Read LA 4.3
1) Main Points
4.3 continues with the concept of using vectors to approximate solutions. The book gives a real life example dealing with approximating a constant that works for two specimens. To calculate this approximation we can use vectors like we did in 4.2 to, knowing that the approximate vector will be distanced from the desired vector by r, a residual vector. We know that for the closest approximation the new multiplied vector and the residual vector will be perpendicular meaning their dot product will be equal to zero.
In equations we are dealing with mx + r = y and x dot r = 0. We can solve this by taking the dot product of both sides of mx + r = y, this way we can find m which is the best approximation for a constant of the original problem. This approximation is where lines of best fit come from.
4.3.2 Applies this method to a more complicated situation. if you are given a table of numbers, these as we have learned can be written in vector form. With this more complicated way of thinking about more three dimensional vectors we can conclude that mu + bv + r = s, and u dot r = 0 and v dot r = 0. This can be solved by taking the dot product of u and the equation, then v and the equation. This gives a solvable system of equations which gives the values of m and b needed to make the closest estimate.
This method is called the fundamental problem of linear modeling and is recapped very well in the end of the LA packet.
2) Challenges
I feel pretty good on the two dimensional stuff, but when it gets the vectors with more elements I foresee some difficulty. Overall though I understand the concept I just need to make sure I print the packet and review/apply the concepts
3) Reflections
I now finally understand where a line of best fit comes from, which is very useful in statistical modeling. Right on!
Monday, November 24, 2008
Wednesday, November 19, 2008
Vector projection
Read LA 4.1 - 4.2.
1) Main Points
4.1 continues on our understanding of finding the solution to a linear combination by finding the solution of a vector equation. This chapter acknowledges that sometimes when solving a system of equations there is not a solution, so it introduces the concept of an approximate solution. The approximate solution is best understood by using vectors and the dot product. The dot product(review) looks like this:
(2,3) dot (4,2) = (2*4) + (3*4) = 8 + 12 = 20
The most important thing to remember about the dot product is that:
If the dot product of two vector is 0 then the vectors are Perpendicular (Orthogonal)
Because of this dot product we can use it and the Pythagorean theorem to learn a lot about the relationships of vectors. The length of vector u is equal to the square root of (u dot u), this is derived from the Pythagorean theorem. Furthermore, with our knowledge of the law of cosines we see that the dot product of u and u is equal to (length of u) * (length of u) * Cos(theta). This means that by using the dot product we can find the angle difference of two vectors.
With this knowledge we see that often we have two vectors (ie. u =(2,1) and a=(6,8)) that are not multiples of each other, meaning that we can't multiply the first vector by anything to get the second vector. We want to know in this case, whats the multiple of the first vector that will bring it to be the closest the second. We know that the vectors will be the most similar when the line that passes between the multiplied vector and the desired vector is parallel with the multiplied one. We denote this vector r and it is called the residual vector. According to the dot product u dot r = 0 because they are perpendicular. We want r to be short as possible which means we want it perpendicular. xu + r = a. Because of this we can solve for x to find the multiple that will make the vector closest to the desired one. To do this we set equal:
u dot (xu + r) = a dot r. And knowing that u dot r = 0 because they are perpendicular we solve for x and get in this case x = 4.
2) Challenges
The major challenged I see with this is with the dot product and how to use it algebraically. When solving for x the book distributed u with parenthesis as if it was a multiplication sign, although they are different. I need more clarification on how to use them in equations to avoid mistakes. Other than that it will be important to understand the equations to be able to apply them correctly.
3) Reflections
This seems to be pretty useful considering often there are not solutions to equations, vectors are an easier way to approximate solutions. I hope to see how to apply this vector knowledge to real world stuff.
1) Main Points
4.1 continues on our understanding of finding the solution to a linear combination by finding the solution of a vector equation. This chapter acknowledges that sometimes when solving a system of equations there is not a solution, so it introduces the concept of an approximate solution. The approximate solution is best understood by using vectors and the dot product. The dot product(review) looks like this:
(2,3) dot (4,2) = (2*4) + (3*4) = 8 + 12 = 20
The most important thing to remember about the dot product is that:
If the dot product of two vector is 0 then the vectors are Perpendicular (Orthogonal)
Because of this dot product we can use it and the Pythagorean theorem to learn a lot about the relationships of vectors. The length of vector u is equal to the square root of (u dot u), this is derived from the Pythagorean theorem. Furthermore, with our knowledge of the law of cosines we see that the dot product of u and u is equal to (length of u) * (length of u) * Cos(theta). This means that by using the dot product we can find the angle difference of two vectors.
With this knowledge we see that often we have two vectors (ie. u =(2,1) and a=(6,8)) that are not multiples of each other, meaning that we can't multiply the first vector by anything to get the second vector. We want to know in this case, whats the multiple of the first vector that will bring it to be the closest the second. We know that the vectors will be the most similar when the line that passes between the multiplied vector and the desired vector is parallel with the multiplied one. We denote this vector r and it is called the residual vector. According to the dot product u dot r = 0 because they are perpendicular. We want r to be short as possible which means we want it perpendicular. xu + r = a. Because of this we can solve for x to find the multiple that will make the vector closest to the desired one. To do this we set equal:
u dot (xu + r) = a dot r. And knowing that u dot r = 0 because they are perpendicular we solve for x and get in this case x = 4.
2) Challenges
The major challenged I see with this is with the dot product and how to use it algebraically. When solving for x the book distributed u with parenthesis as if it was a multiplication sign, although they are different. I need more clarification on how to use them in equations to avoid mistakes. Other than that it will be important to understand the equations to be able to apply them correctly.
3) Reflections
This seems to be pretty useful considering often there are not solutions to equations, vectors are an easier way to approximate solutions. I hope to see how to apply this vector knowledge to real world stuff.
Monday, November 17, 2008
Linear combinations, linear independence, span
Read LA 1.3, 2.0 - 2.3, 3.0 - 3.3
1) Main Points
Example 1.3 reminds us that we can combine three vectors (or more) by using the parallelogram or 'tip to tail' method to find the resultant. Also, by using the concept of linear combination we can combine multiples of vectors. Chapter 2 continues with this concept and asks how to find linear a combination that equals a different desired vector. We can do this by using x and y as variables in an equation. If we have the vectors (2,3),(-1,1) and (0,5) and we want the first two together to equal the third we can write it like this:
x(2,3) + y(-1,1) = (0,5)
At this step it is possible to test solutions for x and y but a more efficient way is to solve for an answer like so:
The previous equation shown above is equal to:
(2x,3x) + (-y,y) = (0,5)
Because vectors add across (remember they are usually written vertically) this is equal to:
(2x-y,3x+y) = (0,5)
This gives us a system of equations like so:
2x-y=0
3x+y=5
This can be solved which gives us x=1, y=2 as our solution to the problem above. This means that if we multiply the first vector by 1 and add it to the second vector multiplied by 2 it will be equal to the third.
It is also possible, using algebra to go backwards from a system of equations to three vectors so make sure you know how to do this. A problem with this method however is that sometimes two vectors will be in the same direction and therefor can't be expressed by using a system of equations because the lines may not intersect. In this case they are most easily thought of by thinking of what variables give the desired vector because the linear set up is easier to think about. Creating a system of equations to find a solution is applicable in dimensions beyond 2, however they will produce a system of however many dimensions there are. Lists of vectors can be listed as matrices. This allows us to rewrite vector equations as matrices:
For example:
x(2,3) + y(-1,1) = (0,5)
can be
(2,3, -1,1)(x,y) = (0,5)
(this doesn't come across well because I can't write it vertically, the point is that matrices can consolidate the equation). This means Ax = b, A is the matrix of vectors, x is the variables matrix and b is the goal.
Chapter 3 introduces a 'span" which is a list of all the linear combinations that can be made from the given vectors.
(6,2) is the span of vectors (2,1),(1,1) because 4(2,1) -2(1,1) = (6,2)
A list of vectors is linearly independent if no vector on the list is a linear combination of the other vectors. Otherwise the list is linearly dependent.
2) Challenges
I followed these examples pretty well until the concept of a "span". I don't understand span and I think I need to know more about what a list of linear combinations looks like. Other than that I can see some difficulties in writing vectors with matrices in mind. This to me seems like it could be a little confusing. Also it seems a little difficult to go backwards from a system of equations, this is something I should practice.
3) Reflections
I remember using vectors a lot in physics, they can really account for forces acting in different (or the same) directions. I think linear algebra will seem more applicable when we learn about where to actually use it.
