The Gross National Income (GNI) Of A Certain Country (in Billions Of U.S. Dollars) Can Be Approximated By $f(t) = 1031 E^{0.173 T}$, Where $t = 0$ Corresponds To The Year 2000.(a) Find $f^{\prime}(t$\].(b) At What Rate Was The

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Introduction

The gross national income (GNI) of a country is a crucial economic indicator that measures the total value of goods and services produced within a country's borders. In this article, we will analyze the growth rate of the GNI of a certain country using the given function f(t)=1031e0.173tf(t) = 1031 e^{0.173 t}, where t=0t = 0 corresponds to the year 2000.

The Function and its Derivative

The given function is f(t)=1031e0.173tf(t) = 1031 e^{0.173 t}, where tt represents the number of years after 2000. To find the growth rate of the GNI, we need to find the derivative of this function with respect to time tt. The derivative of f(t)f(t) is denoted as f′(t)f^{\prime}(t).

Finding the Derivative

To find the derivative of f(t)=1031e0.173tf(t) = 1031 e^{0.173 t}, we will use the chain rule of differentiation. The chain rule states that if we have a composite function of the form f(g(x))f(g(x)), then the derivative of this function is given by f′(g(x))⋅g′(x)f^{\prime}(g(x)) \cdot g^{\prime}(x).

In this case, we have f(t)=1031e0.173tf(t) = 1031 e^{0.173 t}. We can rewrite this function as f(t)=1031â‹…e0.173tf(t) = 1031 \cdot e^{0.173 t}. Now, we can apply the chain rule to find the derivative of this function.

f(t) = 1031 \cdot e^{0.173 t}
f^{\prime}(t) = 1031 \cdot e^{0.173 t} \cdot 0.173
f^{\prime}(t) = 179.079 \cdot e^{0.173 t}

Simplifying the Derivative

We can simplify the derivative by canceling out the common factor of e0.173te^{0.173 t}.

f^{\prime}(t) = 179.079 \cdot e^{0.173 t}
f^{\prime}(t) = 179.079 \cdot f(t)

Interpretation of the Derivative

The derivative f′(t)f^{\prime}(t) represents the rate of change of the GNI with respect to time tt. In other words, it measures the rate at which the GNI is increasing or decreasing over time.

We can interpret the derivative as follows:

  • If f′(t)>0f^{\prime}(t) > 0, then the GNI is increasing over time.
  • If f′(t)<0f^{\prime}(t) < 0, then the GNI is decreasing over time.
  • If f′(t)=0f^{\prime}(t) = 0, then the GNI is not changing over time.

Rate of Change of the GNI

To find the rate of change of the GNI, we need to evaluate the derivative f′(t)f^{\prime}(t) at a specific value of tt. Let's say we want to find the rate of change of the GNI in the year 2005, which corresponds to t=5t = 5.

f^{\prime}(5) = 179.079 \cdot f(5)
f^{\prime}(5) = 179.079 \cdot 1031 e^{0.173 \cdot 5}
f^{\prime}(5) = 179.079 \cdot 1031 e^{0.865}
f^{\prime}(5) = 179.079 \cdot 1031 \cdot 2.373
f^{\prime}(5) = 419,111.19

Therefore, the rate of change of the GNI in the year 2005 is approximately 419,111.19419,111.19 billion U.S. dollars per year.

Conclusion

In this article, we analyzed the growth rate of the GNI of a certain country using the given function f(t)=1031e0.173tf(t) = 1031 e^{0.173 t}. We found the derivative of this function, which represents the rate of change of the GNI with respect to time tt. We then evaluated the derivative at a specific value of tt to find the rate of change of the GNI in the year 2005. Our results show that the GNI is increasing over time, with a rate of change of approximately 419,111.19419,111.19 billion U.S. dollars per year.

References

Introduction

In our previous article, we analyzed the growth rate of the gross national income (GNI) of a certain country using the given function f(t)=1031e0.173tf(t) = 1031 e^{0.173 t}. We found the derivative of this function, which represents the rate of change of the GNI with respect to time tt. In this article, we will answer some frequently asked questions (FAQs) related to the GNI and its growth rate.

Q&A

Q: What is the gross national income (GNI)?

A: The gross national income (GNI) is a measure of the total value of goods and services produced within a country's borders. It is a key economic indicator that measures the country's economic performance.

Q: How is the GNI calculated?

A: The GNI is calculated by adding up the value of all goods and services produced within a country's borders, including wages, salaries, and profits.

Q: What is the difference between GNI and GDP?

A: The gross domestic product (GDP) is a measure of the total value of goods and services produced within a country's borders, while the GNI is a measure of the total value of goods and services produced within a country's borders, including income earned by its citizens abroad.

Q: How does the GNI growth rate affect the economy?

A: The GNI growth rate affects the economy in several ways. A high GNI growth rate indicates a strong economy, while a low GNI growth rate indicates a weak economy. A high GNI growth rate can lead to increased economic activity, job creation, and higher living standards.

Q: What are the factors that affect the GNI growth rate?

A: The GNI growth rate is affected by several factors, including:

  • Economic policies: Fiscal and monetary policies can affect the GNI growth rate.
  • Technological advancements: Technological advancements can lead to increased productivity and economic growth.
  • Global events: Global events, such as wars and natural disasters, can affect the GNI growth rate.
  • Demographic changes: Changes in population demographics, such as aging and urbanization, can affect the GNI growth rate.

Q: How can the GNI growth rate be used to make economic decisions?

A: The GNI growth rate can be used to make economic decisions in several ways. For example:

  • Investment decisions: A high GNI growth rate can indicate a strong economy, making it a good time to invest.
  • Business decisions: A high GNI growth rate can indicate a strong demand for goods and services, making it a good time to start a business.
  • Policy decisions: A low GNI growth rate can indicate a weak economy, making it a good time to implement policies to stimulate economic growth.

Q: What are the limitations of the GNI growth rate?

A: The GNI growth rate has several limitations, including:

  • It does not account for income inequality: The GNI growth rate does not account for income inequality, which can lead to a distorted picture of economic performance.
  • It does not account for non-monetary benefits: The GNI growth rate does not account for non-monetary benefits, such as leisure time and health benefits.
  • It is subject to revisions: The GNI growth rate is subject to revisions, which can affect its accuracy.

Conclusion

In this article, we answered some frequently asked questions (FAQs) related to the gross national income (GNI) and its growth rate. We hope that this article has provided you with a better understanding of the GNI and its importance in economic analysis.

References