Between 2006 and 2016, the number of applications for patents, $N$, grew by about $4.2 \%$ per year. That is, $N^{\prime}(t)=0.042 N(t)$.
a) Find the function that satisfies this equation. Assume that $\mathrm{t}=0$ corresponds to 2006 , when approximately 445,000 patent applications were received.
b) Estimate the number of patent applications in 2022.
c) Estimate the rate of change in the number of patent applications in 2022.
a) $N(t)=445000 e^{0.042 t}$
b) The number of patent applications in 2022 will be (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 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, \(N(0) = 445000\), \(k = 0.042\), and \(t\) is the number of years since 2006.

Step 2 ：To find the number of patent applications in 2022, we need to substitute \(t = 2022 - 2006 = 16\) into the equation.

Step 3 ：Substituting the values into the equation, we get \(N = 445000 * e^{0.042 * 16}\)

Step 4 ：Calculating the above expression, we get \(N = 871376.6193296686\)

Step 5 ：Rounding to the nearest whole number, we get \(N = 871377\)

Step 6 ：Final Answer: The number of patent applications in 2022 will be approximately \(\boxed{871377}\)