4.3 Least Squares

Main Points:
If you have a constant that must be used to satisfy two equations which is impossible, a constant can be found that is the best, but not perfect for both equations using a vector that is perpindicular to the other vector from the original. Mathematically this constant is represented by x dot y divided by x dot x (x is the equation in vector form).

Challenges:
I forgot how to find the perpendicular vector. I realized now that it is the vector that when dotted with the original it =0.

Reflections:
I realize this is a very shitty blog, but I just got done with my consulting project a little while ago and realized I still had to blog.

Linear Algebra 4.1 - 4.2

Main Points:
Dot products can be used to find the lengths of vectors and the angle between them. The length of a vector can be calculated by taking the square root of the sum of the squares of the components of the vector. The cosine of the angle between two vectors is given by dividing the dot product of the two by the product of their lengths. When a vector isn't a multiple of another, one can figure out the closest the vector can get to the intended point. To do this, one must utilize two equations. The first is that the closest the vector can get (which is represented by another vector r) is the point perpendicular to the intended point, so r dotted with the vector trying to be reached equals zero. The second equation is formed from the fact that a multiple times the vector plus the vector r between the two must equal the vector trying to be reached.

Challenges
The biggest hang up of this reading for me was the section describing how to find if two vectors are perpendicular. I understand that the product of the slopes of the vectors equals negative one when the vectors are perpendicular, but the way the reading went about showing this was confusing.

Reflections
This concept of finding an answer that is the best possible, rather than flat out the best brings about some useful applications. We've been finding optimized solutions for problems given restraints, and this seems like it could be used in a similar way.

Linear Algebra 1.3, 2.0 -- 2.3, 3.0 -- 3.3

Main Points
Vectors can be added together by adding the all the x components together and all the y components together. Multiples of vectors can be added together to match another vector. The multiple used can be found by changing the i and j values into constants being multiplied by the variable to equal the final vector value. If the vectors are parallel, then there is no solution. This same method can also be applied in three dimensions and with many different vectors. The span of a vector, or sum of vectors, is represented by all possible multiples of the vector. A subspace is a list of vectors that is the span of another list of vectors. The dimension of a subspace is the minimum number of vectors it takes to span that subspace.

Challenges:
The biggest challenge for me is keeping all of the columns and rows and how they multiply straight in my head. Also I thought some of the applications of spanning were quite confusing. The example using a span to solve for an equation was hard to follow. Hearing the explanation in a human voice will help.

Reflections
I had no idea vectors could be used for so damn much! I guess my only reflection is that I really hope that these concepts have some translation into the real world because vectors aren't my strong point.

10.7 Modeling the Spread of a Disease

Main Points:
Differential equations can be used to describe disease outbreaks and are helpful in finding the necessary vaccination to stop the spread of a disease. The SIR model is used to model this. S stands for the number of people capable of becoming sick. I represents the the number of infected people, and R represents the number of recovered or people who are unable to get sick. The number of people susceptible decreases, so the slope that represents this (dS/dt) = -aSI. The rate of infected increases at the same rate as the rate of susceptible category minus those who have recovered OR DIED. This slope can be represented as (dI/dt)= aSI - bI. The recovered group OR DEAD GROUP has the rate of increase of (dR/dt)= bI. a represents how infectious a disease is, so if you know the initial conditions, you can roughly calculate this constant. b represents the rate at which the infected are removed from the infected population. If you consider vaccination, the vaccinated represent people removed from the infected population. The threshold is the point when the number of infecteds starts to decrease. If enough people are vaccinated that the disease can only infect as many as are represented by the threshold, then it will decrease rapidly just like the graph shows.

Challenges
I honestly think the book explains how to graph this very poorly, I do understand it, but I feel it is a challenging thing to look at this section in the text book and figure out through their jargon how the heck to graph it.

Reflections
This is really really useful stuff. Learning how and when to stop an epidemic or at least an outbreak with calculus is pretty amazing.

10.6 Modeling the Interaction of two Populations

Main Points:
When considering the interaction of two populations, one needs a system of two differential equations. In the predator prey model, the idea that the prey will grow exponentially without the predator can be used to form a definition of the slope. Other ideas that should be considered are that the predator's population would decrease if there is too little prey, and the robins benefit from the worms' presence, but the worms do not. Also the number of prey being preyed upon should be directly proportional to the population of the predator. To form a phase plane, the two slope values for the populations must be multiplied. The chart will usually show a equilibrium at the origin and somewhere else that the slope points flow around. The slope points show a trajectory of the populations. As you move a little farther away from one point the slope changes slightly and you can trace the approximate path starting at one point. You can see that with this predator - prey model, the populations oscillate at regular intervals that are separated by one quarter of a period.

