Engineering Mathematics $1($MAU$11$E$01)$

Homework $2$

This is part of your continuous assessment and contributes to your final grade.

Due date: Thursday, Dec $5$th $2024$ at midnight. No late assignments!

Upload your $($scanned$)$ solutions to Blackboard as a single pdf file. No other formats! WRITE CLEAR AND CONCISE. SCRIBBLED SOLUTIONS WILL NOT BE MARKED.

Problem $1(26$ points$)$ Is the function

$f(x)=\{\begin{array}{cc}{x}^{3}cos(\frac{1}{x}),& x0\\ 0,& x=0\end{array}$

continuous everywhere? Hint: Remember the squeezing theorem.

Problem $2(20$ points$)A5m$ ladder is leaning against the wall at an angle of $=10.$ The bottom of the ladder is pulled along the ground perpendicularly away from the wall at constant rate of $0\mathrm{.}5\frac{(m)}{(s)}.$

a$)$ Draw a sketch of the situation and label all important quantities.

b$)$ How fast will the top of the ladder be moving doun the wall when it is $4$ m and $3$ m above the ground?

c$)$ What is the distance of the top of the ladder to the ground when $=60,$ and $($at the same angle$)$ what is the rate of change of the bottom of the ladder with its distance from the wall?

d$)$ How fast does the angle $$ changes w$.$r$.$t$.$ time when the bottom of the ladder reaches $2\mathrm{.}5$ m from the wall?

Problem $3(24$ points$)$ For the following functions compute $d\frac{y}{d}x$

a$)y=\sqrt[\phantom{2}]{3x^4+12x^2+6}$

b$)y=\sqrt[3]{tan(5x)+3x^2}$

c$)3xy-y^2=cos(xy)$

d$)tan(x^2{y}^3)=3x+y^2$

Problem $4(30$ points$)$ Given the rational function $f(x)=\frac{x^3+x^2+x+1}{x^2-1}.$

a$)$ Determine the roots of $f$ and its domain.

b$)$ Determine the derivative function $f^{\text{'}}$ and its derivative $f^{\text{'}\text{'}}.$

c$)$ For each vertical asymptote of $f,$ calculate the limit from below and above.

d$)$ Determine the slant asymptote of $f.$

e$)$ Calculate the roots of $f^{\text{'}}.$

f$)$ Sketch the function $f$ and its asymptotes.

Hint$-1$: First simplify $f(x)$ by factoring out common parts. Hint$-2$: You are allowed to extend the domain for any removable discontinuity after a$).$