Answer
See the detailed answer below.
Work Step by Step
$$\bf a)$$ We know that the speed of a wave is given by
$$v=\lambda f =(2)(5)$$
where $f$ is the frequency, and $\lambda$ is the wavelength.
$$v= \color{red}{\bf 10}\;\rm m/s$$
$$\bf b)$$
We know that the function of this wave is given by
$$D_{(x,t)}=A\sin\left[ \dfrac{2\pi x}{\lambda}-\dfrac{2\pi t}{T}+\phi_0 \right]$$
And at $t=0$, $x=0$,
$$D_{(0,0)}=A\sin\left[ \dfrac{2\pi (0)}{\lambda}-\dfrac{2\pi (0)}{T}+\phi_0 \right]=A\sin\phi_0$$
where at $t=0$, $D=\frac{1}{2}A=0.5\;\rm mm$, so
$$\phi_0=\sin^{-1}\left(\frac{1}{2}\right)=\color{red}{\bf \dfrac{\pi}{6}}\;\rm rad $$
$$\bf c)$$
The displacement equation for the wave is given by
$$D_{(x,t)}=A\sin\left[ \dfrac{2\pi x}{\lambda}-2\pi ft+\phi_0 \right]$$
Plugging the known;
$$D_{(x,t)}=(1\times 10^{-3})\sin\left[ \dfrac{2\pi }{2}(x)- 2\pi(5) (t)+\dfrac{\pi}{6}\right]$$
$$\boxed{D_{(x,t)}=(0.001)\sin\left[ \pi x -10\pi t+\dfrac{\pi}{6}\right]}$$
