CapStone Seminar (Math)

A Mathematical Model of Bisphenol A pharmocakinetics in the Mouse
By: Stephanie Abascal

Bisphenol A (BPA) is a chemical that is used in epoxy resins and a lot of plastic products ranging from baby bottles to the soda cans. This chemical is a threathening one when taken into the body as it causes health risks from sterilization to being a precursor to breast cancer.
There are mathematical equations namely, ODEs, that can model the levels of BPA in the body. These models are similar if not the same as those used in the SIR model that we studied in class. Abascal used Kawamoto's physiological based pharmacokinetic (PBPK) model which studies what the body of a human or mouse for example does to the BPA drug. This analysis inlcudes 27 equation which she cuts down to 3 and working with the equation that refers to the foetus and nondimensionalised it by dividing by some constant with the same units. This takes away the complexity making it easier to understand. BPa never leaves the body but accumulates.
This is bad as BPA can be found in baby bottles; directly affecting your baby.
Abascal made the final conclusion: BPA is bad! Keep the following in mind: The higher the number of the plastic, the harder it is and the harder it is the more likely is it to release BPA into whatever it holds. (The Macalester water bottles are made of a number 7 plastic!!!!)

10.7: SIR and the Spread of Disease

Main Points
Mathematics has an application in everything in everyday life, including to ability to predict an epidemic and therefore the amount of vaccine that would be needed with respect to a population which is of course growing exponentially; (however for the SIR model to work population is help constant).
The S-I-R model:
S - susceptibles (these are not sick yet but could be, future)
I - Infecteds (these are sick already, present)
R - Recovered (these have already been sick, past)
The 'S' reduce over time as they get sick when they come in contact with the 'I's. The two are thus proportional. (infections and contact) thus giving the formula: dS/dt = - (rate at which S gets sick) = -aSI. (the newly sick are then aSI). Those recovering are proportional to those infected: bI; therefore: dI/dt = rate ('S's get sick - 'I's cured) = aSI - bI.
The recovered are no longer susceptible and therefore also proportional to those infected: dR/dt = bI. ('a' is how infectious the disease is and 'b' is rate at which 'I's become 'R's.)
The threshold population value is computed by b/a.
A vaccination should be administered according to the threshold population value (TPV). The entire population save TPV should be vaccinated to avoid an epidemic.

Challenges
I dont understand how the 192 of the dI/dS equation was obtained. And why is there a peak only went the initial value is greater than 192? Shouldn't all initial values no matter what it is produce a peak with its trajectory? I do not exactly understand the significance of a threshold population value.

10.6: Modeling the interaction of two populations.

Main Points
In reality, entities such as populations tend to interact with others other thna themselves. An analysis of this situation would require two differential equations. Exampels are the Predator-Prey model and the Symbosis model. In the former model, Lotka-Volterra equations are used. In this model, in the absense of predators the prey's population increases and in the absense of prey the predators' population decreases. Multiple population functions can be plotted as a function of time and this information is gotten from the shape of the trajectories of the equations.

Challenges
Is this concept is like those before only with a simultaneous equation twist to it? I do not know how to interprete the slope field or the trajectories. I do not know how the trajectories are obtained . . .

10.5: Applications and modeling.

Main Points
For the differential equation: dy/dx = k(y - A) the general solution is y = A + Ce^kt for any constant C. An equilibrium solution is constant for all the values of the independent variable, at this point the graph of the solution is a horizontal line. This solution is stable when a change in the initial conditions produces a solution that heads towards the equilibrium as the independent variable heads towards positive infinity and is unstable when the solution of the initial conditions veeraway from the equilibrium. A differential equation may have more than one equilibrium.
Newton proposed that the temp of a hot object decreases at a rate proportional to the difference between it and the surroundings and the same with cold objects.

