1. Main Points
In todays reading I learned how to make a projection of vector on a line through the origin. For thus vector there is a multiple ˆmx of x that reaches closest to y. That multiple is called the projection of y on the line spanned by x. The step from the tip of the projection ˆmx to y is the residual vector r. I will never know the value of m; but I can find ˆm which is my best estimate of m. I can evaluate the quality of our estimate by the length of the residual vector r, because IIkII is the distance between ˆmx and y, the two sides of what we had hoped would be an equality between mx and y.
One can also do curve-fitting with two parameters leads to projections onto 2-dimensional subspaces. Allowing one to find the linear combination ˆmu +ˆbv that comes closest to s. Lastly, least-squares curve-fitting with J parameters leads to projection onto subspaces spanned by J
vectors.
2. Confusing
The fundamental problem of linear modeling was hard to grasp.
3. Intruiging
The applications of such linear modeling must be almost limitless. The example of the business who wanted to investigate the return on the money that they put on advertisement and used curve-fitting with two parameters to do so is a good example of this.
Monday, November 24, 2008
Wednesday, November 19, 2008
Vector projection (11/20/2008)
1. Main points
In today’s reading I learned that for equations that have any solution one can make approximate solutions. The main tool for doing this is using the dot product of vectors, obtained by multiplying corresponding coordinates of the vectors and adding. The most important fact about the dot product is that the dot product of two vectors is zero if the vectors are perpendicular and thus nonzero if the vectors are not perpendicular.
2. Challenging
Why is the dot product zero if the vectors are orthogonal?
3.Interesting
I found it interesting learning about methods to get approximate answers to equations and questions that do not have any solutions, I can see how this will can be very useful in statistics etc.
In today’s reading I learned that for equations that have any solution one can make approximate solutions. The main tool for doing this is using the dot product of vectors, obtained by multiplying corresponding coordinates of the vectors and adding. The most important fact about the dot product is that the dot product of two vectors is zero if the vectors are perpendicular and thus nonzero if the vectors are not perpendicular.
2. Challenging
Why is the dot product zero if the vectors are orthogonal?
3.Interesting
I found it interesting learning about methods to get approximate answers to equations and questions that do not have any solutions, I can see how this will can be very useful in statistics etc.
Monday, November 17, 2008
11/18: Linear combinations, linear independence, span
1. Main Points
One can combine the two operations, adding multiples of two vectors. A sum of multiples is a
linear combination.
A system of equations for unknowns x and y
ax + by = e
cx + dy = f
Has two fundamental but different geometric interpretations: the points interpretation and the vector interpretation.
Linear combinations of vectors occur so frequently that a special notation has been developed
using matrices. This is defined as an m × n matrix which is a rectangular array of numbers, arranged in m rows and n columns.
2. Challenging
Linear combinations in higher dimensions was a little challenging to think about.
3. Intruiging
I can see how linear combinations can be applied in many areas of science!
One can combine the two operations, adding multiples of two vectors. A sum of multiples is a
linear combination.
A system of equations for unknowns x and y
ax + by = e
cx + dy = f
Has two fundamental but different geometric interpretations: the points interpretation and the vector interpretation.
Linear combinations of vectors occur so frequently that a special notation has been developed
using matrices. This is defined as an m × n matrix which is a rectangular array of numbers, arranged in m rows and n columns.
2. Challenging
Linear combinations in higher dimensions was a little challenging to think about.
3. Intruiging
I can see how linear combinations can be applied in many areas of science!
Monday, November 10, 2008
SIR and the spread of disease (11/11/2008)
1. Main Points
In todays reading I learned how to apply differential equations to predict epidemic outbreaks just like the CDC. The model used for an epidemic is the S-I-R model, where S is the number susceptible, I the number of infected and R the number of recovered.
dS/dt =-(Rate susceptible get sick)= -aSI
dI/dt=(Rate susceptible get sick)-(Rate of infected get removed=aSI-bI
The constant a measures how infectious the disease is, meaning how quickly it is transmitted from the infected to the susceptible.
a=-(dS/dt)/SI
The constant b represents the rate at which infected people are removed from the infected population.