1) Main Points
Example 1.3 reminds us that we can combine three vectors (or more) by using the parallelogram or 'tip to tail' method to find the resultant. Also, by using the concept of linear combination we can combine multiples of vectors. Chapter 2 continues with this concept and asks how to find linear a combination that equals a different desired vector. We can do this by using x and y as variables in an equation. If we have the vectors (2,3),(-1,1) and (0,5) and we want the first two together to equal the third we can write it like this:
x(2,3) + y(-1,1) = (0,5)
At this step it is possible to test solutions for x and y but a more efficient way is to solve for an answer like so:
The previous equation shown above is equal to:
(2x,3x) + (-y,y) = (0,5)
Because vectors add across (remember they are usually written vertically) this is equal to:
(2x-y,3x+y) = (0,5)
This gives us a system of equations like so:
2x-y=0
3x+y=5
This can be solved which gives us x=1, y=2 as our solution to the problem above. This means that if we multiply the first vector by 1 and add it to the second vector multiplied by 2 it will be equal to the third.
It is also possible, using algebra to go backwards from a system of equations to three vectors so make sure you know how to do this. A problem with this method however is that sometimes two vectors will be in the same direction and therefor can't be expressed by using a system of equations because the lines may not intersect. In this case they are most easily thought of by thinking of what variables give the desired vector because the linear set up is easier to think about. Creating a system of equations to find a solution is applicable in dimensions beyond 2, however they will produce a system of however many dimensions there are. Lists of vectors can be listed as matrices. This allows us to rewrite vector equations as matrices:
For example:
x(2,3) + y(-1,1) = (0,5)
can be
(2,3, -1,1)(x,y) = (0,5)
(this doesn't come across well because I can't write it vertically, the point is that matrices can consolidate the equation). This means Ax = b, A is the matrix of vectors, x is the variables matrix and b is the goal.
Chapter 3 introduces a 'span" which is a list of all the linear combinations that can be made from the given vectors.
(6,2) is the span of vectors (2,1),(1,1) because 4(2,1) -2(1,1) = (6,2)
A list of vectors is linearly independent if no vector on the list is a linear combination of the other vectors. Otherwise the list is linearly dependent.
2) Challenges
I followed these examples pretty well until the concept of a "span". I don't understand span and I think I need to know more about what a list of linear combinations looks like. Other than that I can see some difficulties in writing vectors with matrices in mind. This to me seems like it could be a little confusing. Also it seems a little difficult to go backwards from a system of equations, this is something I should practice.
3) Reflections
I remember using vectors a lot in physics, they can really account for forces acting in different (or the same) directions. I think linear algebra will seem more applicable when we learn about where to actually use it.
Monday, November 10, 2008
Catch up / SIR and the spread of disease
10.7
1) main points
Chapter 10.7 works with modeling the spread of disease. The book uses a model called the SIR model which stands for S-susceptible, I-Infected and R-recovered. In differential equation form this comes out to dS/dt= - (Rate susceptibles get sick) = -aSI. We use the constant a, which is proportional to S and I because we assume that if there are more susceptible people around or more infected people around there will be more encounters. It is also negative because we assume that the rate people get sick will eventually decrease. But we must also account for the infected people that are removed from so we add create a differential equation pertaining to I:
dI/dt = (rate susceptibles get sick) - (rate infecteds get sick) = [aSI - bI].
Next we assume that because the recovered are no longer susceptible they increase so we have:
dR/dt = bI
We know that the total population is not changing so S+I+R= Pop so therefore once we know S and I we can calculate for R, so we are only concerned with dS/dt and dR/dt.
We know that a is a measure of how infectious the disease is and with the knowledge that if I=1 and S=762 we see that dS/dt = apprx. -2. With this info we can solve for a and we get a= .0026
When we use a phase plane to plot the information we see how I increases until S= 192 which is the called the threshold population and is the maximum amount of susceptible people to not have an epidemic. In general this number is b/a. This means that the epidemic could have been avoided if all but 192 people were vaccinated
2) Challenges
This model incorporates everything we have learned about differential equations and put it together so a good knowledge of everything is required. My direct understanding of phase planes/slope curves is minimal so sometimes it is hard for me to follow exactly how they come to graph but I do feel like I can adequately interpret them.
3) Reflections
This obviously has amazing real world applications, especially from the preventative standpoint because it can give a relative quota of vaccines to give to a population. I hope to explore this in a bit more detail because it seems like a really useful in regard to disease prevention.
1) main points
Chapter 10.7 works with modeling the spread of disease. The book uses a model called the SIR model which stands for S-susceptible, I-Infected and R-recovered. In differential equation form this comes out to dS/dt= - (Rate susceptibles get sick) = -aSI. We use the constant a, which is proportional to S and I because we assume that if there are more susceptible people around or more infected people around there will be more encounters. It is also negative because we assume that the rate people get sick will eventually decrease. But we must also account for the infected people that are removed from so we add create a differential equation pertaining to I:
dI/dt = (rate susceptibles get sick) - (rate infecteds get sick) = [aSI - bI].
Next we assume that because the recovered are no longer susceptible they increase so we have:
dR/dt = bI
We know that the total population is not changing so S+I+R= Pop so therefore once we know S and I we can calculate for R, so we are only concerned with dS/dt and dR/dt.
We know that a is a measure of how infectious the disease is and with the knowledge that if I=1 and S=762 we see that dS/dt = apprx. -2. With this info we can solve for a and we get a= .0026
When we use a phase plane to plot the information we see how I increases until S= 192 which is the called the threshold population and is the maximum amount of susceptible people to not have an epidemic. In general this number is b/a. This means that the epidemic could have been avoided if all but 192 people were vaccinated
2) Challenges
This model incorporates everything we have learned about differential equations and put it together so a good knowledge of everything is required. My direct understanding of phase planes/slope curves is minimal so sometimes it is hard for me to follow exactly how they come to graph but I do feel like I can adequately interpret them.
3) Reflections
This obviously has amazing real world applications, especially from the preventative standpoint because it can give a relative quota of vaccines to give to a population. I hope to explore this in a bit more detail because it seems like a really useful in regard to disease prevention.
Wednesday, November 5, 2008
11/6: Interacting systems: modeling, phase plane, and trajectories
10.6
1) Main Points
Chapter 10.6 continues exploring differential equations but this time with two interacting systems. The book uses the example of the relationship between robins and worms. Independently the differential equations regarding their populations are dw/dt = aw (positive because with no robins they increase) and dr/dt=-br (negative because with no food the robins decrease). In relation to each other though these equation look like: dw/dt=aw - Effects of robins on worms and dr/dt = -br + Effects of worms on robins. With these considered as variables with constants the equations come out to be (dw/dt = aw - cwr) and (dr/dt = -br + kwr). We look at this first assuming the constants a=b=c=k=1. We want to see both graphs over time but first it is easier to plot (w,r) the relationship between the two populations. This graph is called a phase plane and the point is called a phase trajectory. Using the chain rule we know that (dr/dt = dr/dw * dw/dt) so to find dr/dw we solve and get (dr/dw =(-r+wr)/(w-wr)). When w and r both equal 1 we see that their differential equations = 0 which means this is an equilibrium and the populations don't change. If we graph this on a slope field we see that the derivatives at the different points create a closed curve. This means that if you plug in a certain number of worms and robins the derivative will point in a certain direction that if continued will lead to the same starting values of w and r.
2) Challenges
This chapter is a little difficult for me because I am unfamiliar with slope fields. I understand this concept graphically but when it comes to using differential equations I fee like it will be hard for me to connect the relationships between the two variables. Think of a graph as the two variables can be misleading. I am used to using time as the x axis so it is very important with these problem to consider the variables represented.
3) Reflections
This concept has obvious applications for real life. Populations often affect each other and their populations can change periodically overtime.
1) Main Points
Chapter 10.6 continues exploring differential equations but this time with two interacting systems. The book uses the example of the relationship between robins and worms. Independently the differential equations regarding their populations are dw/dt = aw (positive because with no robins they increase) and dr/dt=-br (negative because with no food the robins decrease). In relation to each other though these equation look like: dw/dt=aw - Effects of robins on worms and dr/dt = -br + Effects of worms on robins. With these considered as variables with constants the equations come out to be (dw/dt = aw - cwr) and (dr/dt = -br + kwr). We look at this first assuming the constants a=b=c=k=1. We want to see both graphs over time but first it is easier to plot (w,r) the relationship between the two populations. This graph is called a phase plane and the point is called a phase trajectory. Using the chain rule we know that (dr/dt = dr/dw * dw/dt) so to find dr/dw we solve and get (dr/dw =(-r+wr)/(w-wr)). When w and r both equal 1 we see that their differential equations = 0 which means this is an equilibrium and the populations don't change. If we graph this on a slope field we see that the derivatives at the different points create a closed curve. This means that if you plug in a certain number of worms and robins the derivative will point in a certain direction that if continued will lead to the same starting values of w and r.
