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MATH 180\hfill FALL 1995

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10:00 SECTION\hfill FONG

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\centerline{ASSIGNMENT \#2}

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1.3, \#3.  The graph should have the following
features:  i)  a first part which is a line segment of
positive slope since
``the quantity in the blood increases {\it linearly\/}'';
ii)  a second part which is part of an
exponential decay curve since ``the quantity then decays
{\it exponentially\/}''.  So this second part is
decreasing and concave-up, and approaches the $x$-axis
asymptotically from above;  iii)  whether the graph starts
at $(0,0)$ depends on the initial amount of drug already in the blood
when the injection is made, so there is no single right answer here.

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1.3, \#5.  For the inputs $t=1,2,3,4,5,6$, the changes in the values of
$g(t)$ are $1,2,3,4,5$; the changes in the values of $h(t)$ are
10, 9, 8, 7, 6; and the changes in the values of $k(t)$ are
equal to the constant $.3$.
So $g(t)$ best fits the curve (c) since the changes are increasing,
$h(t)$ best fits the curve (b) since the changes are decreasing,
and $k(t)$ fits the curve (a) since the changes are constant.

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1.3, \#7.
$h$, $f$, $g$ are given to be exponential functions.
So consider the ratios $h(s+1)/h(s)$,
$f(s+1)/f(s)$, $g(s+1)/g(s)$ for the given inputs.
The ratios for each function should
equal the base of the exponential function
(up to errors given by approximations).
We find that
$h(s)=c(1.03)^s$, $f(s)=a(1.1)^s$, $g(s)=b(1.05)^s$.
(The constants $a$, $b$, $c$ can also be found by plugging in
$s=2$ and using the given values of $h$, $f$, $g$ at $s=2$.)

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1.3, \#9.  The ratios $g(t+1)/g(t)$ in this
case are all $.8$.  So $g$ behaves like an exponential function
$C(.8)^t$.  To determine $t$, we can use $g(0)=5.50$ to see
that $C=5.50$.

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1.3, \#11.  The exponential function $y=Ca^x$ has two
parameters $C$ and $a$.  In principle knowing two points on
the graph of the function should determine $C$ and $a$.  In
this case the points are $(1,6)$ and $(2,18)$, giving
$6=Ca$ and $18=Ca^2$.  These two equalities force $a=3$ and $C=6$,

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Comments on \#5 and \#7:  It is a good principle to try
simple calculations before doing complicated ones.  So in \#5 one
would consider differences of values to see if this data is enough to
settle the problem.  In fact, this is the case.
In \#7 we are given that
the three functions are exponential functions.  So there is no point
in looking at differences of values, and one should look immediately at
ratios of values.

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A general comment:  An answer without explanations is not an
adequate answer.
No credit (on homework, quizzes, or exams) will be given for such
answers.  Write enough so that the reader can follow your reasons for
the answer.


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