Threshold population = b/a
If the initial number of susceptible is above b/a, then there is an epidemic, if the initial number of susceptible is below b/a, then there is no epidemic.
2. Challenging about the material
Looking at an epidemic in terms of a phase plane was a little confusing I thought, maybe we can go over this example in class?
3. Intriguing
This was one of the more interesting readings for me yet in this book. I am currently taking medical anthropology where we are talking a lot of epidemic outbreaks and containment of such. Therefore I found it very interesting how to interpret epidemics mathematically!
In todays reading I learned how to apply differential equations to predict epidemic outbreaks just like the CDC. The model used for an epidemic is the S-I-R model, where S is the number susceptible, I the number of infected and R the number of recovered.
dS/dt =-(Rate susceptible get sick)= -aSI
dI/dt=(Rate susceptible get sick)-(Rate of infected get removed=aSI-bI
The constant a measures how infectious the disease is, meaning how quickly it is transmitted from the infected to the susceptible.
a=-(dS/dt)/SI
The constant b represents the rate at which infected people are removed from the infected population.
Threshold population = b/a
If the initial number of susceptible is above b/a, then there is an epidemic, if the initial number of susceptible is below b/a, then there is no epidemic.
2. Challenging about the material
Looking at an epidemic in terms of a phase plane was a little confusing I thought, maybe we can go over this example in class?
3. Intriguing
This was one of the more interesting readings for me yet in this book. I am currently taking medical anthropology where we are talking a lot of epidemic outbreaks and containment of such. Therefore I found it very interesting how to interpret epidemics mathematically!
Wednesday, November 5, 2008
Interacting systems: modeling, phase plane, and trajectories (11/6/2008)
1. Main Points
In this reading I learned how to use differential equations to model the growth of two interacting populations, a system requiring two differential equations. One can get an idea of what the solutions for these equations look like from a slope field.
2. Challenging
I had a hard time wrapping my head around the idea of a phase plane, maybe you can explain these in detail in class?
3. Intriguing
Being a biology major and having the opportunity to learn how to model different interactions between populations is really cool! The population size of the two species worms and robins modeled as a function of time I thought was a really interesting example of this!
In this reading I learned how to use differential equations to model the growth of two interacting populations, a system requiring two differential equations. One can get an idea of what the solutions for these equations look like from a slope field.
2. Challenging
I had a hard time wrapping my head around the idea of a phase plane, maybe you can explain these in detail in class?
3. Intriguing
Being a biology major and having the opportunity to learn how to model different interactions between populations is really cool! The population size of the two species worms and robins modeled as a function of time I thought was a really interesting example of this!
Monday, November 3, 2008
Exponential growth and decay / Applications of ODE, equilibria, and stability (11/4/2008)
Main Points
In today’s reading I learned about how to compute and model exponential growth and decay, equilibrium and Newton’s law of heating and cooling with ODEs. The most important concept in doing this is that the general solution to dy/dt=ky is y=Cekt and is considered exponential growth for k>0 and exponential decay for k<0.
I did not quite understand what the difference between a stable/unstable equilibrium was.
Intriguing
A lot of things in our world exhibit exponential growth or decay, therefore it is very interesting to learn about this because of its wide applications to the natural sciences!
Wednesday, October 29, 2008
10/30/2008: Intro. to differential equations, solutions to differential equations, Euler’s method
1. Main Points
In today’s reading I learned about differential equations which are employed when one does not know a key function but has information about its rate of change or its derivative. This information is then used to write a differential equation from which one can get information about the original function. A solution to a differential equation is any function that satisfies the differential equation.
2. Challenging
I did not quite understand how we check a solution to a differential by substituting the solution into the left and right sides of the equation separately.
3. Intriguing
I found the example of a population in confined space growing proportionally to the product of the current population, P, and the difference between the carrying capacity, L, and the current population, to be really cool. This can definitely be applied to bacteria growing in a petrie dish for example!
