4.3: Least Squares

Main Points.

It is possible that one coordinated is not proportional to another, but is very nearly so. Some multiple m will bring a very close value. The vector mx is as close to y as possible  The residual vector r is the distance btwn the tips of those two vectors. The better value of m, the smaller r will be because if mx was to equal y, r would be zero. By taking the dot product, r can be determined.

Data points often show a linear tendency put do not often lie on one single line, or are products of one another. All points can be expressed in three vectors where u contains the x coords, v the   y-intercepts, and s the y coords. The best fit line is the linear combo of u and v that gives s where u and v are multiplied by multiple m, plus r, to make up for the distance btwn that line and s. The fundamental problem of linear modeling is to find a linear combo of vectors that are as close to vector b as possible, with each vector multiplied by a multiple x, and all added to perpendicular vector r, which would ideally be zero. Then, b is in the span of the u vectors.

Conflicts.

Because a line can be drawn through any two points, doesn't that make the two points proportional to each other?

Reflection.

 Because the two vectors meet at the origin, the further they get the farther they are from each other. Therefore this concept seems very similar to local linear approximations. 

This concept seems very relevant because almost always when using data, the points don't fall on one line, making a best fit line necessary.

Main Points.

Linear Combinations is the when there is vector addition and scalar multiplication done simultaneously. When given three vectors, there are multiple ways to determine what operation must be performed on the first two to equal the third. It can be done graphically, or algebraically where x and y represent the unknown, the multiples. 

A system of equations can be points interpreted where you determine at which point the two eqns intersect. In vector interpretation, you determine which multiples are needed to produce a certain vector from two others.

Vectors can be drawn in higher dimensions, with R^3 being 3-D. Matrices are boxes of numbers, with their own special notation, with dimensions m by n. 

The span of vectors are all vectors that can be produced from those vectors via linear combos.

Linear dependency is determined by whether any vectors from the span, is a span from any vectors from that same span.

All vectors in the span of a span or list make up a subpace. The subpace's dimension is the smallest amount of vectos needed to make a subspace.

Reflection.

I don't really understand matrices but they seem to be a much more efficient way of keeping track of and adding or multiplying points then always writing out the vectors. I believe they are important for coding where numbers are used to represent letters and the given multiplier serves as the "key" for the code.

Conflicts.

I don't get linear combos because scalers and vectors are tow different things. I don't understand multiplying the matrices, the numbers seemed to be coming out of nowhere. 

10.7 modeling disease

Main Points

Modeling the spread of disease requires 3 variables S, the # of ppl who can become sick, I, the number that are sick, and R, the number of ppl who have been sick but can't become again. The model is important because it reveals the number of ppl that should be vaccinated, the amount of ppl that will be infected, and the time frame. All of this is important to monitor disease and save lives. As there are more sick, there are more "interactions" btwn I and S, which is proportional to S and I. The change in S over time is equal to the negative rate that members of S become I, which equals SI time a negative constant a. The change in the number of people sick euquals the rate ppl get sick minus the rate they are removed, because as the number of sick is dependent on S and R. The constants represent how quickly the disease spreads or how quickly ppl are removed. They must be constant because these quantities vary depending on disease or population, but are the same for each disease.

Challenges

Because the variables represent ppl from the same original population, i don't see why 3 different variables are needed. 

Reflections.

How are naturally immune people factored in?

10.6 modeling ant the interaction of two populations

Main points

To model the interaction btwn two different populations, 2 differential eqns are needed. In a predator-prey situation. the pops are both effect by the size of themself and each other.

A phrase plane is where the two varriables move with respect to a 3rd variable. The Phase trajectory is the path of a point

Challenges

If there are three variables, why can't a contour diagram be used? I don't understand how you actually graph or model the eqns.

Reflection

I think this is a very interesting topic. The idea that a graph can actually show motion, as opposed to just suggesting it (like how a straight line shows change over time), is really cool, almost 3-d. The models look like diagrams and are therefore easier to comprehend readily. 

Exponential growth and decay / Applications of ODE, equilibria, and stability

Main Points

The general solution to the differentional eqn dy/dt=ky is y=Ce^kt.  This is true becuase the deriv of e^t, and any multiple of it, is itself.  when k is positive there is growth, when negative there is decay. The graphs of these solutions are exponential curves with y intercepts that have the x axis as a horizontal asymptote. Population growth, continuously compounded interest, and other instances of growth/decay follow this model. 

