Find the quadratic approximation of the function \( \sqrt{1}+x \) at \( x \) close to 0 ?

If \( f(x)=\frac{1}{3} x^{3}+c x^{2}+a x+5 \) has alocal manamum value at \( x=1 \), + What is the value of a \( f c \) ?

Evaluate the triple integral. \[ \iiint_{G} x y \sin (y z) d V \text {, } \] where \( G \) is the rectangular box defined by the inequalities \( 0 \leq x \leq \pi, 0 \leq y \leq 17,0 \leq z \leq \frac{\pi}{102} \)

Evaluate the triple integral. \[ \iiint_{G} x y z d V \] where \( G \) is the solid in the first octant that is bounded by the parabolic cylinder \( z=3-x^{2} \) and the planes \( z=0, y=x \) and \( y=0 \).

b) \( y=\frac{x-1}{x^{2}+2 x+3} \)

Find the volume of the solid formed by rotating the area under the curve \( y = \frac{1}{x} \) from \( x = 1 \) to \( x = 3 \) about the y-axis.