In today’s reading I learned about differential equations which are employed when one does not know a key function but has information about its rate of change or its derivative. This information is then used to write a differential equation from which one can get information about the original function. A solution to a differential equation is any function that satisfies the differential equation.
2. Challenging
I did not quite understand how we check a solution to a differential by substituting the solution into the left and right sides of the equation separately.
3. Intriguing
I found the example of a population in confined space growing proportionally to the product of the current population, P, and the difference between the carrying capacity, L, and the current population, to be really cool. This can definitely be applied to bacteria growing in a petrie dish for example!
Monday, October 27, 2008
10/28: Constrained optimization and Lagrange multipliers
1. Main Points
In todays reading I learned how to solve a constrained optimization problem using Lagrange multipliers and Lagrange functions. A lagrange multiplier is an approximation of the change in the optimum value of f when the value of the constraint is increased by 1 unit. In terms of production, it can be seen as the “extra bang for your buck”. Lagranges multiplier is calculated by taking the function to be optimized over the constraining function. Lagrangian functions are used when dealing with constrained optimization problems by first writing down the Lagrangian function and then finding the critical points of the Langrangian Function.
2. Challenging
I did not quite understand why we find the critical points of the Lagrangian function when solving constrained optimization problems nor what they represent.
3. Intriguing
The Lagrangian multiplier and function must be immensely useful when doing calculations for business management. Now I understand what the basis for the expression “getting the most bang for your buck” is!
In todays reading I learned how to solve a constrained optimization problem using Lagrange multipliers and Lagrange functions. A lagrange multiplier is an approximation of the change in the optimum value of f when the value of the constraint is increased by 1 unit. In terms of production, it can be seen as the “extra bang for your buck”. Lagranges multiplier is calculated by taking the function to be optimized over the constraining function. Lagrangian functions are used when dealing with constrained optimization problems by first writing down the Lagrangian function and then finding the critical points of the Langrangian Function.
2. Challenging
I did not quite understand why we find the critical points of the Lagrangian function when solving constrained optimization problems nor what they represent.
3. Intriguing
The Lagrangian multiplier and function must be immensely useful when doing calculations for business management. Now I understand what the basis for the expression “getting the most bang for your buck” is!
Monday, October 20, 2008
Unconstrained Optimization in 1-d and 2-d (10/21/2008)
1. Main Points
The main point of today’s reading was to learn about how to determine global maxima and minima of functions. To find the global maximum and minimum of a continuous function on an interval including endpoints, one compares values of the function at all the critical points in the interval and at the endpoints. When excluding endpoints, one finds the values of the function at all the critical points and sketches a graph.
2. Challenging
I see that they use differentiation, then setting equal to zero and then solving for x in the examples. I do not remember well the tactics used for factoring out when solving for zero, maybe you can go over this quick in class?
3. Intriguing
I realize that being able to maximize and minimize when solving different problems in all natural and social sciences must be very useful. I particularly found the example showing what speed to drive in order to have the smallest gas consumption interesting and very useful. I will drive at about 50 mph as much as I can from now on!
The main point of today’s reading was to learn about how to determine global maxima and minima of functions. To find the global maximum and minimum of a continuous function on an interval including endpoints, one compares values of the function at all the critical points in the interval and at the endpoints. When excluding endpoints, one finds the values of the function at all the critical points and sketches a graph.
2. Challenging
I see that they use differentiation, then setting equal to zero and then solving for x in the examples. I do not remember well the tactics used for factoring out when solving for zero, maybe you can go over this quick in class?
3. Intriguing
I realize that being able to maximize and minimize when solving different problems in all natural and social sciences must be very useful. I particularly found the example showing what speed to drive in order to have the smallest gas consumption interesting and very useful. I will drive at about 50 mph as much as I can from now on!
Monday, October 13, 2008
Second derivative, unconstrained optimization in 1-d (10/14/2008)
1. Main Points
The major new concept in today’s reading was the second derivative (f’’), which is the derivative of dy/dx, and the different ways it can be applied for solving mathematical problems. Some things we can tell from f’’ is that if f’’>0 on an interval it means that f’ is increasing, so that the graph of f is concave up there. The reverse for f’’<0. The second derivative can also be used to test for local maxima and minima for a continuous function f, called critical points. The sign of the f’’ will also change at an inflection point, which is defined as a point at which the graph of a function f changes concavity.
