QUESTION FOUR $(20$ MARKS$)$

$(a)$ The Ventour Athletic Shoe manufacturering Company knows that, for its Stampeder model

basketball shoes, the daily cost function can $be$ modelled $by$

$C(x)=700\sqrt[\phantom{2}]{x}+5000,0x500$

where $xis$ the number $of$ pairs $of$ shoes produced daily and $C(x)is$ the daily cost $in$ dollars.

$(i)$ Determine $AC,$ the average cost function.

$(ii)$ Evaluate and interpret $C(400)$ and $AC(400)$

$(b)$ Differentiate the following functions:

$(i)f(x)=3x^4+2\sqrt[\phantom{2}]{x}-\frac{2}{x^2}$

$(ii)y=ln(x^4-2x)$

$(iii)g(x)=\frac{e^x+2}{x^2+1}$

$(c)$ During its first season, the number $of$ viewers who watched the new television series $"It$

Ain't $Me!"$ can $be$ modelled $by$

$g(x)=\sqrt[3]{(50+2x{)}^2},1x26$

where $x$ represents the number $of$ weeks that the series has been airing and $g(x)is$ the

number $of$ viewers $in$ millions. Evaluate and interpret $g^{\text{'}}(13).$

$(d)$ Evalute the given antiderivatives.

$(i){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(1-x)\sqrt[\phantom{2}]{x}dx.$

$(ii){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}3si{n}^{2}xdx.$