more+sect.+3.2+10-20

The first thing we did today was go over the wikis again, but after that we had a follow-up discussion to yesterday's.

It started with a question: What makes that method of using graphs to find the values of media type="custom" key="11137154" that make the difference between f(x) and f(5) or any other value what we want them to be?

What we said: well, polynomials look like lines when we zoom in on them, except: discontinuous graphs like step graphs graphs with asymptotes.

So, the graphs have to be __continuous__ : informal definition: having no breaks, and you can draw the graph without lifting your pencil formal definition: by restricting how close point x is to point a, you can force f(x) to be arbitrarily close to f(a).

p 172 FYD 6. Name and sketch 3 functions that are continuous:

for example, I used sin(x) Wolfram Alpha LLC. 2009. Wolfram|Alpha. http://www.wolframalpha.com/input/?i=graph+sin(x)(access October 20, 2011).

any value that is plugged in for x will get a y value that is defined.


 * **Mrs. Tyson:** Note that the sentence directly above this just says that the domain of media type="custom" key="11047148" is media type="custom" key="11047154", we need to use the definition above the graph when we are discussing whether this graph is continuous. Informally, we can draw media type="custom" key="11047158" without taking our pencil off the paper. More formally, given (a, f(a)) I can look at two horizontal lines that are symmetric around f(a) and trap the function between them when I am near to a.

7. Sketch 2 functions that are not continuous.

Here's my example: it's a step graph and therefore a piece-wise function (to be inserted by ms tyson)

while tan(x) is actually not really continuous because it has asymptotes, it is considered continuous because it is continuous wherever it is defined. And, piece-wise functions are actually rarer, and harder to name.

We then looked at the Minds in Action on page 172, where Sasha and Tony talk about solving finding the distance between x and 5 if f(x) is a certain distance away from f(5) (or in this case f(2)) algebraically. The algebra involved here is pretty simple, until f(x) is no longer a linear function, when it gets very ugly very quickly. Steps: 1. plug in the actual function for f(x), and the value of f(5) in for f(5). (Five here is only used as an example). 2. use algebra to solve for the absolute value of x minus 5 on one side with a constant on the other, and a < (less than) sign in between them. This will give the greatest possible distance that x can be away from 5.

We then discussed Theorem 3.1 and Theorem 3.2.


 * 3.1 The Change of Sign Theorem states:**
 * Suppose f is a polynomial function and there are two numbers a and b such that f(a) < 0 and f(b) > 0. Then f(c) = 0 for some number c between a and b.**

This means that because f is a polynomial, which by definition are all continuous, it must cross the x axis at some point c that is in between a and b both vertically and horizontally.


 * 3.2 The Intermediate Value Theorem for Polynomials states:**
 * Suppose f is a polynomial function and a and b are two numbers such that f(a) < f(b). Then for any number c between f(a) and f(b), there is at least one number d between a and b such that f(d) = c.**

This means that to go from point (a, f(a)) to point (b,f(b)) you must go through all values between f(a) and f(b) if the function is continuous, as it is here.

Theorems 3.1 and 3.2 can be derived from one another, because each is saying about the same thing. The difference is, 3.1 is specific about which axis the function must cross (the x axis) while 3.2 says only the line y = c, if c is in between a and b. Basically, shifting the axis up or down will prove on theorem from the other.