Challenges:
The hardest part for me was to figure out how they came up with the equation used to plot the phase plane. It looks like they just treated the slopes like functions and multiplied them together. I need some clarification though.

Reflections:
IT'S THE CIRCLE OF LIFE! It really represents that saying perfectly. The slopes of the population trace a circle around the equilibrium, and since we don't live in a perfect world of equilibrium, this graphic representation is really usefull.

10.4 Exponential Growth and Decay & 10.5 Applications and Modeling

Main Points:
The general solution to the differential equation dy/dt = ky is y = Ce^kt (C is a constant). When k is greater than 0, the function is exponentially growing, if k is less than 0, the function is exponentially decaying. Population growth, pollution concentration, and interest are all situations that can be solved by this model. Another common model for a function used in a differential equation is dy/dt = k(y - A). In this form k and A are constants. The general solution for this differential equation is y = A+Ce^kt. Equilibrium solutions are ones that give a constant for all values of the independent variable. This yields a horizontal line. An equilibrium solution is stable if a small change in the initial conditions produces solution that approaches the equilibrium as the independent variable approaches infinity. An equilibrium solution is unstable if the new solution veers away from the equilibrium as the variable approaches infinity.

Challenges:
I think I have a decent grip on the subjects talked about in these chapters. The most challenging thing for me was following book's description of solving the differential equation dy/dt = k(y - A). Ce^kt justs pops in there unexpectedly, so it's a little confusing at first.

Reflections:
Well now I got to see how to find the differential equation, which is good since it was bothering me that we couldn't necessarily solve to find the equation. This new tactic is nice, but it only works in some situations. The situations are pretty broad, so it is usefull.

10.1 Setting up a differential equation & 10.2 Solutions of differential equations

Main Points:
Differential equations can be used to answer questions or find new information about a function when only bits and pieces of the vital information is given about the function. The solution for a differential equation is an equation that satisfies the differential equation which expresses the slope. To solve numerically a differential equation, one must be given a value for a variable in the function. Once you have this you can compute various data points and form a function that reflects that data. Finding a function for the data isn't always possible, but to check if your equation works, substitute it into both sides of the differential equation and see if they equal each other. To find the arbitrary constant of the equation you derived simply substitute in a value given to you and solve for the constant.

Challenges:
I still don't know how to find an equation that satisfies a differential equation. From what I read, it seems that we need to memorize function forms for different differential equation forms. That doesn't seem very mathematical to me. Can't we solve from scratch for the function. How did they come up with those solutions or forms for functions in the first place?

Reflections:
It seems like a great way to solve a problem when only part of the info is given, but I don't know how this can be useful when there isn't a set way to solve for an equation that works. However, I do appreciate the versatility of this method and how it can be used with different info given and in different situations.

9.6 Constrained Optimization

Main Points:
Optimization problem solving is often constrained by certain external things, this is why we need constrained optimization. If the function can be maximized or minimized with the function of a constraint, then that max or min occurs at the point where the constraint function is tangent to the contour of the function being maximized, or at an endpoint of the constraint function. You can also the method of Lagrange Multipliers. The lagrange multiplier represents the change in the optimum value of the function in question when the constraint is increased by one. You can use this fact to construct other equations that relate this fact in relation to each variable (increasing just x or just y) and solve for all three variables with the three equations. The Lagrangian function can also be used, but unfortunately I have no idea how. The best way for me to describe this process is basically finding where the partial derivatives of x, y and Lambda equal 0 (when the function can't increase anymore) . New symbols frighten me.

Challenges:
I don't fully understand the process of solving with Lagrange multipliers. The part I'm having trouble with is how they derive the equations for the Lagrange, and how that relates to x and y of both the original function and the constraint function.

Reflections:
These methods will be very usefull in real world applications, I'm sure. What I need is a person talking through these explanations rather than a book. When in this world of our's are there no constraints?

4.3 Global Maxima and Minima

Main Points:
Global Maxima occure at the greatest value for a function not constrained by a certain interval (except by the endpoints of the function if it has endpoints). Global Minima occure at the lowest value for a function not constrained by a certain interval. To find the global maxima or minima, compare all critical points of the function as well as the endpoints. If the function continues to infinite or -infinite, then there is no global maxima or minima respectively.

