# Dynamical Systems

1. Jun 4, 2005

### sachmo

Q3. Sketch the graph of B(x)

Q4. Prove that B'(0)=0

Q5. Inductively prove that B^n(0)=0 for all n.Conclude that B(x) is a c infiniti function.

Q6. modify B(x) to construct a C infiniti function C(x) whcih satisfies
a. C(x) =0 if x is less than or equal to 0
b. C(x) =1 if x is greater that or equal to 1
c. C'(x)>0 if 0<x<1

Q7. Modify C(x) to construct a C infiniti bump function D(x) on the interval [a,b], where D(x) satisfies
a. D(x) =1 for a<x<b
b. D(x) = 0 for x<alpha and x>beta where a<alpha and beta>b
c. D'(x) not equal 0 on the intervals (alpha,a) and (b,beta)

Q8. Use a bump function to construct a diffeomorphism f;[a,b] goes to [c,d] which satisfies f'(a)=f'(b)=1 and f(a) =c,f(b)=d

Any kind of insight or Help is appreciated.

Last edited: Jun 5, 2005
2. Jun 5, 2005

### matt grime

You might at least consider, when posting homework, to include all of the question. What for instance is B(x)? Not that I personally care, you understand, but someone else may do it for you. The answer to Q2 for instance is kind of trivial isn't it, being a simple check of a definition?

3. Jun 5, 2005

### HallsofIvy

Staff Emeritus
"Q3. Sketch the graph of B(x)

Q4. Prove that B'(0)=0

Q5. Inductively prove that B^n(0)=0 for all n.Conclude that B(x) is a c infiniti function. "

You do understand, don't you, that no one can help you with these if you don't tell us what "B(x)" is?

This looks clearly like homework so I've moved it to the "Homework: College" section.
And, we will expect you to not only explain what "B(x)" is but to show us what you have done on this problem yourself.

4. Jun 5, 2005

### sachmo

Sorry for the delay,
B(x)={exp(-1/x sqaure) if x>0
0 if x< or equal to 0

I have been trying to work on the problem but truthfully I have not achieved anyhting for it
sketch the graph of B(x) and prove that B'(0) = 0