Let $f(x)=x{e}^x.$

$($a$)$ Find the largest interval of the form $[a,)$ for which $f$ is increasing.

$($b$)$ Restrict the domain of $f$ to be the interval you found in part a$).$ Let $W=f^{-1}$

be the inverse of $f$ on this interval and find $W(e).$

$($c$)$ Use implicit differentiation to find the slope of the graph of $W$ at $e.$ Hint: If

$y=W(x),$ then $x=f(y).$

$($d$)$ Find the linear approximation of $W$ at $x=e$ and use this approximation to

estimate $W(3).$

Remark: The function $W$ is sometimes called the Lambert $W-$function.

We invest money at the interest rate of $r$ percent compounded annually.

$($a$)$ Suppose we invest $A_0$ dollars, write a function which gives the value of the

account $A(t)$ after $t$ years.

$($b$)$ How long will it take for the account to double in value? Your answer will be an

expression involving $r.$

$($c$)$ Use your answer in part b$)$ to find the doubling time if the rate is $2\%.$