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Algebra
Pregunta

17-24 Find the Taylor polynomials of orders \( n =...

17-24 Find the Taylor polynomials of orders \( n = 0,1,2,3 , \) and \( 4 \) about \( x = x _ { 0 } \) , and then find the \( n \) th Taylor polynomial for the function in sigma notation. 

17. \( e ^ { x } ; x _ { 0 } = 1 \) 18. \( e ^ { - x } ; x _ { 0 } = \ln 2 \) 

19. \( \frac { 1 } { x } ; x _ { 0 } = - 1 \) 20. \( \frac { 1 } { x + 2 } ; x _ { 0 } = 3 \) 

21. \( \sin \pi x ; x _ { 0 } = \frac { 1 } { 2 } \) 22. \( \cos x ; x _ { 0 } = \frac { \pi } { 2 } \) 

23. \( \ln x ; x _ { 0 } = 1 \) 24. \( \ln x ; x _ { 0 } = e\) 

Answer

\(f(x)= e^{- x}\\f'(x)= - e^{- x}\\f''(x)= e^{- x}\\f'''(x)= - e^{- x}\\f''''(x)= e^{- x}\\ \Rightarrow f(x_{0})= \frac{1}{2} \\ f'(x_{0})= - \frac{1}{2} \\ f''(x_{0})= \frac{1}{2} \\ f'''(x_{0})= - \frac{1}{2} \\ f''''(x_{0})= \frac{1}{2} \\ \Rightarrow e^{- x}\approx \frac{1}{2}- \frac{1}{2}(x- \ln 2)+ \frac{1}{4}(x- \ln 2)^{2}   - \frac{1}{12}(x- \ln 2)^{3} + \frac{1}{48}(x- \ln 2)^{4}   \\ e^{- x}  = \sum_{n= 0}^{\infty } \frac{(- 1)^{n}}{2\cdot n!} (x- \ln 2)^{n}\)

Solución
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