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MATH 180\hfill FALL 1995

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%10:00 SECTION\hfill FONG

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\centerline{ASSIGNMENT \#11}

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2.4, \#8.  If $f'(10)=-29000$, then a small change $h$ in price from
\$10 to \$(10+h) produces a corresponding change in the
quantity sold from $240000$ to $240000-29000h$.

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2.4, \#11.  The units of $g'(55)$ are (miles/gallon)$\big/$(miles/hour).
To say $g'(55)=-0.54$ is to say that at velocity $v=55$ miles/hour,
a small change $h$ in velocity produces a corresponding change
of $-.54h$ miles/gallon in efficiency.  So driving 57 miles/hour
decreases efficiency by 1.08 mile/gallon;  driving
53 miles/hour increases efficiency by 1.08 mile/gallon.

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2.4, \#14.  (a) One would expect $f'$ to be positive, {\it i.e.\/},
an increase in advertising costs produces an
increase in revenues from sales.
If $f'(100)=2$, then a small change $h$ in advertising costs
when the costs are \$100000 produces $2h$ in revenue.
In other words, each \$1000 added to \$100000 produces
\$2000 more in revenue, each
\$1000 subtracted from \$100000 produces \$2000 less in revenue.
On the other hand, if $f'(100)=.5$, then
each additional \$1000 added to \$100000 produces \$500 more in
revenue, each \$1000 subtracted from \$100000 produces
\$500 less in revenue.  Hence the company should spend
slightly more than \$100000 if $f'(100)=2$ and
slightly less if $f'(100)=.5$.

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Comments:  This assignment had lower scores than any of the
previous assignments.  A lot of misunderstandings about the
meaning of $f'(a)$ showed up.
Here are samples of misinterpretations of ``$g'(55)=-0.54$'' in
problem \#11 taken from homework.

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``at 55 miles/hour, it takes just over 1/2 hour to use 1 gallon of gas''

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``When the car is travelling at a rate of 55 mph it is losing
gas at a rate of $-.54$ gallons per mile''

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``That the efficiency decreases every time of .54 of hr$^2$''

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``while driving at the speed of 55 mph, the fuel efficiency gets less''

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``$-0.54$ is the rate of consumption of gasoline''

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``the car is using $-.54$ less gallons at 55 mph''

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Compare these with the answer given above.
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