Today's lecture included the topic of outliers. How is a graph changed when and outlier is included or excluded? Since outliers deal with extremes they often bring up political and social issues. The example from class was of salaries at a hypothetical company where the CEO made more then three times as much as the highest paid employee.
I noticed this graph in the New York Times and wanted to save it for future reference.
Not all YouTube math videos are created equal!! I will post links to the most useful ones here:
Central Limit Theorem:
If you want me to post more post a comment about the topic that you need and I will find the best (and mathematically correct) videos for the topic.
Dear 375 Students:
Here is a calendar showing test dates and other info, it will be updated as the semester rolls along. I have contacted the person in charge of rooms, with some luck we should be moving to more spacious digs soon enough!
Here is the problem:
Prove that the volume of a cone, (defined as the set of points on line segments joining a vertex, v, with a set of points in the same plane called the base) with height H, and base area B is .
What makes this tricky is at first it seems, with so many possible shapes for the base, there is no good way to express all of these 'different' areas with the same A(x). But, since the slices are similar we need only concern ourselves with the ratio of any side, radius, diagonal or width of a given slice to that same side, diagonal radius, or width on the base. With this we can find the area, independent of any specific formula (such as or )
To do this use the fact that the ratio of areas of similar plane figures is equal to the ratio of any one of their dimensions squared. And if we are looking for the area slice at x that ratio will be the same as the ratio of x to H, the total height.
So, this means that . So, now you have a formula for A(x).
You can use this trig identity
Consider , where g and h are the position functions of the two runners.
Induction would be much much harder than L'Hopital.
Divide the square root of x in to each number in the numerator to make it easier to see what the anti derivative should be.
Draw a diagram look at the triangles.
1. Find an equation of the tangent to the curve that is parallel to x−4y = 1.
(It is better to leave the natural log in the equation, don't find a decimal value... Though you may use such value for graphing.)
2. Find an equation of the tangent to the curve that will pass through the origin.
The the tangent to the curve at will have the form ... Plug in (0,0) and you find you are seeking the x-value where the natural log is zero. So the equation is .
How can one deal with ? It appears to be the indeterminate form . The graph shows that it has a nice limit:
(Notice, how this function produces imaginary vales when it is between -1 and 0. )
But, this is the wrong kind of indeterminate form for L'Hopital. If we could somehow show that it was, in fact, the same as we could use the derivative to help us... Here is an extension of L'Hopital that will do just that:
See if you can apply it. We will talk more about this in class. If you simply said "we can't use L'Hopital" that is correct for this problem based on what you knew up until now. If you applied it anyway and tried to say that: ... then that is incorrect! (And I believe it leads to the wrong answer here.)