Questions! Questions! Questions!

S109, Grand Ballroom D, 11/20/15

Eric Hutchinson, College of Southern Nevada

Aminul KM, College of Southern Nevada

Questions?

Purpose of this Presentation

• Topics of this presentation came up during class• Good to talk to colleagues regarding these topics• Good to know history behind what we are teaching• We encourage discussion!

Questions to Consider1. Why do we rationalize denominators?2. What real life applications are there for the

directrix? 3. Can we rewrite the interval a different way?4. Is this correct: “The denominator of a rational

expression cannot equal zero because division by zero is undefined”?

Question 1 Understanding

• Irrational numbers– An irrational number is any real numbers that cannot

be expressed as a ratio of integers. 

Understanding

• How to compute it

–We can check it by, we get =1.5– – 1.41421569

• We see that we are approximating it better.

History of Irrational Numbers• One of the earliest work for this

number was done by Babylonians• Clay drawings date to

approximately 1800 BC to 1600 BC• We see that they are using right

triangle properties to find the diagonal of square.

History (Cont.)• That civilization used a base 60 numbering

system.

• If we look at it, we see this

• This is very good approximation.• Except at 296. It should be 356

So why rationalize the denominator?

• Lets look at the geometric point of view

1

1

√22

√22

• We can understand better than • We can visualize

Why rationalize the denominator?• A primary motivation for rationalizing the

denominator relates to hand calculations versus using a calculator

• If you're computing by hand, once you have a decent approximate value for (such as 1.4142), it is quite easy to divide that by 2: 1.4142/2 = 0.7071. But to find a decimal approximation of 1/1.4142 by long division would be quite a bit more time-consuming.

Why rationalize the denominator?• With the invention of calculator, motivation for

rationalizing the denominator is not as strong.

• At the same time, this algebraic technique is very valuable. It became a tactical choice of what needed to happen at that time.

Question 1: Discussion• We have looked at the historical reasons for

rationalizing the denominator.

• DISCUSS: Why do we still rationalize denominators when we have access to calculators and computers?

• DISCUSS: Can you give an example of something that cannot be rationalized?

Question 2• How would you define the directrix of a

parabola?• To form a parabola according to ancient Greek

definitions, you would start with a line and a point off to one side. The line is called the “directrix”; the point is called the “focus”.

• The parabola is the curve formed from all points (x, y) that are equidistant from the directrix and the focus. (PurpleMath)

Question 2

History of the Focus-Directrix Definition

• Apollonius (262 BC – 190 BC) first postulated that the parabola is a type of conic section in his work Conics

• Diocles (240 BC – 180 BC) was a contemporary of Appollonius who worked on Archimedes’ “burning mirror” problem.

Burning Mirror

Burning Mirror

History of the Focus-Directrix Definition

• Diocles states the problem of constructing a burning-mirror which makes all rays meet in one point was solved by Dositheus who was before Apollonius.

• Diocles implied that no one before himself showed a geometric proof of the focal property of a parabola.

• Apollonius most likely knew of this property but did not mention it in his Conics work.

History of the Focus-Directrix Definition

• Diocles was the first to document the idea of solving the burning-mirror problem by constructing a burning mirror of a given focal length.

• This method involves drawing a parabola by means of the focus and directrix.

History of the Focus-Directrix Definition

• Although other mathematicians probably knew of this property, Diocles was the first to document it.

• Therefore, we can attribute to Diocles the discovery of the focus-directrix property of the parabola.

Question 2

• DISCUSSION: Can you think of a real life application involving the directrix? Have you seen a parabola word problem involving the directrix?

Question 2Each cable of a suspension bridge is suspended in the shape of a parabola between two towers that are 600 feet apart and whose tops are 80 feet above the roadway. The roadway represents the directrix of the parabola created between the two towers. The cables are 20 feet above the roadway midway between the towers. If a Cartesian coordinate system is laid over this figure and the origin is at the intersection of the directrix and the left tower, what is the equation of the parabola?

Question 3• How do we write the range of the following

function in interval notation?

Question 3• Range:

• Instead of {4} can we write [4,4]?

• DISCUSSION: –Which is more correct and why?

Question 3• [4, 4] is considered a degenerate interval• A degenerate interval is any set consisting of a

single real number.• [a, a] = {a}• (a, a) = { }• (a, a] = { }• [a, a) = { }

Question 3• Here is a question you can ask your students:

• Rolle’s Theorem states that a function which is continuous on [a,b] and differentiable on (a,b) will have a point c (a<c<b) such that f’(c)=0, so long as f(a)=f(b).

• Does Rolle’s Theorem apply for a degenerate interval?

Question 4• On pg. 358 of Lial 5th edition “Beginning and

Intermediate Algebra” the following statement appears: “The denominator of a rational expression cannot equal zero because division by zero is undefined”.

• DISCUSS: Is this definition correct or is it missing something?

Question 4• Consider the following revisions:• “The denominator of a rational expression

cannot equal zero because division by zero is undefined or indeterminate.”

• “The denominator of a rational expression in reduced form cannot equal zero because division by zero is undefined”

Example

• Comments?

Comments/Questions?

• Eric Hutchinson: eric.hutchinson@csn.edu• Aminul KM: aminul.km@csn.edu

THANK YOU FOR ATTENDING!!

Question 5

• What is the meaning of life?