How Far Can Math Take Us?

In the first portion of the cover story Math Will Rock Your World, Steven Baker discussed the different applications of math in our world today. The first section discussed the conversion of words into math in order to classify and organize them. He gave the example of Neal Goldman, a math entrepreneur who takes press articles and blog posts and groups them together with related pieces based on their content. The article also discussed mathematical models and their ability to predict a variety of different things such as how to become more productive, make a profit, etc. Through collecting data, they use their findings to establish trends. One potential problem with this system lies with the population and their ability to breech the system. Although this system appears very dependable it also brings its challenges for the United States and the world. As a nation we must work towards creating more mathematicians at home and prevent outsourcing by preparing our students for the business world. Another challenge math entrepreneurs face is individual privacy. They must discover a means of invading privacy to the least extent possible by trying to share group information without sacrificing that of the individual . It surprised me how much math manifests itself in our daily lives. "Some models predict what music we'll buy, others figure out which worker is best equipped for a particular job." I never realized the full potential of math in everything we do. I was also astounded by the extent to which math can invade our privacy and how frequently numbers have begun to replace the individual.
The second portion of the cover story Online Extra: The NSA: Security in Numbers discussed the usage of numbers in terms of security. Baker discussed the NSA's role and how they use math to help figure out terrorist plots and other issues of security. One of the challenges the NSA faces manifests itself in recruiting its mathematicians. They are faced with competition from Google, Yahoo and other organizations. How are they managing? They offer a more quiet, stable lifestyle, which is especially more appealing to women. I thought it was interesting that they were looking for younger employees who did not know the company as well. I assumed they would want older, more experienced workers. I found the selection process quite interesting in that it is a contest which acts as somewhat of an advisory for their decisions.
The third portion Online Extra: Search Advertising by the Numbers discussed bidding on keywords. Baker specifically discussed Imran Khan and his involvement in E-Loan. Kahn makes sure those who are looking for his product will surely find it. Parts of this passage made it seem like math can often get a little bit too invasive, "It (numbers) enables marketers to track customer behavior, and replaces hunches with science." It may have just been the way I read into the article, but that sounds as though it could be problematic in terms of individual privacy issues. I also found the discussion of bidding on keywords and the science behind it a bit confusing.
How Much Math Do We Need to Know? summed up the different uses of Calculus, Algebra and Geometry, Statistics and Probablility, and Math Tools in different careers. Some of the associations are not as obvious as they seem, for example, advanced geometry is used not only in floor tiling but also in designing search engines. With the growing demand for mathematics based, or even just related professions, knowing your math might prove to be helpful. I guess calculus really will pay off in the long run!
How Math Transforms Industries discussed the specific uses of mathematics in many different fields, such as consulting, police and marketing and the media. Who knew a company could "turn written articles into bits of geometry and organize them in a virtual library?" and eventually these automatic systems could make editors obsolete. I found it amazing how much of a difference math has a potential of making on our society. We always talk about technology taking over life as we know it, but what about math?

See Math

Look at this picture and what do you see?


Sometimes the answer is not as evident as it seems...


You might need to put on your calculus glasses for this one...




Almost there...


And now you are officially an expert at recognizing functions

Choices

1) The Derivative of the Inverse of a Function says that for any point (a,b) on an invertible function, the derivative of the function's inverse evaluated at point b is equal to the reciprocal of the derivative of the function evaluated at point a.
2)



3) Volume is an invertible function
4)



5)


a)(4/3)*3.141592654x^3
b)12.566375(x-1)+4.1887902
c)(x/(4*3.141592654/3))^(1/3)
d).09403455(x-4.1887902)+1






6) My point (1, 4.1887902)
slope of tangent line at f(1)=12.566375
slope of tangent line at f'(2.3562)= 0.09403455
12.566375*0.09403455=1

7)I chose to come to class on Virtual Wednesday because I feel as though I learn better in a classroom environment. It provides structure and an atmosphere in which it is easier for me to focus. The resources of Bru and my classmates are of great help when I don't completely understand a concept. Learning from Bru and my classmates in a classroom as opposed to an online classroom is a much more reliable method of learning for me and is a more concrete way to learn the material. I have the opportunity to ask questions and am provided with a structured and complete lesson.

