Bisection Method

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nycmathdad

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What exactly is the Bisection Method?
What's it good for?

Sample:

Use the bisection method three times to
approximate the zero of each function in the given interval.

f(x) = x^3 − 4x + 2; interval: (1, 2)

Note: Michael Sullivan does not explain this method in Section 1.3.
 
Verify that the function has a zero in the indicated
interval. Then use the Intermediate Value Theorem to approximate the zero correct to three decimal places by repeatedly subdividing the
interval containing the zero into 10 subintervals.

f(x) = x^3 − 4x + 2; interval: (1, 2)

I simply don't understand the directions. Can someone get me started?
 
nycmathdad said:
Verify that the function has a zero in the indicated
interval. Then use the Intermediate Value Theorem to approximate the zero correct to three decimal places by repeatedly subdividing the
interval containing the zero into 10 subintervals.

f(x) = x^3 − 4x + 2; interval: (1, 2)

I simply don't understand the directions. Can someone get me started?
Click to expand...
This is the bi-section method. Follow the method described in the reference website in the response above.
 
Subhotosh Khan said:
This is the bi-section method. Follow the method described in the reference website in the response above.
Click to expand...

I noticed that this is very much involved. In one example, 0 was not found until the 7th iteration. It's not something I want to spend too much time learning when there are more important Calculus 1 topics to appreciate. I will visit the link when time allows.
 
nycmathdad said:
I need someone to break this down in layman terms.
Click to expand...
Exactly where in the video, it becomes too complicated for you?

There are only TWO major calculations (operations) involved:

1) Chose two points and calculate the midpoint (xM).​
​
2) Evaluate the function [f(x)] at that point (xM and decide whether to continue...​

Other operations being:

Choosing two close points where the values of f(x) "flip" sign​
​
Deciding when to stop iteration.​

One of the advantages of this method is that it does not propagate ( or accumulate) round-off error from calculations.
 
Beer soaked ramblings follow.
nycmathdad said:
What exactly is the Bisection Method?
What's it good for?
... Note: Michael Sullivan does not explain this method in Section 1.3.
Click to expand...
He didn't?
Take another look.
Do you have a reading problem?
Is Example 10 not clear enough?
You really ought to read your book's relevant section very carefully before making such reckless declarations. And you should do it when you're fresh and full of energy; preferably after you've rested and slept (and presumably had some nourishment with coffee shortly afterwards) so that you can maximize your mental energy into understanding and applying what you've been reading and not when "my brain is tired and I am physically exhausted" as you like to remind everyone to give you a break for being unable to make sense of what you're reading. Regardless of how passionate you are about math, you can't work/study and concentrate as hard at the end of a study session (in your case, the end of a working day) as at the beginning.

In imitation of SK: Exactly where in Example 10, it becomes too complicated for you?

Continuing from:

Intermediate Value Theorem

What exactly is the Intermediate Value Theorem? What's it good for? Do you have a simple geometric interpretation of this theorem?
www.freemathhelp.com
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Subhotosh Khan said:
Exactly where in the video, it becomes too complicated for you?

There are only TWO major calculations (operations) involved:

1) Chose two points and calculate the midpoint (xM).​
​
2) Evaluate the function [f(x)] at that point (xM and decide whether to continue...​

Other operations being:

Choosing two close points where the values of f(x) "flip" sign​
​
Deciding when to stop iteration.​

One of the advantages of this method is that it does not propagate ( or accumulate) round-off error from calculations.
Click to expand...

I found another video that breaks it down a bit further using small graphs. In any case, this Bisection Method is located at the end of Section 1.3 in Chapter 1 of the textbook. I am not too concerned about mastering this idea. I simply wanted to know what it is all about. I am more concerned in terms of learning the more popular Calculus 1 concepts.
 
Yep. That’s what calculus is about: winning the popularity context. Who needs to know anything about the unpopular parts. Every time I go into a bar, I shout out “Hey, anyone want to hear about the latest on the chain rule.”
 
Beer soaked ramblings follow.
nycmathdad said:
What exactly is the Bisection Method?
What's it good for? ...
Note: Michael Sullivan does not explain this method in Section 1.3.
Click to expand...
nycmathdad said:
I found another video that breaks it down a bit further using small graphs. In any case, this Bisection Method is located at the end of Section 1.3 in Chapter 1 of the textbook.
Click to expand...
First, "Michael Sullivan does not explain this method in Section 1.3."
Now, "this Bisection Method is located at the end of Section 1.3 in Chapter 1 of the textbook."
Which one is it?
nycmathdad said:
... I am not too concerned about mastering this idea. I simply wanted to know what it is all about ...
Click to expand...
Google is your friend.
 
JeffM said:
Yep. That’s what calculus is about: winning the popularity context. Who needs to know anything about the unpopular parts. Every time I go into a bar, I shout out “Hey, anyone want to hear about the latest on the chain rule.”
Click to expand...

The best part of self-learning is learning at a slow pace. There's no exam to study for. There's no pressure to make it to class on time. There's no pressure to rush through the material. In fact, I did so poorly in the Limits and Continuity Section 1.3 that I decided to go back to Section 1.2 or Properties of Limits. I will learn Calculus 1 well or simply stay within the realm of Precalculus.
 
nycmathdad said:
What exactly is the Bisection Method?
What's it good for?

Sample:

Use the bisection method three times to
approximate the zero of each function in the given interval.

f(x) = x^3 − 4x + 2; interval: (1, 2)
Click to expand...
I used to know that answer but I forgot. Wait a minute! You said it is used to approximate the zero of a function just as YOU wrote!
 
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