2) Challenges
This chapter is a little difficult for me because I am unfamiliar with slope fields. I understand this concept graphically but when it comes to using differential equations I fee like it will be hard for me to connect the relationships between the two variables. Think of a graph as the two variables can be misleading. I am used to using time as the x axis so it is very important with these problem to consider the variables represented.
3) Reflections
This concept has obvious applications for real life. Populations often affect each other and their populations can change periodically overtime.
Monday, November 3, 2008
Exponential growth and decay / Applications of ODE, equilibria, and stability
10.4 and 10.5
1) Main points
Chapter 10.4 begins to apply our knowledge of differential equations to exponential functions. We know that the solution to the differential equation: dy/dx = y must be a function whose derivative is the same as the function because the derivative = y. We know that the derivative of e^t = e^t so y = e^t must be a possible solution. But we must also account for the constants involved so the family of solutions is y = Ce^kt. It is also important to remember that in the differential equation y can be multiplied by a constant k so: { dy/dx = ky }. This is k is equal to the k in y = Ce^kt because when you take the derivative of y = Ce^kt the constant C is ignored but the constant k matters.
Example:
This means that a differential equation such as dy/dx = .05y has the general solution of y = Ce^kt and k = .05. To find the particular solution you plug in the given constraints, in this example they are y=50 and t=0. So The particular solution of this differential equation would be 50 = Ce^(.05)(0) which gives us C=50 therefore the solution is y=50e^(.05)(t).
The main idea of this chapter is approaching an exponential growth/decay problem from the top down by using what is given about the rate of change to find the actual function.
Chapter 10.5 works off of 10.4 but instead of starting with the differential equation dy/dx = ky it works from the equation dy/dx = k(y-A). We find that the general solution for dy/dt = k(y-A0 is (y = A + Ce^kt). The first example of the chapter introduces a function in which there is an equilibrium solution meaning the function increases or decrease to approach a certain value that creates an equilibrium. This value is the value of the independent variable that makes the differential function equal to zero. If a small change in the initial conditions of a function makes it veer towards the equilibrium solution then it is called a stable equilibrium solution. If it veers away it is unstable.
2) Challenges
When using the dy/dx (Libnitz) notation it is really important to notice which variable (in this case y or x) is represented in the differential equation. For example dy/dx = 2y is different from dy/dx = 2x. The book's examples often jump to the conclusion that from a differential equation such as dP/dt = .02p, C = P0. This comes from the assumption that when t = 0 That the population will be at the initial amount, this information plugged back into the equation gives C = P0. This is an important thing for me to remember. In general it is hard for me to think of these examples backwards. This chapter is centered around using differential equations to find the functions so it is important to distinguish what the different constants mean.
3) Reflections
These concepts allow us to tackle more complicated real life examples. Equilibriums are seen very often in medicine and blood concentration. I think we will find differential equations really useful in analyzing real world things especially because it is often easier to calculate a rate of change than to find an actual function.
1) Main points
Chapter 10.4 begins to apply our knowledge of differential equations to exponential functions. We know that the solution to the differential equation: dy/dx = y must be a function whose derivative is the same as the function because the derivative = y. We know that the derivative of e^t = e^t so y = e^t must be a possible solution. But we must also account for the constants involved so the family of solutions is y = Ce^kt. It is also important to remember that in the differential equation y can be multiplied by a constant k so: { dy/dx = ky }. This is k is equal to the k in y = Ce^kt because when you take the derivative of y = Ce^kt the constant C is ignored but the constant k matters.
Example:
This means that a differential equation such as dy/dx = .05y has the general solution of y = Ce^kt and k = .05. To find the particular solution you plug in the given constraints, in this example they are y=50 and t=0. So The particular solution of this differential equation would be 50 = Ce^(.05)(0) which gives us C=50 therefore the solution is y=50e^(.05)(t).
The main idea of this chapter is approaching an exponential growth/decay problem from the top down by using what is given about the rate of change to find the actual function.
Chapter 10.5 works off of 10.4 but instead of starting with the differential equation dy/dx = ky it works from the equation dy/dx = k(y-A). We find that the general solution for dy/dt = k(y-A0 is (y = A + Ce^kt). The first example of the chapter introduces a function in which there is an equilibrium solution meaning the function increases or decrease to approach a certain value that creates an equilibrium. This value is the value of the independent variable that makes the differential function equal to zero. If a small change in the initial conditions of a function makes it veer towards the equilibrium solution then it is called a stable equilibrium solution. If it veers away it is unstable.
2) Challenges
When using the dy/dx (Libnitz) notation it is really important to notice which variable (in this case y or x) is represented in the differential equation. For example dy/dx = 2y is different from dy/dx = 2x. The book's examples often jump to the conclusion that from a differential equation such as dP/dt = .02p, C = P0. This comes from the assumption that when t = 0 That the population will be at the initial amount, this information plugged back into the equation gives C = P0. This is an important thing for me to remember. In general it is hard for me to think of these examples backwards. This chapter is centered around using differential equations to find the functions so it is important to distinguish what the different constants mean.
3) Reflections
These concepts allow us to tackle more complicated real life examples. Equilibriums are seen very often in medicine and blood concentration. I think we will find differential equations really useful in analyzing real world things especially because it is often easier to calculate a rate of change than to find an actual function.
Wednesday, October 29, 2008
Thurs. 10/30: Intro. to differential equations, solutions to differential equations, Euler’s method
10.1 and 10.2
1) Main points
10.1 introduces the concept of differential equation. A differential equation is an equation of the rate of change of a function, often we don't know the actual function we only know information that comprised the differential equation. This chapter also introduces a logistical differential equation which is a differential equation that explains a graph the satisfies the conditions of a logistic growth curve. 10.2 introduces how to 'solve' a differential equation which means to find an equation that works with the differential equation. To find if a function is the solution to a differential equation you: set the left side of the equation (the derivative of the function you are testing) = to the differential with the equation being tested substituted in as the variable. Because this new equation often involves a constant the determined function may be a family of possibly different functions because the constant is not defined. To find a specific function that works (by defining the constant) you substitute in a y value that you know. This exact function is called the "particular solution".
2) Challenges
It is really easy to mix up these equations when you are checking to see if they work together. It will be important to write and label the equation and function carefully. These problems require a good knowledge of derivates and the chain rule so it is important to differentiate everything carefully.
3) Reflections
Differential equations have a lot of real world applications especially because often you know the rate of change but not the original function. What we have learned so far about differential equations allows us to check whether a rate of change and a proposed equation correlate, this makes our real life data collection more reliable.
1) Main points
10.1 introduces the concept of differential equation. A differential equation is an equation of the rate of change of a function, often we don't know the actual function we only know information that comprised the differential equation. This chapter also introduces a logistical differential equation which is a differential equation that explains a graph the satisfies the conditions of a logistic growth curve. 10.2 introduces how to 'solve' a differential equation which means to find an equation that works with the differential equation. To find if a function is the solution to a differential equation you: set the left side of the equation (the derivative of the function you are testing) = to the differential with the equation being tested substituted in as the variable. Because this new equation often involves a constant the determined function may be a family of possibly different functions because the constant is not defined. To find a specific function that works (by defining the constant) you substitute in a y value that you know. This exact function is called the "particular solution".
2) Challenges
It is really easy to mix up these equations when you are checking to see if they work together. It will be important to write and label the equation and function carefully. These problems require a good knowledge of derivates and the chain rule so it is important to differentiate everything carefully.
3) Reflections
Differential equations have a lot of real world applications especially because often you know the rate of change but not the original function. What we have learned so far about differential equations allows us to check whether a rate of change and a proposed equation correlate, this makes our real life data collection more reliable.
Sunday, October 26, 2008
Tues. 10/28 Constrained optimization and Lagrange multipliers
9.6
1) Main Points
Chapter 9.6 introduces a more real world application of optimization that deals with constraints. An example of a constraint would be a budget, with constrained optimization you are looking the optimum (max or min) out put of a function within the limitations a budget. Constraints can also be graphed. If f(x,y) has a global max/min on the constraint g(x,y)=c, it occurs at the point where the graph of the constraint is tangent to a contour of f, or at an endpoint of the constraint. The method of Lagrange multiplies gives a system of three equations and states that "if f has a constrained global maximum or minimum, then it occurs at one of the solutions (x0,y0) to this system or at an endpoint of the constraint. This basically means that to find the min or max given constraints you have to differentiate fy and fx along with the equation for the constraint and solve these as a system of equations. The lagrange multiplier (the little tepee symbol) is approximately the change in the optimum value of f when the value of the constraint is increased by 1 unit. It represents the rate of change of the optimum value of f as the constraint (budget) increases. The lagrange multiplier can also be used to calculate the price the product must be sold for when the budget is increased by $1. The lagrange multiplier is also incorporated into the Lagrangian function in which critical points of the multivariable function can be found.
