$(1)$
\[
b=\frac{x}{l}b_0
\]
$(2)$
\[
I_x=\frac{1}{12}bh^3=\frac{1}{12}\frac{x}{l}b_0h^3
\]
$(3)$
\[
x=lで\theta=0,\omega=0
\]
$(4)$
\[\begin{align}
\frac{d^2y}{dx^2}
&=-\frac{M^2}{EI}\\
&=-\frac{1}{EI_x}Wx\\
&=\frac{12l}{Exb_0h^3}Wx\\
&=\frac{12lW}{Eb_0h^3}
\end{align}\]
\[
\theta=\frac{dy}{dx}=\frac{12lW}{Eb_0h^3}x+C_1\\
x=lで\theta=0より\\
C_1=-\frac{12l^2W}{Eb_0h^3}\\
\therefore\theta=\frac{12lW}{Eb_0h^3}x-\frac{12l^2W}{Eb_0h^3}
\]
$(5)$
\[
\omega=\frac{6lW}{Eb_0h^3}x^2+C_1x+C_2\\
x=lで\omega=0より\\
C_2=\frac{6l^3W}{Eb_0h^3}\\
\therefore\omega=\frac{6lW}{Eb_0h^3}x^2-\frac{12lW}{Eb_0h^3}x+\frac{6l^3W}{Eb_0h^3}
\]
$(6)$
\[
x=0を代入して\\
\theta_A=-\frac{12l^2W}{Eb_0h^3}
\]
$(7)$
\[
x=0を代入して\\
\omega_A=\frac{6l^3W}{Eb_0h^3}\
\]