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MATH 180\hfill FALL 1995

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%10:00 SECTION\hfill FONG

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\centerline{ASSIGNMENT \#31}

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5.3, \#12.  $y=axe^{-bx}$, $y'=a(1-bx)e^{-bx}$,
$y''=ab(bx-2)e^{-bx}$.  So $y$ has one critical point, namely
$x=1/b$.  The first-derivative test shows $x=1/b$ is a local
maximum.  $y''$ has a root at $x=2/b$, and $y''$ changes sign there.
So $x=2/b$ is an inflection point of $y$.
Changing $a$ in
$y=axe^{-bx}$ rescales the graph vertically.  Changing
$b$ changes the distance $1/b$ between
the origin and the critical point $x=1/b$ as well as the distance
$1/b$ between the critical point $x=1/b$ and the inflection point
$x=2/b$.  Indeed, increasing $b$ decreases that distance,
decreasing $b$ increases it.

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5.3, \#13.  $y=A\sin(Bx+C)$.  The parameters $A$, $B$, $C$ affect
respectively amplitude, period, and phase.  See pages 68--69 of the
text for examples.

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5.4, \#2.  $\pi(q)=R(q)-C(q)$
is negative for $0\le q < q_1$, positive for $q_1< q
<q_2$, negative for $q>q_2$.  In particular, $\pi$ has a
local maximum at a point $q=a$ between $q_1$ and $q_2$.
$C(q)$ is
increasing, so $C'(q)\ge 0$ for $q\ge0$.  The graph of $C(q)$
appears to flatten out at $q=q_1$, but the slopes get steeper
to the right of $q_2$.  So $C'(q)$ falls to 0
on $[0, q_1]$, and rises for $q\ge q_1$.
The graph of $R(q)$ is a line of positive slope.  Thus
$R'(q)$ has a horizontal line $y=b$ with $b>0$ as its graph.
Since $\pi'(a)=R'(a)-C'(a)=0$, the graphs of
$C'$ and $R'$ intersect at $(a,b)$.

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5.4, \#5.
(a)  $C(q)=0.01q^3-0.6q^2+13q$ is zero when
$q=0$, so the fixed cost is zero.

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(b)  The revenue is $R(q)=7q$ when goods are priced at \$7 each.
So profit
$$\pi(q)=R(q)-C(q)=-0.01q^3+0.6q^2-6q.$$
$\pi'(q)=-0.03q^2+1.2q-6$ has roots
$${-1.2\pm\sqrt{(1.2)^2-4(-6)(-0.03)}\over -0.06}
   =5.86,\quad 34.14.$$
34.14 is a local maximum by the second-derivative test
since $\pi''(q)=-0.06q+1.2$ is negative at $q=34.14$.
Since goods are sold in discrete quantities, we should round off
$q$ to 34 or 35.  Now $\pi(34)=\$96.56$, $\pi(35)=\$96.25$,
So profit is maximized when $q=34$.

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(c)  Suppose $q$ is fixed at 34.
Each dollar increase in the price $p$ reduces the
quantity sold by 2.  Since 34 goods are sold when $p=\$7$,
the quantity sold at price $p$ is $q=48-2p$.  We express
revenue as the function $R(p)=qp=48p-2p^2$ of $p$.
The cost is the fixed constant $C(34)$.
Thus the profit as a function of $p$ is $48p-2p^2-C(34)$.
This is a parabola opening downwards.  The profit is maximal
at the vertex.  Since $\pi'(p)=48-4p$, the critical point $p=\$12$
gives the desired price.

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