2) Challenges
I am confused about how to find the endpoints of a function. Constraints add a new step to optimization and involve more computation. Because there are two multivariable equations involved in unconstrained optimization it is easy to mix up what means what. Dealing with the Lagrange multiplier in general is difficult, it is important to be familiar with what it means.
3) Reflections
Constrained optimization seems to have much more real world application than unconstrained. This concept seems like something a business would actually use to calculate optimal production. I find the examples most understandable when they deal with money and production because they are actually used for those applications.
1) Main Points
Chapter 9.6 introduces a more real world application of optimization that deals with constraints. An example of a constraint would be a budget, with constrained optimization you are looking the optimum (max or min) out put of a function within the limitations a budget. Constraints can also be graphed. If f(x,y) has a global max/min on the constraint g(x,y)=c, it occurs at the point where the graph of the constraint is tangent to a contour of f, or at an endpoint of the constraint. The method of Lagrange multiplies gives a system of three equations and states that "if f has a constrained global maximum or minimum, then it occurs at one of the solutions (x0,y0) to this system or at an endpoint of the constraint. This basically means that to find the min or max given constraints you have to differentiate fy and fx along with the equation for the constraint and solve these as a system of equations. The lagrange multiplier (the little tepee symbol) is approximately the change in the optimum value of f when the value of the constraint is increased by 1 unit. It represents the rate of change of the optimum value of f as the constraint (budget) increases. The lagrange multiplier can also be used to calculate the price the product must be sold for when the budget is increased by $1. The lagrange multiplier is also incorporated into the Lagrangian function in which critical points of the multivariable function can be found.
2) Challenges
I am confused about how to find the endpoints of a function. Constraints add a new step to optimization and involve more computation. Because there are two multivariable equations involved in unconstrained optimization it is easy to mix up what means what. Dealing with the Lagrange multiplier in general is difficult, it is important to be familiar with what it means.
3) Reflections
Constrained optimization seems to have much more real world application than unconstrained. This concept seems like something a business would actually use to calculate optimal production. I find the examples most understandable when they deal with money and production because they are actually used for those applications.
Monday, October 20, 2008
Unconstrained optimization in 1-d and 2-d
Chapter 4.3
1) Main Points
Chapter for 4.3 deals with global maxima and minima, where the function is larger or smaller than anywhere else in the function. On an interval that includes end points to find the global max/min you compare the values of the function at all critical points. If a function has no real end points then there are no global maxima or minima because the function is continuous. Maxima and minima occur on either a critical point or an end point, they can occur nowhere else. If you plug in the x values of the critical points and the end points the highest and lowest numbers are the end points.
2) Challenges
The hardest part of finding the global maxima/minima is being able to find the critical points. I am confused why in example two they used a second derivative to find the critical points. Another thing to remember is that once you find the x values of the critical points you plug those back into the original function NOT the function of the derivative to find which is max or min.
3) Reflections
Global maxima and minima seem like they will be most useful in determining real world problems in which you are looking for a maximum or a minimal value. For example when you are looking for the maximum profit you want the global maximum instead of just the local.
1) Main Points
Chapter for 4.3 deals with global maxima and minima, where the function is larger or smaller than anywhere else in the function. On an interval that includes end points to find the global max/min you compare the values of the function at all critical points. If a function has no real end points then there are no global maxima or minima because the function is continuous. Maxima and minima occur on either a critical point or an end point, they can occur nowhere else. If you plug in the x values of the critical points and the end points the highest and lowest numbers are the end points.
2) Challenges
The hardest part of finding the global maxima/minima is being able to find the critical points. I am confused why in example two they used a second derivative to find the critical points. Another thing to remember is that once you find the x values of the critical points you plug those back into the original function NOT the function of the derivative to find which is max or min.
3) Reflections
Global maxima and minima seem like they will be most useful in determining real world problems in which you are looking for a maximum or a minimal value. For example when you are looking for the maximum profit you want the global maximum instead of just the local.
Sunday, October 12, 2008
Second derivative, unconstrained optimization in 1-d
1.3, 2.4, 4.1, 4.2
1) Main Points
Chapter 1.3 is a chapter we have already read but is good to review. The average rate of change f(b) - f(a) / b-a, also a function is increasing if the values of f(x) increase as x increases and vice versa. This chapter also explains concave up vs. down. Another way of thinking about the derivative and average rates of change is the change in distance/ the change in time. 2.4 introduces the concept of the Second Derivative. The second derivative is the derivative of the derivative marked f''. In libnitz d/dx (dy/dx) or d^2y/dx^2. The derivative tells us where the function is increasing or decreasing, the second derivative shows us where the derivative is increase or decreasing (over a certain interval). This also tells us whether a graph is concave up or down over a certain interval: if f''>0 then f is concave up there. 4.1 and 4.2 introduce ways of using derivatives to interpret graphs. We can use the signs of the derivatives to find whether the graph is increase at certain intervals or not. It also introduces local maxima and minima which are the points where f is less than/greater than the nearest areas. A critical point is any point that where f'(p)=0 or f'(p)=undefined, the critical value is the value f(p) at these points. 4.2 introduces inflection points, points where the graph changes concavity. Because the second derivative is related to the concavity, where f''=0 there is an inflection point.
2) Challenges
Reading graphs of second derivatives are difficult. It is important to remember that it is a graph of a graph of a graph, and just because the second derivative is changing one way it doesn't always mean that the function is changing in the same way. Critical points may be confusing to me, it is hard to know when a function has one and/or how many it has. There is a difference between inflection points and critical points.
3)
Being able to interpret a graph using derivatives is really important. The book gave an example of graph that was barely readable but by using derivatives we could learn a lot about how the graph's function. Interpreting graphs with this level of detail will be very important because it allows us to gain more from graphs of real life situations,
1) Main Points
Chapter 1.3 is a chapter we have already read but is good to review. The average rate of change f(b) - f(a) / b-a, also a function is increasing if the values of f(x) increase as x increases and vice versa. This chapter also explains concave up vs. down. Another way of thinking about the derivative and average rates of change is the change in distance/ the change in time. 2.4 introduces the concept of the Second Derivative. The second derivative is the derivative of the derivative marked f''. In libnitz d/dx (dy/dx) or d^2y/dx^2. The derivative tells us where the function is increasing or decreasing, the second derivative shows us where the derivative is increase or decreasing (over a certain interval). This also tells us whether a graph is concave up or down over a certain interval: if f''>0 then f is concave up there. 4.1 and 4.2 introduce ways of using derivatives to interpret graphs. We can use the signs of the derivatives to find whether the graph is increase at certain intervals or not. It also introduces local maxima and minima which are the points where f is less than/greater than the nearest areas. A critical point is any point that where f'(p)=0 or f'(p)=undefined, the critical value is the value f(p) at these points. 4.2 introduces inflection points, points where the graph changes concavity. Because the second derivative is related to the concavity, where f''=0 there is an inflection point.
2) Challenges
Reading graphs of second derivatives are difficult. It is important to remember that it is a graph of a graph of a graph, and just because the second derivative is changing one way it doesn't always mean that the function is changing in the same way. Critical points may be confusing to me, it is hard to know when a function has one and/or how many it has. There is a difference between inflection points and critical points.
3)
Being able to interpret a graph using derivatives is really important. The book gave an example of graph that was barely readable but by using derivatives we could learn a lot about how the graph's function. Interpreting graphs with this level of detail will be very important because it allows us to gain more from graphs of real life situations,
Wednesday, October 8, 2008
Notes on Vectors p 4-8
Pages 4 - 8, more on vectors!!!
1) Main Points
The first section of Page 1 reiterates what we learned in class about the dot product. A "Gradient" is a vector that consists of the partial derivatives f(x,y). It is written as delta f(x,y) or grad(f). To find a gradient you find the partial derivatives of the equation and put them next to each-other separated by a comma, for example delta f(x,y)= <3x, 4x^2>. Then to find the gradient at a certain point you substitute in the desired x and y values to get a vector such as <4, 12>. Next the reading introduces "Directional Derivatives" which go beyond partial derivates and calculates how the function changes when moved in any direction instead of either x or y. The Directional derivative at a given point in given direction is written as Df(x0,y0) = the gradient of f at (x0, y0) times (dot product) the vector u. The reading goes on to justify this algebraically. It shoes a contour diagram of a multivariable function is shown of along with several gradient vectors at different points. The gradient vector always points in the direction of greatest increase, and its length corresponds with how steep the slope is.It helps to think of this as a mountain, with the gradient vector pointing "Up hill".
2) Challenges
I'm Still a bit confused about unit vectors, the reading made some connections about unit vectors that I didn't understand. I am unclear on what the directional derivatives actually tell us. To find gradients and Directional Derivatives it is important to have a really good hold on derivatives in general because messing them up will affect what comes later.
