MathTips

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. 

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.

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) :)