Challenges
I feel as though the book is too specific with its examples and thus understanding the topic outside the example is a little difficult. Obtaining the general solution is fairly easy when we are given the differential equation is given. How do we do anything else with it? And why are there a number of graphs for a solution? Do all differential equations have equilibrium solutions?
(I think I really need to go over that class I missed with you)

10.4: Exponential Growth and Decay

Main Points
For the differential equation dy/dx = ky, its general solution is y = Ce^kt for any constant C. When k > 0 this is known to be exponential growth and when k < 0, this is exponential decay. y = C when t = 0. Exponential growth solution curves become exponential decay curves when they are reflected over the y axis. A differential equation is only an approximation of a representation of a number of values, but for large values this approximation method is pretty accurate.

Challenges
In example 1, I do not understand how the solution for iii was obtained. I'm not sure I understand how to set up a differential equation.

10.2: Solutions to differential equations.

Main Points
A differential equation (DE) is one involving the derivative of an unknown function. The unknown is a function rather than a number. This function is computed by finding the rate of change per some y-value for a number of x values for the differential equation and finding the formula that fits the pattern of values. This formula then becomes the solution of the differential equation. The general solution of a differential equation is the family of functions of the solution of that DE. A DE satisfied by both the solution and initial condition is a particular solution.

Challenges
I know I should probably understand this but how and why is the unknown of a differential equation a functino rather than a number? Does the solution of a differential equation always include a constant?

10.1: Intro to Differential Equations.

Main Points
Even when we do not know a key function, its derivative can still be used to obtain information, such in the marine harvesting example where a formula is created using the rate of change details provided.
The derivative formula is written by multiplying the rate of change by some initial value and adding (if increasing) or subtracting (if decreasing) some stated constant.

Challenges
Why is there a constant of proportionality? what is its function? and why is the derivative always proportional to the equation used? The Logistic Model explanation is a little hard to follow and thus a little more difficult to understand.

9.6: Contrained Optimization.

Main Points
In reality all things have contraints, for example, there is a limit to the amount of money available to build a public transportation system. A maximum value under constraint occurs when the constraint is tangent to a function's contour at a point p. This could also occur at the endpoint of the constraint. The Lagrange Multiplier is like the marginal value in production. ie the change in production levels due to a unit increase in constraint.

Challenges
I do not follow how exactly the Lagrangian Function works. I think all the symbols confuse me.

4.3: Global Maxima and Minima/ 9.5: Critical Points and Optimization

Main Points
Local maxima and minima deal with where a function has greater or smaller values at surrounding points; global maxima and minima deal with the largest and smallest values respectively. In order to find the global maxima and minima one would need to find and graph all the critical points.
In finding the greatest or smallest value of a function is to optimize it.

Challenges
I do not understand how to use the analytical method of finding the local maximum or minimum for more than one variable. I might understand the second derivative test equivalent for this too if I understood how it was done (the analytical method that is).

Reflections
Global extrema seems to me to be a lot more useful than local extrema, in which case I find it difficult to udnerstand the great significance of the local extrema. Global extrema helps you find the "extreme" max or min of any function. This is a lot more useful I would think.

2.4: The Second Derivative

Main Points

The second derivative (2-d) is the derivative of an already derived function and is denoted f '' as well as d^2y/dx^2. When the 2-d > 0, f ' is increasing and the graph of f is concave up and when 2-d <>

Challenges

How do you tell when a 2d is positive or negative? If its just by looking at whether the graph is concave up or down, what happens when we cant see it? and why is a trough concave up and a crest concave down??

Reflections

I'm beginning to lose my grasp on just how applicable calculus is to the real world as it gets more complex. Would a second derivative really be used?

The Gradient and Directional Derivatives

Main Points
An ordered pair of real numbers is a 2-dimnesional vertor which is symbolised by a bold letter. The difference between a point and a vector is that the point is simply a point location on the plane whilst the vector is an arrow. The length of a vector (a,b) is denoted by (a, b). A unit vector has a length of one unit. For any nonzero vector (a, b), you can find a unit vector in the same direction by dividing each coordinate of (a, b) by the length of the vector.
The dot product of two vector is as follows: (a1, a2) · (b1, b2) = a1b1 + a2b2; the result is a single number.
The partial derivative fx(x, y) of a function z = f(x, y), tells us the rate of change in the x-direction just as fy(x, y) tells us the rate of change in the y-direction. A gradient seems to have the following properties:
The gradient vector always points in the direction of greatest increase.
The gradient vector is always perpendicular to the tangent of the curve at which it is rooted.
The opposite direction to that of the gradient is the direction of greatest decrease.