3) Reflections
This chapter helps explain what vectors are useful for, they can visually explain a lot in a graph or a contour diagram, especially about rates of change and derivatives. Seeing them on a contour diagram really helped me see how they work and what they can show.
1) Main Points
The first section of Page 1 reiterates what we learned in class about the dot product. A "Gradient" is a vector that consists of the partial derivatives f(x,y). It is written as delta f(x,y) or grad(f). To find a gradient you find the partial derivatives of the equation and put them next to each-other separated by a comma, for example delta f(x,y)= <3x, 4x^2>. Then to find the gradient at a certain point you substitute in the desired x and y values to get a vector such as <4, 12>. Next the reading introduces "Directional Derivatives" which go beyond partial derivates and calculates how the function changes when moved in any direction instead of either x or y. The Directional derivative at a given point in given direction is written as Df(x0,y0) = the gradient of f at (x0, y0) times (dot product) the vector u. The reading goes on to justify this algebraically. It shoes a contour diagram of a multivariable function is shown of along with several gradient vectors at different points. The gradient vector always points in the direction of greatest increase, and its length corresponds with how steep the slope is.It helps to think of this as a mountain, with the gradient vector pointing "Up hill".
2) Challenges
I'm Still a bit confused about unit vectors, the reading made some connections about unit vectors that I didn't understand. I am unclear on what the directional derivatives actually tell us. To find gradients and Directional Derivatives it is important to have a really good hold on derivatives in general because messing them up will affect what comes later.
3) Reflections
This chapter helps explain what vectors are useful for, they can visually explain a lot in a graph or a contour diagram, especially about rates of change and derivatives. Seeing them on a contour diagram really helped me see how they work and what they can show.
Monday, October 6, 2008
Vectors, dot product, and vector components
Read LA 1.0-1.2, 4.2.0, supplementary notes, pp. 1 - 4.
1) Main points
This chapter introduces the concept of vectors. A vector is a line with a length and direction. They are marked as two numbers over each-other. 3 over 4 would mean a line from (0,0) to (3,4). They can be multiplied by constants to double the length of the line. They can also be added together, the top number plus the top number over the bottom plus the bottom. Graphically they can be added by using the parallelogram or head to tail method. This involves putting the tail of one on the head of the other and wherever it points is the sum. 4.2 introduces the operation of a dot product which is when you are multiplying two vectors together it becomes the sum of the top numbers plus the sum of the second numbers and so on. If the vectors are perpendicular then the dot product is zero. The supplementary notes summarize some of the information in 1.0-1.2. Because of the pythagorean theorem, the length of a vector is the square root of a squared + b squared. Using our existing knowledge of partial derivatives you can find the directional derivative.
2) Challenges
Dot products may present some challenges if I am not careful in writing them out or realizing the answer should be a scalar. I don't really understand the directional derivative, I see where it's coming from but I think I need a better description of it.
3) Reflections
I used vectors and scalars in physics a ton!! They were useful in determining whether an object had a direction or not and whether to pay attention to the sign of a variable. I can see vectors being useful in all sorts of real life applications.
1) Main points
This chapter introduces the concept of vectors. A vector is a line with a length and direction. They are marked as two numbers over each-other. 3 over 4 would mean a line from (0,0) to (3,4). They can be multiplied by constants to double the length of the line. They can also be added together, the top number plus the top number over the bottom plus the bottom. Graphically they can be added by using the parallelogram or head to tail method. This involves putting the tail of one on the head of the other and wherever it points is the sum. 4.2 introduces the operation of a dot product which is when you are multiplying two vectors together it becomes the sum of the top numbers plus the sum of the second numbers and so on. If the vectors are perpendicular then the dot product is zero. The supplementary notes summarize some of the information in 1.0-1.2. Because of the pythagorean theorem, the length of a vector is the square root of a squared + b squared. Using our existing knowledge of partial derivatives you can find the directional derivative.
2) Challenges
Dot products may present some challenges if I am not careful in writing them out or realizing the answer should be a scalar. I don't really understand the directional derivative, I see where it's coming from but I think I need a better description of it.
3) Reflections
I used vectors and scalars in physics a ton!! They were useful in determining whether an object had a direction or not and whether to pay attention to the sign of a variable. I can see vectors being useful in all sorts of real life applications.
Wednesday, October 1, 2008
Derivatives of periodic functions / Partial derivative
3.5, 9.3, 9.4
1) Main Points
Chapter 3.5 continues our exploration of derivatives, this time finding the derivative of periodic functions. The book noticed that the rate of change for a sine function was also in a periodic but reversed manner. They graphed the points and found that the derivative function of a sine function was a cosine function. They proved it by finding the derivative of a certain point on the sine graph and comparing it to the cosine graph. They did this also for the derivative of cosine and found that it was -sin(x). 9.3 and 9.4 introduce the concept of derivatives to multivariable functions. We know that in a multivariable function if you make one variable a constant the function changes in accordance with the other variable, so it is possible to find the rate of change of the function when keeping a variable constant. This is called the partial derivative and if (y is fixed) is shown by: the limit as h approaches zero {f(a+h,b) - f(a,b)}/(h). If x is fixed then its: lim (h app. 0) [f(a+h,b)-f(a,b)]/(h). You can use this model to find the partial derivative of a table or a contour diagram. Partial derivatives can also be computed algebraically by using our knowledge of derivatives. If you consider one variable in a two variable equation a constant you can solve for the derivative in that situation. The book wrote some shortcuts for these as Second-order partial derivatives.
2)Challenges
I am confused about what the table of "Second-Order" partial derivates is about. Another challenge that I foresee is when solving for partial derivatives in multivariable functions you treat the fixed variable as a constant which means that you just multiply the function by it and it doesn't just disappear or become zero.
3)Reflections
Knowing the derivatives for periodic functions seems like it will have plenty of real world applications. So much of the real world changes periodically like day light, tides, population (sometimes), temperature/weather, using the derivative will help us predict those changes. Multivariable functions also have many real world applications and it will be useful to know the rate of change for each part of the equation.
1) Main Points
Chapter 3.5 continues our exploration of derivatives, this time finding the derivative of periodic functions. The book noticed that the rate of change for a sine function was also in a periodic but reversed manner. They graphed the points and found that the derivative function of a sine function was a cosine function. They proved it by finding the derivative of a certain point on the sine graph and comparing it to the cosine graph. They did this also for the derivative of cosine and found that it was -sin(x). 9.3 and 9.4 introduce the concept of derivatives to multivariable functions. We know that in a multivariable function if you make one variable a constant the function changes in accordance with the other variable, so it is possible to find the rate of change of the function when keeping a variable constant. This is called the partial derivative and if (y is fixed) is shown by: the limit as h approaches zero {f(a+h,b) - f(a,b)}/(h). If x is fixed then its: lim (h app. 0) [f(a+h,b)-f(a,b)]/(h). You can use this model to find the partial derivative of a table or a contour diagram. Partial derivatives can also be computed algebraically by using our knowledge of derivatives. If you consider one variable in a two variable equation a constant you can solve for the derivative in that situation. The book wrote some shortcuts for these as Second-order partial derivatives.
2)Challenges
I am confused about what the table of "Second-Order" partial derivates is about. Another challenge that I foresee is when solving for partial derivatives in multivariable functions you treat the fixed variable as a constant which means that you just multiply the function by it and it doesn't just disappear or become zero.
3)Reflections
Knowing the derivatives for periodic functions seems like it will have plenty of real world applications. So much of the real world changes periodically like day light, tides, population (sometimes), temperature/weather, using the derivative will help us predict those changes. Multivariable functions also have many real world applications and it will be useful to know the rate of change for each part of the equation.
Monday, September 29, 2008
3.3, 3.4 Derivatives of composite functions / Derivatives of products and quotients
Derivatives of composite functions / Derivatives of products and quotients, 3.3, 3.4
1)Main Points
Chapter 3.3 introduces a mathematical concept called the chain rule which states that if y=f(z) and z=g(t), then the derivative of the composite function y=f(g(t)) is dy/dx = dy/dz * dz/dt. This in words means that the derivative of a composite function is the derivative of the outside function times the derivative of the inside function. This in function form is y=(t+1)^4, or y=z^4 when z=t+1 (where z is the inside function). This is simply a way of splitting up a function into different parts so you can calculate the derivative of a function like y=(4t^2 + 1)^7. 3.4 introduces how to find the derivative of a product or quotient of functions. We notice that simply multiplying the derivatives does not give us the correct answer. We learn that (fg)'= f'g + fg', which means the derivative of a product is the derivative of the first times the second plus the first times the derivative of the second. We find that through the product rule we can create a rule for quotients as well which is: (f/g)'=f'g-fg'/g^2.
