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Lecture Day 18
Linear Models
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Real World Models
By letting math variables [symbols] correspond to real world
variables [stuff we dont know], a real world problem can be
____________ by something as simple as a function, such a model
is called a ________ __________ model.
Many real world problems are analyzed via models that include a
combination of many different mathematical structures, not just
functions.
A Linear Model is a _____________ function which is used to
model a real world problem, they are used in actual research in
the fields of Business, Medicine, Exercise Physiology, Psychology,
Statistics, and Advanced Mathematics.
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How Banks Make Money
1) Banks make money by charging fees to ___________ money and
fees to ___________ money.
2) Banks can ____________ the money theyre holding to make
more money for themselves.
To do this, the bank account needs to stay _____________.
So ____________ accounts are bad for banks, since money is
always going in and always going out.
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How Banks Make Money, part 2
_____________ accounts are supposedly stable since
1) People want to use them to __________ money.
[Supposedly]
2) Money goes in _____________.
3) Money __________ ever goes out.
To motivate people to use them, Banks pay ____________ as a
percentange of the money kept in the account.
Since these accounts arent really all that stable, banks pay
premium [the best yo!] ___________ on Bonds.
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How Banks Make Money, part 3
When a Bond is purchased its values increases over a time by an
amount _______________ to its initial investment amount, called
the _____________.
Interest that collects only on the _______________ is called
___________.
Bonds are __________ and cant be spent or cashed until they
___________.
Some definitions of a Bond say a Bonds value doesnt change until
it matures, but thats _________ since some investors are willing
to buy bonds before they mature.
The value of a Bond over time is a ______________ Model.
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Linear Model Simple Interest
Give me an amount to invest in an account [I take Credit Cards]:
Suppose a Bank offers a bond which matures in five years and
pays 3 % simple annual interest.
The words Annual means once per ________ or ___________.
Fill out the table
Years after Investing
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Bonds Value
Interest Paid at E.O.Y
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Linear Model Simple Interest 2
What is the absolute change in the Bondss value from year to
year?
What is the relative change in the Bonds Value from year to year?
Plot it.
What shape do we get?
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Linear Models Business
A Linear Model is the simplest business model.
Example: My Lisa wants to start a Photography business and to
start she needs to purchase a new Professional Digital Camera that
costs about $1200.
Question: How often must she make this purchase?
This is an example of a ___________ or ________ up cost.
Question: If she has zero clients, how much money has she then
spent on her business?
Graphically, ___________ costs are the ____-intercept.
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Linear Models Business, part 2
Example Continued: To start, Lisa will provide a digital copy of
photos to clients via a USB flash drive and will use an third party
company to make professional prints.
Question: How often must she make a purchase for a memory key
and prints?
This is an example of a ___________ ___________ which
increases as the amount of product increase.
That is the __________ clients she gets, the ________ shell
spend.
Observe the cost in this case is ________ client, that is, its a
__________ or speed.
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Linear Models Business, part 3
Example Continued: To compete with other local photographers,
Lisa plans to make 8 prints for each customer. Lisa has found a
company that sells USB keys in bulk, a 100 keys at $8 each. She has
also found a company that will make prints on average for $3 a
print. Model Lisas Business Costs.
Fill in the table:
Number of Clients
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Total Cost So Far
Cost Increase
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Linear Models Business, part 4
Sequel Example: She plans to charge $132 per client for her
services. Model her Revenue.
Fill in the table:
Number of Clients
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6
9
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Total Cost So Far
Cost Increase
Sadly there is no such thing as fixed ____________.
If you want it, youve got to ___________ for it.
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Linear Models Business, part 5
Trilogy Example: Graph Lisas Cost and Revenue Functions.
What were saying here is that for this problem, the usual scale is
probably not such a good idea.
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Linear Models Business, part 6
Trilogy Example: Graph Lisas Cost and Revenue Functions on a
better scale.
The where Point the two graphs intersect is known as the
__________ __________ ___________ and is where the costs
equal the revenue coming in.
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Linear Models Business, part 7
An alternative approach is to define the __________ function by
Lets Add its Graph into the mix..
