Main Points
In reality all things have contraints, for example, there is a limit to the amount of money available to build a public transportation system. A maximum value under constraint occurs when the constraint is tangent to a function's contour at a point p. This could also occur at the endpoint of the constraint. The Lagrange Multiplier is like the marginal value in production. ie the change in production levels due to a unit increase in constraint.
Challenges
I do not follow how exactly the Lagrangian Function works. I think all the symbols confuse me.
Thursday, March 27, 2008
Sunday, March 23, 2008
4.3: Global Maxima and Minima/ 9.5: Critical Points and Optimization
Main Points
Local maxima and minima deal with where a function has greater or smaller values at surrounding points; global maxima and minima deal with the largest and smallest values respectively. In order to find the global maxima and minima one would need to find and graph all the critical points.
In finding the greatest or smallest value of a function is to optimize it.
Challenges
I do not understand how to use the analytical method of finding the local maximum or minimum for more than one variable. I might understand the second derivative test equivalent for this too if I understood how it was done (the analytical method that is).
Reflections
Global extrema seems to me to be a lot more useful than local extrema, in which case I find it difficult to udnerstand the great significance of the local extrema. Global extrema helps you find the "extreme" max or min of any function. This is a lot more useful I would think.
Local maxima and minima deal with where a function has greater or smaller values at surrounding points; global maxima and minima deal with the largest and smallest values respectively. In order to find the global maxima and minima one would need to find and graph all the critical points.
In finding the greatest or smallest value of a function is to optimize it.
Challenges
I do not understand how to use the analytical method of finding the local maximum or minimum for more than one variable. I might understand the second derivative test equivalent for this too if I understood how it was done (the analytical method that is).
Reflections
Global extrema seems to me to be a lot more useful than local extrema, in which case I find it difficult to udnerstand the great significance of the local extrema. Global extrema helps you find the "extreme" max or min of any function. This is a lot more useful I would think.
Wednesday, March 12, 2008
2.4: The Second Derivative
Main Points
The second derivative (2-d) is the derivative of an already derived function and is denoted f '' as well as d^2y/dx^2. When the 2-d > 0, f ' is increasing and the graph of f is concave up and when 2-d <>
The second derivative (2-d) is the derivative of an already derived function and is denoted f '' as well as d^2y/dx^2. When the 2-d > 0, f ' is increasing and the graph of f is concave up and when 2-d <>
Challenges
How do you tell when a 2d is positive or negative? If its just by looking at whether the graph is concave up or down, what happens when we cant see it? and why is a trough concave up and a crest concave down??
Reflections
I'm beginning to lose my grasp on just how applicable calculus is to the real world as it gets more complex. Would a second derivative really be used?
Sunday, March 9, 2008
The Gradient and Directional Derivatives
Main Points
An ordered pair of real numbers is a 2-dimnesional vertor which is symbolised by a bold letter. The difference between a point and a vector is that the point is simply a point location on the plane whilst the vector is an arrow. The length of a vector (a,b) is denoted by (a, b). A unit vector has a length of one unit. For any nonzero vector (a, b), you can find a unit vector in the same direction by dividing each coordinate of (a, b) by the length of the vector.
The dot product of two vector is as follows: (a1, a2) · (b1, b2) = a1b1 + a2b2; the result is a single number.
The partial derivative fx(x, y) of a function z = f(x, y), tells us the rate of change in the x-direction just as fy(x, y) tells us the rate of change in the y-direction. A gradient seems to have the following properties:
The gradient vector always points in the direction of greatest increase.
The gradient vector is always perpendicular to the tangent of the curve at which it is rooted.
The opposite direction to that of the gradient is the direction of greatest decrease.
Challeneges
After looking at the properties of a gradient, I dont think I understand exactly how we got to that point because I do not understand wy we have those properties.
Reflections
It looks to me as though vectors might be used to simplify things like a contour diagram to facilitate understanding for those less familiar with the diagram. And they are more useful than just points here, especially since they have a directional factor as well.
An ordered pair of real numbers is a 2-dimnesional vertor which is symbolised by a bold letter. The difference between a point and a vector is that the point is simply a point location on the plane whilst the vector is an arrow. The length of a vector (a,b) is denoted by (a, b). A unit vector has a length of one unit. For any nonzero vector (a, b), you can find a unit vector in the same direction by dividing each coordinate of (a, b) by the length of the vector.
The dot product of two vector is as follows: (a1, a2) · (b1, b2) = a1b1 + a2b2; the result is a single number.