2) Challenges
3.3 introduces some challenges for me. I find it really difficult to keep all the things in the chain rule strait. It is easy to forget that at the end of calculating the derivative for the inside function you must multiply it by the derivative of the outside function. I should practice these a little and be able to use them without getting confused. The product/quotient rules make a little bit more sense to me but could also present a lot of logistical problems. It is easy to get the variables confused or getting the basic derivatives wrong could put off the whole problem. It will be important for me to memorize the derivative rules and make sure that I write clearly.
3)Reflections
Both chapters introduced ways of finding the derivatives of more complicated compound functions. This, I'm sure will be useful when we encounter some real world problems that are more difficult to solve than with just simple derivative rules. I hope that I can master these skills well enough to apply them.
1)Main Points
Chapter 3.3 introduces a mathematical concept called the chain rule which states that if y=f(z) and z=g(t), then the derivative of the composite function y=f(g(t)) is dy/dx = dy/dz * dz/dt. This in words means that the derivative of a composite function is the derivative of the outside function times the derivative of the inside function. This in function form is y=(t+1)^4, or y=z^4 when z=t+1 (where z is the inside function). This is simply a way of splitting up a function into different parts so you can calculate the derivative of a function like y=(4t^2 + 1)^7. 3.4 introduces how to find the derivative of a product or quotient of functions. We notice that simply multiplying the derivatives does not give us the correct answer. We learn that (fg)'= f'g + fg', which means the derivative of a product is the derivative of the first times the second plus the first times the derivative of the second. We find that through the product rule we can create a rule for quotients as well which is: (f/g)'=f'g-fg'/g^2.
2) Challenges
3.3 introduces some challenges for me. I find it really difficult to keep all the things in the chain rule strait. It is easy to forget that at the end of calculating the derivative for the inside function you must multiply it by the derivative of the outside function. I should practice these a little and be able to use them without getting confused. The product/quotient rules make a little bit more sense to me but could also present a lot of logistical problems. It is easy to get the variables confused or getting the basic derivatives wrong could put off the whole problem. It will be important for me to memorize the derivative rules and make sure that I write clearly.
3)Reflections
Both chapters introduced ways of finding the derivatives of more complicated compound functions. This, I'm sure will be useful when we encounter some real world problems that are more difficult to solve than with just simple derivative rules. I hope that I can master these skills well enough to apply them.
Wednesday, September 24, 2008
3.1, 3.2 Derivatives of powers and polynomials / Derivatives of exponentials and logs
Derivatives of powers and polynomials / Derivatives of exponentials and logs, 3.1, 3.2.
1) Main Points
Instead of finding the derivative by way of limits and points on graphs or tables, we can calculate the derivative by using an equation. For a linear function, we know that the rate of change is constant, and in y = mx + b form m = slope = f'(x). We also notice that for a particular x value, the derivative varies directly with the slope ie: y = 3f(x), slope = 3m, y = f(x)/2, slope = m/2. If a function is multiplied by a constant then the derivative is also multiplied by the same constant. When adding derivatives: f'(x) = {f(x) + g(x)} = f'(x) + f'(y). To find the derivative: {d/dx(x^n)= nx^n-1}. To find the slope of a tangent line at a certain point x on a function, you can find the derivative equation of the function and then substitute in the value of x. In 3.2 the book attempts to find the derivative of a power function. The find a value for f'(0), and use this as a multiplier. They also ask when (because both the derivative and the power function are increasing in the same direction) the derivative will be equal to the function. This happens with the function e^x. This helps us realize that : f'(a^x)= (ln a)a^x. For logarithmic functions: f'(lnx)= 1 / x.
2) Challenges
A challenge for me regarding derivatives in general is graphing them. It is important for me to remember that the derivative will have a very different shape than the function itself. And when the derivative is zero it means the graph is changing directions around a curve. Something I noticed about the nature of derivatives that may help me in the future is that the derivative is always one less power than the function (n-1). So if I am trying to find the derivative of a cubed function I know it will be a quadratic type line, etc. I am confused about why the book used f'(0) as a multiplier for the derivative of a power function. In this chapter I was introduced to a several ways of finding the derivative, so I will have to make sure I remember them and don't mix them up.
3) Reflections
Now we have expanded our knowledge of derivatives to encompass many of the types of functions we know today. This will be very useful for calculating the rates of change of those harder-to-model real life equations such as exponential and power functions, and logarithmic functions. Also inflation rates and equations that use natural logarithms.
1) Main Points
Instead of finding the derivative by way of limits and points on graphs or tables, we can calculate the derivative by using an equation. For a linear function, we know that the rate of change is constant, and in y = mx + b form m = slope = f'(x). We also notice that for a particular x value, the derivative varies directly with the slope ie: y = 3f(x), slope = 3m, y = f(x)/2, slope = m/2. If a function is multiplied by a constant then the derivative is also multiplied by the same constant. When adding derivatives: f'(x) = {f(x) + g(x)} = f'(x) + f'(y). To find the derivative: {d/dx(x^n)= nx^n-1}. To find the slope of a tangent line at a certain point x on a function, you can find the derivative equation of the function and then substitute in the value of x. In 3.2 the book attempts to find the derivative of a power function. The find a value for f'(0), and use this as a multiplier. They also ask when (because both the derivative and the power function are increasing in the same direction) the derivative will be equal to the function. This happens with the function e^x. This helps us realize that : f'(a^x)= (ln a)a^x. For logarithmic functions: f'(lnx)= 1 / x.
2) Challenges
A challenge for me regarding derivatives in general is graphing them. It is important for me to remember that the derivative will have a very different shape than the function itself. And when the derivative is zero it means the graph is changing directions around a curve. Something I noticed about the nature of derivatives that may help me in the future is that the derivative is always one less power than the function (n-1). So if I am trying to find the derivative of a cubed function I know it will be a quadratic type line, etc. I am confused about why the book used f'(0) as a multiplier for the derivative of a power function. In this chapter I was introduced to a several ways of finding the derivative, so I will have to make sure I remember them and don't mix them up.
3) Reflections
Now we have expanded our knowledge of derivatives to encompass many of the types of functions we know today. This will be very useful for calculating the rates of change of those harder-to-model real life equations such as exponential and power functions, and logarithmic functions. Also inflation rates and equations that use natural logarithms.
Wednesday, September 17, 2008
2.2, 2.3
2.2 and 2.3
1) Main Points
We can plot a line tangent to the point to find the derivative. When we plot a graph of derivatives we notice that is is different from the graph of the function. If f'> 0 then f is increasing, if f' < 0 then f is decreasing, if f' = 0 then f is constant over that interval. Derivates can also b expressed in tables. From these table we can "guess" a formula for the derivative. The derivative can also be written as dy/dx, which means the difference in y divided by the difference in x. If this was for a graph of distance over time then the derivative would represent a graph of the velocity because distance/time = velocity. The units of the derivative are the units of dy/dx. The derivative of the derivative, dv/dt would be the acceleration, which would graph the changes in the rate of change. The derivative can be used to calculate other things regarding rates of change.
2) Challenges
With derivatives, units can get confused and are also very important. It is easy to functions and their derivatives mixed up. Sometimes it is hard to determine what the derivative represents in a word problem (I need to continue to remind my self that it quantifies a rate of change and can be used to predict what happens next in the function).
3) Reflections
Derivatives are the start of real calculus and are obviously very applicable to and also very useful in real life. You can use the derivative to measure the speed of someone running at constant velocity or you can predict the length of his next couple strides if he is running at an inconstant speed. Derivatives have financial applications. You can use the derivative to predict an increase in sales if you increase a variable such as advertising. The derivative goes beyond just modeling what nature does into predicting and showing how it changes.
1) Main Points
We can plot a line tangent to the point to find the derivative. When we plot a graph of derivatives we notice that is is different from the graph of the function. If f'> 0 then f is increasing, if f' < 0 then f is decreasing, if f' = 0 then f is constant over that interval. Derivates can also b expressed in tables. From these table we can "guess" a formula for the derivative. The derivative can also be written as dy/dx, which means the difference in y divided by the difference in x. If this was for a graph of distance over time then the derivative would represent a graph of the velocity because distance/time = velocity. The units of the derivative are the units of dy/dx. The derivative of the derivative, dv/dt would be the acceleration, which would graph the changes in the rate of change. The derivative can be used to calculate other things regarding rates of change.
2) Challenges
With derivatives, units can get confused and are also very important. It is easy to functions and their derivatives mixed up. Sometimes it is hard to determine what the derivative represents in a word problem (I need to continue to remind my self that it quantifies a rate of change and can be used to predict what happens next in the function).
3) Reflections
Derivatives are the start of real calculus and are obviously very applicable to and also very useful in real life. You can use the derivative to measure the speed of someone running at constant velocity or you can predict the length of his next couple strides if he is running at an inconstant speed. Derivatives have financial applications. You can use the derivative to predict an increase in sales if you increase a variable such as advertising. The derivative goes beyond just modeling what nature does into predicting and showing how it changes.