Challeneges
After looking at the properties of a gradient, I dont think I understand exactly how we got to that point because I do not understand wy we have those properties.

Reflections
It looks to me as though vectors might be used to simplify things like a contour diagram to facilitate understanding for those less familiar with the diagram. And they are more useful than just points here, especially since they have a directional factor as well.

Linear Algebra: 1.0, 1.1, 1.2, 4.1.0

Main Points
1.0
Coordinates can be represented by a vector which starts at the origin (0,0). The coordinates of a vecotr are written vertically, with the x-coordinate above the y. This shows that a vector is being used rather than just the point. a zero vector would simply be a point at the origin since both the head and tail of the vector are in the same place.
1.1
There are 2 operations that can be done with vectors and not points on a plane, one of which is scalar multiplication, where we multiply a vector (both the x and y coordinates) by a constant.
1.2
It is also possible to add two vectors together, called coordinate-wise addition (because corresponding coordinates are added, ie, x with x and y with y). This sum is represented as the diagonal of the parallelogram drawn with the original vectors on two sides.
4.2: Dot Products
Def: The dot product of two vectors is teh real number obtained by multiplying corresponding coordinates of the vectors and adding. Note that this is a single number, a scalar, not a vector. Also the product of 2 vectors is 0 if they are perpendicular to each other.

Challenges
Other than the benefits of having direction included, what are the benefits of having a vector to work with rather than just a point since its just the number that we are often using anyway?

Reflections
Again I can see how calculus applies to life as even with the plain examples that simply include graphs and vectors, I can connect them with problems about wind speed and how it would affect a travelling boat for example.

9.4: Computing partial derivatives algebraically.

Main points
The notation f_x(x, y) represents the partial derivative of the ordinary derivative of f(x, y) wrt y with x fixed and vice versa. Partial derivatives can be differentiated a second time to obtain second-order partial derivatives. A typical function such as z = f(x, y) would have 2 first-order partial derivatives and 4 second order partial derivatives.

Challenges
The book had to complicate things further just as I thought I was getting first-order partial derivatives. I understand what the entire concept is....computation is troublesome.

Reflections
Most of this is beginning to seem rather "far out there." Obviously with the examples used these actually DO have some real life applications but just how relevant is it to people who are not really concerned with how it all works.

9.3: Partial Derivatives.

Main Points
The partial derivative of f wrt x at (a,b) is the derivative of f with y constant. The reverse is also true (ie. wrt y with x constant.) This is computed using the formula: lim (h -> 0) (f(a + h, b) - f(a,b))/h. This is when y is fixed. When x is fixed the formula changes to lim (h -> 0) (f(a, b + h) - f(a,b))/h. Partial derivatives have a different symbol from the normal one. On a contour diagram, the partial derivative is taken as the rate of change of the value of the function on the contours. And as always the units of the partial derivative would help you to identify what exactly it is discussing.

Challenges
I kinda vaguely understood what the partial derivatives were in terms of making a two-variable equation a one-variable one by fixing one of the variables, but then Example 2 lost me. I have no idea how the values are computed or what actually they mean.

Reflections
Calculus and math for that matter can be used for a lot of things from the simple to the complex all in aid of better understanding it...But are these really used? With the airplane example, its pretty obvious that fixing x at 10 would cause the sales value to increase by $350 for each full priced ticket that is bought. Is that not like a "duh factor"? Are we assuming that figured out that increase was using partial derivatives?

3.4: The Product and Quotient Rules/ 3.5 Derivatives of Periodic Functions.

