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MATH 180\hfill FALL 1995

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%10:00 SECTION\hfill FONG

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\centerline{ASSIGNMENT \#3}

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1.3, \#13.
One possible function with the given graph is
the exponential function
$y=4-Ca^x$.  Note $(0,0)$ and $(1,2)$ are points on
the graph.  So plugging these values into $y=4-Ca^x$ gives
$C=4$ and $a={1\over2}$.

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1.4, \#2.
The graphs of $x^{2/3}$ and $x^{1/2}$ cross at
$(1,1)$, so a sketch of the two
should contain a region about this point.
$x^{2/3}$ dominates $x^{1/2}$ as $x\rightarrow\infty$.
The general fact: If $Cx^p$ and $Dx^q$ are power functions such
that $C$, $D$, $p$, $q$ are positive and $p<q$,
then $Dx^q$ dominates $Cx^p$ as $x\rightarrow\infty$.

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1.4, \#17.  The three functions are of the form $ab^t$,
$at^2$, and $bt^3$.  The exponential function can be detected
by computing ratios of consecutive values.
For $h(t)$, these ratios are
$${3.06\over2.04}={4.59\over3.06}=
  {6.89\over4.59}={10.33\over6.89}=
  {15.49\over10.33}=1.50$$
up to 2 decimal places.  So $h(t)$ is exponential with base 1.5.
Since $h(0)=2.04$, $h(t)=2.04(1.5)^t$.
The remaining two functions are
power functions.  A power function
$k(t)=at^p$ has the following property:
$${k(t_{i+1})\over k(t_i)}={at_{i+1}^p\over
  at_i^p}=\left({t_{i+1}\over t_i}\right)^p.$$
So the ratio of outputs is the $p$-th
power of the ratio of inputs.  In the problem
$${f(2.2)\over f(2.0)}={5.32\over4.40}=1.21,\quad
    \left({2.2\over2.0}\right)^2=1.21.$$
So $f(t)$ is quadratic and $g(t)$ is necessarily the cubic function.

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1.5, \#10, \#11.  The sketches tell the story (given appropriate
window parameters, say $-10\le x\le10$, $-50\le y\le50$).
$x^3-5x+10$
has a graph where horizontal lines can meet the graph as many as
3 times;
$x^3+5x+10$ has a graph where every horizontal line meets the graph
only once.  So $x^3-5x+10$ is not invertible, but the
$x^3+5x+10$ is.  We will have other ways of seeing this using
calculus.

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Comments on 1.4, \#17:  Doing calculations is
unavoidable in this problem.  The question is when to stop.
To find the exponential function, one needs ratios of
consecutive values given in the table.
Ratios of values are also needed to distinguish
the quadratic and the cubic power functions.
How many ratios are needed?
The question as posed in the text makes matters easy by saying the
functions {\it are\/} exponential, quadratic, and cubic.
So it is not necessary to calculate all ratios to answer the
question.
But had the question been reworded as ``Are any of these functions
exponential, quadratic, or cubic?'', then one would probably want
to check all ratios.


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