10/30/2008: Intro. to differential equations, solutions to differential equations, Euler’s method

1. Main Points
In today’s reading I learned about differential equations which are employed when one does not know a key function but has information about its rate of change or its derivative. This information is then used to write a differential equation from which one can get information about the original function. A solution to a differential equation is any function that satisfies the differential equation.

2. Challenging
I did not quite understand how we check a solution to a differential by substituting the solution into the left and right sides of the equation separately.

3. Intriguing
I found the example of a population in confined space growing proportionally to the product of the current population, P, and the difference between the carrying capacity, L, and the current population, to be really cool. This can definitely be applied to bacteria growing in a petrie dish for example!

10/28: Constrained optimization and Lagrange multipliers

1. Main Points

In todays reading I learned how to solve a constrained optimization problem using Lagrange multipliers and Lagrange functions. A lagrange multiplier is an approximation of the change in the optimum value of f when the value of the constraint is increased by 1 unit. In terms of production, it can be seen as the “extra bang for your buck”. Lagranges multiplier is calculated by taking the function to be optimized over the constraining function. Lagrangian functions are used when dealing with constrained optimization problems by first writing down the Lagrangian function and then finding the critical points of the Langrangian Function.

2. Challenging

I did not quite understand why we find the critical points of the Lagrangian function when solving constrained optimization problems nor what they represent.

3. Intriguing

The Lagrangian multiplier and function must be immensely useful when doing calculations for business management. Now I understand what the basis for the expression “getting the most bang for your buck” is!

Unconstrained Optimization in 1-d and 2-d (10/21/2008)

1. Main Points

The main point of today’s reading was to learn about how to determine global maxima and minima of functions. To find the global maximum and minimum of a continuous function on an interval including endpoints, one compares values of the function at all the critical points in the interval and at the endpoints. When excluding endpoints, one finds the values of the function at all the critical points and sketches a graph.

2. Challenging

I see that they use differentiation, then setting equal to zero and then solving for x in the examples. I do not remember well the tactics used for factoring out when solving for zero, maybe you can go over this quick in class?

3. Intriguing

I realize that being able to maximize and minimize when solving different problems in all natural and social sciences must be very useful. I particularly found the example showing what speed to drive in order to have the smallest gas consumption interesting and very useful. I will drive at about 50 mph as much as I can from now on!

Second derivative, unconstrained optimization in 1-d (10/14/2008)

1. Main Points

The major new concept in today’s reading was the second derivative (f’’), which is the derivative of dy/dx, and the different ways it can be applied for solving mathematical problems. Some things we can tell from f’’ is that if f’’>0 on an interval it means that f’ is increasing, so that the graph of f is concave up there. The reverse for f’’<0. The second derivative can also be used to test for local maxima and minima for a continuous function f, called critical points. The sign of the f’’ will also change at an inflection point, which is defined as a point at which the graph of a function f changes concavity.

2. What was challenging about this material?

The hardest part of todays rading I think was locating inflection points, maybe you could go through this at greater length during class?

3. To think of the second derivative as the rate of change of a rate of change is a really cool concept I think. The example in the reading about the senate explaining that they had not cut the defense budget, only cut the rate at which it was increasing. Essentially, the derivative of the defense budget was still positive (the budget was increasing), but the second derivative was negative (the budget’s rate of increase had slowed)

Gradient and directional derivatives (10/9/2008)

1. Main Points

In todays reading I learned about the gradient which is a vector consisting of the partial derivatives fy(x,y) and fx(x,y). These vectors tell us the rate of change in the x and y directions, directional derivatives on the other hand can tell us the rate of change in any direction in the (x,y)-plane.

2. Challenging

The justification for the directional derivative formula was a little hard to follow I thought. The algebraic steps especially, are we expected to learn this justification?

3. Interesting

It was interesting to learn how to calculate directional derivatives, no longer being constrained to rates of change in the x and y directions, but now I can find the rate of change in any direction in the (x,y)-plane.

Vectors, dot product, and vector components (10/7/2008)

1. Main Points
Today I learned more about vectors which is a geometric object that has both a magnitude and a direction. A vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B. Vectors can be calculated by multiplication and addition for example.

2. Challenging
The concept of a unit vector and its rationalization was a little hard to understand, maybe you could go over this in class?

3. Intriguing
Vectors is a interesting topic I think because they play an important role in physics: velocity and acceleration of a moving object and forces acting on a body are all described by vectors. Many other physical quantities can be usefully thought of as vectors.

Derivatives of periodic functions / Partial derivatives (10/2/2008)

1. Main Points
In todays reading I read about how periodic functions also have periodic derivatives. In fact d/dx(sinx) = cosx and d/dx = -sinx for example. I also read about partial derivatives, in general a partial derivative of f with respect to x at (a,b) is the derivative of f with y constant. One can use all the techniques from single-variable calculus to find partial derivatives.

2. Challenging
I found the concept of a second order partial derivative to be somewhat confusing. I do not understand why and when they are used, maybe you could go over this in class?

3. Intriguing
It is interesting to learn calculus that feels like it has more than one dimension. I can see how partial derivatives can be useful for predicting the future in many areas of the sciences as shown in the example dealing with rat population surviving after exposure to formaldehyde vapor.