MathTips

[theapproximatepresent]

Suppose you enter a casino offering a very simple game. The croupier tosses a coin, if it shows a tail, you win $1, if it’s heads, he flips again and the pot doubles. If the second flip is a tail, you win $2, if it’s heads, he flips again and the pot doubles. If the...

[robotstudies-deactivated2022111]

Ready to level up your working knowledge of math? Here’s what to read now — and next.

Math 101 with Jennifer Ouellette

First, start with these 5 books…

1. Number: The Language of Science Tobias Dantzig Plume, 2007

“First published in 1930, this classic text traces the evolution of the concept of a number in clear, accessible prose. (None other than Albert Einstein sang its praises.) A Latvian mathematician who studied under Henri Poincare, Dantzig covers all the bases, from counting, negative numbers and fractions, to complex numbers, set theory, infinity and the link between math and time. Above all, he understood that the story of where mathematical ideas come from, how they relate to each other, and evolve over time, is key to a true appreciation of mathematics.”

2. Innumeracy: Mathematical Illiteracy and its Consequences John Allen Paulos Hill and Wang, 2001

“This bestselling book originally published in 1988 remains one of the best introductions to the basics of large numbers, statistics and probabilities with illustrations drawn from everyday life: sports, the stock market, the lottery and dubious medical claims, to name a few.”

3. How Not to Be Wrong: The Power of Mathematical Thinking Jordan Ellenberg Penguin, 2014

“Pair Paulos with the just-released How Not to Be Wrong. Ellenberg deftly introduces the most basic mathematical tools in crisp, lively prose, liberally peppered with real-life examples, to help the reader start thinking like a mathematician. From Francis Galton and regression to the mean, to survivorship bias, uncertainty and the statistical challenges of ferreting out the signal from the noise in increasingly large data sets, he shows us how to use math to avoid potential pitfalls and make smart decisions.”

4. Zero: The Biography of a Dangerous Idea Charles Seife Penguin, 2000

“Most of us take zero for granted, but there was a time when it simply didn’t exist, until some enterprising Babylonian soul inserted it as a placeholder in Eastern counting methods. The ancient Greeks banned its use, the Hindus worshipped it, and eventually it became one of the most important conceptual tools in mathematics. In this lucid, elegantly written history of zero, Charles Seife ably demonstrates that zero is ‘infinity’s twin’ – far more than nothing.”

5. A Tour of the Calculus David Berlinski Vintage, 1997

“The prose gets a bit turgid at times, and some readers might be deterred by the proofs and equations scattered throughout, but Berlinski has some lovely descriptions and turns of phrases. A unique take on a daunting subject, and a longtime favorite among math aficionados.”

(read the full article)

[mathtips-blog]

Why happens if b < a in a Integral?

Hello to you all,

Hope we can see each other more now that I'm studying for my finals. Today I'll discuss what happens if you change the order of the limits in an Integral.

You now that According to Riemann the definition of a (definite) Integral is:

Now, that works out really well when a is less than b (a < b). What happens if b is less than a (b < a)?

I took that answer from Renata Tordai, someone taking Calculus One in coursera. She also added that:

"I think you shouldn't look at Riemann integration as just a method for calculating areas. Yes, it's definitely one feature it can offer, but it can do much more for us which are not so close to areas. At this point I would say that the Riemann integral is a function that maps from the set of continuous functions to the set of real numbers, and when the application requires this real number can be considered as the measure of an area."

I find hard to understand the last sentence of that paragraph. Can someone please help understand what she meant?

Anyway, Renata, thanks for the help. Never forget the integral means SIGNED area, not just area. 

Good bye and happy learning. 

[mathtips-blog]

Understanding Hyperbolic Functions (Basics and derivatives)

This week we learned about Hyperbolic Functions and the basics of maximum and minimum values of functions.

Our teacher gave us no explanation of the origin of the hyperbolic functions, so here is a little history of the origins of them. (http://scimath.unl.edu/MIM/files/MATExamFiles/Schutz_MATpaper_FINAL.pdf)

Its a thesis, so don’t worry if you don’t understand something. Just focus on the history and then you’ll be able to understand the rest.

Now, you must know that these functions exist thanks to the Hyperbola, a really cool geometric curve.

http://en.wikipedia.org/wiki/Hyperbolic_function#mediaviewer/File:Hyperbolic_functions-2.svg

Have you seen regular trigonometric functions? well, hyperbolic functions, but they are defined in terms of the exponential function.

In order to understand these functions, I recommend first to watch the fiirst video of this Khan Academy’s series:  

https://www.khanacademy.org/math/precalculus/hyperbolic_trig_topic/hyperbolic_trig_intro/v/hyperbolic-trig-function-inspiration

Then, read this pdf document, in which they define the three main hyperbolic functions, and sketch their graphs. They also discuss some identities relating these functions, and mention their inverse functions and reciprocal functions: http://www.mathcentre.ac.uk/resources/workbooks/mathcentre/hyperbolicfunctions.pdf

For those learning calculus,do not forget to learn the identities, and also, the inverse functions:

and also, their derivatives:

(taken from Calculus, Early Trascendentals by Stewart)

To do limits, multiply and divide by e^(-x). Also, do not forget: f’( f(x) ) = x. This is useful when you need to find the value of the functions given one of them.

Hope you like it guys. I think I will do another post regarding hyperbolic functions in a deeper level. Until the next time, happy learning!

[mathtips-blog]

Applying ln to both sides to solve! (Derivatives using logarithms)

I've struggled myself with this  a few weeks ago. It may look intimidating, but it's really easy! 

According to a comment by Aish, a user of Khan Academy "Logarithms have been used to simplify exponents. Basically, exponential growth is often written in terms of log and it can now be modeled as a linear function. We use a logarithm because it is taken to the x power meaning an ever-increasing exponential change which is what logarithms are used for."

https://www.khanacademy.org/math/differential-calculus/taking-derivatives/derivatives-inverse-functions/v/calculus-derivative-of-x-x-x

I want to ask a question? What happens with x^x when x is negative? well, when the values of x are negative and even, it seems that nothing has happened; nonetheless, if you analyze negative, odd numbers, something has happened. 

That is the graph of the function (made with Desmos). Why do we only take into account positive numbers and cero?

Homework: Find the derivative of y = 2^(sec x) * (sin x)^(2^x) :)

Happy learning!