Q2. Identify which of the following subsets of R are closed,open or neither
a. (x,where x is an interger)
b. (x,where x is a rational number)
c. (x,where x=1/n for some natural number n)
d. (x,sin(1/x)=0)
e. (x,xsin(1/x)=0)
f. (x,sin(1/x)>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
Please any kind of help would be appreciated.
SK
a. (x,where x is an interger)
b. (x,where x is a rational number)
c. (x,where x=1/n for some natural number n)
d. (x,sin(1/x)=0)
e. (x,xsin(1/x)=0)
f. (x,sin(1/x)>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
Please any kind of help would be appreciated.
SK