Main Points
3.4:
The derivative of the product is nto equal to the product of the derivatives! That is
f(x)*g(x) = x*x^2 = x^3 whose derivative is 3x^2 which is not equal to f '(x)*g'(x) = (1)(2x) = 2x. 3x^2 is not equal to 2x.
The product rule is stated as: u = f(x) and v = g(x); (fg)' = f 'g + fg'
also stated as d(uv)/dx = du/dx*v + dv/dx*u
The quotient rule is stated as: u = f(x) and v = g(x); (f/g)' = (f 'g - fg')/g^2
also stated as d/dx(u/v) = (du/dx*v - dv/dx * u)/v^2
3.5:
Derivatives of periodic functions are also periodic. d/dx(sin x) = cos x and d/dx(cos x) = -sin x

Challenges
The Liebniz notation is alot more confusing than the other. if both forms of the derivative function were not given I would have a lot more difficulty differentiating. Why would you want to multiply or divide derivatives?

Reflections
Is it possible that between sine and cosine, one or the other simply exists because it happens to be the derivative of the one of the two. ie. for example, cosine exists because it is the derivative of sine. Therefore cosine curve only exists because sine and its derivative were discovered first.

3.3: The Chain Rule.

Main Points
For a function y = f(z) and z = g(t), { y = f(g(t)) }, a change in 't' affects z in turn affecting y and the derivative can thus be computed with the formula: dy/dt = dy/dz * dz/dt. In other words: d/dt(f(g(t))) = f '(g(t))*g '(t).

Challenges
I do not understand how we move from this formula: dy/dt = dy/dz * dz/dt, to this one: d/dt(f(g(t))) = f '(g(t))*g '(t). What exactly is the connection between the two?

Reflections
Being able to differentiate compositre functions allows one to take into consideration the further effects of an event rather than only the immediate one. Again facilitating the possiblity of having future plans. Applied calc seems so far to simply be facilitating predictions.

3.2: Exponential and Logarithmic Functions.

Main Points
The derivative graph of an exponential function looks exactly like the original graph. For any function in the form a^x, their derivative graphs are proportional to the original, but for the function e^x the derivative graph is exactly the same. Thus, the exponential rule is d/dx(a^x) = (ln a)a^x. Finally why is e is useful base is explained: the fact that the constant of proportionality is 1 when a = e. The derivative of ln x is 1/x.

Challenges
Why is the derivative at the point x=0 of the 2^x function approx. 0.693 rather than 0? How exactly are functions where x is the power differentiated?

Reflections
With the examples that we have been doing in class it is becoming more and more obvious how differentiation fits seamlessly into everyday life though it may of course be very theoretical with hypothesized figures. I never really thought differentiation was what was used to compute the demand and supply functions and thus interpretations in economics.

3.1: Derivative Formulas for Powers and Polynomials.

Main Points
The derivative of a constant is always 0 as it is simply a horizontal line and the slope of a linear function (a non-horizontal line) is always constant. When a constant, c, multiples a function, the derivative is: cf '(x). The derivative of a sum or difference is the sum or difference of each functions derivative respectively. The power rule of derivatives is: d/dx(x^n) = nx^n-1. To differentiate polynomial, the same rules for the sum, constant and powers apply. The derivative of a function using formulas would always give the function of the tangent at a point.

Challenges
The differences between the two methods used in obtaining a derivative are large. how was the connection between the two discovered? Because the honestly the derivative using the formula is very simple.

Reflections
Starting with lessons with learning to get derivatives by calculating the slope of the point in questions over smaller and smaller intervals does a lot with explaing just how differentiation works and what exactly it is and what it is meant for.

2.3: Interpretations of the derivative

Main Points
The derivative can also be denoted by f '(x) = dy/dx, where the 'd's is useful in reminding the user that the derivative is a difference in y-values over a difference in x-values. A derivative tells us how fast a function is changins thus enabling us to estimate the values of that function at points close to the derivative point.

Challenges
Under the "Using Units to interpret the Derivative," I do not understand the second point in the box.

