Linear Algaebra Section 4: Dot Products; Projection on a subspace; Least squares curve-fitting.

I had the most difficulty with example 4.4; I feel I may need a crash course in trigonometry! But I do appreciate the dot product is revisited later on in this class. But Least squares are very confusing, how can a single constant satisfy both equations that have no solution; is this approximation again? But if possible can we go over how to use the multiple of x and r (the residual vector) to determine the closest correct answer? also I would like to know what exactly the variables in this section represent; like what they are telling us about the vectors.

I suppose the idea of approximating solutions is better than no solution at all. As difficult as the calculations can be, they are useful. There are many situations that do not have an exact answer but approximations or the best educated guesses are the best available information we have.

Linear Algaebra Section 3: Span of a List of a Vectors

A span of a set of vectors can be linearly dependent or independent and there is the concept of the dimension of a the span of a set of vectors. What I found difficult was the idea of redundant vectors. If a combination of matrixes cancels a variable out of the final solution, than why have it in the matrix to begin with?

My concern with this section is the importance of whether a matrix is linearly dependent or independent. How does it apply to a real world situation? OR is this section suppose to show us other ways to manipulate vectors, I am having trouble realizing what the purpose of spanning matrixes is.

Linear Algebra.Section2.Linear Equations: two inerpretations

It has been so long since I have last seen vectors!!!! That is with the exception of the gradient vector. The most difficult idea was the matrix notation. I understand that an m*n matrix represents m rows and n columns, but i don't see how the Ax=b notation is logical. Otherwise Section 1 did a great job of explaining how vectors can be manipulated to show dilations, contractions, and multipliers.

I think matrixes would be useful because it allows you to solve multiple equations by calculating multiple systems simultaneously. Efficient calculations are appreciated in many places of business!

Modeling the Spread of Diseases

What is most difficult about this section are all the variables needed to follow the S-I-R model for diseases. S are those susceptible, I those infected, R those recovered/removed. Than there is a,that measures how infectious the disease is and b, is a constant that represents the rate that infected people are removed from the infected population. This gets confusing! but what does make sense is that the threshold population would be equal b/a. If S is greater than the threshold than there is an epidemic, if S is less there is no epidemic.

What is important from the S-I-R model is than it works for more than hypothetical situations. There have been historical incidents of infections diseases and it is important to know the likelyhood of if a population of people will be infected in order to control outbreaks and for public safety.

Modeling the Interaction of Two Populations

The idea behind the interactions of two populations is well explained in the robin and worms example. Essentially one population can only exist with the population of the other group, there are limits to the quantity of the groups. It is an example of sustainability and co-dependence. The hardest part though was looking at trajectories in the wr phase plane. when w,r, and t were given for dr/dt it = 1.2>0 what is the significance of this numerical value? Is this the value for the closed solution curve on the slope field?

Using math that involves two different entities is a good progression. I feel like it is bringing the "applied" portion back into calculus because the predator prey system is a real situation that exhibits a cyclic behavior. It is important to see math in more ways than just numbers and this is a connection I wanted to get out this class.

Exponential Growth and Decay. Applications and Models

All of the examples for Exponential growth and decay in 10.4 were great, but the one I had difficulty with was Continuously compounded interest. I understand that dB/dt is the rate (0.05) times the amount in the savings account, but the book says the intial amount of $1000 doesn't matter? why though, since the amount in the account would be the initial $1000. Why does the ODE not need the initial value? What is confusing is that in the solution of the ODE they do use the initial $1000 as B(sub0). Could you explain initial values and their involvement with ODE in class?

What I really enjoyed were all the applications of ODE's in section 10.5. I felt as if the book used an example from every possible situation so that every one could relate to at least one example. Similar to how economic examples will confuse me to no end, there is opportunity in drug quantity problems or physics (Newton's Law)!

I think that the only confusing situation that the class may have trouble with is the equilibrium solutions. It begins like a drug elimination ODE but the idea of stability/instability depending on if the reaction goes to infinity needs clarification. I personally have worked with biological and chemistry equilibriums and I have never come across equilibriums in math before.

By the way Chad... Amazing job on your swarming presentation, it was really interesting. I must admit though I chuckled when I saw a differential equation on the power point!

Slope Fields

The most difficult part about slope fields was really visualizing. Is a slope field really suppose to simplify things? How I see it is that at each point you have to calculate the slope of each y coordinate in the plane. It just appears to be inconvenient. What I had difficulty with was solving differential equations to find families of solution curves, I'm not sure but if there are multiple solutions, are there limits to what the solutions can be? I guess I am trying to figure why slope fields are useful and what they can show us besides the rate of change at specific points for functions; or is that the point and I am not seeing its importance?


What I did find interesting was how the idea of differential equations with initial values are incorporated into slope fields. Initial values equations represent real situations that have a real (unique) answer. Its gratifying to have math presented in reality so there is substance to the ideas we discuss.