An equilibrium solution holds true for all y values. The graph is therefore a horizontal line. They are found by setting deriv=0. The equilib func is stable when a small change of initail condions gives a solution that  stays close to the equilib as y gets increasingly bigger, but is unstable it veers from the equilib. Newton's law of heating and cooling state that a differently tempertured object will approach room temperature at a rate proportional to thier initial temp, therefore 2 diff temperatured objects will after a given period be very close in temp.

Challenges.

The text states that a solution is a function that is its own derivative. How is this so?

Reflection

Does Newton's law relate to the law of entropy? 

10.1 and 10.2: Differential eqns

Main Points

Differential equations are those which express information about the deriv. or rate of change of the original eqn. Instead of starting with the eqn and computing the deriv, you can use the deriv and work backwards. This method can be applied to many different type of real world problems sucha s marine harvesting, the net worth of a company, and pollution in a lake. The rate of change of a population can be expressed by the constant of proportionality times the current population times the difference btwn. the carrying capacity and the current population. The solution to a differential eqn is in fact another equation, it is the function with that deriv. The general solution is the family of functions that satisfies it, while the particular solution is the function with variables set at specific variables.

Challenges

How can an equation be the answer to another equation?

Reflection

Because we start with a deriv and work off that, isn't using differential eqns like finding the indefinite integral? 

9.6. constrained optimization

Main points

When the function has a limit, to express the "constrained" max or min you set it equal to the upper or lower level, respectively. Graphically, the limit is represented by a line that intersects with the contours of the function line. The max lies on this line because anything bellow is not reaching full potential by not hitting the limit, and everything above is disallowed because it exceeds the limit. Lagrange multipliers can also be applied to a constrained eqn to optimize it. Local max.s occur at a particular point that are greater than all immediate points and meets the requirements of the constrainment, while the global max this is true for all points of the function, vice versa for min.s. The optimization is found by plugging the function into three specific eqns. The Legrange multiplier used in these eqns is defined by the optim value of a funtion as the constrained value is increased by one unit. The Legrange multiplier is the rate of change of the increase in a function per unit. Furthermore a constrained function can be optimized by plugging it into the Legrange function and determining the critical points.

Challenges

Why do you set the limit equal to the limit instead of it having be less than or equal to? If there is a limit, like the 378,000 used in the book, how can that possibly represent a line as opposed to a point? so the legrange multiplier is not like e right, in that it is a special number? is it a special ratio? i dont understand it's significance. 

Reflections

Because the Legrange multiplier represents a rate of change, is it closely connected with the derivative function? Is it the deriv.?

4.3: global max and min

Main Points

While local max and min are the function values lowest or highest on a specific curve of the graph, the global max and mins are the highest and lowest points of the entire function, respectively. Because those points are the lowest and highest of the entire function, they are more extreme than any points on the same curve as well. Therefore they are also local max and mins,  and the lowest local min and highest local max are the global mins and maxs, respectively.

To find the global max and mins you compute all critical points and find the most extreme ones. 

Challenges

I understand that the global min would also be the local min. Because all animal are not cats while all cats are animals, then why does the local min have to be a global min. I just realized that its not, but one has to be. The contrapostive is that if its not a local min it's not a global min. So i guess i no longer have a challenge but the whole point of the blog is to teach yourself so i hope that's okay.

Reflections. 

It makes sense that the global max and min are important because this can show you how to obtain the greatest yield or minimize damage for a real world problem. Now I don't understand why the local mins and max's are important. Is it just to find to global ones? Perhaps i should switch my challenges and reflections paragraph.

Main Points

Because f'(x) is function, evident by its graph, it too has a deriv. f"(x), like any deriv, shows certain properties for f'(x), which in turn indicates certain things for f(x).  When f" is positive, the function f' is increasing, therefore f is concave up. The opposite is true for when f" is negative. The local maximum is the highest value on the part or the curve, while the local minimum is the lowest. A critical value is a point where the deriv is undefined or =0. It is also the maximum or minimum of a function. For a function f, the min occurs when a function changes at p when the function changes from decreasing to increasing. When the change is form increasing to decreasing, it is a max. Because concavity is related to a function increasing or decreasing, when x is concave down there is local max, when it concaves up there is a local min.

Changes in convavity result in inflection points.

Challenges. 

How can p be a point. Isn't a point an x and y value, there for how can there be f(p)?

