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It was a surprise that when any circumference of a circle is divided by its diameter, the answer was always the same; the number we know know as Π. This was known many years BCE; but, it wasn't named then.

No one created pi; it's just there when calculating using our base 10 number system.

The first person to use the symbol Π for this number was William Jones in the eary 1700's. It became popular when Leonhard Euler, in the mid 1700's began to use it. (Euler was one of the greatest mathematicians of all times; and probably wrote more articles and books on mathematics than anyone else; so, when he began to use it, it came into standard use.)

4x + 44 = 100

4x = 100 - 44

4x = 56

x = 56 / 4

x = 14

Yes, I followed it to its conclusion. There isn't much to choose between them as far as length is concerned is there ? I do think though that my method is the one that the examiner expected (?).

-5t² + 20t + 50

-5(t² - 4t - 10)

This cannot be factored any farther using whole numbers.

By completing the square, you can get:  -5(t - 2 + √14)(t - 2 - √14).  (Don't wory about this answer, unless you have been doing advanced algebra.)

-5t^2 + 20t + 50    .....taking out the greatest common factor, we have

-5(t^2 - 4t - 10)

And that's it......!!! (the trinomial inside the parenhteses isn't factorable)

There are 16 ounces in a pound...so  6 x 16 = 96 ounces

And  96/5 = 19 full packages

P.E.  =  mass x g x height

P.E.  =  54 kg x 9.8 N/kg x 8.0 m

P.E.  =  4200 J   (rounded)

Taking each different factor along with the highest power occurring on that factor, we have

2^2 * 3^3 * 5 * 7 =

4 * 27 * 35  =

3780

The domain consists of all the x-values:  domain = {-1}

The range consists of all the y-values:  range = {1}

Usually, the domain and range is defined for a more extensive function.

Perhaps you mean "a score??".....a "score" = 20...so.....a sum of "a score" + 13 = 33

nah man you can, 

Suppose . Then . But .

You could choose to make 0/0 = 1 but you'd have to give up on the idea that a*(b/c) = (a*b)/c which would be more inconvenient than any benefit you might get.

 Some people dismiss these kinds of questions as stupid. I take them seriously. The rules of algebra that we have today are a result of a couple of thousand years of experimentation to find what works best. I think that generally there has been a motion towards having the simplest rules possible. In this case simply disallowing division by zero is a much simpler rule than listing all of the myriad places where 0/0=1 would require special rules like "(a*b)/c=a*(b/c) except when b and c are zero". But it is a choice. You could use the more complicated rules if you want. But millennia of experimentation has shown the benefit not to be worthwhile. And you'd also cause confusion when talking to people who stick with the usual conventions.

10^(-1)  =  1/(10^1)  = 1/10

Also:  3^(-1)  =  1/3

and:  4^(-2)  =  1/(4^2)  =  1/16

         2^(-5)  =  1/(2^5)  =  1/32

If you look here you will find your answer.

there is some serious discussion of this here if you are really interested. :)

Thanks anon,

Have you followed this through to its conclusion?  Is it shorter?

An alternative approach, (to Melody's method), is to remove the parameter from the co-ordinates of G.

We have $$x=2ap,$$ from which $$p=x/2a$$, and on substitution, $$y=-2a-a(x/2a)^{2}$$, so $$x^{2}=-4ay-8a^{2}$$ etc..

Thanks, Anonymous....you are correct......I forgot about that!!!

The correct interval is (-√(2/3), √(2/3))

Let me add one proviso to what Anonymous said.......setting the second derivative to 0 and solving only indicates possible  inflection points. For instance...the second derivative of x^4 is 12x^2. Setting this to 0 gives us that x = 0. But this isn't an inflection point because this curve is always concave "up". To see this, pick a point on either side of 0 and "plug" them into the second derivative. Both results will be positive....indicating that the curve is concave "up" on both sides. Thus, there is no inflection point !!!

Concavity changes at points of inflexion. To find such  points, equate the second derivative to zero.

I think that leads to $$x=\pm\sqrt{2/3}.$$

Is the score + 13; can you tell us the rest of the problem?

Authors have agreed to  define the set of "natural numbers" to be the same set as the set of "counting numbers", which is:  {1, 2, 3, 4, 5, ...}.

They then define the set of "whole numbers" to be: {0, 1, 2, 3, 4, ...}.

Thus, yes, 0 is a whole number.

@@ End of Day Wrap          Mon 27/10/14           Sydney, Australia            Time 11:40pm      ♬

Hello everyone,

I forgot to put the credits   - I will add them now.   

There were many great answers from Alan, CPhill, Geno3141, Robtre, Robert, LancelotLink, TakahiroMaeda, Kitty<3 and last but definitely not least Reinout-g.  Thanks all.  

It was really busy today.  Perhaps the busiest Sun/Mon that we have ever had.     I find busy days frustrating because I do not have a chance to answer questions like I want to or to learn from other people's questions and answers.  Still it is usually better to be too busy than to be not busy enough.  

I put aside a great many posts today.  Most of them I have not even looked at properly, but they all look interesting all the same.  We also had a couple visits from LancelotLink.  He was behaving himself relatively well.  He made a couple of hilarious posts.  

1) This one looks really interesting.  Thanks Chris and anonymous.

Tues 28/10/14

1) Some bright (school) seniors might like to think about this one.

Yes, actually I can probably do integration by parts.  I need lessons in matrices and vectors.

Yes we should do the swap.  It is all a matter of time.  You know when Christmas comes this places slows to a snails pace.  We should put effort into teaching each other then.  I shall look forward to it.  :)

At the bottom left of the calculator next to the "pi(e) a la mode"  button.....

I'll trade you one "Integration By Parts" lesson for one "Parametric Equations as they apply to Parabolas" lesson.....

Deal???