Setting Up Differential Equations

Essentially a differential equation can be written from the information provided. All of the examples in section 10.1 were from verbal description. The most difficult part was the logistic model, because I personally have difficulty imaging a graph from a verbal description, for example when dP/dt =kP(L-P) the graph showed delta p small's Im not sure how that was known to satisfy the curve.

My favorite example was about the dug in the body, I like bio-pharmaceutical problems, they are easier to visualize... much more favorable than economics problems... I know its a personally preference, but hey if it helps the math skills all the better. Also, since I work in a pharmacy it was funny to actually recognize the drug used in the example, Vancomycin.

Constrained Optimization

This was sort of a strange concept but page 381 broke down how each step was obtained was amazing! I believe the constraint on a function is actually a second function that limits the first. The hardest part was the Lagrange multiplier.It was interesting that it is a rate of the change in the optimum value of f over the change in g. but i don't understand the purpose of the Lagrange multiplier, what does it have to do with constrained optimization?

On a random note... are constraints represented graphically as contour maps?

What was neat was how useful constrained optimization is, it really applies to real world situations. Like businesses and budgets. Seriously if you do not put constraints on functions than you can see all possible values, but not the ones that are important to a situation.

Seriously we are in college we need to budget just about everything from time to money to life!

Global Maxima and Minima

It was really interesting to see that global maximum and minimum differ if the endpoints of a function or included or not. But it makes sense that if there are no endpoints there can not be a global min or max because the function is continuing, you may not be able to see the real max or min. I had some difficulty with example #2; the question asks for the time when photosynthesis is fastest, but the rate at that time so is time the first derivative and the rate of at the time ,t, the second rate? Its the units I am trying to get to that's confusing.

What I found interesting was the graphical example, about minimizing gas consumption because I commute to my job about 45 minutes away every weekend and that driving speeds effect gas consumption and that is helpful to know since I unfortunately drive and SUV... yeah 14 miles per gal....

The Second Derivative, Local Maxima and Minima, and Inflection Points

Its really cool to think that f",f', and f are related. f" lets us know depending on whether it is less than or greater than zero, which describes if f' is concave up or down! I enjoy clear connections between ideas. However I did not understand the government defense budget example, perhaps that was worded funny?

Local Maxima and Minimum were not too hard of an idea to grasp, but one clarification is the critical point f'(p)=0 reflected in the original function as crossing an axis? I want to make sure I am drawing the correct conclusions from the reading. I had the most difficulty with the idea that critical points could divide the domain of f into intervals. If there are critical points, must we look at the function f in sections defined between two critical points or can it still be looked at as a whole?

The idea of an inflection point was pretty easy, its just when the concavity changes, but its neat to realize inflections occur when f" changes signs.

I think the class will have the most difficulty with keeping all the relationships together. Also perhaps the units of the second derivative. Understanding f" as a rate of change, I'm not quite sure if it is as simple if the second derivative is positive then the rate of change is increasing and vice versa. It seems to simple to be right... Or am I over thinking it?

The mid-term exhausted mind is a funny thing!

Partial Derivatives and Computing Them Algebraically

I have actually never been exposed to the concept of partial derivatives before and was confused when the books definition of a partial derivative contained limits. I thought this class wasn't planning on using limits??? Also i don't think the purpose of a partial derivative is explained either? Is it just a fraction of a rate of change?

Perhaps the class will have the most difficulty with second-order partial derivatives, especially if the concept of partial derivatives was difficult to begin with. But what I did like about these sections were familiar examples used to try and explain this type of function.

Derivatives of Periodic Functions

This section was not too terribly difficult, and rather short too. But its good to know that the derivative of sine is cosine. But what I found the most difficult is that the reverse is not true. The derivative of cosine is not sine, it is -sine. Why is that?

I think the most difficult thing the class will work with is looking at a function and having to figure out which of all the available rules we were introduced to in this chapter to use!

The Product and Quotient Rules

After Monday's class the chain rule was much clearer and section 3.4 introduces to new rules to help with derivation; the product and quotient rules. I didn't find this section to terribly difficult, but I wished a couple more q's were assigned for homework for more practice. Honestly the most difficult part was remembering that for the quotient rule the denominator is squared. By the way why is it that the derivative is squared. Can you explain that in class?

I believe that the most difficult idea for the entire class would be the order of which derivative should be taken for each piece. For example the product rule, (fg)'= f'g+fg'
It looks simple but sometimes when you forget, it can mess it all up. So the class has to remember the derivative of f multiplied by g, then add that to f multiplied by the derivative of g.

