Between 2006 and​ 2016, the number of applications for​ patents, N, grew by about 3.4​% per year. That​ is, Upper N prime left parenthesis t right parenthesisequals0.034Upper N left parenthesis t right parenthesis.
​a)
Find the function that satisfies this equation. Assume that tequals0 corresponds to 2006​, when approximately 449 comma 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.

We are given the differential equation \( N'(t) = 0.034 N(t) \) and the initial condition \( N(0) = 449,000 \), where \( t = 0 \) corresponds to the year 2006. Let's solve this step by step.

### Part (a) – Finding the Function that Satisfies the Differential Equation

We are tasked with finding the function \( N(t) \) that satisfies the differential equation:

\[
N'(t) = 0.034 N(t)
\]

This is a separable differential equation, which can be written as:

\[
\frac{dN(t)}{N(t)} = 0.034 \, dt
\]

Integrating both sides:

\[
\int \frac{dN(t)}{N(t)} = \int 0.034 \, dt
\]

The left side integrates to \( \ln|N(t)| \), and the right side integrates to \( 0.034t + C \), where \( C \) is the constant of integration:

\[
\ln|N(t)| = 0.034t + C
\]

Exponentiating both sides to remove the natural logarithm:

\[
N(t) = e^{0.034t + C} = e^C e^{0.034t}
\]

Let \( e^C = C_1 \), a new constant. Thus, we have:

\[
N(t) = C_1 e^{0.034t}
\]

Now, using the initial condition \( N(0) = 449,000 \), we can find \( C_1 \):

\[
N(0) = C_1 e^{0} = C_1 = 449,000
\]

Thus, the function that satisfies the equation is:

\[
N(t) = 449,000 e^{0.034t}
\]

### Part (b) – Estimating the Number of Patent Applications in 2020

To estimate the number of patent applications in 2020, we need to calculate \( N(t) \) at \( t = 2020 - 2006 = 14 \). Using the function \( N(t) = 449,000 e^{0.034t} \):

\[
N(14) = 449,000 e^{0.034 \times 14}
\]

Now we can compute this value.

### Part (c) – Estimating the Rate of Change in 2020

The rate of change of the number of patent applications is given by \( N'(t) = 0.034 N(t) \). To estimate the rate of change in 2020, we substitute \( t = 14 \) into \( N'(t) \):

\[
N'(14) = 0.034 \times N(14)
\]

Since we already computed \( N(14) \), we can use this value to find \( N'(14) \). Let's compute both of these values.### Part (b) – Estimate of Patent Applications in 2020:
The estimated number of patent applications in 2020 is approximately **722,721**.

### Part (c) – Rate of Change in 2020:
The estimated rate of change in the number of patent applications in 2020 is approximately **24,573 applications per year**.

Do you need further details or have any questions?

Here are 5 related questions to explore further:
1. How would the model change if the growth rate were 4% instead of 3.4%?
2. Can we determine how many patent applications there will be in 2030 using the same method?
3. What would the doubling time be for the patent applications with this growth rate?
4. How does exponential growth differ from linear growth in terms of patent application growth?
5. How sensitive is the result to small changes in the initial number of patents?

**Tip**: Exponential growth models like this one can be used to approximate growth in various fields such as population studies, finance, and technology advancements.

Between 2006 and 2016, the number of applications for patents, N, grew by about 3.4% per year. That is, N'(t) = 0.034N(t). a) Find the function that satisfies this equation. Assume that t = 0 corresponds to 2006, when approximately 449,000 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.

Learn how to solve the exponential growth problem involving the number of patent applications from 2006 to 2020. This solution involves solving the differential equation N'(t) = 0.034N(t) with initial conditions and provides estimates for 2020. Suitable for advanced high school or early college students studying calculus.

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