Reflections
The derivative serves as an investigative tool that helps the user to compute certain figures pertinent to whatever situation they are in. I especially like the cost of building a house example (Example 2, p. 113)

2.2 The Derivative Function

Main Points
The derivative is the slope of a tangent line to a point in a curve. This is derived from a secant where the point are so close together they are almost one. When a derivative (tangent) is positive (sloping up) the graph is increasing, when 0 (horizontal) the graph is constant and when negative (sloping down) the graph is decreasing. To estimate a derivative numerically we simply use a small difference between two points for the numerator of a slope equation.

Challenges
Considering I have dealt with differentiation before without the explanations on how to get to it by taking the differences of two points over a smaller and smaller range, it makes it a little difficult and frustrating to follow. It feels as though what I learnt before is conflicting with what I'm learnign now even thought they are the same thing.

Reflections
Considering derivatives are instantaeous rates of change for different point over a curve, it would be possible to calculate at which points over a certain car's journey for example that it stopped, (rest stop) sped up (high way) or slowed down (traffic jam.)

2.1: Instantaneous Rate of Change

Main Points
This chapter considers the rate of change of ONE POINT of a function rather than a range of points. Instantanoues velocity is the velocity of an object at a specific point in time rather than an average over a period of time. The derivative of a function at a point is the instantaneous rate of change which in turn is the limit of average rates of change of short intervals about the point.

Challenges
Summarizing the entire chapter was difficult considering I did not understand most of it especially the bit about the limits. And thus reflecting also is difficult.

Reflections
I feel as though finding out what, say, the speed of a car was at a specific second in its journey would be something that pertains to investigations and the like where specific detail like that is needed. It might also be useful for projectiles, so that the necessary thrust is enough to get an object to its desired destination; and this could be with the instantaneous velocity for example.

1.3 Rates of Change

Main Points
Rates of change could also be calculated when changes in a certain value are not constant. and the formula used is: dy/dx = f(b) -f (a)/b -a. This formula thus calculate the average rates of change. When the average rate of change is the same at all intervals the functions is linear. A positive rate of change suggests an increase whilst a negative one suggests a decrease. The secant line is that with draws out the average rate of change of two points on a curve. This is also their slope. A concave up graph indicates that a function is increasing whilst a concave down graph has a decreasing function. In effect a line is constant.
The average rate of change of height per time is velocity this is different from speed, as speed is a magnitude that is either positive of zero and velocity is a vector. (ie. it can be negative as well.)

Challenges
Example 5 (a): between -1 and 2, I have no idea whether to say the function is increasing because as x increases and so does y, or to say it is decreasing because it is concave down.
And for the graph between x= 2 and 6, how would you describe wrt to (b) the first half is concave downa nd teh second is concave up...

Reflections
Again, being able to compute certain values using applied calculus and in this case rates of change one is able to predict future values for certain situations and thus act accordingly as they would be theorically prepared. But nothing that happens in theory ever happens exactly the same in real life.

9.1: Understanding Functions of two variables/ 9.2: Contour Diagrams

Main Points
A function with 2 variables can be represented using a table of values, with a formula done algebraically or pictorially with a contour diagram. The best way to learn something about a function with 2 variable is to vary one at a time.
Contour diagrams connect points with similar properties, such as isotherms over a certains geographical area. These diagrams could be used to compare different values and also help in the decision making process for a number of situations.

Challenges
How exactly are contour diagrams obtained? Is there a fomula that is used?
What is the significance of the Cobb-Douglas Production Model other than with the US economy? It seems to be pretty inportant since it has its own title and all.

Reflections
ALMOST EVERY PROBLEM IT SEEMS NOW CAN BE SOLVED BY MATH. Of course the nature of the problem has to be considered. And further more it helps with making predictions about certain situations.

1.10: Periodic Functions.

Main Points
Periodic functions have graphs that look like waves as their values repeat after fixed time periods and this goes on forever. The sine and cosine graphs are the most common periodic functions there are and the graph of the cosine function is the same as that of the sine graph only shifted "pi"/2 to the left. Both have an amplitude of 1 and a period of 2*"pi". Periodic functions come with a family with the general term: y = A sin(Bt) where A and B are parameters. "A" affects the amplitude of the graph and "B" affects the period. (the smaller B is the bigger the period.)