The partial derivative fx(x, y) of a function z = f(x, y), tells us the rate of change in the x-direction just as fy(x, y) tells us the rate of change in the y-direction. A gradient seems to have the following properties:
The gradient vector always points in the direction of greatest increase.
The gradient vector is always perpendicular to the tangent of the curve at which it is rooted.
The opposite direction to that of the gradient is the direction of greatest decrease.
Challeneges
After looking at the properties of a gradient, I dont think I understand exactly how we got to that point because I do not understand wy we have those properties.
Reflections
It looks to me as though vectors might be used to simplify things like a contour diagram to facilitate understanding for those less familiar with the diagram. And they are more useful than just points here, especially since they have a directional factor as well.
Thursday, March 6, 2008
Linear Algebra: 1.0, 1.1, 1.2, 4.1.0
Main Points
1.0
Coordinates can be represented by a vector which starts at the origin (0,0). The coordinates of a vecotr are written vertically, with the x-coordinate above the y. This shows that a vector is being used rather than just the point. a zero vector would simply be a point at the origin since both the head and tail of the vector are in the same place.
1.1
There are 2 operations that can be done with vectors and not points on a plane, one of which is scalar multiplication, where we multiply a vector (both the x and y coordinates) by a constant.
1.2
It is also possible to add two vectors together, called coordinate-wise addition (because corresponding coordinates are added, ie, x with x and y with y). This sum is represented as the diagonal of the parallelogram drawn with the original vectors on two sides.
4.2: Dot Products
Def: The dot product of two vectors is teh real number obtained by multiplying corresponding coordinates of the vectors and adding. Note that this is a single number, a scalar, not a vector. Also the product of 2 vectors is 0 if they are perpendicular to each other.
Challenges
Other than the benefits of having direction included, what are the benefits of having a vector to work with rather than just a point since its just the number that we are often using anyway?
Reflections
Again I can see how calculus applies to life as even with the plain examples that simply include graphs and vectors, I can connect them with problems about wind speed and how it would affect a travelling boat for example.
1.0
Coordinates can be represented by a vector which starts at the origin (0,0). The coordinates of a vecotr are written vertically, with the x-coordinate above the y. This shows that a vector is being used rather than just the point. a zero vector would simply be a point at the origin since both the head and tail of the vector are in the same place.
1.1
There are 2 operations that can be done with vectors and not points on a plane, one of which is scalar multiplication, where we multiply a vector (both the x and y coordinates) by a constant.
1.2
It is also possible to add two vectors together, called coordinate-wise addition (because corresponding coordinates are added, ie, x with x and y with y). This sum is represented as the diagonal of the parallelogram drawn with the original vectors on two sides.
4.2: Dot Products
Def: The dot product of two vectors is teh real number obtained by multiplying corresponding coordinates of the vectors and adding. Note that this is a single number, a scalar, not a vector. Also the product of 2 vectors is 0 if they are perpendicular to each other.
Challenges
Other than the benefits of having direction included, what are the benefits of having a vector to work with rather than just a point since its just the number that we are often using anyway?
Reflections
Again I can see how calculus applies to life as even with the plain examples that simply include graphs and vectors, I can connect them with problems about wind speed and how it would affect a travelling boat for example.
Tuesday, March 4, 2008
9.4: Computing partial derivatives algebraically.
Main points
The notation f_x(x, y) represents the partial derivative of the ordinary derivative of f(x, y) wrt y with x fixed and vice versa. Partial derivatives can be differentiated a second time to obtain second-order partial derivatives. A typical function such as z = f(x, y) would have 2 first-order partial derivatives and 4 second order partial derivatives.
Challenges
The book had to complicate things further just as I thought I was getting first-order partial derivatives. I understand what the entire concept is....computation is troublesome.
Reflections
Most of this is beginning to seem rather "far out there." Obviously with the examples used these actually DO have some real life applications but just how relevant is it to people who are not really concerned with how it all works.
The notation f_x(x, y) represents the partial derivative of the ordinary derivative of f(x, y) wrt y with x fixed and vice versa. Partial derivatives can be differentiated a second time to obtain second-order partial derivatives. A typical function such as z = f(x, y) would have 2 first-order partial derivatives and 4 second order partial derivatives.
Challenges
The book had to complicate things further just as I thought I was getting first-order partial derivatives. I understand what the entire concept is....computation is troublesome.
Reflections
Most of this is beginning to seem rather "far out there." Obviously with the examples used these actually DO have some real life applications but just how relevant is it to people who are not really concerned with how it all works.
9.3: Partial Derivatives.