The Chain Rule

The eight different examples used to explain the chain rule were very helpful. I found the car situation the best at explaining this rule. To find the rate gas is being consumed is rationalized by seeing how units cancel eachother out. What I found most difficult was example 2, I felt like the chain rule was trying to use composite functions... Maybe it was just fuzzy, because I understand the idea of inside and outside functions but just reading on my own i don't feel confident in it.


The class may have the most difficulty with using the chain rule without units, such as the bank account example (#8) where you cannot double check you answer by seeing if the units are correct. On the bright side there is familiarity in the idea its a question about rates.

Short Cuts to Derivatives and Exponential and Logarithmic Functions

Finally a direct rule! The power rule is officially my favorite. Section 3.1 wasn't too terribly difficult. It was more exciting to have more rules to use so that taking derivatives is more efficient. But what i did have some difficulty with was trying to understand how derivative of ln(x) = 1/x. Is there a proof to show how this is true or came about? I just don't like to use rules if i don't understand how it works.

As a whole the class may have trouble with the behaviors of the original function and the derivative. for example Monday's NYT about what f'(x) can tell about f(x) not everyone could understand what happens from one range of values to another. (increase,decrease, no change) So perhaps before we step forward into analyzing more complex functions.

until next time

The Derivative Function and Interpretations of the Derivative

The most difficult part of sections 2.2 and 2.3 was Lebniz's notation. In high school I had seen derivatives as f'(x)= dy/dx but i had never known that the notation was to indicate that the derivative is a limit of ratios. Just by using multiple notations to imply the same point [the derivative of a function] gets cofusing. I am kind of wondering why Gottfried Wilhelm Leibniz felt the need to use d instead of delta?

I think the class will have to work through estimating derivative values from looking at a graph. I mean the derivative is the slope of the tangent line of a point on that graph, but its not as easy as it sounds. Also looking at graphs everybody will need to be able to predict how a graph of f'(x) will look depending on f(x)

By the way, as we progress though the examples in the book, i am finding more biological uses for math and that's pretty exciting since its connecting math to something very important/interesting to me. See example 8- health care costs or problem 37-diaphragm pressure.

Instantaneous Rates of Change

What i found most confusing was how the book tried to use two different approaches when trying to explain that a derivative is the average rate of change at a point. I was personally confused when instantaneous velocity was defined to be the limit of the average velocity of the object over shorter and shorter periods of time intervals. Table 2.1 was just not that effective.

As a class I believe the most difficult area will be visualizing the derivative of the graph. At a point the derivative is the slope of the tangent line of the point on a graph. Again I believe we will need practice finding the derivative from data, a graph, and from an equation. But i think as we do some more examples it will become clearer.

Functions of Two Variables and Contour Diagrams

The most difficult idea from sections 9.1 and 9.2 was probably the Cobb-Douglass Production Model. It is a completely new concept to me, but the example about a small company going through expansion, was very helpful. There is the idea of holding one variable constant while seeing how the other changes. The other example modeling the production of the US economy was precise. apparently the equation P= 1.01L^0.75K^0.25 is very accurate, but how was this equation developed?

I see the class having difficulty with the contours on a topographical map. Contour diagrams are common in maps and weather charts, but honestly i never analyzed the functions before. For example figure 9.10 I have never seen how a pass between two mountains was represented.

Periodic Function

So I was a bit thrown of when the pages of 28 -32 did not correspond to periodic functions. so I stuck with reading section 1.10.

Essentially periodic functions have values that repeat and consistent intervals, and its really cool how there are loads of real life examples of periodic functions such as music or blood pressure.

The Most confusing or difficult part of a periodic function that I found was which mode to be in; radians or degrees. The book says to assume work in radians, but what is a radian??? I know degrees is 360 in a full circle, but where does radians come into play again? I think i knew the where and why once upon a time but its hard to use something you don't quite understand.

I think the most difficult part of periodic functions will be coming up with an equation just from looking at the graph. Its not as obvious as say a linear function. There are more pieces of the graph to consider; there are parameters that fluctuate as amplitudes, periods, and shifting occurs.

As for R, the second lab appears to be going better or I should say more smoothly. I think its due to the fact that we've now had exposure to HTML entering.

More to come....

Exponential Growth and Decay

It was kinda cool to think that a lot of events in life are actually exponential growth or decay. What I found to be most difficult was present and future values. Specifically inflation represented as the future value that is dependent on the interest rate. Its hard to keep track of interest when it is being compounded.

I would think that most of the class, myself included, would have the most difficulty with deciding which of the given functions (compounded annually v. compounded continuously) would be appropriate for a problem. For instance example 8 about the lottery. For the option of being paid out assuming a 6%interest rate, i didn't understand why r=-.06, is that compounded interest?

I just feel as if it were alluding to economics, which is in my opinion a difficult area of study.

I can relate better to the environmental examples.... i seem to have a better grasp on a concept when i can relate.

