QUESTION FIVE $(20$ MARKS$)$

$(a)$ The FrezMore Company has determined that its cost $of$ producing $x$ refrigerators can $be$

modeled $by$

$C(x)=2x^2+15x+1500,0x200$

where $xis$ the number $of$ refrigerators produced each week and $C(x)$ represent the weekly

$costin$ dollars. The company also determines that the price$-$demand function for the refrig$-$

erators $is$

$p(x)=-0\mathrm{.}3x+460$

$(i)$ Determine the revenue function $R(x).$

$(ii)$ Determine the profit function $P(x)$ for the refrigerators.

$(iii)$ Determine the marginal profit function $MP(x).$

$(iv)$ Compute $MP(50)$ and interpret the result.

$(b)$ Investigate the behaviour $of$

$f(x)=\{\begin{array}{ccc}x& if& x>0\\ x+1& if& x0\end{array}$

$asx0.$ Draw a graph $of$ the function $f(x)$ for xin$[-3,3].$

$(c)$ When a new product $is$ introduced $on$ the market, sales often follow a pattern $of$ rapid initial

growth followed $by$ some levelling off. For a certain new candy bar, this pattern $is$ modelled

$by$

$S(t)=4000-4000e^{-0\mathrm{.}25t}$

where $S(t)is$ the number $of$ units sold after $t$ months.

$(i)$ How many units were sold after $2$ months?

$(ii)$ Determine $\underset{t}{lim}S(t)$ and interpret.

$(d)$ Evaluate

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{x^3+5x^2-4}{\sqrt[\phantom{2}]{x}}dx.$