Between $2006$ and $2016,$ the number of applications for patents, N$,$ grew by about $3\mathrm{.}6\%$ per year. That is$,N^{\text{'}}(t)=0\mathrm{.}036N(t).$

a$)$ Find the function that salisfies this equation. Assume that $t=0$ corresponds to $2006,$ when approximately $454,000$ patent applications were received.

b$)$ Estimate the number of patent applications in $2020.$

c$)$ Estimate the rate of change in the number el palent applications in $2020.$