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\markright{Chapter 2: Worksheet 1}


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\Bf{\Star\ Chapter 2: Worksheet 1  \hfill Jack K. Cohen \hfill Colorado School of Mines}

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\Cb{Introduction to Slopes and Tangents}
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\noindent \Bf{Suggested Problems}
Section 2.1: 1, 4, 8, 14, 18, 20, 28, 32, 34.

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\begin{enumerate}

\begin{figure}[htb]
\epsfysize 160pt
\centerline{\epsffile{ws1p1.eps}}
\caption{Ups and downs in a balloon.} 
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\item Figure 1 shows the time history $y = H(t)$ of a balloon flight for $0 \leq t \leq 75$.  Note that the curve is \Em{not} necessarily related to the spatial path of the balloon, since the horizontal axis is time $t$, not horizontal distance $x$. 
	\begin{enumerate}
	\item There are three \Em{local minimum} points on the graph (one is at the start of the flight), label them $a$, $b$ and $c$ and estimate these values as well as $H(a), H(b), H(c)$.  Do the same for the three \Em{local maximum} points, $A, B, C$.
	\item Give the best  definition you can for the concept \Em{local minimum point} introduced so casually in the previous question.
	\item Since our curve is smooth, we can intuitively assign a slope at each point of it---except for maybe the endpoints.  Using the letters $a, b, c, A, B, C$ introduced above, give the intervals where the slope is positive.  Are these the same as the intervals where the function is increasing?
	\item Consider the interior extrema (i.e. maximum points and minimum points).  What is the value of the slope function at these points?
	\item We want a condition that guarantees that a point $p$ in the interior of the domain of a smooth function gives a \Em{maximum} value $H(p)$.  State such a condition in terms of the behavior of the slope \Em{near} $p$ (i.e. just to the left and right of $p$, \Em{not} at $p$ itself).  
	\item When is  $H(t)$ the biggest?  (The biggest value of $H(t)$ is called the \Em{global maximum} or sometimes the \Em{absolute maximum}).   Similarly, when does $H(t)$ achieve its the global minimum?
	\item Since the independent variable is time $t$, the slope function here is actually velocity (right?).   When is the velocity 0?  When is the velocity (locally and globally) the greatest?   The least?  Explain your reasoning and give approximate numerical values from Figure 1.
	\end{enumerate}

\item \Star\ (2.1.36) Prove that the line tangent to the parabola $y = x^2$ at the point $(x_0, y_0)$ intersects the $x$-axis at the point $(x_0/2, 0)$.  \Bf{Hints}:  Draw a picture.  No, not a few scraggly lines and curves---draw a \Em{good} picture!  Label your picture.  Temporarily use a distinct label, say $(x_1, 0)$, for the target point on the $x$-axis.  Now show that $x_1 = x_0/2$ by constructing a suitable tangent line.

\item (2.1.42) Write equations for the two straight lines through the point (2, 5) that are tangent to the parabola $y = 4x - x^2$.   Does it make sense that there are \Em{two} tangents through the point (2, 5)?  \Bf{Advice}:  Don't panic.  Draw a picture (use \Tt{Plot} if you need it).   Label a general point on the parabola as $(a, y(a)) = (a, 4a - a^2)$.  You can figure out the slope at the general point $a$ and hence write an equation for the tangent line in point slope form (with $a$'s in it all over the place).  There is one fact you haven't used---use it to get an equation for $a$.  Then solve for $a$ and 
 plug back into the tangent equation.

\noindent For the really brave: replace the point (2, 5) by a general point $(x_0, y_0)$.

\item (2.1.43) Find the point on the parabola $y = x^2$ that is closest to the point (3, 0) by using the insight that for this point, the line from (3,  0) to the parabola will be \Em{normal} to the parabola.  Note:  you have to solve a cubic equation to find the closest point---here, you can probably guess the relevant root, but if not use \Mma.

\item Following up on previous problems:
	\begin{enumerate}
	\item Are there always exactly \Em{two} tangent lines from a general point $(x_0, y_0)$ to the parabola $y = x^2$?
	\item Is there always exactly \Em{one} normal line from a general point $(x_0, y_0)$ to the parabola $y = x^2$?
	\end{enumerate}


\item \Bf{Honor Problem}:  In finding the ``physically sensible'' solution to the asteroid problem you may have noticed that there are a lot (infinitely many actually) of roots in the interval $[0, s = 1000]$.  Find the largest two roots less than $s = 1000$ and explain their significance.   Further investigations and discussions along this line are welcome.


\end{enumerate}
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