We will start with a question about polynomials!

Task 1: What degree could this polynomial be?

fourth degree poly

Why do you think what you think?

Task 1A: Why does the degree have to be even?

We all seem to agree that the polynomial in the picture would have to be an even degree polynomial. Why is this the case?

Task 2: Define “root” or “zero” of a polynomial

Let’s agree on a definition of “root” which is a key concept when you are talking about real valued functions.

Suppose that $f: \mathbb{R} \rightarrow \mathbb{R}$ is a function. A (real) root of $f$ is a number $x \in \mathbb{R}$ such that $f(x) = 0$.

Task 3: Does every 5th degree polynomial have a real root?

Consider the conjecture that every 5th degree polynomial has at least one (real) root.

Is this true or false? Why?

We decided yes! An odd degree polynomial will have different end behavior in the two directions (because of how negative numbers work with even and odd powers) AND polynomials are continuous, so a fifth degree polynomial will have to cross the $x$-axis.

Task 4: Formulating our BIG CONJECTURE for the term

Our justification that a 5th degree polynomial must have at least one real root: such a polynomial will be continuous and will go to infinity in one direction and negative infinity in the other. So it must cross the x-axis.

We all seem convinced by this argument. Maybe we can get a general conjecture/theorem out of this idea!

1. Does this argument only work for polynomials?

2. Does it only work for functions that go to $+ \infty$ in one direction and $- \infty$ in the other?

3. Can we generalize this idea to make a conjecture about conditions that would guarantee a function had a root?

ANSWER: We decided that it could work for any continuous function. And we don't need it to go to positive and negative infinity, we just need it to take on positive and negative values.

We came up with an informal version of a conjecture: Any function that is continuous and takes on positive and negative values will have at least one real root.

Since we missed two days of class due to winter storm shenanigans, I took the liberty of formalizing the conjecture myself:

Conjecture (IVT): Suppose $f: \mathbb{R} \to \mathbb{R}$ is a continuous function. If there exists inputs $a, b$ where $ a < b $ and $f(a)$ and $f(b)$ have opposite signs, then there exits $r \in (a, b)$ such that $f(r) = 0$

Our BIG GOAL for the rest of the term will be to prove this conjecture. We will start the same way that Cauchy did back in the day - by becoming experts on approximation methods.

Task 5: Approximating Roots

Find an approximation of a root of $f(x) = x^5 – 5x -5$ that is correct to four decimal places.

Task 6: Approximation Rules of Engagement

What can we use a calculator for given that we need to have complete control and understanding of our approximation method?
Answer: Pretty much the only thing we can trust is plugging in nice values and getting outputs (that might be approximation after some number of decimals).

What exactly does it mean to be “within four decimals”?

There are a number of reasonable ways to interpret this! The one we are going to choose is that: We are correct to n decimals when we know that the first n digits after the decimal are the exact same as those of the exact value.

Given that are ultimately interested in the existence of an EXACT root, the picture below indicates why don't want to focus on how close the outputs are to zero:


Task 7: Approximation Instruction Manual

Write an instruction manual that anyone could use to approximate a root of any function that satisfies the hypotheses of our IVT conjecture.

Explain the steps of the method as clearly as you can.

Use your work on Task 5 give give an example of how to use your method

Make sure your manual explains how you would get an approximation accurate to any number of decimal places.

Here is a fun video showing the method in action!

Task 8: What is a Sequence?

These letters with subscripts remind me of sequences from calculus. What exactly is a sequence?

Definition (sequence): A sequence of real numbers is a function with domain $\mathbb{N}$ and codomain $\mathbb{R}$

Task 9: Sequences Generated by our Approximation Method

Identify at least five sequences that are generated if you iterate the approximation method forever.

We came up with three input sequence, $a_n$, $b_n$,$c_n$, three output sequences, $f(a_n)$, $f(b_n)$, $f(c_n)$ and the sequence of interval lengths, $e_n = b_n - a_n = \frac{b_1 - a_1}{2^{n-1}}$

Task 10: Conjectures about the Sequences Generated by our Approximation Method

Make some conjectures about the sequences we have identified:

  1. What properties do they have?
  2. Do the sequences have limits? If so, what are they?
  3. Are there any interesting relationships between any of the sequences? Any other thoughts?

