Evaluate the line integral \( \int_{C} x^{5} z d s \), where \( C \) is the line segment from \( (0,7,7) \) to \( (6,1,2) \).

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

(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.

(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

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)

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