d. \( \quad \lim _{x \rightarrow 1}\left(\frac{x^{2}-1}{x-1}\right)=2 \)

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

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)

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

(b) i. If \( F \) is a conservative field, prove that curl \( \vec{F}=\nabla \times \vec{F}=\overrightarrow{0} \) (i.e., \( \vec{F} \) is irrotational). ii. Conversely, if \( \nabla \times \vec{F}=\overrightarrow{0} \) (i.e., \( \vec{F} \) is irrotational), prove that \( \vec{F} \) is conservative. (c) If \( \phi=2 x y z^{2}, \quad \vec{F}=x y \vec{i}-z \vec{j}+x^{2} \vec{k} \) and C is the curve \( x=t^{2}, \quad y=2 t, \quad z=t^{3} \) from \( t=0 \) to \( t=1 \), evaluate the line integrals: i. \( \int_{C} \phi d \vec{r} \). ii. \( \int_{C} \vec{F} \times d \vec{r} \). "Mathematics Can Smile!" ( 2 mar

Encuentre \( y^{\prime} \) si \( y^{5}+3 y^{2}-2 x^{4}=-4 \)