Monday, September 15, 2008
1.3 and 2.1.
1.3 and 2.1.
1)Main Points
Chapter 1.3 deals with rates of change and how to calculate them, especially if they are not constant. The average rate of change can be modeled as the Change in the dependent variable / the change in the independent variable. For a function to be increasing, the values of F(x) must increase as x increases. Chapter 2.1 discusses using the derivative to calculate the instantaneous rate of change as opposed to the rate of change over a long period time. The instantaneous velocity of an object can be determined by taking the limit of the average velocity of the object at smaller and smaller time intervals. The instantaneous rate of change of a function f at a is called the "derivative" of f at a, this is written as f'(a). The derivative is the same thing as a) the slope of the graph at that point and b) the slope of a line tangent to that exact point. The derivative can be calculated mathematically by subtracting a very small change in the dependent variable by a the same change in the independent variable. For example: you know that the rate of change is the change in y divided by the change in x. And you know that y=f(x). So with two points, a and (a+.0001) you can in the derivative of f(x) at a by doing {f(a+.0001)- f(a)/ (a+.0001) - a}.
2)Challenges
On a distance over time graph it is easy to forget that the slope between two points is the average velocity. The concept of limits is something that I've only had a little bit of exposure to but it makes sense when calculating something at in instant in time. It will be important for me to keep in mind that it is not an exact number. Also it is sometimes difficult to recognize the tangent line of a curve in an exact spot.
3)Reflections
The use of derivative seems to have lots of practical applications regarding speeds of objects and many inconstant rates of change. The book used the example of a grape fruit thrown up into the air and calculated its average speed. The derivative could have been applied in this situation calculate the instantaneous speed of the grapefruit instead of just the average speed. There are many real world applications for this chapter, probably more so than linear functions because usually things in the world don't change at constant rates.
1)Main Points
Chapter 1.3 deals with rates of change and how to calculate them, especially if they are not constant. The average rate of change can be modeled as the Change in the dependent variable / the change in the independent variable. For a function to be increasing, the values of F(x) must increase as x increases. Chapter 2.1 discusses using the derivative to calculate the instantaneous rate of change as opposed to the rate of change over a long period time. The instantaneous velocity of an object can be determined by taking the limit of the average velocity of the object at smaller and smaller time intervals. The instantaneous rate of change of a function f at a is called the "derivative" of f at a, this is written as f'(a). The derivative is the same thing as a) the slope of the graph at that point and b) the slope of a line tangent to that exact point. The derivative can be calculated mathematically by subtracting a very small change in the dependent variable by a the same change in the independent variable. For example: you know that the rate of change is the change in y divided by the change in x. And you know that y=f(x). So with two points, a and (a+.0001) you can in the derivative of f(x) at a by doing {f(a+.0001)- f(a)/ (a+.0001) - a}.
2)Challenges
On a distance over time graph it is easy to forget that the slope between two points is the average velocity. The concept of limits is something that I've only had a little bit of exposure to but it makes sense when calculating something at in instant in time. It will be important for me to keep in mind that it is not an exact number. Also it is sometimes difficult to recognize the tangent line of a curve in an exact spot.
3)Reflections
The use of derivative seems to have lots of practical applications regarding speeds of objects and many inconstant rates of change. The book used the example of a grape fruit thrown up into the air and calculated its average speed. The derivative could have been applied in this situation calculate the instantaneous speed of the grapefruit instead of just the average speed. There are many real world applications for this chapter, probably more so than linear functions because usually things in the world don't change at constant rates.
Wednesday, September 10, 2008
9.1 and 9.2
9.1 and 9.2.
1)Main points
This chapter has to do with functions that have two variables. All possible inputs ie; (x,y) is called the domain of f. By keeping one variable constant in a two variable function you can interpret useful data. For example, if you have an equation for the time of diffusion of a medicine in your body, you can keep the amount injected at a constant and notice the time it takes to diffuse, or you can keep the time constant and calculate the remaining medicine in the blood stream for a given injection. Two variable functions can also be modeled by a contour diagram (9.2). When depicting altitude on a contour map, the closer together the contour lines, the steeper the terrain. Also the numbers that label the lines usually stand for altitude on a topographical map.
In contour diagrams such as the corn example given in the book, information can also be collected by keep one of the variables constant and noticing how the effects of the other variable changing. The Cobb-Douglas function can be used to model economic output. A contour consists of all the points (x,y) where f(x,y) has a constant value, c.
2) Challenges
I find drawing contour lines very difficult. It is hard for me to find where the line should be from the data. In general thinking about a function with two variables is confusing, but I find it easier when I can use a real world example such as the office example about increasing staff & equipment. I will also have to remember which variable is which in a problem. For example, if there is a difference between the effects of rainfall than temperature on corn and it is easy to mix up those variables which would give some wrong info.
3) Reflections
Contour diagrams and two variable functions are used in all sorts of applications today. From maps to weather charts, and business or economic models. They model very real world situations and are extremely easy and quick to read. A map with contour lines can show you the altitude and shape of a mountain in an incredibly useful way.
1)Main points
This chapter has to do with functions that have two variables. All possible inputs ie; (x,y) is called the domain of f. By keeping one variable constant in a two variable function you can interpret useful data. For example, if you have an equation for the time of diffusion of a medicine in your body, you can keep the amount injected at a constant and notice the time it takes to diffuse, or you can keep the time constant and calculate the remaining medicine in the blood stream for a given injection. Two variable functions can also be modeled by a contour diagram (9.2). When depicting altitude on a contour map, the closer together the contour lines, the steeper the terrain. Also the numbers that label the lines usually stand for altitude on a topographical map.
In contour diagrams such as the corn example given in the book, information can also be collected by keep one of the variables constant and noticing how the effects of the other variable changing. The Cobb-Douglas function can be used to model economic output. A contour consists of all the points (x,y) where f(x,y) has a constant value, c.
2) Challenges
I find drawing contour lines very difficult. It is hard for me to find where the line should be from the data. In general thinking about a function with two variables is confusing, but I find it easier when I can use a real world example such as the office example about increasing staff & equipment. I will also have to remember which variable is which in a problem. For example, if there is a difference between the effects of rainfall than temperature on corn and it is easy to mix up those variables which would give some wrong info.
3) Reflections
Contour diagrams and two variable functions are used in all sorts of applications today. From maps to weather charts, and business or economic models. They model very real world situations and are extremely easy and quick to read. A map with contour lines can show you the altitude and shape of a mountain in an incredibly useful way.
Monday, September 8, 2008
1.10
1.10
1) Main Points
Periodic means repeating at regular intervals. Amplitude is half the distances between the max and min. The period is the time for the function to complete one whole cycle. y=sin(t), the period of a sine function is 2 pi. The number that multiplies the sin(t) part of the function affects the amplitude. If a number is added or subtracted to sin(t) it is shifted up or down accordingly. If a number is added to t then the graph is shifted left, subtracted it is shifted right. Period = 2pi/B and B is the number that will multiply t (y=Asin(Bt).
2) Challenges
1 cycle is when the function goes down then up then back to the starting point. It will be important for me to keep in mind what a whole cycle looks like. It is counter intuitive that when you add a number to t the graph shifts to the negative direction. I will also have to make sure i recognize that this relationship is backward. I will also need to burn into my memory period= 2pi/B. This is a very important for finding the relationship between the parameter B and the period, which are NOT the same.
3) Reflections
Periodic functions are very applicable to the real world. I can imagine using them to describe temperature/whether patterns, tides, or changes in the stock market. So many things occur periodically in the world and a knowledge in periodical functions can help describe them.
1) Main Points
Periodic means repeating at regular intervals. Amplitude is half the distances between the max and min. The period is the time for the function to complete one whole cycle. y=sin(t), the period of a sine function is 2 pi. The number that multiplies the sin(t) part of the function affects the amplitude. If a number is added or subtracted to sin(t) it is shifted up or down accordingly. If a number is added to t then the graph is shifted left, subtracted it is shifted right. Period = 2pi/B and B is the number that will multiply t (y=Asin(Bt).
2) Challenges
1 cycle is when the function goes down then up then back to the starting point. It will be important for me to keep in mind what a whole cycle looks like. It is counter intuitive that when you add a number to t the graph shifts to the negative direction. I will also have to make sure i recognize that this relationship is backward. I will also need to burn into my memory period= 2pi/B. This is a very important for finding the relationship between the parameter B and the period, which are NOT the same.
3) Reflections
Periodic functions are very applicable to the real world. I can imagine using them to describe temperature/whether patterns, tides, or changes in the stock market. So many things occur periodically in the world and a knowledge in periodical functions can help describe them.
