Sure, let's go through each part of the question step-by-step.
### Part a: Find the function that satisfies the differential equation [tex]\( N'(t) = 0.035 \, N(t) \)[/tex]
Given the differential equation [tex]\( N'(t) = 0.035 \, N(t) \)[/tex], we can recognize this as a first-order linear differential equation. The general solution to such an equation is of the form [tex]\( N(t) = N(0) e^{rt} \)[/tex], where [tex]\( r \)[/tex] is the growth rate (in this case 0.035), and [tex]\( N(0) \)[/tex] is the initial number of patent applications.
Since [tex]\( N(0) = 459000 \)[/tex] (the number of patent applications at [tex]\( t = 0 \)[/tex] or the year 2006), we can write the solution as:
[tex]\[ N(t) = 459000 \, e^{0.035t} \][/tex]
### Part b: Estimate the number of patent applications in 2021
To find the number of patent applications in 2021, we need to evaluate [tex]\( N(t) \)[/tex] at [tex]\( t = 2021 - 2006 = 15 \)[/tex] years after 2006.
Using the function we derived in part a:
[tex]\[ N(15) = 459000 \, e^{0.035 \times 15} \][/tex]
Evaluating this expression gives us the estimated number of patent applications in 2021:
[tex]\[ N(15) \approx 775920.61 \][/tex]
Thus, the estimated number of patent applications in 2021 is approximately 775,920.
### Part c: Estimate the rate of change in the number of patent applications in 2021
The rate of change in the number of patent applications is given by the derivative [tex]\( N'(t) \)[/tex]. From the differential equation, we know:
[tex]\[ N'(t) = 0.035 \, N(t) \][/tex]
To find the rate of change in 2021, we need to evaluate [tex]\( N'(t) \)[/tex] at [tex]\( t = 15 \)[/tex]:
[tex]\[ N'(15) = 0.035 \times N(15) \][/tex]
We already know [tex]\( N(15) \approx 775920.61 \)[/tex]:
[tex]\[ N'(15) \approx 0.035 \times 775920.61 \][/tex]
[tex]\[ N'(15) \approx 27157.22 \][/tex]
Thus, the estimated rate of change in the number of patent applications in 2021 is approximately 27,157 applications per year.
### Summary:
a) The function that satisfies the differential equation is:
[tex]\[ N(t) = 459000 \, e^{0.035t} \][/tex]
b) The estimated number of patent applications in 2021 is approximately:
[tex]\[ 775,920 \][/tex]
c) The estimated rate of change in the number of patent applications in 2021 is approximately:
[tex]\[ 27,157 \, \text{applications per year} \][/tex]
