Along the way to Bernstein (1912), Fej´er (1933), Curtis and Rabinowitz (1972), Riess and Johnson (1972), Trefethen (2008, 2013) etc., by building on the aliasing errors on integration of Chebyshev polynomials and using the asymptotic formulae on the coefficients of Chebyshev expansions, in this presentation, we will consider optimal general convergence rates for n-point Gauss, Clenshaw-Curtis and Fej´er’s first and second rules for Jacobi weights. All are of approximately equal accuracy. The convergence rate of these quadrature rules is up to one power of n better than polynomial best approximation. Further, we will introduce the optimal general convergence rates for Lagrange interpolation polynomials deriving from Gauss or Chebyshev points, and fast implementation of these polynomials by barycentric formulae. In addition, we will compare Lagrange interpolation with Hermilte-Fej´er interpolation for continuous functions. Finally, we consider some applications in acoustic scattering problems.