2. What was challenging about this material?
The hardest part of todays rading I think was locating inflection points, maybe you could go through this at greater length during class?
3. To think of the second derivative as the rate of change of a rate of change is a really cool concept I think. The example in the reading about the senate explaining that they had not cut the defense budget, only cut the rate at which it was increasing. Essentially, the derivative of the defense budget was still positive (the budget was increasing), but the second derivative was negative (the budget’s rate of increase had slowed)
The major new concept in today’s reading was the second derivative (f’’), which is the derivative of dy/dx, and the different ways it can be applied for solving mathematical problems. Some things we can tell from f’’ is that if f’’>0 on an interval it means that f’ is increasing, so that the graph of f is concave up there. The reverse for f’’<0. The second derivative can also be used to test for local maxima and minima for a continuous function f, called critical points. The sign of the f’’ will also change at an inflection point, which is defined as a point at which the graph of a function f changes concavity.
2. What was challenging about this material?
The hardest part of todays rading I think was locating inflection points, maybe you could go through this at greater length during class?
3. To think of the second derivative as the rate of change of a rate of change is a really cool concept I think. The example in the reading about the senate explaining that they had not cut the defense budget, only cut the rate at which it was increasing. Essentially, the derivative of the defense budget was still positive (the budget was increasing), but the second derivative was negative (the budget’s rate of increase had slowed)
Wednesday, October 8, 2008
Gradient and directional derivatives (10/9/2008)
1. Main Points
In todays reading I learned about the gradient which is a vector consisting of the partial derivatives fy(x,y) and fx(x,y). These vectors tell us the rate of change in the x and y directions, directional derivatives on the other hand can tell us the rate of change in any direction in the (x,y)-plane.
2. Challenging
The justification for the directional derivative formula was a little hard to follow I thought. The algebraic steps especially, are we expected to learn this justification?
3. Interesting
It was interesting to learn how to calculate directional derivatives, no longer being constrained to rates of change in the x and y directions, but now I can find the rate of change in any direction in the (x,y)-plane.
In todays reading I learned about the gradient which is a vector consisting of the partial derivatives fy(x,y) and fx(x,y). These vectors tell us the rate of change in the x and y directions, directional derivatives on the other hand can tell us the rate of change in any direction in the (x,y)-plane.
2. Challenging
The justification for the directional derivative formula was a little hard to follow I thought. The algebraic steps especially, are we expected to learn this justification?
3. Interesting
It was interesting to learn how to calculate directional derivatives, no longer being constrained to rates of change in the x and y directions, but now I can find the rate of change in any direction in the (x,y)-plane.
Monday, October 6, 2008
Vectors, dot product, and vector components (10/7/2008)
1. Main Points
Today I learned more about vectors which is a geometric object that has both a magnitude and a direction. A vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B. Vectors can be calculated by multiplication and addition for example.
2. Challenging
The concept of a unit vector and its rationalization was a little hard to understand, maybe you could go over this in class?
3. Intriguing
Vectors is a interesting topic I think because they play an important role in physics: velocity and acceleration of a moving object and forces acting on a body are all described by vectors. Many other physical quantities can be usefully thought of as vectors.
Today I learned more about vectors which is a geometric object that has both a magnitude and a direction. A vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B. Vectors can be calculated by multiplication and addition for example.
2. Challenging
The concept of a unit vector and its rationalization was a little hard to understand, maybe you could go over this in class?
3. Intriguing
Vectors is a interesting topic I think because they play an important role in physics: velocity and acceleration of a moving object and forces acting on a body are all described by vectors. Many other physical quantities can be usefully thought of as vectors.
Derivatives of periodic functions / Partial derivatives (10/2/2008)
1. Main Points
In todays reading I read about how periodic functions also have periodic derivatives. In fact d/dx(sinx) = cosx and d/dx = -sinx for example. I also read about partial derivatives, in general a partial derivative of f with respect to x at (a,b) is the derivative of f with y constant. One can use all the techniques from single-variable calculus to find partial derivatives.