Reflection.

I have gone over maximum and minimums and inflection points in my previous calc course.  I think the better understanding of derivs i now have allows me to understand the implications of concavity and therefore max and mins, because they occur at points of concavity.

4-8

Main Points

The gradient is the vector that included the derivative of both the x and y coordinates of a point.

While partial derivatives state the rate of change in either a straight vertical or horizontal point, a directional derivative shows the rate of change in any direction. Gradients have certain defined traits. They are always directed toward that of the largest increase and point away from greatest decrease. Length and slope steepness are directly proportional. For contour diagrams, the gradient vector is perpendicular to the curve it starts at. All gradient vectors are directly related to and there fore correspond to curves,

Challenges

Don't we normally find the deriv at a certain point. Why is this now a partial deriv? I don't understand the notation for the directional deriv. What does it mean that the gradient vector points in the direction of greatest increase? Greatest increase of what?What are the curves in the gradient contour diagrams?

Reflection

The idea of gradients and directional derivs builds nicely on my knowledge of derivs and partial derivs. However I cannot think of them as contour diagrams.

3.5, 9.3, 9.4.

Main Points

The derivatives of sine and cosine are closely related to the functions themselves. The derivative of sine is cosine, and the deriv. of cosine is sine. This is true regardless of if you are finding the deriv. of x in radians or an actual function. To find the deriv of a function times a constant, the constant is taken out and multiplied by the sin/cosin of the function. 

To find the deriv of a multi-variable function, the same rules apply for when finding the deriv.s of regular functions. To determine the partial deriv of f(x,y) either x or y is held constant. They can be determined through tables, equations, and contour diagrams. The deriv. of a partial deriv is called a second-order partial deriv. Any function with two variables will have 4 of these type of derivs.

Challenges

 Why is the deriv of sin=cos, but the deriv of cos=-sin, not sin? What is a mixed partial deriv.?

Reflection

I have never heard of second order partial deriviatives, but the concept makes sense. if each partial derivative has 2 derivs, it would make sense that they each have 2 derivs, resulting in 4 second order partial derivs. I don't really understand them, but I think that you can leave one variable constant each time, resulting in the 4 posibilities. it is interesting that leaving y constant for the partial deriv and x constant for the second-order is equal to if you made x constant and then y. I hope to learn more about why this is true.

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

Main Points

3.1 lists several formulas for finding derivatives that i had to memorize last year: the derivative of a constant=0, ect... These rules allow us to find short cuts when determining a function's derivative. The derivative of a^x is closely related to the original function. As values of a change, the graph of the derivative varies in its similarity to the original. When a is a certain number e, it naturally has a derivative that equals the function.

Challenges

why is f'(x) of f(x), m in the equation? I thought the derivative was the slope of the tangent, not the slope of the actual line. I do not understand logarithms in general, their relationship to derivatives,and to e. 

Reflections

I memorized all these rules last year so they are familiar to me. However now that we've done some graphing first i understand why the derivative of a constant is zero. The deriv of a line is a constant, so it makes sense that if u have a constant, it would have percentage change of zero and a deriv. of zero. I was wondering if e had special implications for nature as well as logs and ln. Is it a number that is found in the ration of plants, animals, stars, ect...?

2.2 and 2.3: the derivative of a function and interpretations of a derivative

 Main Points

Because the slopes of the points of a function have numerical values, these numbers can be plotted to create both another graph and another function. The derivative function is the derivative of a y value at an x value.  Because the derivative is the instantaneous change which merely looks at the average rate of change for 2 points infinitely close, the f' can also be written as change in y over the change in x. The equation for this is delta y is approximately f' times delta x.

Challenges.

How can a a bunch of tangent lines (the derivative) form a graph? Why can f'=delta y/delta x. F' is the slope of the tangent line, not f(x). Why does the derivative show how quickly the function is chaning?

Reflections

When first reading the section about derivatives having derivatives, it made no sense to me. I just remembered though that in calc last year we found f' and f'' and f'''. It makes sense because you can graph velocity and acceleration and jerk.

1.3 and 2.1: rates of change and instantaneous rate of change

main points

When giving a a list of data, their average can always be determined. Therefore even when there is no constant rate of change there always an average rate of change. A function is said to be increasing if y increases proportionally to x. The function is decreasing if y increases disproportionally to x. Because y changes with respect to x, as the x values increase the change in y is the change between the y coordinates those two x values. Periodic functions go through increases and decreases and there for concave up and down, respectively. 

