going to do is talk about functions mostly linear functions will talk about systems of equations yes yes yes and can you tell me your name when you know I will why

won’t be offering it for the second semester but I am being recorded so you will have the you’ll have the video we’ll talk about linear functions equations systems of linear equations

we’ll talk a little bit about special nonlinear functions that we often use in some of the science classes and in economics and we’ll wrap up by talking about inverse functions and implicit

functions it will probably spend quite a bit of time on a couple of examples okay so what’s a function it’s a rule a rule that assigns an output to an input so you could think of

any rule as essentially a function for example the function 2x takes any number an arbitrary number or an arbitrary quantity X multiplies it by 2 and gives you the output which is 2x given that if

X is equal to five then 2x is equal to 10 right everybody feel comfortable with the idea of function right we owe leonhard euler the name you’ll hear many times for the notion of a function or

the mathematical definition of a function so an example of a function this is this this was when I taught the math camp last year so this was fresh this was in the news last years perhaps

less apropos today but so around this time last year of course they were actually maybe even two years ago there was the the Fed had a program to write checks to large banks that lost money so

you might imagine a policy function that says if a bank loses X dollars then the Fed will write them a check for 2 X dollars okay and so that would be a function so any bank which loses X

billion will receive 2x billion in in low-interest loans okay so you can represent a function in a tabular way so when you see tables very often you’re really seeing a function

you may not see all the points in the domain of that function in a table but you are seeing an underlying function in in that table you’ll have a row with inputs and a row with outputs remember

the function is just a rule that associates inputs to outputs okay okay this is probably a little hard to see but let’s okay so how many people here have not met the Cartesian plane okay so

let’s let’s talk briefly about the Cartesian plane it is simply a method of graphically displaying two sets of numbers paired sets of numbers so the function remember gives me an output for

an input a function is a rule that assigns an output to an input the Cartesian plane is a way of displaying paired sets of numbers we use the Cartesian plane to display the graph of

a function and the graph is constructed by taking as paired points the input and output we usually think of the x axis as the input as the metric on which the input is displayed and the y axis as the

metric on which the output is displayed so if play the pair set of numbers 23 we assign the value 2 to the x axis so we move two squares of the right and we

assign the value 3 to the y axis we move 3 units up along the wire yeah okay if you if you didn’t know what a Cartesian plane was before I’m afraid that my explanation probably did not help you

but hopefully it it reminded you okay okay so here’s a graphical representation of the function we just discussed we have the x-axis the input loss amount for the bank the y-axis the

output the loan amount received by the bank given to them by the federalism we could draw for every potential loss an associated level of loan for example a loss of five and associated level of

loan of ten join all the points together that would be called a locus so that is every pair of numbers that satisfies the rule would be the definition of the graph of a function every pair of

numbers that satisfy the rule okay this of course only works this Cartesian plane here the way I’ve drawn it only works where you have one input and one output there you can imagine a Cartesian

plane in many dimensions and we’ll talk about that next time when we when we talk about multivariate function okay let’s get to a little bit of notation y equals FX I’m sure you’ve

seen this obviously in the primer so many times indicates that the variable Y is a function f of X so X is an input and y is an output both x and y are variables that is their concepts that

are measurable and can vary f is the rule that assigns f of X 2 X okay exit the input variable so known as the independent variable Y is the output variable known as the dependent variable

F is the rule now a common source of confusion is to think of f of X as a function and this is a common source of confusion because everybody makes that confusion but note that the rule is

actually s an f of X is actually the output so f is the rule and f of X is the output because of course y is equal to f of X and Y is the output and this is a useful distinction to keep in mind

when you do calculus because one of the things you’ll do is change of variables and so you want to make sure you understand what’s the function and what’s the optimum an example function

commonly seen manufacturing cost is often thought of in economics as a function of the quantity produced and so you would have C as a function of Q desired residential square footage as a

function of family size but for example okay so suppose that I give you the following slightly complicated function in that it is a function of mouth two variables suppose that profit is a

function of product development and advertising expenditure and it is 36 thousand plus 40 X plus 30 y plus X times y over a thousand what’s the profit of the firm spends a certain

amount on product development two thousand two thousand dollars on product development and a certain amount on advertising five thousand dollars on advertising anybody here with a

calculator I don’t expect you to do this in your head okay quick it we can just do it we can we can just do it we can work it out yes Richard okay so that’s a good question Richard the x

and y are separated by a comma this does any book does everybody else know what that signifies yes what’s your name Jenny yeah so x and y are both input variables actually I jumped a step

because of course here I showed you a function with a single input variable and here i have a function with two input variables so my apologies for that ok so here this is the output Y is not

an output everybody see that oh ok so in this example why is not an output f of X comma Y is the output there are actually two inputs x and y so let’s just let’s just do this so

we’ve got 36 thousand plus eighty thousand plus plus XY 10 million / it up okay so that’s three knows it yeah so 36 it’s got to be 300 right 150 so it’s got yeah it’s going to be more than yeah

1902 66 yeah sorry yeah 276 though perfect okay okay so everybody see how to apply functions simply applying a rule on one or more inputs to get up