The Chebyshev polynomials \(T_n\) are defined by \[ T_n(x) = \cos\big(n \arccos(x)\big)\quad n=0,1,2,3,\dotsc \,.\]
- What are the domain and range of these functions?
- We know that \(T_0(x) = 1\) and \(T_1(x) = x.\) Express \(T_2\) explicitly as a quadratic polynomial and \(T_3\) as a cubic polynomial.
- Show that, for \(n \gt 1\), \[T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)\,.\]
- Use the previous parts to show that \(T_n\) is a polynomial of degree \(n\).
- Use the previous parts to express \(T_4,\) \(T_5,\) \(T_6,\) and \(T_7\) explicitly as polynomials.
- What are the zeros of \(T_n?\) At what numbers does \(T_n\) have local maximum and minimum values?
- Graph \(T_2,\) \(T_3,\) … \(T_7\) on a common screen. How do the zeros of \(T_n\) relate to the zeros of \(T_{n+1}?\) How do the \(x\)-coordinates of the minimums/maximums of \(T_n\) relate to the minimums/maximums of \(T_{n+1}?\)
- Based on the graphs, what can you say about \(\int_{-1}^1 T_n(x)\,\mathrm{d}x\) when \(n\) is odd and when \(n\) is even?
- Use the substitution \(u = \arccos(x)\) to evaluate the integral in the previous part.
- The family of functions \(f(x) = \cos\big(c \arccos(x)\big)\) are defined even when \(c\) is not an integer (but then \(f\) is not a polynomial). Describe how the graph of \(f\) changes as \(c\) increases.