Taylor Swift - "Blank Space"

Edited from http://en.wikipedia.org/wiki/Blank_Space

"Blank Space" was a commercial success, having reached number one on the US Billboard Hot 100 following "Shake It Off".

The song debuted at number 18 on the Billboard Hot 100.  In its third week, the song sold 328,000 digital copies (up 100%), spurred by the release of its accompanying music video, and jumped to number one, becoming her third number one single and making Swift the first female artist to replace herself at number one (her number one song of the previous week was "Shake It Off") and tenth overall among all artists.^[25] 

RM 12.90 Digital Multimeter from Mr DIY

I bought a digital multimeter from Mr DIY which is  in   Tesco Ipoh building  a few weeks ago . It is probably the cheapest one I have ever bought.

I remembered it wasn't long ago when the analog ones cost around RM50.

I bought a handheld digital one a few years back but that one did not last long. I have got a feeling that this one would last longer even though it has only a few days warranty.

 So far I have bought quite a  few things from Mr DIY. The other day someone was asking me at Mr DIY what is good there and I pointed at the multimeter. He sort of looked at me blankly and asked
" What is that ?"

Well, what he was looking for was a remote control for his DVD/VCD player. He wasn't very familiar with the English language as we were actually speaking in Cantonese. So I showed him some of the remotes  which seem to cost around the range of RM 6 to RM10. He wasn't convinced as he thought it could destroy his set.

Anyway I am quite satisfied with the multimeter  I bought  - except for one time when it took some time for the 20V range measurement to work. But it seems to work fine now - just tested it with a 1.5V battery.

Identifying sequence convergence and divergence

When the degree of the numerator is the same as that of the
 denominator , the sequence converges.

When the degree of the numerator is bigger as that of the 
denominator , the sequence diverges.

A sequence that alternates with positive and negative values 
such as +1 and -1 is not convergent.



   = e





    Identifying sequence convergence and divergence:

Radius of convergence

The radius of convergence is 1/2 the length of the interval of convergence.

 http://www.sosmath.com/calculus/radcon/radcon02/radcon02.html

When ratio test =0, the series converges for all real numbers so it has an infinite radius of convergence.


    Radius of convergence using Ratio Test: 

OneRepublic - "I Lived"

This was the number one song in one of the radio channels in Malaysia last week.

Three way tug of war vectors

If the ring is not moving, vectors a+b+c=0


    Three way tug of war vectors:

Geometric series convergence and divergence examples

Convergence occurs when 0< |r| <1
Geometric series convergence and divergence examples:

One-sided limits from graphs

When approaching from the -ve side, there is a negative superscript for the limit number


    One-sided limits from graphs:

Function as geometric series

Find the common ratio r.

convergence occurs if  -1 < r < 1

For such an interval of convergence, S=a/(1-r)

  Function as geometric series:

Tangent slope as limiting value of secant slope example 1

Draw out the graph to see what is needed.
It may be possible to substitute algebraically to get the solution.


    Tangent slope as limiting value of secant slope example 1:

Relatively prime numbers

In number theory, two integers a and b are said to be relatively prime, mutually prime, or coprime (also spelled co-prime)^[1] if the only positive integer that evenly divides both of them is 1. That is, the only common positive factor of the two numbers is 1. 

For example, 14 and 15 are coprime, being commonly divisible by only 1, but 14 and 21 are not, because they are both divisible by 7. 

Trapezoidal Rule to find the average value of f(x)

From http://www.intmath.com/integration/5-trapezoidal-rule.php

Area≈​2​​1​​(y​0​​+y​1​​)Δx+​2​​1​​(y​1​​+y​2​​)Δx+​2​​1​​(y​2​​+y​3​​)Δx+

Area≈Δx(​2​​y​0​​​​+y​1​​+y​2​​+y​3​​+…+​2​​y​n​​​​) for uniform grid


The trapezoidal rule finds the area. To get the average 
value, you will need to divide the area by b-a which is the
 the difference between the final and initial x values.

Inverse of a straight line

The straight line graphs shown below are inverses of one another.

 

Given the invertible function f(x), we determine the inverse by:

• replacing every x with y and y with x;

Moving left and right, speeding up or slowing down?

From Khan Academy

Let's deal with moving left and right to start.

The particle is moving to the right when its velocity is positively valued; that is, when v(t)>0.

The particle is moving to the left when its velocity is negatively valued; that is, when v(t)<0.

Now we consider when the particle is speeding up and slowing down.

The particle is speeding up for those times when the product of its velocity and its acceleration is greater than zero. Therefore we want

v(t)⋅a(t)>0.

For this condition to be satisfied, we want those times when v(t) and a(t) are both positively valued or both negatively valued.

In contrast, he particle is slowing down for those times when the product of the velocity and the acceleration is less than zero. Therefore we want

v(t)⋅a(t)<0.