f) \( f(x)=3 / 5 x^{4}-2 / 9 x+5 / 2 x^{2} \) 9) \( f(x)=100 \) \( \begin{array}{lll}\text { - Encuentra sus puntos criticos e identificalos como minimos y máximos } \\ \text { a) } f(x)=1 / 3 x^{3}+7 / 2 x^{2}+6 x & \text { b) } f(x)=\frac{x^{3}}{x-2} & \text { c) } f(x)=x^{3}+3 x^{2}-4 \\ \text { d) } f(x)=-2 x^{6}+4 x^{3}\end{array} \)

Find the Laplace transform \( F(s)=\mathcal{L}\{f(t)\} \) of the function \( f(t)=(7-t)(h(t-2)-h(t-6) \), for \( s \neq 0 \). \( F(s)=\mathcal{L}\{f(t)\}=1 \) help (formulas)

1) Calcular las derivadas de las siguientes funciones en los puntos indicados, aplicando la definición de derivadas. \( \begin{array}{ll}\text { 1. } y=3 x^{2}-5 x \text { en } x_{0}=-2 & \text { 5. } y=\cos x \text { en } x_{0}=1 \\ \text { 2. } y=2 x^{3}+x \text { en } x_{0}=1 & \text { 6. } y=2 x^{2}-3 x+4 \text { en } x_{0}=1 \\ \text { 3. } y=\sqrt{x} \text { en } x_{0}=2 & \text { 7. } y=6 x-7 \text { en } x_{0}=-1 \\ \text { 4. } y=\operatorname{sen} x \text { en } x_{0}=\frac{\pi}{2} & \text { 8. } y=9 x-8 \text { en } x_{0}=-1\end{array} \)

Write a triple integral including limits of integration that gives the volume of the cap of the solid sphere \( x^{2}+y^{2}+z^{2} \leq 5 \) cut off by the plane \( z=2 \) and restricted to the first octant. (In your integral, use theta, rho, and phi for \( \theta, \rho \) and \( \phi \), as needed.) What coordinates are you using? spherical (Enter cartesian, cylindrical, or spherical)

25. \( N=\frac{p+q}{p+r}, \quad p=u+v w, \quad q=v+u w, \quad r=w+u v \) \( \frac{\partial N}{\partial u}, \frac{\partial N}{\partial v}, \frac{\partial N}{\partial w} \quad \) donde \( u=2, v=3, w=4 \)

2. (4 pts) What is the derivative of \( y=x^{\pi}+e^{3}+\ln \left(3^{x}\right)+\pi^{x} \). Be sure to answer with a derivative notation. \[ \text { ANSWER: } \quad y^{\prime}=\pi x^{\pi-1}+\ln (3)+\pi^{x} \ln (\pi) \]