Exponential Functions and Natural Logarithms

R- hmm can't be sure if I am in complete awe yet. My partner, Katie and I, have spent some more time working with R and its a fun program, but very picky about how many parenthesis are necessary in a command. We are also having issues with adjusting limits on our window. But more on that later since we are not done with the lab as of yet.

The most difficult part about this section was from 1.6. There are a lot of properties of the e that can be used to solve word problems, but its actually difficult to remember that the actions are valid ways to solve a problem. Kind of like a tool in a toolbox that you didn't know you had. I was working through problem # 41 about when there would be 1 vehicle per person and in order to solve it i had to bring down the exponent. I eventually solved it, but i had to figure out which property was going to help me get to the next step. Turned out that it was ln (A^p)= p ln A.

I think for the class as a whole the most difficult thing would be setting up the word problems. First, you need to figure out what the question is, then how to solve it. A lot of times there are hints to whether or not you are looking into an exponetial growth or decay problem. Most times in was a percentage of growth where in the function it would be used as (1.00+ x%) or if it were decay in would appear as (1.00- x%). once you figured out the rate of change, you had to remember the base was to be when time=0 or the initial quantity. So i geuss its best to just disect the problem until you have found all the pieces to put into the function.

more on feeling about R later...

linear functions and rates of change

So I am trying to get back on the right schedule with blogs, but to make sure all basis are covered I'll begin with sections 1.2 and 1.3.

The most difficult part from these sections was calculating the intercept (b) in order to fulfill the linear function of Y= f(x) = b + mx. The slope m was easier to calculate (rise over run) from using two different points. but b was more difficult because sometimes its confusing which equation to set equal to 0 in order to find the needed intercept. It is often necessary for me to re-read the question in order to see what the question was asking.

Being interested in medicine, my favorite question from the problem set from 1.3 # 26- 31 pertaining to maximum heart rates. By using two different equations to draw conclusions from; the MHR is actually not linear using the more accurate equation of MHR= 208-0.7a

Since next class is a lab, there isn't suppose to be a Blog... so stay tuned to hear about R

Functions and Composite Functions

The most difficult part about section 1.1 and 1.8 was opening the book! No, I am just kidding from section 1.1 I was having difficulty determining from the tables, graphs, formulas, and words which variables were the dependents and which were the independents. I had to actually stop and think aloud, if y were to happen, it would depend on x. Section 1.8 on compositions weren't too bad, i understand which function should be performed first, and then have that answer used in the next function. Graph transformations were a bit more difficult. I get the idea that graphs can be repositioned but multiple transformation or stretches may be more difficult.

What I found interesting was that and EKG pattern was used as an example. I volunteer at the ED at a hospital and I had seen the graphs before. I knew they measured electrical activity as a function of time, but it put functions in a more meaningful (and real-life example) light.

lecture 9.5.2007

Nothing, is not an acceptable answer. But today was more of an introduction to the course and material wasn't really discussed.

Scratch that thought; the two main concepts are functions and differential equations. I have had experience with functions, so i suppose differential equations would be the most difficult out of the two.

In class today when were brain-storming examples of how differential equations cab be used to answer questions, I think with the time restraint it was hard for everyone to come up with different variables or factors that alter and outcome of a situation.

In my small group we thought of obesity and the variables we could come up with were diet, exercise, genetics, culture, mother's cooking talents, and location. It sort of served as an ice breaker to be able to get out of our seats and talk about it.

Who is Lucky?

Well, my name is Lucky Homesombath; and yes, that is my given name. It has grown on me and now I love it.

At the moment I am beginning my sophomore year at Macalester.

I intend on earning my degree in biology with a minor in anthropology.

Applied Calculus would be my first mathematics course at Macalester, however I have taken calculus in high school, but that was over two years ago!

From my mathematical experiences, I would have to say that trigonometry (sine, cosine, etc.) was most challenging.

I asked to add applied calculus, because I like math, but it is also required for my biology major.

From taking calculus, I hope to have a deeper understanding of where calculus applies to biology. In my opinion, calculus is prevalent in physics and chemistry... but i haven't seen too much in biology yet.

As exciting as math is, I am also interested in music (listening, not playing unfortunately), dance, reading, I am very open to trying new activities.

My worst math teacher.... was back during Algebra 2, unless you were going to be his next prodigy he didn't give the time of day to help those who were struggling. As long as your were going to pass into the next class, he wasn't concerned whether or not the material was learned.

My best math teacher... was actually my high school calculus teacher. She was approachable, yet at the same time very professional. She valued our opinions on the topic and took into consideration how the students wanted to learn calculus. We were enjoying and learning. And for some reason I can vividly remember her crazy cackling laughter.

I think I am going to like applied calculus, and I like your laid back approach. and I agree that having the general understanding that mistakes will be made when learning calculus makes it easier to try and try again.

See you in class Friday,

Lucky H