Between 2006 and 2016, the number of applications for patents, N, grew by about $4.8 \%$ per year. That is, $N^{\prime}(t)=0.048 N(t)$. 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 \(N(t) = 451000 e^{0.048t}\), 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 \(N(t) = 451000 e^{0.048t}\), we get \(N(14) = 451000 e^{0.048*14} = 883125.5175678214\).

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