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Linear Models Business, part 8
Lets look closer at these graphs. In fact, lets expand the picture
so that we see the intersection of any line with any other line.
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Generalizing Direction
These arrows travel along the same directions as the __________
colored lines on the previous graph. They have lengths
proportional to the snippets between intersections.
Observe that if you _________ the red
and green arrows tails together then
the blue arrow is the __________ that
runs from red arrows tail to the
green arrows head.
Name the Arrows.
Since
the relationship
we can think of the arrows as having
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Generalizing Direction, part 2
If your paying attention then you should be thinking
Did he just say Subtract Arrows. What the .
And yes, I did. This isnt so strange, yet another way to model
addition and subtraction on the number line is with _________.
Example: 5 3 = 5 + -3 = 2
Addition: 5 + -3 = 2
Subtraction: 5 3 = 2
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Vectors
Arrows have a definite __________ and a definite ___________
Any object that has both of these is called a Vector.
Vectors can be added
by gluing head to tail in the arrows case.
Vectors can be subtracted
by gluing tail to tail in the arrows case.
Every Vector had an additive inverse or opposite
the backwards arrows in the arrows case.
Every Vector can be multiplied by a Real Number.
stretch and shrink the arrows
Sometimes Vectors can be multiplied together to make vectors or
just real numbers.
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Vectors, part 2
Arrows in the plane are called _______________ vectors because
they are pieces of a 2 dimensional space. [Thats one reason]
If we move the tail of any arrow in the plane to the ________,
then that arrow will point at a specific point in space.
Since every point in the plane has exactly one arrow that points
from the origin to it, there is a one to one _____________
between vectors and points and vectors and ordered pairs.
The _______________ of a vector are the _______________ of
the point in space they point to when their tail is placed on the
origin.
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Vectors, part 3
As a result, Vector Operations can be completely studied with
numbers.
Addition
Multiplication
Opposites
When operations are performed on Ordered Pairs in this way we
say the operations are being performed ______________ wise.
Another reason why vectors in the plane are called 2 dimensional
is because these vectors can be described by _____ coordinates.
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Advanced Linear Modeling Curve Fitting
Lines suck for interpolating since they can only intersect at most ____ points.
They are good at and commonly used to perform regression curve fitting.
Data is sampled from a real
phenomenon.
And a guess line might be made.
We can measure the closeness
of any line drawn in the plane
by drawing _____________ lines
from the data points to the line.
The Best Fit Regression line is chosen by making the sum of the
_______________ line lengths or their ____________ as small as possible.
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Which Line Fits the Data Better?
Top Line
#
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2
3
4
Point
(0, 3)
(1, 1)
(2, 2)
(3, 2)
|R|
1.33
0.00
0.08
0.17
R2
1.7689
0.0000
0.0064
0.0289
Bottom Line
|R|
0.67
0.42
0.08
0.17
R2
0.4489
0.1764
0.0064
0.0289
5 (4, 2) 0.58 0.3364 0.42 0.1764
6 (4, 4) 0.33 0.1089 0.58 0.3364
7 (5, 3) 0.50 0.2500 0.00 0.0000
8 (5, 4) 0.00 0.0000 0.42 0.1764
Sums 2.99 2.4995 2.76 1.1910
Roots
1.5810
1.0913
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Advanced Linear Modeling Curve Fitting 2
Lines can be used to make an interpolate but not a linear one
A ________ _________ linear function is a function whose graph
consists of pieces of lines.
As long as the data selected is a
__________ function,
We can connect the dots with
strait lines, creating a function that
is linear on its pieces.
It is used extensively in ________ imaging technology. As you read this
the images on this page are probably being rendered by such a function.
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Sampling Down & Interpolating Up
Suppose you took a digital picture but the file was too large so you
shrink it using some computer program. Can you reverse this?
No! But Linear Interpolation can help
Too large,
So selectively
Discard pixels,
Or Down
Sample
Lin
Interpolation
provides an
approximation
to Original.
Linear Interpolation has limits.
Take Bobs severed head.
The more we stretch it,
the worse it looks.
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