b. \( \lim _{x \rightarrow a}\left(\frac{1}{x}\right)=\frac{1}{a} \), con \( a \neq 0 \)

Use Newton's method to find an approximate solution to \( x^{2}-2=0 \), starting from an initia estimate \( x_{1}=2 \). Iterate until 6 iterations. Hints: \( \boldsymbol{f}^{\prime}(\boldsymbol{x})=2 \boldsymbol{x} \) Marks)

\( y ^ { \prime \prime \prime } + y = e ^ { x } + e ^ { - x } \cos ( 3 x ) + x ^ { 3 } \)

\( f_{x}=\frac{2}{3} x^{2}-x \) en el punto de abscisa \( x=a \) ParA \( a=0 \).

(f) Find the constants \( a, b, c \), so that \[ \vec{F}=(x+2 y+a z) \vec{i}+(b x-3 y-z) \vec{j}+(4 x+c y+2 z) \vec{k} \] is irrotational and hence find function \( \phi \) such that \( \vec{F}=\nabla \phi \). (g) Show that the vector field \( \vec{F}=\left(e^{x} \cos y+y z\right) \vec{i}+\left(x z-e^{x} \sin y\right) \vec{j}+(x y+z) \vec{k} \) is conservative by (3 marks) finding a potential for it.

Calcular la derivada de las a) \( F_{(x)}=\left(x^{5} \cdot \operatorname{sen} x\right) \)