Monday, September 1, 2008
1.5 and 1.7
Blog for reading due Tuesday 9/2, on 1.5 and 1.7
1) Main Points:
Whenever you have a constant percent increase it's exponential growth. If the rate of change number is greater than zero and less than one the function is a decreasing exponential function. Growth and decay can be modeled by P=P(e) to the (k x t) power. Every exponentially growing function has a fixed doubling time. For interest: annually P=P(1+r) to the t, continuously P=Pert. B, the future value of a payment is the amount P would have grown if in an interest-bearing bank account.
2) Challenges:
I was challenged by the example given of a decreasing exponential function. For something that is decreased by 40% for every
unit of time, it took me a while to understand that 60% of it would be left each time. Another confusing aspect of exponential functions is that the absolute rate of change is not constant but the relative rate of change is. For solving exponential growth and decay problems I will have to remember to use natural log, Ln, to isolate the variable. It took me a while to understand the two ways of calculating the future values of sums of money (re: interest), you can either calculate what the actual present value of it is or you can calculate what it will be worth in the future.
3) Reflections:
Exponential functions apply very well to population data, because most often populations grow faster as they get bigger. I also notice this mathematical relationship in cells reproduction. Future and Present values were concepts that were new to me but they show many practical uses for determining not only the changing value of money but of other things in nature the exhibit predictable change.
1) Main Points:
Whenever you have a constant percent increase it's exponential growth. If the rate of change number is greater than zero and less than one the function is a decreasing exponential function. Growth and decay can be modeled by P=P(e) to the (k x t) power. Every exponentially growing function has a fixed doubling time. For interest: annually P=P(1+r) to the t, continuously P=Pert. B, the future value of a payment is the amount P would have grown if in an interest-bearing bank account.
2) Challenges:
I was challenged by the example given of a decreasing exponential function. For something that is decreased by 40% for every
unit of time, it took me a while to understand that 60% of it would be left each time. Another confusing aspect of exponential functions is that the absolute rate of change is not constant but the relative rate of change is. For solving exponential growth and decay problems I will have to remember to use natural log, Ln, to isolate the variable. It took me a while to understand the two ways of calculating the future values of sums of money (re: interest), you can either calculate what the actual present value of it is or you can calculate what it will be worth in the future.
3) Reflections:
Exponential functions apply very well to population data, because most often populations grow faster as they get bigger. I also notice this mathematical relationship in cells reproduction. Future and Present values were concepts that were new to me but they show many practical uses for determining not only the changing value of money but of other things in nature the exhibit predictable change.
Thursday, August 28, 2008
Math 135: Applied calculus
Guidelines for blog postings
Purpose. The purposes of the blog posting are:
• to give you multiple exposures to the course material,
• to help you identify your specific points of difficulty, and eventually, to become your own teacher,
• to help me allocate lecture time effectively,
• to encourage transfer of mathematics to other areas of your academic/professional/personal life.
In short, these blogging assignments can be a tremendously valuable learning tool for you. In addition,
before your final exam, you can go back through your blogs to review your specific points of difficulty.
Finally, note that data collected in other classes1 shows that blog grades correlate with final grades
much more strongly than midterm exam grades do, even though in that class midterms were worth 40%
of the final grade and blogs were only worth 10%.
Content. Each blog posting has three parts:
1. Answer the question “What are the main points of the reading assignment for tomorrow?” You
should explain these in your own words, as if you were explaining them to a friend or family member.
2. Answer the question “What about the material challenged you?” “Nothing” is not an acceptable
answer. If nothing challenges you, then you should think about the material at a deeper level and
generate some honest questions. If you still cannot generate a genuine challenge, then answer the
question “What do you think was the most difficult part of the material for most people in the class?”
3. Write anything reflective about the material. Some suggestions are “what was the most interesting
part of the material and why?” or “how does this material connect to something else you have
learned in mathematics?” or “how is this material useful/relevant to your intellectual or career
interests?” but anything genuine, honest, and original will suffice.
Format. Question #1, #2, and #3 should each be answered in a separate paragraph. The paragraphs
should be given the labels “Main points,” “Challenges,” and “Reflections” respectively. You do not need
to write essays. A couple of sentences answering each question may be sufficient.
Deadline. You must post by 11:59 p.m. on the night before each lecture that has a new reading
assignment. If there is a technical problem with blogger.com, you can always email your posting to me
as a last resort and add it to your blog when blogger.com is fixed.
Grading. Blogging is worth 10% of your final grade. I randomly select which blogs to read for each
lecture. On average, I will grade your blog approximately once per week. You will not know in advance
when I will grade your blog. If your blog posting is not online when I sit down to read it (sometime
between midnight and lecture time) you will receive a 0 for that post. Blogs are graded according to the
rubric below. Each blog posting is graded out of a total of three points.
Question #1 Question #2 Question #3
0 pts None or some of the main
points summarized in your own
words.
No genuine specific/actionable
challenges identified. No original reflection provided.
1 pt Most or all of the main points
summarized in your own words. At least one genuine specific/
actionable challenge identified. Original reflection provided.
Examples. A bit later in the term I will distribute examples of blog entries that satisfy the criterial above.
Guidelines for blog postings
Purpose. The purposes of the blog posting are:
• to give you multiple exposures to the course material,
• to help you identify your specific points of difficulty, and eventually, to become your own teacher,
• to help me allocate lecture time effectively,
• to encourage transfer of mathematics to other areas of your academic/professional/personal life.
In short, these blogging assignments can be a tremendously valuable learning tool for you. In addition,
before your final exam, you can go back through your blogs to review your specific points of difficulty.
Finally, note that data collected in other classes1 shows that blog grades correlate with final grades
much more strongly than midterm exam grades do, even though in that class midterms were worth 40%
of the final grade and blogs were only worth 10%.
Content. Each blog posting has three parts:
1. Answer the question “What are the main points of the reading assignment for tomorrow?” You
should explain these in your own words, as if you were explaining them to a friend or family member.
2. Answer the question “What about the material challenged you?” “Nothing” is not an acceptable
answer. If nothing challenges you, then you should think about the material at a deeper level and
generate some honest questions. If you still cannot generate a genuine challenge, then answer the
question “What do you think was the most difficult part of the material for most people in the class?”
3. Write anything reflective about the material. Some suggestions are “what was the most interesting
part of the material and why?” or “how does this material connect to something else you have
learned in mathematics?” or “how is this material useful/relevant to your intellectual or career
interests?” but anything genuine, honest, and original will suffice.
Format. Question #1, #2, and #3 should each be answered in a separate paragraph. The paragraphs
should be given the labels “Main points,” “Challenges,” and “Reflections” respectively. You do not need
to write essays. A couple of sentences answering each question may be sufficient.
Deadline. You must post by 11:59 p.m. on the night before each lecture that has a new reading
assignment. If there is a technical problem with blogger.com, you can always email your posting to me
as a last resort and add it to your blog when blogger.com is fixed.
Grading. Blogging is worth 10% of your final grade. I randomly select which blogs to read for each
lecture. On average, I will grade your blog approximately once per week. You will not know in advance
when I will grade your blog. If your blog posting is not online when I sit down to read it (sometime
between midnight and lecture time) you will receive a 0 for that post. Blogs are graded according to the
rubric below. Each blog posting is graded out of a total of three points.
Question #1 Question #2 Question #3
0 pts None or some of the main
points summarized in your own
words.
No genuine specific/actionable
challenges identified. No original reflection provided.
1 pt Most or all of the main points
summarized in your own words. At least one genuine specific/
actionable challenge identified. Original reflection provided.
Examples. A bit later in the term I will distribute examples of blog entries that satisfy the criterial above.
My name is Brian Hurlow and I'm a first year at Mac. I am still deciding on my major/minor, I'm considering a Polisci major with a minor in music but at this point I'm experimenting with different things to see what I really like.
This semester I'm taking:
Music Theory 1
The Psychology of Presidential Politics
The Principles of Art
And Applied Calculus.
The weakest part of my math background is probably that my math experience has involved a lot of routine computation and not a whole lot of challenging problems. I'm very good at paying attention to what I learn in class and being able to reproduce it on the test, but I don't think my math skills have really been challenged so there's no telling how comfortable I'm going to feel in a harder math class.
I'm taking this class because I really want to take calculus. I feel like I want to make sure I leave college with and understanding of higher level math, I think it is an important part of being a well rounded, informed individual.
I want to be challenged and put my math skills to the test. I want to expand my mind by learning something I never have before.
My interests consist of: Traveling, Sailing, Playing music in any form, meeting people, goofing around, and reading/writing short stories.
The worst math teacher I've ever had was in my senior year of high school, and he was incredibly lazy and mostly unwilling to help his students.
The best math teacher I've ever had was in my freshman year, he was just very approachable.
I would love to hear a bit of rent in the morning, just to get me even more sick to my stomach. no seriously I am in love with radiohead.
I'm excited for math, I actually really like and hope to work hard and enjoy calculus.
Thanks
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