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.
Wednesday, October 29, 2008
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.
Subscribe to:
Posts (Atom)