To make things simpler, let's name those sequences:

$a_n$ is the sequence of left endpoints $b_n$ is the sequence of right endpoints $c_n$ is the sequence of midpoints $e_n = b_n - a_n$ is the sequence of error bounds (AKA the sequence of interval lengths) $f(a_n)$ is the sequence of outputs of the left endpoints $f(b_n)$ is the sequence of outputs of the right endpoints $f(c_n)$ is the sequence of outputs of the midpoints

Here are some of the conjectures you came up with. Let's discuss them on the way to making a new list that everyone is happy with.

Conjectures NOT about limits:

  1. $e_n = b_n - a_n = \frac{b_1-a_1}{2^{n-1}}$ for every $n \in \mathbb{N}$

  2. $a_{n+1} \geq a_n$ for every $n \in \mathbb{N}$. In other words, $a_n$ is an increasing sequence

  3. $b_{n}$ is decreasing

  4. $e_n$ is strictly decreasing

  5. $a_n$ is bounded above by $b_1$

  6. $b_n$ is bounded below by $a_1$

  7. $a_n < b_n$ for every $n \in \mathbb{N}$

  8. $f(a_n) < 0$ for every $n \in \mathbb{N}$ (NOTE: Assumes $f(a_1) < 0$ and $f(b_1) > 0$)

  9. $f(b_n) > 0$ for every $n \in \mathbb{N}$ (NOTE: Assumes $f(a_1) < 0$ and $f(b_1) > 0$)

Conjectures about limits:

  1. $e_n$ converge to 0. Using limit notation: $\displaystyle{\lim_{n \to \infty}(b_n - a_n)} = 0$

  2. $a_n$ converges to a root (from the left)

  3. $\displaystyle{\lim_{n \to \infty}(a_n)} < b_k$ for every $k \in \mathbb{N}$

  4. $a_n$ does converge

  5. $\displaystyle{\lim_{n \to \infty}(a_n)} = \displaystyle{\lim_{n \to \infty}(b_n)} = \displaystyle{\lim_{n \to \infty}(c_n)}$.

  6. $f(a_n)$ converges to 0

  7. $f(b_n)$ converges to 0

Task 11: What happens if we try our method on a function with no sign change?

What happens if you apply your approximation method to a function that does not have a sign change? Try it on this function:

No Sign Change

Task 12: What if you try it with a discontinuous function?

It is impossible to even start the approximation method when your function does not have a sign change. Now let’s consider the other part of our IVT Conjecture hypothesis!

What happens if you apply your method to a discontinuous function? Try it with these two functions:

NOTE: You aren’t trying to find numbers just trying to figure out what will happen if you apply the method.



Answer: We can definitely apply the method for these functions! The only difference is that the left and right endpoints appear to converge to the point where the function is discontinuous instead of to a root!

Task 13: Sorting Conjectures

Sort our list of conjectures into two groups. Group 1 is the conjectures that will be true for any function that has a sign change. Group 2 is the conjectures that will be true for any continuous function that has a sign change.

KEEP IN MIND: We are ASSUMING that 1) the sign change goes from negative to positive and 2) the approximation process goes on forever (our goal is to prove there is a root, and if we happen to hit it on the number with one of our approximations - great! - so we don't need to worry about that).

(Group 1) True as long as there is a sign change:

Everything except the three below

(Group 2) True ONLY if the function has a sign change AND is continuous:

  1. $a_n$ converges to a root (from the left)

  2. $f(a_n)$ converges to 0

  3. $f(b_n)$ converges to 0

One of our big goals the next 5 class sessions will be to PROVE that $a_n$ does converges. That turns out to be a big first step in proving IVT and in figuring out some of the foundational ideas of real analysis. But first we are going to build up some expertise about sequences.