Ejercicio 1. Determine la quinta derivada de cada una de las siguientes funciones \( \begin{array}{ll}\text { 1. } f(x)=\frac{1}{x^{4}} \\ \text { 2. } f(x)=x^{5}+11 \\ \text { 3. } f(x)=5 \cos x+4 e^{x} \\ \text { 4. } f(x)=-3 \tan x+4 x^{3}-34-21 e^{x} \\ \text { 5. } f(x)=x^{4}+2 \csc x\end{array} \)

\( \left. \begin{array} { l } { f ^ { \prime } ( x ) = } \\ { f ( x ) = \sqrt { \cos ( e ^ { x ^ { 5 } \sin ( x ) } ) } } \\ { \frac { 5 x ^ { 4 } e ^ { x ^ { 5 } \cdot \sin ( x ) } \cdot - \sin ( e ^ { x ^ { 5 } \cdot \sin ( x ) } ) \cdot - \sin ( x ) - x ^ { 5 } e ^ { x ^ { 5 } \cdot \sin ( x ) } \cdot \sin ( e ^ { x ^ { 5 } \cdot \sin ( x ) } ) \cos ( x ) } { 2 \sqrt { \cos ( e ^ { x ^ { 5 } \sin ( x ) } ) } } } \end{array} \right. \)

(f) \( \lim _{x \rightarrow \frac{\pi}{4}} \frac{\sin x-\cos x}{1-\tan x} \) (g) \( \lim _{x \rightarrow 0} \frac{x-\sin 2 x}{x+\sin 3 x} \) (h) \( \lim _{x \rightarrow 0} \frac{1-\sqrt{\cos x}}{x^{2}} \) (i) \( \lim _{x \rightarrow \infty}\left(\frac{x-1}{x+3}\right)^{x+2} \)

CHECKING COMPREMENSION (press pause for more time) Find the volume of the solid of revolution that results from rotating \( y=x^{2}+3 \) around the \( x \) axis, over the interval from 0 to 1 .

Let \( f(x)=\frac{x}{x^{2}+13} \). Find (a) the intervals on which \( f \) is increasing, (b) the intervals on which \( f \) is decreasing, (c) the open intervals on which \( f \) is concave up, (d) the open intervals on which \( f \) is concave down, and (e) the \( x \)-coordinates of all inflection points. (a) \( f \) is increasing on the interval(s) \( [-\sqrt{13}, \sqrt{13}] \) (b) \( f \) is decreasing on the interval(s) \( (- \) infinity, 0\( ) \) (c) \( f \) is concave up on the open interval(s) (-infinity , 0\( ) \) (d) \( f \) is concave down on the open interval(s) \( (0 \), infinity) (e) the \( x \) coordinate(s) of the points of inflection are \( -\sqrt{39}, 0, \sqrt{39} \) Notes: In the first four boxes, your answer should either be a single interval, such as \( [0,1) \), a comma separated list of intervals, such as (-inf, 2), ( \( (3,4] \), or the word "none". In the last box, your answer should be a comma separated list of \( x \) values or the word "none".

\( y=e^{2 \cos 2 x} \) deviuates