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Introduction to Linear Algebra with Mathematica

Glossary

Mean Value Theorems

If function f is contnuous amd has continuous derivatives up to n + 1 in some open interval U, we abbreviate is as f ∈ ℭn+1(U, then for any x₀ ∈ U we have

\[ f(x) = f(x_0 ) + f'(x_0 ) \left( x - x_0 \right) + \frac{f'' (x_0 )}{2!} \left( x - x_0 \right)^2 + \cdots + \frac{f^{(n)} (x_0 )}{n!} \left( x - x_0 \right)^n + R_n , \]
where the remainder
\[ R_n (f) = \int_{x_0}^x f^{(n+1)} (t)\, \frac{(x -t )^{n+1}}{(n+1)!} \,{\text d}t = f^{(n+1)} (x^{\ast})\, \frac{(x - x_0 )^{n+1}}{(n+1)!} , \qquad x^{\ast} \in (a, b) . \]

 

  1. Boyd, J.P. and Flyer, N., Compatibility conditions for time-dependent partial differential equations and the rate of convergence of Chebyshev and Fourier spectral methods, Computer Methods in Applied Mechanics and Engineering, 1999, Volume 175, Issues 3–4, Pages 281--309. https://doi.org/10.1016/S0045-7825(98)00358-2

 

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