\begin{align} f(b) = f(a) + {f ' (a)\over 1!} (b-a) + {f '' (a)\over 2!} (b-a)^2 + {f ''' (a)\over 3!} (b-a)^3 + ... {f^{(n-1)} (a)\over (n-1) !} (b-a)^{n-1} + {f^{(n)} (c)\over n !} (b-a)^n \end{align} |
\begin{align} f(b) = f(a) + {f ' (a)\over 1!} (b-a) + {f '' (a)\over 2!} (b-a)^2 + {f ''' (a)\over 3!} (b-a)^3 + ... {f^{(n-1)} (a)\over (n-1) !} (b-a)^{n-1}+ {K \over n !} (b-a)^n \end{align} |
\begin{align} f(b) -\{ f(a) + {f ' (a)\over 1!} (b-a) + {f '' (a)\over 2!} (b-a)^2 + {f ''' (a)\over 3!} (b-a)^3 + ... {f^{(n-1)} (a)\over (n-1) !} (b-a)^{n-1}+ {K \over n !} (b-a)^n\} = 0 \end{align} |
\begin{align} f(b) -\{ f(x) + {f ' (x)\over 1!} (b-x) + {f '' (x)\over 2!} (b-x)^2 + {f ''' (x)\over 3!} (b-x)^3 + ... {f^{(n-1)} (x)\over (n-1) !} (b-x)^{n-1}+ {K \over n !} (b-x)^n\} = F(x) \end{align} |
\begin{align} F'(x) = 0 -\{ f'(x) &+ {f ' (x)\over 1!} (b-x) ' + {f '' (x)\over 1!} (b-x) \\ &+ {f '' (x)\over 2!} ( (b-x)^2 )' + {f ''' (x)\over 2!} (b-x)^2 \\ &+ {f ''' (x)\over 3!}( (b-x)^3)' + {f '''' (x)\over 3!} (b-x)^3 +...\\ &+ {f^{(n-1)} (x)\over (n-1) !} ((b-x)^{n-1})' + {f^{(n)} (x)\over (n-1) !} (b-x)^{n-1}+\\ &+ {K \over n !} ((b-x)^{n})' \} \end{align} |
\begin{align} F'(x) = 0 -\{ f'(x) &+ {f ' (x)\over 1!} (-1) + {f '' (x)\over 1!} (b-x) \\ &+ {f '' (x)\over 2!} ( -2(b-x) ) + {f ''' (x)\over 2!} (b-x)^2 \\ &+ {f ''' (x)\over 3!}( -3(b-x)^2) + {f '''' (x)\over 3!} (b-x)^3 +...\\ &+ {f^{(n-1)} (x)\over (n-1) !} (-(n-1)(b-x)^{n}) + {f^{(n)} (x)\over (n-1) !} (b-x)^{n-1}+\\ &+ {K \over n !} (-n(b-x)^{n-1}) \} \end{align} |
\begin{align} F'(x) = 0 -\{ f'(x) &- {f ' (x)\over 1!} + {f '' (x)\over 1!} (b-x) \\ &- {f '' (x)\over 1!} (b-x) + {f ''' (x)\over 2!} (b-x)^2 \\ &- {f ''' (x)\over 2!}(b-x)^2 + {f '''' (x)\over 3!} (b-x)^3 +...\\ &- {f^{(n-1)} (x)\over (n-2) !} (b-x)^{n} + {f^{(n)} (x)\over (n-1) !} (b-x)^{n-1}+\\ &- {K \over (n-1) !} (b-x)^{n-1}\} \end{align} |
\begin{align} F'(x) = 0 &-\{ {f^{(n)} (x)\over (n-1) !} (b-x)^{n-1}- {K \over (n-1) !} (b-x)^{n-1}\} \\ =&-\{ {(f^{(n)} (x)-K ) \over (n-1) !} (b-x)^{n-1}\} \end{align} |