2. Challenging
I found the concept of a second order partial derivative to be somewhat confusing. I do not understand why and when they are used, maybe you could go over this in class?
3. Intriguing
It is interesting to learn calculus that feels like it has more than one dimension. I can see how partial derivatives can be useful for predicting the future in many areas of the sciences as shown in the example dealing with rat population surviving after exposure to formaldehyde vapor.
In todays reading I read about how periodic functions also have periodic derivatives. In fact d/dx(sinx) = cosx and d/dx = -sinx for example. I also read about partial derivatives, in general a partial derivative of f with respect to x at (a,b) is the derivative of f with y constant. One can use all the techniques from single-variable calculus to find partial derivatives.
2. Challenging
I found the concept of a second order partial derivative to be somewhat confusing. I do not understand why and when they are used, maybe you could go over this in class?
3. Intriguing
It is interesting to learn calculus that feels like it has more than one dimension. I can see how partial derivatives can be useful for predicting the future in many areas of the sciences as shown in the example dealing with rat population surviving after exposure to formaldehyde vapor.
Monday, September 29, 2008
Derivatives of composite functions/Derivatives of products and quotients (9/30/2008)
1. Main points
In today’s reading we learned how to find the derivative of a composite function by taking the derivative of the outside function times the derivative of the inside function, called the chain rule. To use this rule one must first rewrite the composite function using a new variable to represent the inside function. We also learn how to find the derivative of a product, which is the derivative of the first times the second, plus the first times the derivative of the second, this is the product rule. Lastly the derivative of a quotient, which is the derivative of the numerator times the denominator minus the numerator times the derivative of the denominator, all over the denominator squared, the quotient rule.
2. What was challenging
There was not a lot of hints in the book to why we use these rules which I would be very useful to be at least elementary be able to do. The examples you do in class to do this is much appreciated!
3. Interesting
I now have a deeper understanding of the many uses of a derivative within calculus and its many uses by learning these rules and how they are applied to make sense of the world around us.
In today’s reading we learned how to find the derivative of a composite function by taking the derivative of the outside function times the derivative of the inside function, called the chain rule. To use this rule one must first rewrite the composite function using a new variable to represent the inside function. We also learn how to find the derivative of a product, which is the derivative of the first times the second, plus the first times the derivative of the second, this is the product rule. Lastly the derivative of a quotient, which is the derivative of the numerator times the denominator minus the numerator times the derivative of the denominator, all over the denominator squared, the quotient rule.
2. What was challenging
There was not a lot of hints in the book to why we use these rules which I would be very useful to be at least elementary be able to do. The examples you do in class to do this is much appreciated!
3. Interesting
I now have a deeper understanding of the many uses of a derivative within calculus and its many uses by learning these rules and how they are applied to make sense of the world around us.
Wednesday, September 24, 2008
Derivatives of powers and polynomials / Derivatives of exponentials and logs (9/25/2008)
1. Main Points
The main points of the reading assignment for tomorrow was to learn more about the derivative and its different uses. For example we learn how to calculate the derivative of a linear function, of polynomials, exponential functions and of ln x.
2. Challenging
The explanation regarding the derivative of the number e was a little bit hard to follow. Maybe you can go over this in class?
3. Most Interesting
The example dealing with the instantaneous growth rate of a population I thought was interesting because of its possible applications within public health.
The main points of the reading assignment for tomorrow was to learn more about the derivative and its different uses. For example we learn how to calculate the derivative of a linear function, of polynomials, exponential functions and of ln x.
2. Challenging
The explanation regarding the derivative of the number e was a little bit hard to follow. Maybe you can go over this in class?
3. Most Interesting
The example dealing with the instantaneous growth rate of a population I thought was interesting because of its possible applications within public health.
