Si \[ f(x)=\left\{\begin{array}{ll}x^{2} \sin \left(x^{-1}\right)+\frac{1}{2} x & \text { si } x \neq 0 \\ 0 & \text { si } x=0\end{array}\right. \] (a) Demostrar que \( f^{\prime}(0)>0 \) (b) Demostrar que \( f(x) \) no es creciente en cualquier intervalo abierto que contiene al cero.

e) Slope of the curve \( y=\sqrt{x} \) at \( (1,1) \) \( \begin{array}{ll}\text { i) } 0 & \text { ii) } 1 \\ \text { iii) }-1 & \text {-iv) } 1 / 2\end{array} \)

A hotdog vendor must pay a monthly fee to operate a food cart in the city park. The cost in dollars of selling \( x \) hundred hotdogs in a month can be approximated by \( \mathrm{C}(\mathrm{x})=150+60 \mathrm{x} \). Complete parts (a) through (d) below. (a) Find the average-cost function. The average-cost function is \( \overline{\mathrm{C}}(\mathrm{x})=\square \) (b) Find and interpret the average cost at sales levels of 1000 hotdogs and 5500 hotdogs. The average cost at sales levels of 1000 hotdogs is \( \$ \square / 100 \) hotdogs. (Round to the nearest cent as needed.) The average cost at sales levels of 5500 hotdogs is \( \$ \square / 100 \) hotdogs. (Round to the nearest cent as needed.) Interpret these results, choose the correct answer below. Take the sales level of 1000 hotdogs as an example. Select the correct choice below and fill in the answer box to complete your choice. A. At a sales level of 1000 hotdogs, the average cost is \( \$ \square \) per 100 hotdogs. B. At a sales level of 1000 hotdogs, if an additional 100 hotdogs are sold, the average cost increases by approximately \( \$ \square \) per 100 hotdogs.

\( \int e ^ { \frac { - 1 } { x } } d x \)

(c) Use Bisection method to solve \( x^{3}-9 x+1=0 \) for the root between \( x=2 \) and \( x=4 \). 5 marks) (d) Using \( \sin (0.1)=0.09983 \) and \( \sin (0.2)=0.198 \), find an appropriate value of \( \sin (0.15) \) by using Langranges interpolation.

Si la posición de un objeto está dada por la función \( s(t) = 5t^3 - 2t^2 + 3t \), ¿cuál es la expresión para su velocidad en términos de \( t \)?