Problem 8 - Active Calculus

Question

Suppose that the population, \(P\), of China (in billions) can be approximated by the function \(P(t)=1.15(1.014)^{t}\) where \(t\) is the number of years since the start of \(1993 .\) a. According to the model, what was the total change in the population of China between January 1,1993 and January \(1,2000 ?\) What will be the average rate of change of the population over this time period? Is this average rate of change greater or less than the instantaneous rate of change of the population on January \(1,2000 ?\) Explain and justify, being sure to include proper units on all your answers. b. According to the model, what is the average rate of change of the population of China in the ten-year period starting on January 1, \(2012 ?\) c. Write an expression involving limits that, if evaluated, would give the exact instantaneous rate of change of the population on today's date. Then estimate the value of this limit (discuss how you chose to do so) and explain the meaning (including units) of the value you have found. d. Find an equation for the tangent line to the function \(y=P(t)\) at the point where the \(t\) -value is given by today's date.

Step-by-Step Solution

Answer

The total change in the population was 0.12 billion. The average rate of change from 1993 to 2000 was approximately 0.01714 billion/year. Today's instantaneous rate can be evaluated using the derivative function at the current t-value.

Step 1: Calculate Population in 1993

Given the function is \(P(t) = 1.15(1.014)^t\), where \(t\) represents the number of years since 1993, calculate \(P(0)\) to find the population at the start of 1993. \[ P(0) = 1.15(1.014)^0 = 1.15 \text{ billion} \]

Step 2: Calculate Population in 2000

Calculate \(P(7)\), because 2000 is 7 years after 1993. \[ P(7) = 1.15(1.014)^7 \] Plug in the values to get the result. Using a calculator we have: \[ P(7) = 1.15(1.1043) \approx 1.27 \text{ billion} \]

Step 3: Calculate Total Change in Population (1993 to 2000)

Subtract the population at the start of 1993 from the population at the start of 2000. \[ \Delta P = P(7) - P(0) = 1.27 - 1.15 = 0.12 \text{ billion} \]

Step 4: Calculate Average Rate of Change (1993 to 2000)

Average rate of change is given by \[ \text{Average Rate} = \frac{\Delta P}{\Delta t} = \frac{0.12}{7} \approx 0.01714 \text{ billion/year} \]

Step 5: Compare with Instantaneous Rate of Change (January 1, 2000)

The instantaneous rate of change of the population at any time \(t\) can be found using the derivative of \(P(t)\). First find \(P'(t)\): \[ P'(t) = 1.15 \cdot \ln(1.014) \cdot (1.014)^t \] Then evaluate it at \(t = 7\): \[ P'(7) \approx 1.15 \cdot 0.0139 \cdot 1.1043 \approx 0.0177 \text{ billion/year} \] So, the instantaneous rate of change on January 1, 2000 is slightly greater than the average rate of change.

Step 6: Calculate Average Rate of Change (2012 to 2022)

For the period starting on January 1, 2012, calculate the populations at \(t = 19\) and \(t = 29\) because 2012 is 19 years after 1993 and 2022 is 29 years after 1993. \[ P(19) \approx 1.15(1.014)^{19} \approx 1.46 \text{ billion} \] \[ P(29) \approx 1.15(1.014)^{29} \approx 1.64 \text{ billion} \] Calculate the change in population: \[ \Delta P = P(29) - P(19) = 1.64 - 1.46 = 0.18 \text{ billion} \] Calculate the average rate of change: \[ \text{Average Rate} = \frac{0.18}{10} = 0.018 \text{ billion/year} \]

Step 7: Expression for Instantaneous Rate of Change Today

The exact instantaneous rate of change of the population on today's date is obtained from the derivative: \[ \lim_{{\Delta t \to 0}} \frac{P(t+\Delta t) - P(t)}{\Delta t} \] For today, with a given year since 1993 of \(t\): \[ \text{Estimate} \approx P'(t) \approx 1.15 \cdot 0.0139 \cdot (1.014)^t \]

Step 8: Tangent Line to the Function at Today's Date

The equation of the tangent line is given by \[ y - P(t) = P'(t)(x - t) \] where \(P(t)\) is the population at time \(t\) and \(P'(t)\) is the derivative at time \(t\). Plug in the values to get the equation.

Key Concepts

Population Change

Population change refers to the difference in population size between two distinct time periods. In this exercise, we are looking at the total change in China's population from January 1, 1993, to January 1, 2000. Initially, the population is calculated using the given function for the years 1993 and 2000. The change is then found by subtracting the initial population from the final population.
This concept is crucial in understanding how populations grow or shrink over time, and it includes both natural increase (births minus deaths) and net migration (immigrants minus emigrants).

Average Rate of Change

The average rate of change of a population over a given time period shows how much the population changes per unit of time on average.
In mathematical terms, it's the total change in population divided by the time interval over which this change occurred. For instance, the average rate of change between 1993 and 2000 is calculated by dividing the total change in population (0.12 billion) by the number of years (7).
This gives an average rate of approximately 0.01714 billion people per year, which is useful for comparing long-term trends.

Instantaneous Rate of Change

The instantaneous rate of change examines how quickly the population is changing at a specific point in time. It's akin to the population's 'velocity' at one particular moment.
In this exercise, the instantaneous rate of change on January 1, 2000, is found by calculating the derivative of the population function at the specific time, which gives a rate of approximately 0.0177 billion people per year.
Unlike the average rate of change, this measure provides a snapshot of the population growth's speed at a single instant. This is typically more accurate for policy-making and short-term planning.

Tangent Line

The tangent line to the population function at any given point represents the direction and rate at which the population is changing at that particular moment.
Mathematically, the equation of the tangent line at a point is derived using the derivative (instantaneous rate of change) at that point. It helps visualize and understand the local behavior of the population model.
For today's date, the general form is y - P(t) = P'(t)(x - t) This provides a linear approximation of the population function near the point.
It is especially useful for making short-term predictions and understanding the growth trend at a specific date.