Challenges:
When I looked at the gas consumption example in the book, and I saw the graph, my first reaction was, "hey 20 mph must be most efficient since they're talking about global maxima and minima, but I quickley realised the problem is slightly more complicated. They were looking for most efficient mpg's not gallons per hour, the book made a good point that maximizing or minimizing a function definately goes beyond finding the highest point on a graph.

Reflections:
I think this will help more so than just a local max or min in solving real world problems that require maximizing or minimizing something. I could be wrong, but this concept seems to be more applicable in more realistic problem solving.

1.3 Rates of Change 2.4 The Second Derivative 4.1 Local Maxima and Minima & 4.2 Inflection Points

Main Points:
The average rate of change is defined as the change in y divided by the change in x. The average rate of change for a linear function is its slope. A function is linear if the rate of change is constant on all intervals. A function is increasing if the values of y increases with x, the opposite is true if y decreases as x increases. A function is concave up if it bends upward from left to right, the opposite is true if the function bends downward from left to right. If the second derivative of a function is greater than 0 on an interval, then f ' is increasing, so the graph is concave up. If the second derivative is less than 0, then the opposite is true. The second derivative can be thought as the rate of change of the slope. A critical point is one where f '(x) = 0. This means that the tangent line has a slope of 0 at this point (indicating a possible minimum or maximum). You can test for a local maxima or minima by using the first derivative test. If f changes from increasing to decreasing, then the critical point is a maxima, if the function changes from decreasing to increasing, then the critical point is a minima. If the second derivative is positive at the critical point, then it is a minima. If the second derivative is negative, then the critical point is a maxima. The point at which a function changes concavity is called an inflection point. A point of inflection is where the second derivative equals zero. The sign of the second derivative changes from one side of an inflection point to the other.

Challenges:
The only thing that was challenging was trying to wrap my head around the idea of a rate of change for a rate of change. I understand what it means, but it's a little hard to visiualize at first. One can think of an exponential function when trying to understand this.

Reflections:
This material seems like it could have many practical applications in the real world, especially finding the maximum or minimum of a function. I'm wondering how finding an inflection point of a function could be usefull. Second derivatives could be usefull for finding information about exponential functions that represent real world problems.

Supplementary notes

Main Points:
The gradient of a function is a vector that consists of the partial derivative of the function in respect to x and the partial derivative in respect to y. The partial derivative in the x direction makes up the x component of the vector and the partial derivative in the y direction makes up the y component of the vector. Directional derivatives are a representation of the change in x and y in any direction given by a unit vector. To find the directional derivative one must find the gradient of the function and then multiply the x and y components of the gradient at a given point by the x and y components of the unit vector being used. If you look at a graph of gradient vectors of a function you'll notice that they are perpendicular to the function and they point in the direction of greatest increase and the direction opposite would represent the greatest decrease. As the slope of a function increases, so does the gradient vector's magnitude.

Challenges:
The statements made about the graphical representation of gradient vectors were a little confusing. The direction that they point is stated as the direction of greatest increase for the function, but when I tried to visualize this rule with the given graph, I couldn't figure out how this was true.

Reflections:
It seems to me that there are real world applications for math that can solve for the rate of change of a function in any direction. I hope we can see some of the applications of this math, so it will be more than just a bunch of numbers and coordinates.

Linear Algebra for math 135 - Scalar Multiplication, Vector Addition, and The Dot Product

Main Points:
Vectors are comprised of a point with a magnitude from the origin to that point. The notation can be simply written as the point would such as (2,3). Vectors can be multiplied too. If multiplied by a scalar, you simply multiply the x and y by the number and the new coordinates represent the new vector. If you add two or more vectors together you simply add all the x values together and all the y values together to get the resultant vector. To find a dot product, you multiply the x values together and add the product of the y, z etc. values (x*x + y*y + z*z etc.). The result is a scalar. If two lines are perpindicular, the dot product will be 0. The length of a vector (a) is equal to the square root of (a^2). To find the angle between two vectors (u and v), use the equation u*v=(length of u)*(length of v)*cos(the angle between the two vectors).

Challenges:
I think it may be a typo that is screwing me up right now. The reading says that to find the angle between u and u you use the equation u * v = (length of u)*(length of u)*cos(angle). Where did that v come in? Is it a typo?, because I can't make any sense of it.

Reflections:
I thought the dot product section of the reading was the most interesting. I thought it was interesting how we could calculate the angle between two vectors with this simple equation they gave involving simply the length and magnitude of the vectors. I just thought that was neat.

