Between 2006 and 2016, the number of applications for patents, $N$, grew by about $3.9\mathrm{\%}$ per year. That is, $N^{\prime }(t)=0.039N(t)$.
a) Find the function that satisfies this equation. Assume that $\mathrm{t}=0$ corresponds to 2006 , when approximately 461,000 patent applications were received.
b) Estimate the number of patent applications in 2021.
c) Estimate the rate of change in the number of patent applications in 2021.
a) $N(t)=$
b) The number of patent applications in 2021 will be (Round to the nearest whole number as needed.)
c) The rate of change in the number of patent applications in 2021 is about (Round to the nearest whole number as needed.)

Step 1 ：The given differential equation is a first order linear differential equation. The general solution to this type of 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, $N(0)=461000$, $k=0.039$, and $t$ is the number of years since 2006.

Step 2 ：Substituting the given values into the equation, we get the function that satisfies the equation as $N(t)=461000\ast e^{0.039t}$.

Step 3 ：To estimate the number of patent applications in 2021, we substitute $t=15$ (since 2021 is 15 years after 2006) into the equation to get $N(2021)=827490.8443799939$. Rounding to the nearest whole number, we get approximately $\boxed{{\displaystyle 827491}}$ patent applications in 2021.

Step 4 ：To estimate the rate of change in the number of patent applications in 2021, we differentiate the function $N(t)$ with respect to $t$ to get $N^{\prime }(t)=0.039\ast N(t)$. Substituting $t=15$ and $N(15)=827490.8443799939$ into the equation, we get $N^{\prime }(2021)=32272.14293081976$. Rounding to the nearest whole number, we get approximately $\boxed{{\displaystyle 32272}}$.