Between 2006 and 2016, the number of applications for patents, N, grew by about $4.8\mathrm{\%}$ per year. That is, $N^{\prime }(t)=0.048N(t)$.
a) Find the function that satisfies this equation. Assume that $\mathrm{t}=0$ corresponds to 2006 , when approximately 451,000 patent applications were received.
b) Estimate the number of patent applications in 2020 .
c) Estimate the rate of change in the number of patent applications in 2020.

Step 1 ：The given differential equation is a first order linear differential equation. The general solution to this type of differential equation is given by $N(t)=N(0)e^{kt}$, where $N(0)$ is the initial condition, $k$ is the rate of growth, and $t$ is the time. In this case, $N(0)=451000$, $k=0.048$, and $t$ is the number of years after 2006.

Step 2 ：The function that satisfies the given differential equation is $N(t)=451000e^{0.048t}$, where $t$ is the number of years after 2006. This function represents the number of patent applications as a function of time.

Step 3 ：To find the number of patent applications in 2020, we need to find the value of $N(t)$ when $t=2020-2006=14$.

Step 4 ：Substituting $t=14$ into the function $N(t)=451000e^{0.048t}$, we get $N(14)=451000e^{0.048\ast 14}=883125.5175678214$.

Step 5 ：Final Answer: The number of patent applications in 2020 is approximately $\boxed{{\displaystyle 883,125}}$.