Challenges
Are the functions called periodic simply because they have a constant period? Does that mean that if the period were constant but the amplitude of the graph changed it would still be a perfectly good periodic function? Period is equivalent to wavelength right? Therefore light can be considered periodic until it moves from one medium to another and the wavelength changes...
Is this right?

Reflections
Considering that periodic functions are supposed to repeat basically forever applying the functions to real life situations allows the user to predict and therefore prepare for the future. Granted the data would not fit perfectly on a sin curve but it would suggest such as for the housing project that the builders would be able to project would rough amounts of concrete they would be needing.

1.7: Exponential Growth and Decay

Main Points
Exponential growth and decay occur often in nature such as with radioactive decay and compound interest. Every growth has a time when it has doubled in value (doubling time) and a decay has a time when it is half its value (half-life). Interest can be compounded in two ways: annually and continuously, where interest is compounded a number of times in a year (known as compounding). Up to a certain point compounding can be more beneficial had doing it annually as one collects interest on an interest already paid. Annual interest is calculated using the formula: P = Po(1 + r)^t and compounding interest has: P = Poe^(rt). Most products have a present a future value.

Challenges
I do not understand why the two different compound interest formulas are the way they are. I do not understand how they were derived. Thus the same goes for present and future value.

Reflections
Its cool to think that because we know the effects of radioactivity and the half-lifes of the various materials that give it off, the time until which the area of the iodine spill could be occupied again can be calculated. This has then led to higher and better security and safety methods.

1.5: Exponential functions.

Main Points
The exponential function was defined as being in the form f(x) = a^x, with a being a positive constant. With exponential functions the dependent value increases faster per the independent values and have a constant ratio. A typical example of a exponential function is a population growth.
Another form for the exponential function is P = Poa^t where Po is the initial amount at t = 0. Exponential growth occurs when a > 1 and decay occurs when 0 < a < 1. Also considering "a" is a factor of P, the bigger "a" is the faster P grows or decays.
For both a growth (increase) and decay (decrease) the graphs of the functions both are concave up. The difference between the linear and exponential functions is that the linear functions have an absolute rate of change whilst the exponential function has a percentage rate of change.
e is the most commonly used base.

Challenges
A decay always has the x-axis as its asymptote because it never gets to zero and even if it could it could not go beyond it because it cannot go into the negatives. But this is not the same for a growth, it has no asymptote because there are an infinite number of numbers and therefore P can increase forever. This still does not make sense to me though...

Reflections
Whatever medication you take technically never leaves your bloodstream because it never decays to a value of 0! The fact it never gets to zero is possibly the theory behind immunizations. This also explains why there is always a certain amount of time that must pass between two doses of a certain medication. If this time frame was not observed an overdose is more likely to occur. This is good to know, I nearly took my anti-malaria pill 24hrs too early.

1.2: Linear Functions

Main Points.
This chapter discussed linear functions. They are data representations that produce straight lines as their graphs because the differences between two y-values and their corresponding x-values remain the same. The rate of this comparative increase in their values (ratio) is called the slope (m) of a function and is calculated by dividing the differences in the y-values by that of the x-values. This value along with a vertical intercept (b) is used to create the general formular for linear functions: y = b + mx. The equation of the line could also be written in poin-slope form: y – y0 = m(x – x0). A linear function which can have different m and b values belongs to a family and each different function shares a similiar trait: a straight line as a graph.

Challenges
What happens when you do not have a starting point (a "b")? Would you have to simply randomly pick one? Such as with time...there is no sure start of time; but for certain purposes we have to simply pick a place to start. Does that not make the data somewhat inaccurate?
Also the families of linear functions is confusing...according to the section, linear functions are their own relatives.

Reflections
Its interesting to think that there is really nothing in real life that fits perfectly in a linear functions (at least none I can think of); well yes when the data is restricted, but when a larger amount is considered such as with the Olympic and World Records the slope is no longer constant and thus the graph is not a straight and the function not linear. Thus in essence...linear functions could possibly be entirely theoretical.