10) Use Newton's Method to approximate the real zero of the function in the interval \( [1 \), \( f(x)=x^{3}-x-2 \), use \( x_{1}=1.5 \). Find \( x_{2}, x_{3} \) and \( x_{4} \). Keep 5 decimal places for each \( x \) value.

Dada la siguiente función: \[ f(x)=3 x^{x^{4}} \] Calcule el valor, redondeando a cuatro decimales, de \( f^{\prime}(x) \) en el punto \( x=1,07 \).

12) [10 points] Determine which of the following definite integrals equals \( \lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{1}{n} \sqrt{\frac{i}{n}} \).

\( \int \frac { x ^ { 2 } + 3 x - 12 } { ( x - 2 ) ^ { 2 } ( x + 3 ) ( x ^ { 2 } + 1 ) } d x = \)

) 8. Al sustituir \( u=\operatorname{sen} \theta \), ¿qué resultado se obtiene al evaluar \( \int \operatorname{sen}^{2} \theta \cos \theta d \theta \) ? \( \begin{array}{llll}\text { A) }-\cos ^{2} \theta \operatorname{sen} \theta+C & \text { B) } \frac{1}{6} \operatorname{sen}^{3} \theta \cos ^{2} \theta+C & \text { C) } 2 \operatorname{sen}^{2} \theta \cos ^{2} \theta-\operatorname{sen}^{3} \theta & \text { D) } \frac{1}{3}\end{array} \)

Given the differential equation \( \frac{dy}{dx} = 3y \) and the initial condition \( y(0) = 2 \), find the particular solution.