SquareRootofAwesome – DiNapoli's Math Blog

For having the best participation rate in the Math Formal, you have earned a party next Tuesday!  This party will most definitely consist of food, so think about what kind of stuff you’d like to make/bring in to share. 

In years past, this party usually involved a movie, however we will probably not be able to finish it in the 85 minutes we have together that day.  I also thought we could play a game that involved fun (non-math) questions.   So this year I am entertaining new ideas about how to celebrate our victory.  If you have a cool idea for what we could do in class next Tuesday, please post a comment here! 

I would also like to use the blog as a sign up sheet for foods, drinks, and supplies for the party.  Post a comment here with what you are bringing.  I can put things in the fridge or heat things up as necessary.

Make sure you are commenting on your correct precalc block!

For having the best participation rate in the Math Formal, you have earned a party next Tuesday!  This party will most definitely consist of food, so think about what kind of stuff you’d like to make/bring in to share. 

In years past, this party usually involved a movie, however we will probably not be able to finish it in the 85 minutes we have together that day.  I also thought we could play a game that involved fun (non-math) questions.   So this year I am entertaining new ideas about how to celebrate our victory.  If you have a cool idea for what we could do in class next Tuesday, please post a comment here! 

I would also like to use the blog as a sign up sheet for foods, drinks, and supplies for the party.  Post a comment here with what you are bringing.  I can put things in the fridge or heat things up as necessary.

Make sure you are commenting on your correct precalc block!

Comment on this post with your Cooling example problem.  Be sure to include the solution!

Posts are due by the end of the weekend, or midnight Sunday night.  You will be graded on participation, correctness of solution, and creativity.

This counts as your first homework grade of the new marking period.

Easier than you expected?  Harder?  How come?

In response to Sam’s question today, about why the rational zero theorem works the way it does, the best explanation comes from the proof.  I attached the proof below, and I am sure you cried.  Basically, this proof is using some laws of divisability (laws you would learn in an upper level college course) between the leading coefficient and the constant of any polynomial.

PROOF:

Suppose p/q is a root to our integer coefficient polynomial, f(x), of degree n, where p and q are integers. Suppose p/q is fully reduced so that p and q are co-prime (i.e., the greatest common divisor of p and q is 1; in short gcd(p, q) = 1).

Then,

ƒ(p/q) = 0.

Multiplying both sides of this by qⁿ⁻¹, yields

qⁿ⁻¹ · ƒ(p/q) = a_n · pⁿ / q + {sum of integers} = 0.

Hence, a_n · pⁿ / q is an integer. Because p and q are co-prime, q divides a_n.

Similarly, suppose we had multiplied by qⁿ/p instead. Then we have

qⁿ · ƒ(p/q) / p = {sum of integers} + a_o · qⁿ / p = 0.

Rearranging, we find a_o · qⁿ / p is an integer. Because p and q are co-prime p divides a_o. Thus p is an integer factor of the constant term, q is an integer factor of the leading coefficient, and p/q is a root of ƒ(x).

Here’s the Verizon Math Fail video on youtube: http://www.youtube.com/watch?v=zN9LZ3ojnxY&feature=related

Can anyone find any other funny, math related videos similar to this?

BONUS!  http://opinionator.blogs.nytimes.com/2010/03/07/finding-your-roots/?emc=eta1

Read the article and post answers to these questions:

1.) What are some applications of complex numbers?
2.) Where have you seen “fractals” before?
3.) Why are complex numbers necessary to learn?

Today we revisted the quadratic formula.  I uploaded the CUBIC formula (solves all cubic functions set equal to zero) to my Moodle site.  Check it out – I dare you not to cry.

Today in class we discussed what my “selling headshots on eBay” business model would look like.  It’s revenue graph was an upside-down parabola because there was an exact, central point (the vertex or maximum) where it told me what price to charge to maximize my profits. 

What kind of a business model would have  a revenue graph that was a rightside-up parabola?  Explain!

Sam had a great question in class today: are we allowed to distribute the 3 in an expression like 3|x + 2| so we would get |3x + 6|?  Does absolute value work that way?  I found a pretty good explanation….

****

Yes and no. You can’t blindly apply it as if the absolute value symbol
were just a pair of parentheses; but you can prove a sort of
distributive property using the properties of the absolute value.

Remember that

    |a| * |b| = |ab|

Now let’s try to manipulate |a| * |b + c|:

    |a| * |b + c| = |a(b + c)| = |ab + ac|

This is the absolute value equivalent of the distributive property. If
a is non-negative, you can drop the absolute value around it, and you
get

    a * |b + c| = |ab + ac|   for a >= 0

But don’t forget the condition on a. If a is negative, the left side
of this is negative, and the right side is positive, which won’t work
well.

*****

Great question, Sam!  Hope that helps.