Answer the following questions rising the fact that the volume of a sphere with radius \( r \) is (a) Mike is a carking the helium out of a spherical balloon (see Figure 1). At a cartain time, the radius of the balloon is 2 cm and the volume of air j) the balloon is decreasing at a rate of \( 8 \mathrm{~cm}^{3} \) per second. Use the chain rale to find the rate of change of the balloon's radius. at a certain time we know the radius of the balloon is 3 cm , and the radius is decreasing by 0.5 cm per second. Use the chain rule to find how fast he is sucking out the helium (the rate of change of the volume).

Find an equation of the tangent plane to the given parametric surface at the specifie point. \( \begin{array}{l}\text { (a) } r(u, v)=\left(u^{2}+1\right) \mathbf{i}+\left(v^{3}+1\right) \mathbf{j}+(u+v) \mathbf{k} .(5,2,3) \\ \text { (b) } r(u, v)=\sin u \mathbf{i}+\cos u \sin v \mathbf{j}+\sin v \mathbf{k},(0.5, \sqrt{3} / 4,0.5)\end{array} \)

\( \frac{d}{dx}(x^{2}+lnx-e^{3x)}) \)

Evaluate the line integral \( \int_{C} \mathbf{F} \cdot d \mathbf{r} \), where \( \mathbf{F}(x, y, z)=5 \sin (x) \mathbf{i}-4 \cos (y) \mathbf{j}-2 x z \mathbf{k} \) and C is given by the vector function \( \mathbf{r}(t)=t^{5} \mathbf{i}-t^{4} \mathbf{j}+t^{3} \mathbf{k}, \quad 0 \leq t \leq 1 \).

9. Determinar la solución general del sistema: \[ \left\{\begin{array}{c}x_{1}^{\prime}=3 x_{1}+x_{2} \\ x_{2}^{\prime}=2 x_{1}+3 x_{2}-x_{3} \\ x_{3}^{\prime}=2 x_{2}+3 x_{3}\end{array}\right. \]

Find \( D_{x} y \) if \( y=\left(\frac{2}{x^{4}+1}+\frac{3}{x}\right) \)