Derivatives of Other Trigonometric Functions (Scribe Post for Monday's Class)

Welcome to a snowy day in Carbondale, Colorado. Today's topic of interest: The derivatives of trigonometric functions.

In the beginning of class, we reviewed a few topics from trigonometry including the reciprocal properties, the quotient properties and the Pythagorean Properties. Here are the slides from the notes we took in class. It also includes the derivations of the four other trig functions. Hope it helps!

CRMS Calculus 2010 January 18, 2010

View more documents from Colorado Rocky Mountain School.

I found the unit circle diagram for the Pythagorean Properties very helpful, but in case this is hard to read, here is a summary of the three types of properties:

#1: THE RECIPROCAL PROPERTIES:



#2: THE QUOTIENT PROPERTIES:



#3: THE PYTHAGOREAN PROPERTIES:


Ok, so now onto the fun part, the derivatives of the other trig functions. In order to discover these derivatives, we used the quotient rule.
(Just in case you forgot, the quotient rule is the derivative of numerator times the denominator minus the numerator times the derivative of the denominator all divided by the denominator squared.)

So here are our the results of our derivations for the six trigonometric functions:













In order for this to be true though, x must be in radians

Here are some of the memory aids we came up with in class:
- Derivatives of "co" functions have negative signs
-from Pythagorean Properties
+tanx goes with secx
+cotx goes with cscx
+sinx goes with cosx


Here is a short video I found helpful. It has a few examples:

3.5 Derivatives Of Trig Functions

View more presentations from ricmac25.
http://www.slideshare.net/share/blogspot/116892

REMEMBER TOMORROW IS VIRTUAL WEDNESDAY. YOU CAN CHOSE WHETHER TO COME TO CLASS OR TO DEVELOP YOUR OWN VIRTUAL LESSON. The topic is: the derivative of inverse functions.

If you have any questions at all please feel free to leave a comment.

And the next scribe is..... drum roll please.... Beston

Here is the sample derivation for secant:

Anyone can learn anything, from anyone, anywhere, at anytime

To some extent, because of Web 2.0 it is true that anyone can learn anything, from anyone, anywhere, at anytime. Nowadays, a person can find anything online. Whether you have a question on homework or would like to learn how to knit, the Internet will provide the answer. Instead of purchasing textbooks, they are becoming increasingly available to read online, such as our calculus book. Web 2.0 allows us to read the opinions of others, which can significantly help a person understand a problem or a topic through blogs, forums and even videos. For example, in physics we were asked to build a spaghetti bridge. Having the Internet available to research successful structures in terms of videos and blueprints proved extremely resourceful.
Although I believe Web 2.0 is a useful resource in finding many different types of information, I do not believe it should be the sole informant. Schools and formal education, in my opinion are very important components of education. Teachers and classrooms encourage learning and focus. In addition, a student faculty relationship is created, predominantly one that pushes the students to complete their work to their best ability. Internet sources, on the other hand, can often provide distractions. Focus while on the Internet often requires control and restraint in terms of the distractions. I believe the Internet can and does provide a source to find anything from anyone anywhere at anytime, but schools and a formal education are also a necessity in order to create a balanced education.

Product Rule

Or textbook clearly explained the definition of the derivative of a product of two functions. It also gave three examples in order to fully understand the product rule. The book includes both a mathematical and verbal explanation. The online explanation was similar in that it mathematically explained the product rule and gave multiple examples. I found the online explanation useful in that it explained the product rule in terms of f(x), g(x), and h(x), mathematically in multiple ways. I found the textbook explanation a bit easier to understand and visualize because of its verbal explanation.

Product Rule: Mathematical: If h(x)=f(x)g(x), then h'(x)= f'(x)g(x)+f(x)g'(x),

Verbally: The derivative of the function of the product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second.

Example:

f(x)=x⁴cos6x

f’(x)=4x³cos6x+x⁴(-sin6x)(6)

f’(x)=4x³cos6x-6x⁴sin6x

Calculus Semester 1 Concept Map