Forum Question: Quantifying Humanity

In January, 2006 the cover story of BusinessWeek was about numbers. According to the article, "the world is moving into a new age of numbers," where the “mathematical modeling of humanity promises to be one of the great undertakings of the 21st century.”

This article is another sign of the need for a more quantitatively-literate society. As more of us live, work, and play online, the more our digital profiles are open to analysis and susceptible to exploitation. People have been using mathematics to extract hidden patterns from data for centuries. However the increased volume of data in modern times has elevated the knowledge of math and computer technology to a 21st century life skill. In order to be smart consumers and informed citizens we need the ability to sort through data to identify patterns and establish relationships. Giving over that authority to the software behind our computers does not relieve us from the responsibility to be critical thinkers. Companies like Google and Yahoo are charting these oceans of data navigated by quants and math geeks who are becoming the new elite of the global economy. In the information age of the 21st century the best job in the world is that of data miner.

This week's Forum Question is to read, or listen to (a podcast is available) the BusinessWeek cover story. Of particular interest to you will be the last page which contains the overviews, "How Much Math Do We Need to Know?" and “How Math Transforms Industries”. Write a short summary of the article and then share your “take-aways” from the story.

Derivative of Inverse Trig Functions

Calculus… Episode 2/9/10: The Scribe Post Strikes Back.

Monday we discussed the topic of Derivatives of Inverse Trigonometric Functions
Let’s begin. We started out with a quick refresher on the different trigonometric functions and their individual inverse. Since none of the trig functions pass the horizontal line test because they are periodic, none of them are actually invertible functions or one-to-one functions. So in actuality they are inverse trig relations. Therefore in order to actually create an inverse of the functions you must restrict the domain of each so that they do become one-to-one functions.
This is simply done by taking only half of their corresponding unit circles.

This chart shows all of the inverse trig functions and their derivatives. Illustrating how they are to be used is shown above in those few example equations from class. Below I provided a few links that hopefully will be helpful. Lastly the next scribe post is dammitimmad.


http://en.wikipedia.org/wiki/Inverse_trigonometric_functions
http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/invtrigderivdirectory/InvTrigDeriv.html

Today's Slides: February 8, 2010

Hello All,
There are two sets of slides from today's class. Here's the first set on our review of inverse trig functions:

CRMS Calculus 2010 February 8, 2010_A

View more presentations from Colorado Rocky Mountain School.

The second set is on discovering the derivatives of these inverse trig functions:

CRMS Calculus 2010 February 8, 2010_B

View more presentations from Colorado Rocky Mountain School.

Cheers, Bru

Choices

Derivative of the Inverse of a Function:

1. If f(x) is an invertible function, then for any point on an invertible function, the derivative of the inverse of the function evaluated at b is equal to the reciprocal of the derivative of the function evaluated at a.

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

3. The volume of a sphere is a function of its radius:

Is volume an invertible function?

”Why yes indeed it is”

4. Inverse Function

5. Graph of:

(2) Choose any point (1,4.188) on V(r) and graph the tangent line through this point (Blue)(1) The volume function, V(r) (black)

(3) The inverse function r(V) (red)

(4) The tangent line through the “mirror” point on the graph of the inverse function. (Green)

6. OH MY! The tangents are reciprocals!

Function of the tangent line for V(r) at point (1,4.188)

Function of the tangent line for r(V) at point (4.188,1)

7. I am a strong believer in attending Calculus. I like having the opportunity to ask questions an see what specifically I have trouble understanding as well as listening to a variety of explanations from Bru and other students. I also seem to struggle with the Internet and technology in general so good old-fashioned pencil and paper helps me take in the material much better. Lastly, I am a procrastinator/ minimalist, so having the time to sit down in class provides a much more productive learning experience for me.

I See Functions!

Invertible

LINK: http://www.flickr.com/photos/47183167@N03/4325838211/

Continuous & Differentiable

LINK:http://www.flickr.com/photos/47183167@N03/4326566468/in/photostream/

Non One-to-One

LINK:http://www.flickr.com/photos/47183167@N03/4326552974/

Continuous, Non Differentiable

LINK:http://www.flickr.com/photos/47183167@N03/4325766571/in/photostream/

Not Continuous or Differentiable

LINK:http://www.flickr.com/photos/47183167@N03/4326444560/in/photostream/

I see, and I see Math.

A not one-to-one function

A continuous and Differentiable function

An invertible function

Not Continuous and not Differentiable

Continuous but not Differentiable

I Can See Clearly Now...

Welcome to my photos!
Just in case the titles and hot-spots aren't with the photos, I've put the link to Flickr underneath all my photos.


Here's the Link:
http://www.flickr.com/photos/47077084@N08/4310871956/



Here's the Link:
http://www.flickr.com/photos/47077084@N08/4310876542/



Here's the Link:

http://www.flickr.com/photos/47077084@N08/4310137659/



Here's the Link:
http://www.flickr.com/photos/47077084@N08/4310872732/



Here's the Link:
http://www.flickr.com/photos/47077084@N08/4320611099/