Wednesday, September 17, 2008
The derivative function and interpretations of the derivative (9/18/2008)
1. Main Points
The main point of today’s reading was to learn different aspects of the derivative. For example how to find the derivative of a function graphically, what the graph of the derivative tells us, how we can use units to interpret the derivative and how to use the derivative to estimate the values of a function.
2. Challenging
I found the explanation of how to estimate the derivative of a function given numerically a little hard to follow, maybe you could elaborate on this part.
3. Interesting
It’s great to be learning the foundations on which calculus is built on! The derivative is a very useful tool and can be applied in all of the natural sciences.
The main point of today’s reading was to learn different aspects of the derivative. For example how to find the derivative of a function graphically, what the graph of the derivative tells us, how we can use units to interpret the derivative and how to use the derivative to estimate the values of a function.
2. Challenging
I found the explanation of how to estimate the derivative of a function given numerically a little hard to follow, maybe you could elaborate on this part.
3. Interesting
It’s great to be learning the foundations on which calculus is built on! The derivative is a very useful tool and can be applied in all of the natural sciences.
Sunday, September 14, 2008
Rates of Change and the Derivative (9/16/2008)
1. Main Points
In this reading assignment we were introduced to what is behind the concept of the derivative of a function. Basically, the derivative of a function is defined as the instantaneous rate of change of that function at that particular point.
2. Challenging
What I found challenging about this passage was the concept of the limit of average rates of change when defining instantaneous rate of change. I have always thought that the use of limits and the reason why to be a little hard, maybe you could go over that in class?
3. Most Interesting
Instantaneous velocity has always been pretty cool to think about for me. I took physics this summer and did many such calculations under different conditions, again it's interesting to see how the sciences always seem to blend together! The concepts of instantaneous rate of change I realize have far-reaching applications.
In this reading assignment we were introduced to what is behind the concept of the derivative of a function. Basically, the derivative of a function is defined as the instantaneous rate of change of that function at that particular point.
2. Challenging
What I found challenging about this passage was the concept of the limit of average rates of change when defining instantaneous rate of change. I have always thought that the use of limits and the reason why to be a little hard, maybe you could go over that in class?
3. Most Interesting
Instantaneous velocity has always been pretty cool to think about for me. I took physics this summer and did many such calculations under different conditions, again it's interesting to see how the sciences always seem to blend together! The concepts of instantaneous rate of change I realize have far-reaching applications.
Wednesday, September 10, 2008
Functions of two variables & Contour Diagrams (9/11/2008)
1. Main Points of reading
The main point of today’s reading was to get familiar with functions of two variables and their applications. These functions have one dependent variables and two independent variables. They can be represented numerically, algebraically or pictorially (often by contour diagrams). Topographical maps for example are based on contour diagrams.
2. Challenges in Material
It was somewhat challenging to follow the examples involving contour diagrams, especially the terrain examples, how does one goes about making them?
3. Interesting
I especially found the Cobb-Douglas Production function to be very interesting for its applicability to the real world and it is something which one learns about in introductory economics courses. It is interesting to me how all the sciences often intervene.
The main point of today’s reading was to get familiar with functions of two variables and their applications. These functions have one dependent variables and two independent variables. They can be represented numerically, algebraically or pictorially (often by contour diagrams). Topographical maps for example are based on contour diagrams.
2. Challenges in Material
It was somewhat challenging to follow the examples involving contour diagrams, especially the terrain examples, how does one goes about making them?
3. Interesting
I especially found the Cobb-Douglas Production function to be very interesting for its applicability to the real world and it is something which one learns about in introductory economics courses. It is interesting to me how all the sciences often intervene.
Monday, September 8, 2008
Periodic Functions (09/08/2008)
1. The main point of the reading was to learn about periodic functions and their oscillating nature, often represented using the functions sine and cosine. The amplitude and period define these functions, and they are affected by different variables in the function:
In y = Acos(Bt)+C
A affects amplitude
B affects period
C affects the vertical shift of the function
2. Something that was a bit challenging for me in the reading was to conceptualize how to define the specific periodic function from just looking at the graph. However I am sure I will have plenty of practice doing that in the problems later on.
