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\markright{Chapter 2: Worksheet 4}
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\Bf{Chapter 2: Worksheet 4  \hfill Jack K. Cohen \hfill Colorado School of Mines}

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\Cb{Intermediate Value Theorem and the Method of Bisection}
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\noindent \Bf{Suggested Problems}
Section 2.4: 48, 52, 54
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\begin{enumerate}
\item Prove that every cubic equation, $a x^3 + b x^2 + c x + d = 0 \quad (a \ne 0)$, has at least one real root.

\item The folklore says that it takes about 10 bisection steps to gain 3 decimal places of accuracy.  For example, if we know the root to within an integer, then 10 bisections will determine 3 places after the decimal point.  The folklore is right---explain why.

\item It takes about  4 bisections to gain \Em{one} decimal place.  Given that the asteroid equation, $\cos(1000/r) = r/(r + 20)$, has a root in the interval $[50008.325, 50008.335]$,
do 4 steps of bisection by hand (by reasoning similar to that in the previous problem, 4 steps is enough to pin down the third digit after the decimal point accurately).  You get \Em{no} credit for a series of cryptic equations!  Explain, using clear sentences, your successive calculations and results.

\item We've made available a \Mma\ code called \Tt{Bisect}.  Here is the syntax:
\begin{verbatim}
Bisect[f[x], {x, a, b}, eps]
\end{verbatim}
To get that third decimal place in the previous exercise, you would type:
\begin{verbatim}
Bisect[Cos[1000/r] - r/(r + h), {r, 50008.325, 50008.335}, .0005]
\end{verbatim}
This should produce the output:
\begin{verbatim}
50008.33
50008.3325
50008.33125
50008.331875
50008.3321875

50008.3320312
\end{verbatim}

\noindent Using \Tt{Bisect}, obtain the asteroid root $r$ to six places after the decimal point starting from the approximation we have so far (50008.332).  \Tt{Bisect} doesn't let you directly specify the folklore 10 iterations, so you have to think about the \Tt{eps} value that will accomplish the task.  Tell what your selection  for \Tt{a, b} and \Tt{eps} was with your answer.

\item If we knew only that the asteroid root $r$ was in the interval $[5000, 100000]$, would the method of bisection be a practical tool?
\end{enumerate}
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