Light Bulb Project

Ski Movie Final Project

The purpose of this final project is to help us understand related rates through teaching them ourselves. We are also hoping that we can help to teach others and challenge them to a new Related Rates problem. About all everyday scenarios, such as skiing, contain some bit of calculus. We are proposing a scenario, in which two skiers are skiing away from each other. We are going to calculate the rate at which the distance between them is changing. This problem was created by skirdude, secret and Flying Slug.

Two skiers are having an extreme day on Snowmass Mountain. They have been skiing together all day, dropping fatty cliffs and schralping mad gnar. Its the end of the day, and they are getting tired, so they now decide they both want to go down different runs. They are total math buffs and decide to turn this into an awesome Related Rates problem!

You can find our solution here!

And a big shout out to the crew at Aspen for being very patient with us and giving us tons of help!

Math is Everywhere

The article was about how math is used; it is used in everything now. It is used to track the way people buy products, where they buy products, and the way they react to ads. Google uses it to see which sites people are clicking on, and what sites they are buying things from. All of this monitoring, does have it’s downsides though, even though companies will be saving money what about people’s personal privacy, do they still deserve that?

I think that the main problem with this whole thing is that idea of lost privacy. I don’t necessarily want people watching all of the things that I buy, and all of the things that I click on the Internet even if it isn’t inappropriate. So as the article pointed out I would probably take measures to see that the companies couldn’t track what I said, bought and clicked on. The article also pointed out that the government is using the same data to find terrorists, and track diseases. So it is a mixed bag whether we should trust this whole database of information about us. What is interesting and more concrete though is that people who are going into college right now should probably major in something math related if they are at all into it because that is where the market is. At least that is what the article said. I disagree. I think that it would be sad if all these bright people decided to go into math simply because that is where the jobs lie. People should follow their passion. This was an interesting article but I disagree with it in some aspects, I don’t think that our lives should be so closely monitored even if it does help homeland security or our health, it isn’t natural, or healthy in my opinion. And I don’t want my life to be run by either a government or by industries.

SeeMath

Derivatives and Inverse Derivatives

1. For the point (a,b) on the x,y plane the derivative of the inverse function derived at point b is the reciprocal of the derivative of the function at point a.

2. If F(x) is an invertible function, then for any point (a,b) on F(x):


3. The volume equation is an invertible function because its inverse is a function.

4. This is my thought process:




5.
Black is the original volume function, red is the inverse of that, blue is the derivative of the original function at x = 1, and green is the derivative of the inverse at point y = 1
The functions I used were as follows:
Y = (4/3)*pi*x^3
Y^-1 = ((3x)/(4*pi))^(1/3)
Y' = 4*pi*(x-1)+((4/3)*pi)
Y'^-1 = ((1/(4*pi))(x-((4/3)*pi)))+1

6. 1/(4*pi) = the inverse of 4*pi

7. I came to class on Wednesday because I would not have learned the same amount on my own. I needed the review that we did in class and I learn better with interaction. I also probably don't have the initiative to do the work on my own, so it is good that I came to class because otherwise I might not have studied thoroughly.

Differentiability Implies Continuity

Good day to all.

Today we studied Differentiability Implies Continuity a property that will help in simplifying the process of deciding whether a function is differentiable or continuous.

WHO IS THIS???

(*answer at the end of the post along with citation)

Continuity

Opening up class today we reviewed Continuity at a point, on an interval, and even in a function. Continuity at a point has three requirements; do you remember them?

1. There must exist a value for f(c)

2. There must be a limit for f(x) at x = c

3. The limit must equal the value at x = c

Continuity at an interval is not far off from that: f(x) is continuous on an interval only if it is continuous at each x value in the interval. And for a function to be continuous it must be continuous for every x value in it’s domain.

Differentiability

We then continued reviewing by covering differentiability. We started with the definition of derivative at a point:

We also reviewed the graphical and physical meaning of derivative, which are the slope of the tangent line and the instantaneous rate of change of the function respectively.

Then, like with continuity we reviewed differentiability of f(x): at a point x = c if there is a derivative at x = c; on an interval if it is differentiable for

every x-value in the interval; and in a function if it is differentiable of each x-value in it’s domain.

