Problem 7 - Calculus and its applications

Question

U.S. patents. The number of applications for patents, \(N,\) grew dramatically in recent years, with growth averaging about \(5.8 \%\) per year. That is, \(N^{\prime}(t)=0.058 N(t)\) a) Find the function that satisfies this equation Assume that \(t=0\) corresponds to \(2009,\) when approximately 483,000 patent applications were received. b) Estimate the number of patent applications in 2020 c) Estimate the doubling time for \(N(t)\).

Step-by-Step Solution

Answer

Function: \( N(t) = 483,000 e^{0.058t} \); 2020: Approx. 973,149 applications; Doubling time: Approx. 11.95 years.

Step 1: Identify the Differential Equation

The differential equation given is \( N^{\prime}(t) = 0.058 N(t) \). This is a first-order linear differential equation in the form \( y^{\prime} = ky \) where \( k = 0.058 \).

Step 2: Solve the Differential Equation

To solve \( N^{\prime}(t) = 0.058 N(t) \), we recognize it as a separable equation. The solution is found by integrating both sides: \[\int \frac{dN}{N} = \int 0.058\, dt\]which yields:\[\ln |N| = 0.058t + C\]Exponentiating both sides, we get:\[N(t) = e^{C} e^{0.058t} = C^{\prime} e^{0.058t}\]where \( C^{\prime} = e^C \) is a constant.

Step 3: Determine the Constant Using Initial Condition

Using the initial condition \( N(0) = 483,000 \):\[483,000 = C^{\prime} e^{0.058 \times 0}\]\[C^{\prime} = 483,000\]Thus, the function becomes:\[N(t) = 483,000 e^{0.058t}\]

Step 4: Estimate the Number of Patents in 2020

To find the number of patents in 2020, calculate \( N(t) \) for \( t = 2020 - 2009 = 11 \):\[N(11) = 483,000 e^{0.058 \times 11}\]Calculating the exponential part and multiplying, we estimate the number of patent applications.

Step 5: Calculate the Doubling Time

The doubling time \( T \) can be calculated using the formula:\[T = \frac{\ln 2}{k}\]where \( k = 0.058 \). Therefore:\[T = \frac{\ln 2}{0.058} \]

Key Concepts

Exponential Growth

Exponential growth is a process where a quantity increases at a rate proportional to its current value. This means that as the quantity grows, the rate of growth accelerates. A common characteristic of exponential growth is the rapid increase over time. In mathematics, it is often described by an equation of the form \( y' = ky \), where \( y \) represents the quantity, \( y' \) is the rate of change, and \( k \) is a positive constant representing the growth rate. An example from the real world can be the growth of investments or population growth.

In the context of the given exercise, the number of patent applications is described as growing at an exponential rate. The equation \( N'(t) = 0.058 N(t) \) shows that patent applications increase at a rate of 5.8% per year. As time progresses, the number of applications increases more and more quickly. This characteristic makes exponential growth a powerful tool for modeling processes that multiply based on their size.

Separable Equations

Separable equations are a class of differential equations where variables can be separated on different sides of the equation, facilitating easy integration. When dealing with separable equations, the first step is to rearrange the terms to isolate the derivatives and dependent variables.

In the example from the given exercise, the equation \( N'(t) = 0.058 N(t) \) is separable. We can rewrite it as \( \frac{dN}{N} = 0.058 \, dt \). By integrating both sides, we can solve for the function \( N(t) \). Integration of the left side gives us a logarithmic function, whereas the right side integration results in a linear function of \( t \).

• Left side: \( \int \frac{dN}{N} = \ln |N| \)
• Right side: \( \int 0.058 \, dt = 0.058t + C \)

Solving these equations yields \( N(t) = C' e^{0.058t} \), demonstrating the power of separable equations to simplify and solve differential equations.

Doubling Time

Doubling time is a concept related to exponential growth. It refers to the period required for a quantity to double in size at its current growth rate. The doubling time can give valuable insights into how quickly a system is growing. For exponential growth described by \( y(t) = y_0 e^{kt} \), the doubling time \( T \) can be calculated using the formula \( T = \frac{\ln 2}{k} \).

In the case of patent applications, with a growth rate \( k = 0.058 \), the doubling time is calculated as \( T = \frac{\ln 2}{0.058} \). This calculation shows how long it will take for the number of patent applications to double from its initial amount at the established growth rate.
Understanding doubling time helps in planning and forecasting future growth, making it a crucial factor in business and economics. It also offers a simplified way to comprehend the effect of sustained exponential development over time.