%\centerline{\bf Homework on Inverse Trig Functions}
\begin{homework}\
\label{homework:7} 

\medskip
\exercise
Find $d/dx$ of: 


(a) $x^2\,\arctan(x/2)$, 

(b) $(x+1)/\arcsin(x^2)$, 

(c) $\arcsin(\sqrt{1-x})$, 

(d) $x\,\arcsin(x)+\sqrt{1-x^2}$.

\smallskip
\exercise
Find the anti-derivative of 

(a) $(100-16x^2)^{-1/2}$, 

(b) $(1+2x^2)^{-1}$.

\smallskip
\exercise
Using the tangent line approximation and the fact that
$\sin(30^\circ)=0.5$, find the angle in degrees whose sine is 0.52.

\smallskip
\exercise
Find the volume of the solid formed when the shape bounded by
$y=\sqrt{x^{-2}-1}$, the line $y=1$, and the $x$- and $y$-axes is rotated
around the $y$-axis.

\smallskip
\exercise
Find the center of mass of the uniform flat plate bounded by the curve
$1/\sqrt{1-x^2}$, the curve $-1/\sqrt{1-x^2}$, the $y$-axis, and the line
$x=0.5$.

\smallskip
\exercise
Graph and find max/min points of 

(a) $y=(\arctan\,x)-0.5x$, 

(b) $y=\bigl(\arcsin(x/2)\bigr)-x$.



\smallskip
\exercise
Sketch and find the area between 

(a) $y=2/(1+x^2)$ and $y=x^2$, 

(b) $y=1/\sqrt{1-x^2}$ and $y=2$.

\smallskip
\exercise
A rocket launched vertically from a point 2 km from an observer is
ascending at $0.20$ km/sec at the instant when it is 1 km above the ground.
Find the rate in degrees/sec at which its angle of inclination (as seen by
the observer) is increasing at that moment.

\smallskip
\exercise
A triangle has two fixed sides of length $a$ and $b$ with included angle
$\theta$ that is free to widen.  Let $x$ be the side opposite $\theta$.
(a) Use the law of cosines to express $\theta$ as a function of $x$.  (b)
At the instant when $x=c$, what is the ratio of the rate at which the angle
$\theta$ is widening (in rad/sec) to the rate at which the side $x$ is
increasing?

\end{homework} 


