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\markright{Chapter 3: Worksheet 3}
\begin{document}

\Bf{\Star\ Chapter 3: Worksheet 3  \hfill Jack K. Cohen \hfill Colorado School of Mines}

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\Cb{Techniques of Differentiation: Product and Quotient Rules}
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\noindent \Bf{Suggested Problems}
Section 3.2:  See below.

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\noindent For the next few days, your job is get some mechanics well in hand.  It is not practical to use a computer for every routine calculation!   However, in the Colorado School of Mines calculus environment, you can make your learning more efficient by getting instant feedback on the accuracy of hand calculation of derivatives.

\begin{verbatim}
(* Take a simple derivative with Mma *)
D[x^3 + 3x, x]
           2
    3 + 3 x

(* Put in problem 3.2.25, but change variable to "t" *)
g[t_] := (1/t - 2/t^2) / (2/t^3 - 3/t^4)
D[g[t], t]           (* Ask for derivative *)
    4     -2    12   6    -2   1
    -- - t     (-- - --) (-- + -)
     3           5    4    2   t
    t           t    t    t
    -------- - ------------------
    -3   2          -3   2  2
    -- + --        (-- + --)
     4    3          4    3
    t    t          t    t

(* What a mess!  Fortunately Mma has a Simplify function.
   The symbol % used in the next line means "previous output" *)
Simplify[%]
                      2
    t (12 - 13 t + 4 t )
    --------------------
                  2
        (-3 + 2 t)

(* If we anticipate the need for Simplify, we can do it all in one step.
   Reminder: g[] has been defined as a true function, so we
   can change the variable to x if we like *)
Simplify[ D[g[x], x] ]
                      2
    x (12 - 13 x + 4 x )
    --------------------
                  2
        (-3 + 2 x)

(* Mma also understands the prime notation for derivative. *)
Simplify[ g'[t] ]
                      2
    t (12 - 13 t + 4 t )
    --------------------
                  2
        (-3 + 2 t)
\end{verbatim}

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\begin{enumerate}
\item Do the even numbered problems (2-50) in section 3.2.  Be sure to vary your notation between $\frac{dy}{dx}, \frac{df}{dx}, f'(x)$ and $Df(x)$.  Make sure that there is no sense of panic when the independent variable is (say) $t$ instead of $x$.  Check your answers using \Tt{D[f[x], x], f'[x], Simplify} (answers will \Em{not} be given for this exercise, ask your instructor about anything that stumps you) .  Dispensation: if you get 3 in a row correct with no help from anyone or from \Mma\  you can skip to harder looking exercises.

\item \Star\ According to the inverse square law, the weight of an object near the Earth is given by \[ W = \frac{GMm}{r^2} \] where $G$ is the universal constant of gravitation, $M$ is the mass of the Earth, $m$ is the mass of the object and $r$ is the distance from the Earth's center.  For problems concerning objects near the Earth's surface, it is convenient to introduce the surface gravitational constant $g = GM/R^2$ where $R$ is the mean radius of the Earth.  This lets us write the inverse square law as \[ W = mg \left( \frac{R}{r} \right)^2. \]
	\begin{enumerate}
	\item What is the rate of change of $W$ with respect to $r$?
	\item Evaluate $W(r)$ and $W'(r)$ when $r = R = 3960$ miles and $g = 32$ ft/sec$^2$ for a mass of 5.5 pounds.
	\item Near the surface of the Earth, the approximation $r = R$ is commonly used to give $W \approx mg$.  Redo part (b) with this approximation and comment on its validity.
	\item Repeat the last part, but with $r$ replaced by 5500 miles.
	\end{enumerate}

\end{enumerate}
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