The instantaneous rate of change is the the velocity at a specific point. Its value must be calculated, because this value is based on average velocity, which gets more precise for a given point as the intervals decrease. Instantaneous velocity is also the smallest interval of the average velocity of the surrounding points, because two points are required to find an average.

As a tangent line becomes more horizontal, the interval between the two points it hits on the function line decreases, therefore indicating the derivative.

challenges

If values increase proportionally to each other, does that mean that they both increase or does is mean that they both increase by the same factor?

I know they are related because the points for corners of a triangle, but I don't really understand why the change in the function between two x values is delta y.

Why can't the actual velocity of the grapefruit be calculates, as opposed to estimated?

How exactly is the tangent the derivative. How can another line represent the velocity?

reflections

I think that the concept of average rate of change is very important because in real world situations, I assume that data actually literally increases/decreases with a constant rate of change. However a pattern an be gauged and assumptions can still be drawn about the function by using the average rate of change.

The visualizing rate of change section looks very similar to a diagram my old math teacher showed us when explaining derivatives, so I'm guessing they are closely related.

9.1 + 9.2

Main Points

Certain values (variables) can dependent on multiple factors. Therefore the dependent variable R can have both x any y as its independent variable. Increasing functions are those which have one value increasing while the independent variables remain the same. Contour diagrams can be used to visually represent 2 variable functions. Taking a map of the U.S. for example, the map can serve as a graph because the horizontal and vertical increases represent latitude and longitude, the independent variables, while the third is represented by written numerical values  and shown pictorially through a diagram. Contour diagrams can be thought of as a birds-eye-view of a certain geographical feature, when appropriate. The Cobb-Douglas Production Function has important real world implications. This function allows one to determine the output of two different methods, or both combined. Contour tables, and diagrams (which utilize 3 axes). The algebraic formula for contours is f(x,y)=c.

Challenges

Is the definition of an increasing function that R increases while either x or y increases and the other stays constant?

How is dividing a country into zones mean that there is a graph of 2 variables. I am used to thinking of graphs as a line or curve and do not comprehend how a bunch of diving lines constitutes a graph. I don't understand the graphs of the Cobb-Douglas function, or how it can be used to determine how much of both, for example, the number of workers and value of the machines, would increase out-put. I also don't understand what cross contouring is.

Reflections

By looking at one variable at a time, aren't you just essentially making it a normal problem with one dependent and one independent. I therefore don't see the point in it. 

I have used topographical maps for earth science classes but it never occurred to me that they are actually graphs of 3 variable functions. Its pretty crazy that contour diagrams are both diagrams and graphs at the same time. The plugged in formula for contour function seems very similar to that of a circle. 

exponential functions, growth and decay (1.5+1.7)

Back round

Hazel Schaeffer, Freshman

I intend to major in Political Science, and perhaps minor in English or International Studies.

In high school I took trig, pre-calc, and calc.

Trig is my least favorite, what I least remember, and is probably my weakest part.

My strongest part is probably calculus because I just took it.

I am taking this class because I do enjoy math when I understand it, because I wanted to take a variety of classes in my first semester, and because I wanted to satisfy my math requirement while my calculus is freshest.

I hope to pacify my fears that I cannot take a college math class. Because I am intending to major is poli sci and because the world is filled with numbers, I hope to learn skills I can apply to my real life.

My worst teacher was boring so paying attention in class was difficult. However I value clarity over excitement because math naturally flusters me.

My favorite math teachers are engaged and are humorous, therefore encouraging me to pay attention and work hard. If I do not understand something (which is often), they are patient and do not make me feel bad. However they always try to be clear.

Personally listening to music makes it harder for me to study. However supposedly, according to some study, listening to Bach or Mozart or Beethoven (or some classical musician that I do not remember) improves studying.  In general I like the  Red Hot Chili Peppers and Fleetwood Mac.

During class I found the clicker system a hindrance. I felt under pressure to click in before you ended the time, there forfeiting 85% of the participation grade, so I felt flustered and was too busy paying attention to the number of people who had answered to properly concentrate. When I noticed a majority of people had answered I merely guessed so as to not lose the points. I know this is silly and that hopefully I will get used to the system, but I would appreciate if for the first few classes you could go slower so I will feel less pressured.