\begin{homework}\
\label{homework:9}

\medskip

\exercise %1
You know the exact value $\tan(45^\circ)=1$.  Find $\tan(44^\circ)$
using the \index{ tangent line approximation} tangent line approximation, and compare with the exact value
(found by calculator).

\smallskip

\exercise %2
Find the derivative of (a) $y=$cot$\,x$, (b) $y=$csc$\,x$ (csc is
the abbreviation of cosecant). 

\smallskip

\exercise %3
Use the tangent line approximation to find (a) $\cos(62^\circ)$,
(b) $\sin(29^\circ)$, (c) $\text{cot}(47^\circ)$.

\smallskip

\exercise %4.
Find the derivative of each of the functions in Problem 8(a)--(e) of the
last homework set.  In each case find the first positive time $t$ for which
the derivative is 0 and the first positive $t$ for which the derivative
reaches its maximum value; then explain in practical terms what is
happening at each of these moments.

\smallskip

\exercise %5.
Find the derivative of
(a) $y=\sin x\cos x$, (b) $y=\sin(\cos x)$, (c) $y=\sqrt{x\tan x}$,
(d) $\disp y=\frac{\tan x}{ 1+\sin x}$.

\smallskip

\exercise %6.
Suppose that the point $(\pi/3,\pi/4)$ is on the \index{ implicitly defined curve} implicitly defined curve 
$\disp\frac{\sin x}{ x}+\frac{\sin y}{ y}=\text{const}$.  Use the tangent line
approximation to determine the $y$-coordinate of the point on the curve
with $x$-coordinate $\disp\frac{\pi}{ 3}+\frac{\pi}{ 180}$.

\smallskip

\exercise %7.
Suppose that the point $(1,1.5)$ is on the curve $x\sin(\pi y/3)+y\sin(\pi
x/4)=$const.  Use the tangent line approximation to determine the
$y$-coordinate of the point on the curve with $x$-coordinate 1.1.

\smallskip

\exercise %8.
A \index{ rope} rope is tied at its left end, and its right end is moved up and down to
create a sinusoidal wave of period 6 ft.  The equation of the shape of the
rope (``standing wave'') is then
\[
   y=A\sin\left(\frac{2\pi}{ 6}x\right)=A\sin\left(\frac{\pi}{ 3}x\right)\,,
\]
where we are taking the origin to be the point where the rope is fastened,
and $A$ is the amplitude.  Now actually, the amplitude $A$ varies with time
as the rope oscillates.  Suppose that the maximum value of $A$ is 2 ft, and
it varies sinusoidally with time $t$ with period 2 sec:
\[
A=2\sin(\pi t).
\]
Thus, the vertical coordinate $y$ of a point on the rope depends on both
its $x$-coordinate and the time $t$:
\[
y=2\sin(\pi t)\cdot\sin\left(\frac{\pi}{ 3}x\right).
\]

(a)~Sketch the appearance of the rope at quarter-second intervals from
$t=0$ to $t=2$ sec (all on the same graph).

(b)~Suppose your vision is blocked by a \index{ horizontal ledge} horizontal ledge at a height of 6
inches above the $x$-axis, so you see only the tips of the \index{ waves} waves.  Write an
equation that gives the $x$-value where the wave emerges above the ledge
implicitly as a function of $t$ (i.e., write an equation of the form
$F(x,t)=$const).

(c)~What is the first $x$-value where the rope emerges above the ledge when
$t=1/6$ sec?

(d)~How rapidly is that first $x$-value moving leftward at the instant
$t=1/6$ sec?

(e)~Approximately where will that first point where you see the rope be
located when $t=0.2$ sec?  (Use parts (c) and (d) and the tangent line
approximation.)


\end{homework}