Syllogisms

After we had reviewed the basics needed for the day we dove into the important information of the day. Definition (from the dictionary on my computer) : syllogism |’silə,jizəm| Noun • deductive reasoning as distinct from induction. Do you remember these from geometry?

Ex. of the positive: If pajamas are flannel then they are comfortable -- P => Q

Ex. of the contrapositive: If pajamas are not comfortable then they are not flannel -- ˜Q => ˜P

Ex. of the inverse: If pajamas are not flannel then they are not comfortable -- ˜P => ˜Q

Ex. of the converse: If pajamas are comfor

table then they are flannel -- Q => P

Think about which ones of these are true, then look at the mathematical examples and ask yourself the same question.

If function f is differentiable at x = c then f is continuous at x = c

Ex: just think of any continuous line, and

it’s derivative

If the function f is not continuous at x = c then f is not differentiable at x = c

Ex: any sort of discontinuity, leap, and infinite, removable… will fill this example

If the function f is not differentiable at x = c then f is not continuous at c

Counter Ex: a cusp is continuous, but not differentiable. The reason it is not differentiable is because there are infinite different derivatives at a cusp.

If the function f is continuous at x = c then it is differentiable at x = c

Counter Ex: again a cusp fits this counter example because it is continuous, but has infinite tangent lines, and thus is not differentiable.

Take a look at these graphs, Blue represents that a function that is not continuous is not differentiable. the reason that this is not continuous is because of the step, the limits from the two different sides are different, thus there is no limit, and it cannot be continuous. The red represents a cusp. The cusp is continuous, there is a limit, and a value, and they are equal, but it is not differentiable because at the cusp there are an infinite number of lines tangent to the tip of the cusp.

On a side note, the easiest way to graph a cusp is to use the absolute values, for example: y = |x| + 1 would be a V-shaped graph with a y intercept of 1. To make it negative you can just add a negative sign in front of the absolute value sign because that will flip the whole thing into the negative.

And what we draw from that experience is that:

Differentiability implies Continuity

If a function f is differentiable at x = c, then f is continuous at x = c

If a function f is not continuous at x = c, then f is not differentiable at x = c

Example:
If the equation is :
Is the function continuous at the point x = 2?
Not it is not because when x = 2 the numerator and denominator are zero, and thus there is a hole at x = 2 because 0/0 can be any number.
Is the function differentiable at the point x = 2?
No because the function is not continuous at the point.

*The picture was a statue of Socrates

http://heroworkshop.files.wordpress.com/2009/11/socrates.jpg

The next scribe will be (drum roll please): Mc Casper

An Online Education

Internet Schooling

The internet has a bountiful amount of information that exceeds all other sources. Take for instance this blog, any person in the world can read this blog and learn the second semester of calculus without ever setting foot in a classroom. If I wanted to figure out how to use the slide rule, if I wanted to learn Spanish, even if I wanted an in depth analysis of Hamlet, I could find that online. In fact I have used Google to figure out how to make soda from a guy who lives in Indiana and made it as a kid with his parents. It has never been so easy to gather or share information. No longer do you have to go to the library to get a book, it is there on Google Books, three feet away at your desktop computer.

There is one simple flaw to this avalanche of information called the internet. Students are not interested in simply learning, most kids go to school because they have to. On the internet what will stop them from playing games, checking Facebook, or looking at a more interesting site? Students already have a hard enough time completing the assigned homework, part of the force behind doing homework is that the students will see the teacher and their disappointment the next day when they don't finish their homework. Anything that a student doesn't understand in class but needs to know to complete the homework can be found online, I don't think that is should go any farther. The internet is not as good at teaching as a human, and not as demanding.

Product Rule

Book vs. Internet

Looking in the book, I found that the definition was very concise and easy to understand. Unlike the site that I found which took time to watch and absorb all of the information as it flashed by on an animation. In the Presentation that was dragged on for several minutes I learned the same amount as in the two line definition box in the book. It is also a lot easier to understand an elementary explanation than a huge production with lots of distractions.

Definition:
if f(x) = u•v then f'(x) = u'•v + u•v'

Examples:
f(x) = 3x^2 • lnx
f'(x) = (6x • lnx) + (3x^2 • 1/x)
f'(x) = (6x • lnx) + (3x)

First Semester Calc Mind Map