3.5 Derivatives of Periodic Functions, 9.3 Partial Derivatives & 9.4 Computing Partial Derivatives Algebraically

Main Points:
The derivative of the sin function is the cos function. This is true because the slopes of these functions are periodic. The partial derivative is the slope of a function dependent on one variable (x or y). The partial derivative can be written as (f (a+h, b) - f (a, b))/h this can be used to find the partial derivative from a data table. This method can also be used on a contour diagram. All one must do is find the interval between contours and the difference between them in the x or y direction. To find the partial derivative of a function algebraically, one must plug in the value for x or y that is static, and then find the derivative of the function after that value has been substituted in. Since the result of a partial derivative is a function itself, you can find the 2nd order partial derivative of that function too. Also, f ' (x,y) (a,b) = f ' (y,x) (a,b).

Challenges:
The second order partial derivative is a little bit confusing. I understand the concept of a derivative of a derivative or how fast the slope is increasing. It's kind of like an acceleration. The part that I was having slight troubles with was just how that concept applies to a second derivative. I just don't understand if this is saying how fast the x increases in terms of y or what. I just need some clarifying on what it basically means.

Reflections:
When thinking about 2nd derivatives while I was writing the Challenges section, I thought that a really good way to think of a 2nd derivative is that it's like an acceleration (yes I'm taking physics). Acceleration is exactly what a 2nd derivative is because it's how fast your position changing is changing.

3.3 The Chain Rule & 3.4 The Product and Quotient Rules

Main Points:
The derivative of a composite function (f (g(t)) can be calculated as the derivative of the outside function multiplied by the derivative of the inside function (f ' (g(t)) * g ' (t)). This rule can also be used on a composite function. Just substitute the function in for g (t). The product rule states that (fg)' = f ' g + f g'. In other words, the derivative of a product is the derivative of the first times the second plus the derivative of the second times the first. The Quotient rule states that (f/g)' = f ' times g minus f times g ' all divided by g^2.

Challenges:
The first explanation of the chain rule where the book defined it in terms of dy/dt was confusing for me, I guess the form of notation that they put it in was the only part I had trouble with because I understand the concept, but it would be helpful if you went over that way of describing the chain rule in class.

Reflections:
The chain rule is really helpfull in solving derivatives, but I wonder what a graph of the function of a function and then the derivative of that would look like. I can't picture it. I wonder if you could give us a graphic sense of the chain rule in class. This class seems to be all about the why aspect of calculus, but it seems like the book glosses over the proofs for these rules.

3.1 Derivative Formulas for Powers and Polynomials & 3.2 Exponential and Logarithmic Functions

Main Points
The derivative of a constant function is always the same. The same is true of a Linear Function. If a function is multiplied by a constant, so is the derivative. The derivative of two functions added together is equal to the derivative of each separate function added together. The power rule states that d/dx (x^n) = nx^n-1. Differentiation is used to find the equation of a line tangent to a slope. If you graph the derivatives of 2^x or 3^x etc. you will see that the derivative graph looks like the original function, but stretched upward. This means that the derivative must be simply the function multiplied by a number. e^x = e^x. e approximately equals 2.718. The exponential rule states that d/dx (a^x) = (ln a)a^x. The derivative of ln(x) = 1/x.

Challenges
The exponential rule is a little mysterious, the solution is more complex than what the book gave, but maybe that's because they don't want to confuse us too much. Anyway, e and natural log just overall I don't know much about, so it's a little confusing. My problem is just not knowing enough about log and e and how they originated.

Reflections
Well now we can solve those pesky derivative problems quicker. I'm understanding everything. I've seen all of this material before, so it's not a problem. I really enjoyed going over the manual ways of finding derivatives and other things, because it makes it relavant, but now we forge on to the land of straight up math, but I think we'll still be able to apply it to the real world easily. I just liked to see how the math methods were made in the first place.

2.2 The Derivative function & 2.3 Interpretations of the derivative

Main Points:
The derivative function is defined as the Instantaneous rate of change of f at x. The derivative function is positive when the slope of the original function is positive. The opposite is true when the original function has a negative slope. The sign of f ' corresponds with the direction of the slope of f (increasing, decreasing, or constant). Finding the slope at a given point from numerical values given is one way of finding the derivative. To get a more accurate result from this method average the slope from the values on either side of the target point. To find the derivative from a formula, you can take two very close values, plug them into the equation, find their difference and divide that by the difference between the two x values. The derivative can also be described as (delta y/delta x). This can be written (dy/dx). The derivative can be written to represent values such as velocity (v=ds(position)/dt). A local linear approximation can be made when the derivative is given. Delta y approximately equals f '(x) * delta x.

