Question

between 2006 and 2016, the number of applications for patents, n, grew by about 4.4% per year. that is, n(t)=0.044n(t)
a) find the function that satisfies this equation. assume that t = 0 corresponds to 2006, when approximately 444,000 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
a) n(t)=444000e^{0.044t}
b) the number of patent applications in 2020 will be 822,069 (round to the nearest whole number as needed.)
c) the rate of change in the number of patent applications in 2020 is about
(round to the nearest whole number as needed.)
year(s) per application
application(s) per year

Explanation:

Step1: Recall the function and find $t$ for 2020

We know $N(t)=444000e^{0.044t}$. For 2020, since $t = 0$ is 2006, then $t=2020 - 2006=14$.

Step2: Differentiate the function

The derivative of $N(t)=444000e^{0.044t}$ with respect to $t$ is $N'(t)=444000\times0.044e^{0.044t}=19536e^{0.044t}$ according to the chain - rule for exponential functions $\frac{d}{dt}(e^{ax})=ae^{ax}$.

Step3: Evaluate $N'(t)$ at $t = 14$

Substitute $t = 14$ into $N'(t)$: $N'(14)=19536e^{0.044\times14}=19536e^{0.616}$.
We know that $e^{0.616}\approx1.851$. Then $N'(14)=19536\times1.851\approx36161$.

Answer:

36161 applications per year