Solving Dynamical Systems Q3-Q8: Sketch Graph and Construct C Infiniti Functions

In summary, the conversation involves discussing the function B(x) and its properties. The first question (Q3) asks to sketch the graph of B(x), while the second question (Q4) asks to prove that the derivative of B(x) at 0 is equal to 0. The third question (Q5) asks to prove inductively that the nth derivative of B(x) at 0 is also equal to 0, leading to the conclusion that B(x) is a c-infinity function. The conversation then moves on to modifying B(x) to construct a new c-infinity function called C(x) which satisfies certain conditions and then modifying C(x) further to construct a bump function called D(x) on a
  • #1
sachmo
2
0
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.
 
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  • #2
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
"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
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
 

Related to Solving Dynamical Systems Q3-Q8: Sketch Graph and Construct C Infiniti Functions

1. What is a dynamical system?

A dynamical system is a mathematical model that describes the behavior of a system over time. It is composed of a set of variables, equations, and rules that govern how those variables change over time. It is often used to study complex systems in physics, biology, economics, and other fields.

2. What is the purpose of solving dynamical systems?

The purpose of solving dynamical systems is to understand the behavior of a system over time and predict its future states. This can help in making decisions, designing experiments, and developing solutions to problems. It is also used to study the stability and sensitivity of systems, as well as to model real-world phenomena.

3. How do you sketch a graph for a dynamical system?

To sketch a graph for a dynamical system, you need to identify the variables and their relationships in the system. Then, plot the variables on the x and y-axes and use the equations to determine how the variables change over time. You can also use software or online tools to help with graphing and visualizing the system.

4. What is a C Infinity function in relation to dynamical systems?

A C Infinity function is a mathematical function that is infinitely differentiable, meaning it has derivatives of all orders. In the context of dynamical systems, C Infinity functions are often used to model the behavior of a system over time. They are continuous and smooth, making them useful in studying the stability and sensitivity of systems.

5. How do you construct a C Infinity function for a dynamical system?

To construct a C Infinity function for a dynamical system, you need to determine the equations and variables that describe the system. Then, use mathematical techniques such as integration, differentiation, and power series to create a continuous and smooth function. It is important to ensure that the function accurately represents the behavior of the system and is stable and sensitive to changes in the variables.

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