Main Points
The partial derivative of f wrt x at (a,b) is the derivative of f with y constant. The reverse is also true (ie. wrt y with x constant.) This is computed using the formula: lim (h -> 0) (f(a + h, b) - f(a,b))/h. This is when y is fixed. When x is fixed the formula changes to lim (h -> 0) (f(a, b + h) - f(a,b))/h. Partial derivatives have a different symbol from the normal one. On a contour diagram, the partial derivative is taken as the rate of change of the value of the function on the contours. And as always the units of the partial derivative would help you to identify what exactly it is discussing.
Challenges
I kinda vaguely understood what the partial derivatives were in terms of making a two-variable equation a one-variable one by fixing one of the variables, but then Example 2 lost me. I have no idea how the values are computed or what actually they mean.
Reflections
Calculus and math for that matter can be used for a lot of things from the simple to the complex all in aid of better understanding it...But are these really used? With the airplane example, its pretty obvious that fixing x at 10 would cause the sales value to increase by $350 for each full priced ticket that is bought. Is that not like a "duh factor"? Are we assuming that figured out that increase was using partial derivatives?
The partial derivative of f wrt x at (a,b) is the derivative of f with y constant. The reverse is also true (ie. wrt y with x constant.) This is computed using the formula: lim (h -> 0) (f(a + h, b) - f(a,b))/h. This is when y is fixed. When x is fixed the formula changes to lim (h -> 0) (f(a, b + h) - f(a,b))/h. Partial derivatives have a different symbol from the normal one. On a contour diagram, the partial derivative is taken as the rate of change of the value of the function on the contours. And as always the units of the partial derivative would help you to identify what exactly it is discussing.
Challenges
I kinda vaguely understood what the partial derivatives were in terms of making a two-variable equation a one-variable one by fixing one of the variables, but then Example 2 lost me. I have no idea how the values are computed or what actually they mean.
Reflections
Calculus and math for that matter can be used for a lot of things from the simple to the complex all in aid of better understanding it...But are these really used? With the airplane example, its pretty obvious that fixing x at 10 would cause the sales value to increase by $350 for each full priced ticket that is bought. Is that not like a "duh factor"? Are we assuming that figured out that increase was using partial derivatives?
Sunday, March 2, 2008
3.4: The Product and Quotient Rules/ 3.5 Derivatives of Periodic Functions.
Main Points
3.4:
The derivative of the product is nto equal to the product of the derivatives! That is
f(x)*g(x) = x*x^2 = x^3 whose derivative is 3x^2 which is not equal to f '(x)*g'(x) = (1)(2x) = 2x. 3x^2 is not equal to 2x.
The product rule is stated as: u = f(x) and v = g(x); (fg)' = f 'g + fg'
also stated as d(uv)/dx = du/dx*v + dv/dx*u
The quotient rule is stated as: u = f(x) and v = g(x); (f/g)' = (f 'g - fg')/g^2
also stated as d/dx(u/v) = (du/dx*v - dv/dx * u)/v^2
3.5:
Derivatives of periodic functions are also periodic. d/dx(sin x) = cos x and d/dx(cos x) = -sin x
Challenges
The Liebniz notation is alot more confusing than the other. if both forms of the derivative function were not given I would have a lot more difficulty differentiating. Why would you want to multiply or divide derivatives?
Reflections
Is it possible that between sine and cosine, one or the other simply exists because it happens to be the derivative of the one of the two. ie. for example, cosine exists because it is the derivative of sine. Therefore cosine curve only exists because sine and its derivative were discovered first.
3.4:
The derivative of the product is nto equal to the product of the derivatives! That is
f(x)*g(x) = x*x^2 = x^3 whose derivative is 3x^2 which is not equal to f '(x)*g'(x) = (1)(2x) = 2x. 3x^2 is not equal to 2x.
The product rule is stated as: u = f(x) and v = g(x); (fg)' = f 'g + fg'
also stated as d(uv)/dx = du/dx*v + dv/dx*u
The quotient rule is stated as: u = f(x) and v = g(x); (f/g)' = (f 'g - fg')/g^2
also stated as d/dx(u/v) = (du/dx*v - dv/dx * u)/v^2
3.5:
Derivatives of periodic functions are also periodic. d/dx(sin x) = cos x and d/dx(cos x) = -sin x
Challenges
The Liebniz notation is alot more confusing than the other. if both forms of the derivative function were not given I would have a lot more difficulty differentiating. Why would you want to multiply or divide derivatives?
Reflections
Is it possible that between sine and cosine, one or the other simply exists because it happens to be the derivative of the one of the two. ie. for example, cosine exists because it is the derivative of sine. Therefore cosine curve only exists because sine and its derivative were discovered first.
Subscribe to:
Posts (Atom)