3. Most interesting to me about periodic functions is the ability to make predictions of the future if a trend has been observed. I imagine that there must be many applications for this kind of functions.
In y = Acos(Bt)+C
A affects amplitude
B affects period
C affects the vertical shift of the function
2. Something that was a bit challenging for me in the reading was to conceptualize how to define the specific periodic function from just looking at the graph. However I am sure I will have plenty of practice doing that in the problems later on.
3. Most interesting to me about periodic functions is the ability to make predictions of the future if a trend has been observed. I imagine that there must be many applications for this kind of functions.
Monday, September 1, 2008
Exponential Functions (1.5, 1.7)
The main point of the two reading assignments for tomorrow (9/2/2008) is to understand how exponential functions work and learn about possible applications in natural and social sciences.
I found some aspects of the exponential growth and decay in section 1.7 to be somewhat challenging, yet very interesting. Particularly I thought that the present and future value of money and the compounding effect was at first a little hard to grasp. One thing that I still do not quite understand was why the number e (2.71828..) was used when interest is compounded continuously and not when compounded annually, what is special about e?
Most interesting was the relevance to everyday life this math has! I am excited to learn how to manage my future earnings with this type of math knowledgeJ It was also great to refresh how to calculate population growth rates and the half life of compounds which I can use in other social and natural science classes.
I found some aspects of the exponential growth and decay in section 1.7 to be somewhat challenging, yet very interesting. Particularly I thought that the present and future value of money and the compounding effect was at first a little hard to grasp. One thing that I still do not quite understand was why the number e (2.71828..) was used when interest is compounded continuously and not when compounded annually, what is special about e?
Most interesting was the relevance to everyday life this math has! I am excited to learn how to manage my future earnings with this type of math knowledgeJ It was also great to refresh how to calculate population growth rates and the half life of compounds which I can use in other social and natural science classes.
Sunday, August 31, 2008
First Blog Post!
Name: Emil Folke Mellgren
Year: Junior
Major: Biology (World Health Concentration hopefully)
Minor: Hispanic Studies
Previous math classes: None at Macalester, took pre-calc and Calculus I in highschool (long time ago!)
Math Weakness: Anything above highschool calc I
Math Strength: Enjoy analythical thinking and problem solving
Taking class because: Required by major and cannot go through college without taking math!! What kind of education would that be? I enjoy math, excellent mental exercise. I am excited to learn math that I will be able to apply in the real world. It is also a medical school pre-req.
My hopes for this class: To improve my problem solving skills and learning lots of applicable math!
Interests: Avid downhill skier, enjoy playing club soccer and ice-hockey at macalester, camping in the great outdoors, travelling, reading, running (training for TC marathon) & more.
Worst Math teacher: No particular bad experiences, been lucky on that front.
Best Math teacher: My pre-calc teacher junior year in high school, very cheerful, knowledgeable, helpful, enthusiastic and encouraging. Really made me enjoy math class.
Song: The Rising, Bruce Springsteen
Other: Looking forward to what seems to be a very good class!
Year: Junior
Major: Biology (World Health Concentration hopefully)
Minor: Hispanic Studies
Previous math classes: None at Macalester, took pre-calc and Calculus I in highschool (long time ago!)
Math Weakness: Anything above highschool calc I
Math Strength: Enjoy analythical thinking and problem solving
Taking class because: Required by major and cannot go through college without taking math!! What kind of education would that be? I enjoy math, excellent mental exercise. I am excited to learn math that I will be able to apply in the real world. It is also a medical school pre-req.
My hopes for this class: To improve my problem solving skills and learning lots of applicable math!
Interests: Avid downhill skier, enjoy playing club soccer and ice-hockey at macalester, camping in the great outdoors, travelling, reading, running (training for TC marathon) & more.
Worst Math teacher: No particular bad experiences, been lucky on that front.
Best Math teacher: My pre-calc teacher junior year in high school, very cheerful, knowledgeable, helpful, enthusiastic and encouraging. Really made me enjoy math class.
Song: The Rising, Bruce Springsteen
Other: Looking forward to what seems to be a very good class!
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