- #1
mathnoob
- 3
- 0
Question
A Chebyshev polynomial is Tn(x) = cos(arccos^(-1)(x))
My questions are:
1. what are the domain(s) and range(s) of this function?
2. Give equivalent polynomial definitions for Tn(x) when n = 0; 1; 2; 3. That
is: show that the definition for Tn above really is a polynomial.
3. Compute integral(-1 to 1) Tn(x)dx
4. Compute integral (-1 to 1) [Tn(x)Tm(x) / sqrt(1-x^2)] dx, when
a.) n = m = 0
b.) n = m =/= 0
c.) n =/= m
My attempt at a solution
1. I graphed it on a graphing calculator and I know for sure that both its domain and range = [-1,1]. I'm guessing this is because the domain and range of cos(x) = [-1,1]?
2. I found an equation online that T(n+1)(x) = (2x)Tn(x) - T(n-1)(x) when n >= 1. Hence:
T(o)x = 1
T(1)x = x
T(2)x = 2x^2-1
T(3)x = 4x^3-3x
Would this be enough to prove that Tn(x) is a polynomial?
Lastly, for #3 and #4, How are you supposed to find the integral when there are two variables (n and x)? I got this question for my Calculus II class (mostly integration). Thanks!
A Chebyshev polynomial is Tn(x) = cos(arccos^(-1)(x))
My questions are:
1. what are the domain(s) and range(s) of this function?
2. Give equivalent polynomial definitions for Tn(x) when n = 0; 1; 2; 3. That
is: show that the definition for Tn above really is a polynomial.
3. Compute integral(-1 to 1) Tn(x)dx
4. Compute integral (-1 to 1) [Tn(x)Tm(x) / sqrt(1-x^2)] dx, when
a.) n = m = 0
b.) n = m =/= 0
c.) n =/= m
My attempt at a solution
1. I graphed it on a graphing calculator and I know for sure that both its domain and range = [-1,1]. I'm guessing this is because the domain and range of cos(x) = [-1,1]?
2. I found an equation online that T(n+1)(x) = (2x)Tn(x) - T(n-1)(x) when n >= 1. Hence:
T(o)x = 1
T(1)x = x
T(2)x = 2x^2-1
T(3)x = 4x^3-3x
Would this be enough to prove that Tn(x) is a polynomial?
Lastly, for #3 and #4, How are you supposed to find the integral when there are two variables (n and x)? I got this question for my Calculus II class (mostly integration). Thanks!