Math in our everyday life!!

“There has never been a good time to be a mathematician.”

In today’s society, everything falls on math. All the careers have some math concept involved in somehow. This has been the case for several years, but nowadays, as evolutions keeps on going farther, the need of math increases. Statistics is used in the medical field to determinate the improvement of health conditions, in business to improve the quality of work done, in security agencies to insure the safety of the citizens.

First, this article explains the pros of math and its development. The world is now dominated by Web2.0. Education, entertainment, shopping, careers, etc. everything can be done online, and this won’t be possible without the contributions of mathematicians. Since they are much needed in today’s society, the article suggests that students, especially in America, should be encouraged to pursue their careers in math or math related major fields. The example is given in the case of the NSA which employs only US citizens. From here, it can be concluded that the security of Americans is limited, since it is less than what it could be.

The other side of the article explains how these ongoing improvements in the math’s world are quite disturbing. Since people are concerned about the privacy problem, it’s hard to make improvement in medicine for example. Since much of these math based programs use people’s private information, there have been some complaints about this, and they are refusing to give up their information to the public. This pushed mathematicians, doctors and other concerned people to look for other alternatives in their problem solving strategies

Overall, this article made me think about all the connections that can be made between the world of math, which is always perfect (with exact answers), and the real world, which is always a cumulative result of more or less estimations and probabilities. There have been some times when, in the middle of class or while doing a math problem, I said to myself “when am I ever going to use this?” This article reminded me that’s always a rhetorical question I have in mind, that somewhere in the back of my conscious mind I know that I will use it. It also reminded me the saying that says “Math is related to Physics, Physics related to Chemistry, Chemistry related to Biology, and Biology related to Life.” So, in the end, math is related to our lives, either directly or indirectly.

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

Choices

1. For every invertible function, the derivative of its inverse function is the reciprocal of the derivative of the initial function.

2.

3. Yes it is an invertible function, because:

(in black on the graph, and its derivative at x=1is in blue)

(in red on the graph and its derivative at x=1 is in green)

6. The slope of the tangent line at x=1 on V(r) is 12.56 and the slope of the tangent line at x=1 on r(V) is 1/12.56, and these two numbers are reciprocals.
Then, since the derivative at a point is the same as the slope of the tangent line at that point, the derivatives of the two functions V(r) and r(V) are reciprocals.

7. On "Virtual Wednesday" I chose not to go in the face-to-face class because I wanted to try something new. I had the intention of going in class probably for the last half of class, just in case I found difficulties learning on my own. I used http://www.analyzemath.com/calculus/Differentiation/derivative_inverse.html .
This website was helpful and easy to follow in the beginning, but as I kept doing exercises, I figured out that this rule about the inverse functions doesn't apply to trig functions, and then I stopped, and used the in-class exploration.

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

I disagree with this statement to some extent.

This is true, as long as everyone has access on the internet. Here in the US and in other developed countries, people can access web2.0 any time they want. They don’t have to sit in class to learn, nor do they have to be stuck in one place while using the internet, since there are a lot of wireless internet devices in place nowadays. So, if everyone has access on the internet, everyone can share his/her knowledge to the whole world. For these reasons, everyone can learn anything, from anyone, anywhere, at anytime.

However, in some parts of the world people don’t have access on the internet, and only wealthy families can afford the cost of the internet. Most of the times those people are preoccupied with some other activities in their daily lives, and they may or may not know that web2.0 provides all the information they need. Some of them do all they can to go on the internet once and then, but this is not on the daily bases. Therefore, web2.0 can’t be considered as the major source of education to “everyone” since we all don’t have access on the internet anytime and anywhere. Thus, schools are needed.

Product Rule

As in the process of learning, we should be able to "watch", "do", and "teach". The online definition was easier for me to "watch" because it had both visual and audio non complicated explanations. This helped me to understand what is the "product rule." On the other hand, the book's definition increased my understanding of the product rule because it has different examples. Therefore, if i was to chose between the text book and the internet, I would rather check on the internet first, since it's easier at the first glance, and then follow the text book later. If I didn't do it this way, I would certainly be confused by the text book, and it would take me longer to understand because the book is not direct - it proved the product rule, and then stated it later.

The product rule:

If f(x) = g(x) . h(x),

then f'(x) = g'(x) . h(x) + g(x) . h'(x)

In words: The derivative of a product of two functions equals the derivative of the first function times the second function, plus the derivative of the second function times the first function.

Example:

f(x) = 3x.cos(5x)

f’(x) = 3cos(5x) + 3x . – sin(5x).5

f’(x) = 3cos(5x) – 15xsin(5x)

My 1st semester mind map