9 EXERCICE 05 : Soit la fonction \( f \) définie sur \( \mathbb{R} \) par: \( f(x)=x \sqrt{x^{2}+1} \) 1) Montrer que \( f \) admet une fonction réciproque \( f^{-1} \) définie sur son domaine de définition qu'on déterminera 2) a-Montrer que \( f^{-1} \) est dérivable en o et en \( \sqrt{2} \). b-Calculer \( \left(f^{-1}\right)^{\prime}(0) \) et \( \left(f^{-1}\right)^{\prime}(\sqrt{2}) \) (on remarque que: \( f(0)=0 \) et \( f(1)=\sqrt{2} \) )

Use an appropriate test to determine whether the following series converges. \( \sum_{\mathrm{k}=1}^{\infty} \frac{5}{\sqrt{\mathrm{k}^{3}}} \) Select the correct choice below and fill in the answer box to complete your choice. A. The series diverges. It is a p-series with \( \mathrm{p}=\square \). The series converges. It is a p-series with \( \mathrm{p}=\square \). C. The series diverges by the Integral Test. The value of \( \int_{1}^{\infty} \frac{5}{\sqrt{\mathrm{x}^{3}}} \mathrm{dx} \) is \( \square \). D. The series converges by the Divergence Test. The value of \( \lim _{\mathrm{k} \rightarrow \infty} \frac{5}{\sqrt{\mathrm{k}^{3}}} \) is \( \square \). E. The series diverges by the Divergence Test. The value of \( \lim _{\mathrm{k} \rightarrow \infty} \frac{5}{\sqrt{\mathrm{k}^{3}}} \) is \( \square \).

Use the given function to complete parts (a) through (e) below. \( f(x)=x^{4}-36 x^{2} \) At which zeros does the graph of the function touch the \( x \)-axis and turn around? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( x=\square \) (Type an integer or a decimal. Use a comma to separate answers as needed.) B. There are no \( x \)-intercepts at which the graph touches the \( x \)-axis and turns around.

\begin{tabular}{l} (1) \( f(x)=3 x^{3}+2 x^{2}-5 x-10 \) \\ a) \( \int_{3}^{8} f(b)-f(a)= \) \\ b) Derivada \\ c) Velocidad \( \quad{ }_{m}=\frac{f\left(x_{i}\right)-f(x i)}{x_{f}-x_{i}} \quad[3,5] \) \\ \hline\end{tabular}

(2) \( f(x)=x^{3}-10 x^{2}-5 x+20 \) a) \( \int_{4}^{7} f(b)-f(a) \) b) \( 1^{\text {ra }} \) derivada c) Velocidad \( [5,8] \quad V_{m}=\frac{f\left(x_{f}\right)-f\left(x_{i}\right)}{V_{f}-x_{i}} \)

Math 60 Exam \#4 1) Use Right-endpoint Rule with \( n=5 \) to approximate \( \int_{2}^{3} x \sqrt{x+1} d x \) to 4 decimal places.