For the function $f ( x ) = 3 ( x - 1 ) ^ { 2 } + 5$, define the function $F ( x )$ as $F ( x ) = \int _ { 0 } ^ { x } f ( t ) \, dt$. A differentiable function $g ( x )$ satisfies the following for all real numbers $x$:
$$F ( g ( x ) ) = \frac { 1 } { 2 } F ( x )$$
When $g ^ { \prime } ( 2 ) = p$, find the value of $30 p$. [4 points]

Let $\gamma:[a,b]\rightarrow\mathcal{H}$ be a curve that is continuous and piecewise $\mathcal{C}^1$. The hyperbolic length of $\gamma$ is defined by $$\ell(\gamma) = \int_a^b \sqrt{B(\gamma'(t),\gamma'(t))}\,\mathrm{d}t.$$ Show that if $h:[c,d]\rightarrow[a,b]$ is a diffeomorphism, then $\ell(\gamma) = \ell(\gamma\circ h)$.

Let $\gamma:[a,b]\rightarrow\mathcal{H}$ be a curve that is continuous and piecewise $\mathcal{C}^1$. The hyperbolic length of $\gamma$ is defined by $$\ell(\gamma) = \int_a^b \sqrt{B(\gamma'(t),\gamma'(t))}\,\mathrm{d}t.$$ Let $f(t) = -B(\gamma(a),\gamma(t))$ and $n(t) = \sqrt{B(\gamma'(t),\gamma'(t))}$. Show that $$f'(t) \leq \sqrt{f(t)^2-1}\, n(t).$$

Let $\gamma:[a,b]\rightarrow\mathcal{H}$ be a curve that is continuous and piecewise $\mathcal{C}^1$, with $f(t) = -B(\gamma(a),\gamma(t))$ and $n(t) = \sqrt{B(\gamma'(t),\gamma'(t))}$ satisfying $f'(t) \leq \sqrt{f(t)^2-1}\,n(t)$.
Deduce that $$-B(\gamma(a),\gamma(b)) \leq \operatorname{ch}(\ell(\gamma)).$$

Let $u$ and $v$ be two points of $\mathcal{H}$. The hyperbolic distance between $u$ and $v$ is defined by $$d(u,v) = \inf_\gamma \ell(\gamma)$$ where the infimum is taken over the set of continuous and piecewise $\mathcal{C}^1$ paths $\gamma:[a,b]\rightarrow\mathcal{H}$ such that $\gamma(a)=u$ and $\gamma(b)=v$.
Show that $d$ is a distance on $\mathcal{H}$, that is,

• $d(u,v) = d(v,u)$,
• $d(u,w) \leq d(u,v)+d(v,w)$, and
• $d(u,v)=0 \Leftrightarrow u=v$

for all $u,v,w\in\mathcal{H}$.

Consider a twice differentiable function $y ( x )$ in an $x y$ plane which connects two points $A ( - 1,2 )$ and $B ( 1,2 )$. Let $S$ be outer surface area of the cylindrical object created by rotation of the curve $y ( x )$ about the $x$ axis. Answer the following questions.
(1) Prove that the surface area $S$ is given by
$$\begin{aligned} S & = 2 \pi \int _ { - 1 } ^ { 1 } F \left( y , y ^ { \prime } \right) \mathrm { d } x \\ F \left( y , y ^ { \prime } \right) & = y \sqrt { 1 + \left( y ^ { \prime } \right) ^ { 2 } } \end{aligned}$$
where $y ^ { \prime } = \frac { \mathrm { d } y } { \mathrm {~d} x }$.
(2) Let the curve $y ( x )$ satisfy the following Euler-Lagrange equation for arbitrary $x$:
$$\frac { \partial F } { \partial y } - \frac { \mathrm { d } } { \mathrm {~d} x } \frac { \partial F } { \partial y ^ { \prime } } = 0$$
Considering Eq. (2.3) along with $\frac { \mathrm { d} F } { \mathrm {~d} x }$, prove that the following relation holds:
$$F - y ^ { \prime } \frac { \partial F } { \partial y ^ { \prime } } = c$$
Here $c$ is a constant.
(3) Express a differential equation satisfied by the curve $y ( x )$ using $y , y ^ { \prime } , c$.
(4) Represent the curve $y ( x )$ as a function of $x$ and $c$.
Obtain an equation which should be satisfied by the constant $c$.