Finding All Roots of e^x=3-3x Using Newton's Method

[skyturnred]

Homework Statement

Use Newton's method to find ALL roots of e^x=3-3x

Homework Equations

The Attempt at a Solution

I know how to use Newton's method, but how is it possible to use it to find ALL the roots of the function? Just by looking at the function however, I THINK that there should be only one root (because it is an exponential function). Whether it does or not, however, I need to know how to do this for the upcoming final. How can I use Newton's method to find ALL the roots of a function f(x)?

Thanks so much in advance!

 

[2milehi]

If you use the graphical way of finding intersections of the functions that will give you an idea of where the real roots exist.

This is just a hunch but say there are two roots to some equation. If you guessed lower than the lowest root, the method should converge to the lowest root. If you guessed higher than the highest root, the method should converge to the highest root

 

[SammyS]

Hello skyturnred. Welcome to PF.

e^x is an increasing function and 3 - 3x is a decreasing function. What does that tell you?

 

[skyturnred]

If you use the graphical way of finding intersections of the functions that will give you an idea of where the real roots exist.

This is just a hunch but say there are two roots to some equation. If you guessed lower than the lowest root, the method should converge to the lowest root. If you guessed higher than the highest root, the method should converge to the highest root

Yes, I can definitely do it that way. But I am just trying to prepare myself for the final in which I will have no calculator with me.

ex is an increasing function and 3 - 3x is a decreasing function. What does that tell you?

If e^x is always increasing and 3-3x is always decreasing and both are continuous, then they will intersect at only one spot, meaning there will be only one solution to x. I understand this, my concern is if I don't get such a simple function. If I get a function where f(x)=g(x) where either function is discontinuous and/or increases and decreases on different intervals, how can I use Newton's method in the way described in the original post?

Thanks in advance!

 

[Ray Vickson]

Yes, I can definitely do it that way. But I am just trying to prepare myself for the final in which I will have no calculator with me.



If e^x is always increasing and 3-3x is always decreasing and both are continuous, then they will intersect at only one spot, meaning there will be only one solution to x. I understand this, my concern is if I don't get such a simple function. If I get a function where f(x)=g(x) where either function is discontinuous and/or increases and decreases on different intervals, how can I use Newton's method in the way described in the original post?

Thanks in advance!

Basically, the way is to isolate the different roots in intervals, then apply root-finding techniques within those intervals. Newton's Method may not be applicable directly; instead, you may need to apply one of the so-called "safeguarded" methods, such as secant, safeguarded regula falsi, bisection, or a safeguarded Newton method. As for the problem of isolating all the real roots in separate intervals: it is difficult, and I am not sure there is any generally-applicable method (i.e., a method that always works).

RGV

 

[2milehi]

Yes, I can definitely do it that way. But I am just trying to prepare myself for the final in which I will have no calculator with me.



If e^x is always increasing and 3-3x is always decreasing and both are continuous, then they will intersect at only one spot, meaning there will be only one solution to x. I understand this, my concern is if I don't get such a simple function. If I get a function where f(x)=g(x) where either function is discontinuous and/or increases and decreases on different intervals, how can I use Newton's method in the way described in the original post?

Thanks in advance!

If you are required to use Newton's method, I would bet that you will be given continuous functions.

You are using your noodle, do the same thing during the test and write down your assumptions. Also you can roughly draw out a lot of functions.

 

[Office_Shredder]

This is just a hunch but say there are two roots to some equation. If you guessed lower than the lowest root, the method should converge to the lowest root. If you guessed higher than the highest root, the method should converge to the highest root

This only works under certain concavity conditions