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
$C(x) = 700\sqrt{x} + 5000$, $0 \leq x \leq 500$
where $x$ is the number of pairs of shoes produced daily and $C(x)$ is the daily cost in dollars.
(i) Determine $AC$, the average cost function. [1 Marks]
(ii) Evaluate and interpret $C(400)$ and $AC(400)$ [4 Marks]
(b) Differentiate the following functions:
(i) $f(x) = 3x^4 + 2\sqrt{x} - \frac{2}{x^2}$ [2 Marks]
(ii) $y = \ln(x^4 - 2x)$ [2 Marks]
(iii) $g(x) = \frac{e^{x^2+2}}{x^2+1}$ [2 Marks]
(c) During its first season, the number of viewers who watched the new television series "It Ain't Me!" can be modelled by
$g(x) = \sqrt[3]{(50 + 2x)^2}$, $1 \leq x \leq 26$
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 $g'(13)$. [5 Marks]
(d) Evalute the given antiderivatives.
(i) $\int (1 - x)\sqrt{x} \,dx$. [2 Mark]
(ii) $\int 3\sin^2 x \,dx$. [2 Mark]
