Problem 34
Question
The world's population is about \(f(t)=6.8 e^{0.012 t}\) billion, \(^{3}\) where \(t\) is time in years since \(2009 .\) Find \(f(0)\), \(f^{\prime}(0), f(10)\), and \(f^{\prime}(10)\). Using units, interpret your answers in terms of population.
Step-by-Step Solution
Verified Answer
In 2009, the population was 6.8 billion, increasing by 81.6 million per year. By 2019, it was 7.66 billion, growing by 92 million per year.
1Step 1: Find \(f(0)\)
The function given is \(f(t) = 6.8 e^{0.012t}\). To find \(f(0)\), substitute 0 for \(t\) in the equation:\[f(0) = 6.8 e^{0.012 \times 0} = 6.8 e^0 = 6.8\]So, \(f(0) = 6.8\) billion. This means that in the year 2009, the population was 6.8 billion.
2Step 2: Find \(f^{\prime}(t)\)
The derivative of the function \(f(t) = 6.8 e^{0.012t}\) will give us the rate of change of the population. Use the chain rule to differentiate:\[f^{\prime}(t) = 6.8 \times 0.012 e^{0.012t} = 0.0816 e^{0.012t}\]This represents the rate of population growth in billions per year.
3Step 3: Find \(f^{\prime}(0)\)
To find \(f^{\prime}(0)\), substitute 0 for \(t\) in the derivative:\[f^{\prime}(0) = 0.0816 e^{0.012 \times 0} = 0.0816 e^0 = 0.0816\]So, \(f^{\prime}(0) = 0.0816\) billion per year. This indicates that in 2009, the population was increasing at a rate of 81.6 million per year.
4Step 4: Find \(f(10)\)
To find \(f(10)\), substitute 10 for \(t\) into the original function:\[f(10) = 6.8 e^{0.012 \times 10} = 6.8 e^{0.12}\]Using a calculator, compute \(e^{0.12} \approx 1.1275\), so:\[f(10) \approx 6.8 \times 1.1275 \approx 7.66\]Thus, \(f(10) = 7.66\) billion, meaning the population in 2019 was approximately 7.66 billion.
5Step 5: Find \(f^{\prime}(10)\)
To find \(f^{\prime}(10)\), substitute 10 for \(t\) in the derivative:\[f^{\prime}(10) = 0.0816 e^{0.012 \times 10} = 0.0816 e^{0.12}\]Using a calculator, \(e^{0.12} \approx 1.1275\), so:\[f^{\prime}(10) \approx 0.0816 \times 1.1275 \approx 0.092\]Thus, \(f^{\prime}(10) = 0.092\) billion per year, indicating that in 2019, the population was increasing at a rate of about 92 million per year.
Key Concepts
Exponential FunctionDifferentiationPopulation ModelingRate of Change
Exponential Function
An exponential function is a mathematical expression in which a constant base is raised to a variable exponent, typically written as \( f(t) = a e^{bt} \). In population modeling, exponential functions are often used to represent growth due to their consistent rate of increase relative to the current population size.
In the formula \( f(t) = 6.8 e^{0.012t} \), the value \( 6.8 \) represents the initial population when \( t = 0 \), which corresponds to the year 2009. Here, \( e \) is the mathematical constant approximately equal to 2.71828, and it is the base of the natural logarithm.
The exponent \( 0.012t \) indicates that the population grows continuously at a rate proportional to its current size. This characteristic makes exponential functions a very useful tool for modeling populations that grow without restrictions.
In the formula \( f(t) = 6.8 e^{0.012t} \), the value \( 6.8 \) represents the initial population when \( t = 0 \), which corresponds to the year 2009. Here, \( e \) is the mathematical constant approximately equal to 2.71828, and it is the base of the natural logarithm.
The exponent \( 0.012t \) indicates that the population grows continuously at a rate proportional to its current size. This characteristic makes exponential functions a very useful tool for modeling populations that grow without restrictions.
Differentiation
Differentiation is a calculus concept used to determine the rate at which one quantity changes with respect to another. It involves finding the derivative of a function, a process that allows us to understand how values change instantaneously.
To differentiate an exponential function like \( f(t) = 6.8 e^{0.012t} \), we apply the chain rule. The derivative, denoted \( f'(t) \), is calculated by multiplying the derivative of the exponent \( 0.012t \) (which is \( 0.012 \)) with the original function:
\[ f'(t) = 6.8 \times 0.012 e^{0.012t} = 0.0816 e^{0.012t} \]
This derivative gives us the rate of change of the population over time, showing how quickly or slowly the population is increasing.
To differentiate an exponential function like \( f(t) = 6.8 e^{0.012t} \), we apply the chain rule. The derivative, denoted \( f'(t) \), is calculated by multiplying the derivative of the exponent \( 0.012t \) (which is \( 0.012 \)) with the original function:
\[ f'(t) = 6.8 \times 0.012 e^{0.012t} = 0.0816 e^{0.012t} \]
This derivative gives us the rate of change of the population over time, showing how quickly or slowly the population is increasing.
Population Modeling
Population modeling is a technique used to represent the growth, decline, or stability of a population over time. It involves creating a mathematical representation that describes how a population changes, allowing researchers and policymakers to predict future trends.
The function \( f(t) = 6.8 e^{0.012t} \) is a model for the world's population growth from the year 2009 onward. By substituting different values of \( t \), we can estimate the population at various points in time. For example, \( f(0) = 6.8 \) billion indicates the population in 2009, while \( f(10) \approx 7.66 \) billion shows the population in 2019.
Models like these help estimate future population sizes and help plan for resources, such as food, water, and housing. Understanding the implications of these models is crucial for decision-making on both local and global scales.
The function \( f(t) = 6.8 e^{0.012t} \) is a model for the world's population growth from the year 2009 onward. By substituting different values of \( t \), we can estimate the population at various points in time. For example, \( f(0) = 6.8 \) billion indicates the population in 2009, while \( f(10) \approx 7.66 \) billion shows the population in 2019.
Models like these help estimate future population sizes and help plan for resources, such as food, water, and housing. Understanding the implications of these models is crucial for decision-making on both local and global scales.
Rate of Change
The rate of change in a population model indicates how quickly the population size increases or decreases over time. It is typically derived from the derivative of the function representing the population.
For the function \( f(t) = 6.8 e^{0.012t} \), the derivative \( f'(t) = 0.0816 e^{0.012t} \) provides information about the population's growth rate. In this case, \( f'(0) = 0.0816 \) billion per year shows the rate at which the population was increasing in 2009.
Simultaneously, \( f'(10) \approx 0.092 \) billion per year represents the population growth rate in 2019. This suggests that the population is growing faster over time due to the compounding nature of exponential functions, where each increase builds on the current total.
Understanding the rate of change helps policymakers be proactive instead of reactive in planning for societal needs.
For the function \( f(t) = 6.8 e^{0.012t} \), the derivative \( f'(t) = 0.0816 e^{0.012t} \) provides information about the population's growth rate. In this case, \( f'(0) = 0.0816 \) billion per year shows the rate at which the population was increasing in 2009.
Simultaneously, \( f'(10) \approx 0.092 \) billion per year represents the population growth rate in 2019. This suggests that the population is growing faster over time due to the compounding nature of exponential functions, where each increase builds on the current total.
Understanding the rate of change helps policymakers be proactive instead of reactive in planning for societal needs.
Other exercises in this chapter
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