Challenges:
I was confused as to why Leibniz's notation is difficult to use to express what f '(x) equals. The whole infitesimally small change in x and y way of describing derivative was also a bit confounding. I just thought it was a poor way of explaining it.

Reflections:
I never got this extensive background of how a derivative is essentially derived, so it is interesting to see all of the very painfull methods of finding something so simple. In my calculus class last year, the teacher didn't bother explaining the how or why part of calculus, we just did it, so I appreciate this........kind of.

1.3 Rates of Change and 2.1 instantaneous rate of change

Main points
The average rate of change is determined by the change in y divided by the change in time for a given data set. A function is increasing if f(x) increases with x. The opposite is true for decreasing functions. The average rate of change between two points of a function is represented by a line connecting the two points. This line is called the secant line. concave up and down refer to which way a function "opens up." The average change in position with respect to time is also known as velocity. Speed is the magnitude of a velocity without respect to direction. Instantaneous velocity is the limit of the average velocity of the object over shorter and shorter time intervals containing (t) time. Instantaneous rate of change is similar except that it is basically a measure of the slope of a function over shorter and shorter intervals. The derivative of a function at point a is the instantaneous rate of change.

Challenges
I had a hard time distinguishing between instantaneous velocity and instantaneous rate of change. I realized that they are actually the same. Yeah I feel stupid. Velocity is a rate of change or basically the slope of a function.

Reflections
I'm used to finding the derivative in the good ol bring the exponent out front and subtract one from that method. It's good to see this method again to see how to "manually" come up with a derivative, it connects the process with its roots, at least for me.

section 9.1 & 9.2

Main Points
The functions of two variables described in this section have two independent variables which affect the dependent variable instead of just one independent. A table of values can be used to demonstrate the function or an algebraic function itself can be used. If you want to see how the variables affect the dependent variable independently (though it's not that useful in every situation), you can define one of the variables to see how that affects the function. Contour diagrams are one graphical way to show information from a two variable function. Contour diagrams show equal values of the function connected by lines. This helps to visualize the idea. Cobb and Douglas came up with a two variable function for worker productivity and capital investment. P= f(N,V)=cN^aV^b. Contours on the diagrams are found by setting the dependent variable equal to a value and then plotting the resulting line equation.

Challenges
The whole setting one variable equal to something and then graphing the two variable function is confusing for me. I would like to see/do an example of isolating one variable in class. I'm pretty sure it's easier than I think, but the two graphs the book shows are slightly confusing.

Reflections
I thought the contour mapping section was really interesting. I never considered that those temperature maps were just lines connecting places with a temp on a ten degree interval.

Main Points:
Linear functions are straight lines. The slope of that given line is the the rise of the y value over a certain period divided by the change in x value over that same period. The main form for linear functions is y=mx+b. b represents the y intercept. Exponential equations often are used for functions involving growht and decay. They take the form of y=a^x Exponential growth and decay functions are very similar to exponential functions, but the rate of growth is also exponential in these problems. Doubling time, half-life, financial interest compounding, and present and future value functions all use the exponential growth form.
Challenges:
I had trouble with the idea of compounding interest continuously, the whole concept baffled me for a while, but I believe the idea is basically that the bank is just splitting up the interest payment so that they are constantly paying a tiny fraction of the whole, but that whole sum of money is constantly creeping higher.
Reflections:
This reading helped me understand my bank's operations more, which is usefull. Anytime I gain an understanding of why something is or how something works is a good feeling for me, so I appreciated learning that.

1st entry

My name is Rye Carlson, I'm a freshmen this year and I plan on majoring in physics. I took one CIS (college in the schools) calculus course during my senior year, so I'm familiar with some calculus. I would say the weakest part of my math background is probably the aspect of applying calculus concepts in the real world. I always felt that we learned equations and fancy graphing skills, but we never had a good understanding of how the concepts could be used practically. The strongest aspect of my math education is simply the fact that I'm familiar with calculus prior to taking this course, so everything isn't totally new to me, at least right now. I'm taking applied calculus for it's application in my physics courses. I want to gain a firm understanding of calculus, and more importantly a good feel for applying that knowledge in the world around me. I'm also really interested in the humanities and international studies. My worst math teacher taught my calculus course, my only complaint about him was that he never answered the why question. If I ever approached him about a problem he would simply do it for me and not really show me the fault in my method. My best math teacher taught geometry, and I liked him because of his physical expressions for geometry, he didn't just write on the white board about geometric concepts, he wrote on the walls, ceiling, and used any object at hand to explain something. I'd appreciate hearing "So He Won't Break" by the Black Keys.